博士論文
Study of the Oscillation of Neutrinos and Anti-Neutrinos and of
Non-Standard Neutrino Interactions with the
Atmospheric Data in Super-Kamiokande(スーパーカミオカンデの大気ニュートリノデータによるニュートリノと反ニュートリノ振動の研究と非標準ニュートリノ相互作用の研究)

平成２７年３月博士(理学)申請

東京大学大学院理学系研究科
物理学専攻
マクラクラン・福英・トーマス
Mclachlan Fukuei Thomas
March 2015

Abstract
In atmospheric neutrinos, there has been a long established deficit of νµ events that enter frombelow the detector after traversing long distances. This phenomena has been well described bya theory called Neutrino Oscillations, which allows neutrinos to oscillate from one flavour toanother. The probability of these transitions is characterised by the ratio of the flight path of
the neutrino to its energy, L/E. The L/E Analysis aims to accurately determine the oscillation
parameters by reconstructing the L/E dependence of the flavour transitions. It achieves this byselecting events with a high resolution in L/E.
Atmospheric neutrino experiments have typically measured oscillations with combined neutrino and anti-neutrino data. In recent years, accelerator experiments have been reversing thepolarity of their focussing magnets, greatly increasing the ν̄ composition in their beams. Early
oscillation analyses of this ν̄ data by MINOS (2011) resulted in a much larger ∆m2 measurement than the established value for neutrinos. This lead to experiments checking if distinct
anti-neutrino oscillation parameters can be measured by oscillating ν̄ independently from ν. In
addition, Neutral-Current (NC) Non-Standard Interactions (NSI) of neutrinos with matter havebeen proposed to explain the anomaly. In the years following 2011, MINOS reported on ν andν̄ oscillation parameters that were in agreement.
However, it is important to make independent checks of any physical phenomena, particularly
with an experiment with a different scope of sensitivity. Super-Kamiokande (SK) covers a wider
range of the parameter space with baselines extending from ∼10 km to over 104 km; and energies
ranging from 30 MeV to 50 GeV in this analysis. These baselines traverse extensive amounts ofmatter, crossing the entire Earth and include the mantle and core. While SK cannot distinguishbetween ν and ν̄ on an event-by-event basis, it has the potential for discovery of unexpected
effects and unparalleled sensitivity to matter effects, such as in NC NSI.
In this thesis, both of the above hypotheses have been investigated. After extending theStandard L/E Analysis to cover over 10 years of atmospheric neutrino data, further improvements
were made to increase the sensitivity to ν̄ parameters and sub-dominant NSI. These included aseparately binned second resolution sample to considerably increase the available statistics, andan improved χ2 calculation with greater stability in bins with few events. The standard analysis
was in 2-flavours, which was extended to a 3-flavour model and then again to include µτ –sector
NSI. Included in this thesis is the first L/E analysis performed with 3-flavour oscillations, and thefirst analysis of atmospheric neutrino data with 3-flavour oscillations and NSI in the µτ –sector.
No distinction could be observed between ν and ν̄ parameters, and the results were consistent
with no observation of NSI. The 90% C.L. limits on the NSI parameters were−1.97 × 10−2 < εµτ < 1.45 × 10−2 and −4.54 × 10−2 < ε0 ≡ (ετ τ − εµµ ) < 4.47 × 10−2 , assuming
a normalisation of neutrino NSI with down-quarks in the matter traversed, and εµµ = εeτ = 0.
With proper consideration of 3-flavour effects, and the best sensitivity to matter effects, these
constraints are currently the best in the world. The oscillation parameters measured in the 3+0.07
flavour L/E Analysis for Normal (Inverted) Hierarchy were sin2 θ23 = 0.55+0.07
0.54−0.12 and
−0.13

+0.22
+0.25
2
−3
2
∆m31 = 2.60−0.21 2.45−0.16 × 10 eV . The data showed no preference between Normal andInverted Hierarchy.

Acknowledgements
It has been a true challenge seeing this thesis through to completion, and there are surely
too many people to be named in fully extending my gratitude. Firstly, I would like to thankProf. Takaaki Kajita, who provided the wonderful opportunity to carry out my post-graduate
studies at the University of Tokyo and to work in a historical international collaboration, SuperKamiokande. He has been more kind, patient, and supportive beyond any reasonable expectation
for a supervisor. For giving me the freedom to explore the many possibilities for an interesting
topic of study, and for helping me find my way again when things did not go so well. His guidancehas continued beyond duty, allowing this thesis to finally reach a satisfying conclusion. I know
I have not been an easy student, so for everything written here, and much more, I am truly
grateful.
I must thank my family for their perseverance and unending patience with me, especiallythrough the final years of my PhD. This thesis has survived through some trying circumstances,and surely would not exist without the support of my family. In particular to my mother, whoprovided persistent motivation whenever my own was weak. This gratitude is extended to mychildhood friend, Kenny, who is like a brother and has been incredibly patient and supportive
through the many difficult moments I have faced. For always being there, for our rejuvenating
long walks in nature, and for helping me find my way again when I stray off-course.
I am forever thankful to the amazing friends I met on this journey, who have made my timein Japan an unforgettable experience. There are too many wonderful people to mention, butparticularly I would like to thank Sarah A. for being so inspirational, for her acceptance andcompanionship, and for helping me believe I can tackle the most challenging dreams. Thank you
to Rie I. for our many open-minded conversations, for her belief and support, and for being there
during the most difficult times I have ever faced. My thanks to Farid S. for our explorations in
culture and technology, for constantly challenging my opinions, and for being one of the mostcaring friends I have ever had. Cheers to Sebastian, Tony and Elliot for our many conversations
as we faced the wonders and challenges of being a student in Japan. Thanks also to Mayuko,
Sekine, Yukina, Yanju, Saki, Rie K., Megumi, Magdalenda, and Akabane for making me feel athome.
On campus, my gratitude goes to Yamamoto-san for running a wonderful homely cafe, where
I have enjoyed many conversations and fun times. And to the friends I met there, particularly
Yoshi, Eiji, Sasha, Grace, Bruno, Shashank, Fid, Regi, Hui, Yoon Lin, Koji, Yuuka, Mariko, Nick,
Mark, Paul, Goran, Will and Sourav for making campus life enjoyable. I am also very thankful to
Yoshida-san, Nishikawa-san, Yamaguchi-san, Kitsugi-san, and Watanabe-san for their kindness
and support.
I am grateful to the Osaka group, where I had my first taste of neutrino research. In particular
to Prof. Y. Kuno, Yoshida-san, Prof. Sato, Prof. Aoki, Itahashi-san, Nori, Naosuke, Zono, Yano,iii

Nakanishi, Taichou, Kyapuchin, Atsushi, and Wakako who taught me much about research andthe wonders of life in Japan. Also to Dr. Soler who gave me the opportunity to continue to workwith neutrinos in Glasgow.
I am thankful to the students who helped me get started when I arrived at RCCN, in particular I would like to thank C. Ishihara, who introduced me to research life in the collaboration,and to H. Nishino, G. Mitsuka, M. Ikeda for their help early on.
I am tremendously grateful to Prof. S. Moriyama who brought me up to speed with researchin the collaboration, and who was always happy to answer my many questions. I would like tothank Prof. M. Nakahata, who often answered my curiosities while working in the SKDETSIM
group. And to Prof. K. Kaneyuki and Dr. J. Kameda who helped me get started in analysis.
I would like to thank the many ICRR and Kamioka staff who have assisted me at various times, including Prof. M. Shiozawa, Prof. Y. Hayato, Dr. Y. Obayashi, Dr. M. Miura,
Dr. Y. Koshio, and Kai. Particularly, I would like to thank Jan and Lluis, for their many
discussions and 3 a.m. coffee breaks while working night-shifts.
I am indebted to the members of the RCCN group, for their company, conversation andadvice, in particular to Okumura-san, Honda-san, Andrew, Maggie, Tristan, Euan, Kaji-san,Tanimoto-san, Shimizu-san, Shinohara-san, and Nishimura-san. Thank you for everything you
have taught me, for our chats; coffee, lunch and dinner breaks; and for making research much
more pleasant. I know I can be difficult to work with, so thank you also for your patience andunderstanding.
I am grateful for the comments and advice from the atmospheric neutrino analysis group,
including Prof. E. Kearns, Prof. C. Walter, Prof. K. Sholberg, Dr. J. Raff, Dr. S. Mine, Piotr,
Tarek, and particularly to Dr. R. Wendell and Okumura-san for their advice and assistance forsolving challenging problems during analysis. Of course, my thanks go to Dr. M. Ishitsuka and
Dr. F. Dufour for their previous work on the Standard L/E Analysis.
I am grateful to the other students S. Hazama, Y. Kozuma, T. Yokozawa, A. Shinozaki,
H. Nishiie, K. Iyogi, T. Tanaka, D. Motoki, K. Ueshima, T. Iida, Byeongsu, Josh, Patrick,Michael D., Koun, Haibing, and Yang who helped make research life more enjoyable.
I am especially indebted to all those who helped me during the final stretch of my thesis.
I am very grateful to Watanabe-san, who went out of her way to assist me when I could not
be present in Japan, and for her ever-present encouragement. Thank you to Yamamoto-sanand her husband for their exceptional compassion and care, and for helping me stay in Japanto finalise my thesis. I am truly humbled by the compassion of Kana and Scott, Sei-chan andChi-tan, Sibylle and Roger, as well as Farid for accommodating and supporting me during thistime. I will pass on this compassion to others, whenever I have the chance. I am very gratefulto Nakazawa-sensei, for his patience and kindness, and for helping me raise the standard of thisthesis considerably to reach its final form. Of course, any remaining errors are my own.
I am thankful also to the countless others who have made each moment interesting, delightful,
and a learning experience.
Finally, I am eternally thankful to my father, who could not see the completion of this thesis,but has subtly guided me throughout my life.

Contents
1 Introduction
1.1 Introduction to Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Thesis Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 The L/E Analysis with Atmospheric Neutrinos at Super-Kamiokande . . .
1.2.2 Neutrino Anti-Neutrino L/E Analysis . . . . . . . . . . . . . . . . . . . .
1.2.3 3-Flavour L/E Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.4 NSI L/E Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Atmospheric Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Neutrino Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.1 3-Flavour Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.2 2-Flavour Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.3 Matter Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.4 Status of the Oscillation Parameter Research . . . . . . . . . . . . . . . .
1.5 Non-Standard Neutrino Interactions (NSI) . . . . . . . . . . . . . . . . . . . . . .
1.5.1 Treatment of NSI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.2 Previous Work with Atmospheric Neutrinos and NSI . . . . . . . . . . . .
1.6 Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
3
3
4
4
4
4
6
6
8
9
11
13
13
16
22

2 Experimental Evidence for Neutrino Oscillations
2.1 Solar Neutrino Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Homestake Chlorine Experiment . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Gallium Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 Kamiokande . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.4 Super-Kamiokande . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.5 Sudbury Neutrino Observatory . . . . . . . . . . . . . . . . . . . . . . . .
2.1.6 Borexino . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.7 Solar Neutrino Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Atmospheric Neutrino Experiments . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Kamiokande and IMB . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Fréjus and NUSEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Super-Kamiokande Atmospheric Neutrinos . . . . . . . . . . . . . . . . . .
2.2.4 Soudan2 and MACRO . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.5 Atmospheric Neutrino Summary . . . . . . . . . . . . . . . . . . . . . . .
2.3 Reactor Neutrino Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 CHOOZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24
24
25
27
29
31
34
38
40
40
41
42
43
45
47
48
48

v

2.3.2 KamLAND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Daya Bay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.4 Double CHOOZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.5 RENO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.6 Reactor Neutrino Summary . . . . . . . . . . . . . . . . . . . . . . . . . .
Accelerator Neutrino Experiments . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 K2K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 MINOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.3 T2K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.4 MiniBooNE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.5 OPERA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.6 Accelerator Neutrino Summary . . . . . . . . . . . . . . . . . . . . . . . .
Future Neutrino Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 Neutrino Factory and Beta Beams . . . . . . . . . . . . . . . . . . . . . .
2.5.2 Megaton Detectors: Hyper-Kamiokande, DUSEL, and LAGUNA . . . . .
2.5.3 Off-Axis Super-beams: NOνA and T2HK . . . . . . . . . . . . . . . . . .
Neutrino Experiments Summary . . . . . . . . . . . . . . . . . . . . . . . . . . .

49
50
52
53
54
55
55
56
58
61
63
65
65
66
67
69
70

3 Super-Kamiokande Detector
3.1 Cherenkov Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Detector Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Super-Kamiokande Tank . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Photo-Multiplier Tubes (PMTs) . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Water Purification System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Air Purification System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Electronics and Data Aquisition System . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 Electronics and Data Aquisition: SK-I to SK-III . . . . . . . . . . . . . .
3.5.2 Hardware Trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.3 Upgraded Electronics and Data Aquisition for SK-IV . . . . . . . . . . . .
3.5.4 SK-IV New Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.5 SK-IV New Online System . . . . . . . . . . . . . . . . . . . . . . . . . . .

72
73
74
74
76
79
79
81
81
84
84
86
87

4 Monte Carlo Simulation
4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Atmospheric Neutrino Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Hadronic Interaction Models . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 3-Dimensional Flux Calculation . . . . . . . . . . . . . . . . . . . . . . . .
4.2.3 The Neutrino Flux at Super-Kamiokande . . . . . . . . . . . . . . . . . .
4.3 Neutrino Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Quasi-Elastic Neutrino-Nucleon Scattering . . . . . . . . . . . . . . . . . .
4.3.2 Single Meson Production . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.3 Deep Ineslastic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.4 Coherent Pion Production . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.5 Nuclear Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Detector Simulation (SKDETSIM) . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Particle Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88
88
88
89
90
92
94
95
96
97
97
98
98
99

2.4

2.5

2.6

4.4.2
4.4.3

Cherenkov Photon Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . 99
PMT Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5 Calibration
102
5.1 Detector Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.1.1 Relative Gain Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.1.2 Absolute Gain Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.1.3 Relative Timing Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.1.4 Water Transparency from Cosmic-Ray Muons . . . . . . . . . . . . . . . . 104
5.2 Water Parameter Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.2.1 Rayleigh and Mie Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.3 Energy Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.3.1 Decay Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.3.2 Neutrino Induced π 0 Events . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.3.3 Low Energy Stopping Muons . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.3.4 High Energy Stopping Muons . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.3.5 Time Variation of Energy Scale . . . . . . . . . . . . . . . . . . . . . . . . 114
5.3.6 Summary of Energy Scale Calibrations . . . . . . . . . . . . . . . . . . . . 117
5.3.7 Uniformity of Energy Scale . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6 Data Reduction
120
6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.2 Reduction for Fully-Contained Sample . . . . . . . . . . . . . . . . . . . . . . . . 121
6.2.1 First Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.2.2 Second Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.2.3 Third Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.2.4 Fourth Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.2.5 Fifth Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.2.6 FC Reduction Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.3 Reduction for Partially-Contained Sample . . . . . . . . . . . . . . . . . . . . . . 136
6.3.1 First Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.3.2 Second Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.3.3 Third Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.3.4 Fourth Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.3.5 Fifth Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.3.6 PC Reduction Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
7 Event Reconstruction
150
7.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
7.2 Vertex Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.2.1 Point-Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.2.2 Ring Edge Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.2.3 TDC-Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
7.3 Ring Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.3.1 Ring Candidate Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.3.2 Ring Candidate Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

7.4

Particle Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.1 Estimation of the Particle Type . . . . . . . . . . . . . . . . . . . . . . . .
7.4.2 Expected Charge Distribution: Electrons . . . . . . . . . . . . . . . . . . .
7.4.3 Expected Charge Distribution: Muons . . . . . . . . . . . . . . . . . . . .
7.4.4 Expected Charge Distribution: Scattered Light . . . . . . . . . . . . . . .
7.4.5 Performance of PID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5 Precise Vertex Reconstruction (MS-Fit) . . . . . . . . . . . . . . . . . . . . . . .
7.6 Decay Electron Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.7 Momentum Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.8 Ring Number Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.9 π 0 Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.10 Upward-Going Muons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

157
157
158
159
160
160
160
162
163
164
165
167

8 An Update of the 2-Flavour L/E Analysis
169
8.1 Dataset and Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
8.1.1 Fully-Contained: Single-Ring (FC1R) and Multi-Ring (FCmR) . . . . . . 170
8.1.2 Partially-Contained: PC Stopping And PC Through-Going . . . . . . . . 171
8.1.3 PC Stopping and Through-Going Separation . . . . . . . . . . . . . . . . 172
8.2 Reconstruction of L/E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
8.2.1 Neutrino Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
8.2.2 Neutrino Flight Path Length . . . . . . . . . . . . . . . . . . . . . . . . . 176
8.3 L/E Event Resolution Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
8.3.1 Estimation of the L/E Resolution . . . . . . . . . . . . . . . . . . . . . . . 182
8.3.2 Definition of the L/E Resolution . . . . . . . . . . . . . . . . . . . . . . . 183
8.3.3 L/E Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
8.3.4 L/E Resolution Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
8.4 Likelihood Analysis and χ2 Minimisation . . . . . . . . . . . . . . . . . . . . . . . 186
8.5 Systematic Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
8.5.1 Neutrino Flux Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . 190
8.5.2 Neutrino Interaction Uncertainties . . . . . . . . . . . . . . . . . . . . . . 193
8.5.3 Event Reconstruction Uncertainties . . . . . . . . . . . . . . . . . . . . . . 195
8.5.4 Reduction Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
8.6 2-Flavour Oscillation L/E Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
8.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
9 Analysis of Neutrino and Anti-Neutrino Oscillation
206
9.1 Analysis Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
9.1.1 Reconstruction of L/E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
9.1.2 Neutrino and Anti-Neutrino Oscillations . . . . . . . . . . . . . . . . . . . 207
9.1.3 Neutrino and Anti-Neutrino Flux Ratio . . . . . . . . . . . . . . . . . . . 209
9.2 Sensitivity Study for ν and ν̄ Parameters . . . . . . . . . . . . . . . . . . . . . . . 211
9.2.1 Increasing Statistics with a Second Resolution Sample . . . . . . . . . . . 211
9.2.2 Analysis Bins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
9.2.3 Sensitivity with the Second Resolution Sample . . . . . . . . . . . . . . . 213
9.3 Likelihood Analysis and χ2 Minimisation . . . . . . . . . . . . . . . . . . . . . . . 214
9.4 Neutrino Anti-Neutrino Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

9.5
9.6

9.7

Systematic Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Neutrino Anti-Neutrino Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.6.1 Neutrino Anti-Neutrino L/E Analysis Best Fit . . . . . . . . . . . . . . .
9.6.2 Analysis With Forced Normal Oscillations . . . . . . . . . . . . . . . . . .
9.6.3 Systematic Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.6.4 Analysis Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.7.1 Comparisons Between The L/E Analyses . . . . . . . . . . . . . . . . . . .
9.7.2 Fit in The Non-Physical Region . . . . . . . . . . . . . . . . . . . . . . . .
9.7.3 Comparisons With Other Analyses . . . . . . . . . . . . . . . . . . . . . .
9.7.4 The Result and Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . .

216
216
218
222
224
224
225
225
225
228
230

10 3-Flavour L/E Analysis
231
10.1 Moving from 2-Flavours to 3-Flavours . . . . . . . . . . . . . . . . . . . . . . . . 232
10.1.1 Treatment of Oscillation Probabilities . . . . . . . . . . . . . . . . . . . . 232
10.1.2 Flavour Flux Re-Weighting . . . . . . . . . . . . . . . . . . . . . . . . . . 233
10.2 Neutrino Mass Hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
10.3 Analysis Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
10.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
10.4.1 3-Flavour Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
10.4.2 Comparison With The 2-Flavour L/E Results . . . . . . . . . . . . . . . . 238
10.4.3 3-Flavour L/E Systematic Uncertainties . . . . . . . . . . . . . . . . . . . 239
10.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
10.5.1 Comparisons Between L/E Analyses . . . . . . . . . . . . . . . . . . . . . 240
10.5.2 Comparisons With Other Analyses . . . . . . . . . . . . . . . . . . . . . . 243
10.5.3 Improved Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
10.5.4 The 3-Flavour L/E Result . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
11 Analysis of NSI with Oscillation in the νµ –ντ Sector
252
11.1 NSI Background and Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
11.2 Assumptions in This Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
11.3 3-Flavour Oscillation and NSI Results . . . . . . . . . . . . . . . . . . . . . . . . 255
11.3.1 Best Fit and Allowed Regions . . . . . . . . . . . . . . . . . . . . . . . . . 255
11.3.2 The εµτ and ∆m231 Dependence . . . . . . . . . . . . . . . . . . . . . . . . 259
11.3.3 3F Oscillation and NSI Systematic Uncertainties . . . . . . . . . . . . . . 261
11.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
11.4.1 Comparisson With Atmospheric NSI Measurements . . . . . . . . . . . . . 261
11.4.2 Comparisson With MINOS Measurements . . . . . . . . . . . . . . . . . . 264
11.4.3 Improving the Sensitivity to NSI . . . . . . . . . . . . . . . . . . . . . . . 266
11.4.4 The 3-Flavour Oscillation and NSI L/E Result . . . . . . . . . . . . . . . 266
12 Conclusion

270

A L/E Analysis Distributions

272

Bibliography

280

List of Figures
1.1 Anti-Neutrino Oscillation Parameter Measurement from MINOS . . . . . . . . .
1.2 Atmospheric Neutrino Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Neutrino Oscillations Survival Probability . . . . . . . . . . . . . . . . . . . . . .
1.4 Feynman Diagram: W-exchange Matter Effect . . . . . . . . . . . . . . . . . . . .
1.5 Oscillation Parameter Allowed Regions . . . . . . . . . . . . . . . . . . . . . . . .
1.6 θ13 Allowed Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7 Feynman Diagram: FCNC NSI . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.8 NSI Survival Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.9 NSI Allowed Regions From 2-Flavour Atmospheric Analyses . . . . . . . . . . . .
1.10 Effect of νe Mixing With Electron Flavour ε NSI in Atmospheric Neutrinos . . .
1.11 Fit of NC-like Matter NSI to Explain The 2010 MINOS Data . . . . . . . . . . .
1.12 NC-like NSI Allowed Regions From a GLoBES Fit to The 2010 MINOS Data . .
1.13 SK Allowed Regions From 2-Flavour Oscillation+NSI Zenith Analysis . . . . . .
1.14 MINOS Collaboration Allowed Regions For NC-like NSI . . . . . . . . . . . . . .

2
5
8
10
12
12
13
16
17
18
19
20
20
21

2.1 Fluxes of Solar Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Chlorine Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Gallium Solar Neutrino Rate at SAGE, GALLEX and GNO . . . . . . . . . . . .
2.4 Kamiokande Solar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 SK Solar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 SK Solar Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 SNO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8 SNO NC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.9 Borexino . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.10 Kamiokande Zenith Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.11 Super-Kamiokande Zenith Distributions . . . . . . . . . . . . . . . . . . . . . . .
2.12 Super-Kamiokande L/E Distributions and Atmospheric Oscillation Parameter Allowed Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.13 Soudan 2 and MACRO Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.14 Soudan2 L/E Distributions and Soudan2 and MACRO Oscillation Regions . . . .
2.15 CHOOZ Detector and ν̄e ↔ ν̄x oscillation excluded parameter space . . . . . . .
2.16 KamLAND Detector and Spectral Distortion L/E . . . . . . . . . . . . . . . . . .
2.17 KamLAND 2008 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.18 Daya Bay Experimental Arrangement and Results . . . . . . . . . . . . . . . . .
2.19 Double CHOOZ Detector and Disappearance With θ13 Fit . . . . . . . . . . . . .
2.20 RENO Detector and θ13 Measurement From Rate . . . . . . . . . . . . . . . . . .

25
26
28
30
32
33
35
36
39
42
43

x

44
46
47
49
50
51
52
53
54

2.21 K2K Horn Magnets and Near Detector . . . . . . . . . . . . . . . . . . . . . . . .
2.22 K2K Neutrino Energy Spectrum and Oscillation Parameter Space . . . . . . . . .
2.23 MINOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.24 T2K Beamline and Near Detector . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.25 T2K νµ Disappearance Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.26 T2K νe Appearance Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.27 MiniBooNE Distributions and Allowed Regions . . . . . . . . . . . . . . . . . . .
2.28 OPERA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.29 Neutrino Factory and Beta-Beam Facility Layout . . . . . . . . . . . . . . . . . .
2.30 Megaton Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.31 NOνA and T2HK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56
57
58
59
60
61
63
64
66
68
69

3.1 The Super-Kamiokande Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.2 Event display of a typical neutrino event . . . . . . . . . . . . . . . . . . . . . . . 74
3.3 Quantum Efficiency of the PMTs together with the Cherenkov wavelength spectrum 75
3.4 Technical drawing of the SK PMT Support Structure . . . . . . . . . . . . . . . . 75
3.5 Inner Detector PMT technical drawing . . . . . . . . . . . . . . . . . . . . . . . . 77
3.6 ID PMT quantum efficiency and single p.e. distribution . . . . . . . . . . . . . . 77
3.7 Cover of PMTs since SK-II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.8 PMT cover transparency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.9 Water System [156]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.10 Air purification system [156]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.11 Radon levels in the air and control room . . . . . . . . . . . . . . . . . . . . . . . 81
3.12 Input proportion of the ATM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.13 Inner Detector DAQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.14 Outer Detector DAQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.15 Inner Detector Trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.16 SK-IV New Charge to Time Converter (QTC) . . . . . . . . . . . . . . . . . . . . 86
3.17 SK-IV New Online System [163]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.1 Measurements of primary cosmic ray proton flux and the Honda flux . . . . . . .
4.2 Secondary cosmic-ray muon fluxes . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 HARP JAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Illustration of the difference of the 3D cosmic-ray flux . . . . . . . . . . . . . . .
4.5 Neutrino production height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Zenith dependence of the atmospheric flux . . . . . . . . . . . . . . . . . . . . . .
4.7 Neutrino flux and flavour ratio prediction from cosmic-ray flux . . . . . . . . . .
4.8 Quasi-elastic cross-sections of νµ and ν̄µ for experimental data and NEUT . . . .
4.9 Cross-sections for CC DIS νµ and ν̄µ interactions . . . . . . . . . . . . . . . . . .
4.10 Cross-sections for π + −16 O scattering as a function of π + momentum . . . . . . .

89
90
91
91
92
93
94
95
97
99

5.1
5.2
5.3
5.4
5.5
5.6

103
104
105
106
106
107

Relative Gain Calibration with the Xe lamp . . . . . . . . . . . . . . . . . . . . .
Absolute Gain Calibration with the Ni source . . . . . . . . . . . . . . . . . . . .
Relative Timing Calibration with the diffuser ball . . . . . . . . . . . . . . . . . .
Water transparency from cosmic-ray muons . . . . . . . . . . . . . . . . . . . . .
Time variation of the water transparency from cosmic-ray muons . . . . . . . . .
Time variation of the water transparency from cosmic-ray muons . . . . . . . . .

5.7 Water parameter calibration by laser setup . . . . . . . . . . . . . . . . . . . . .
5.8 Water parameter calibration analysis distributions . . . . . . . . . . . . . . . . .
5.9 Attenuation Coefficient L−1 from the water parameter calibration . . . . . . . . .
5.10 Momentum spectra for the decay electrons . . . . . . . . . . . . . . . . . . . . . .
5.11 Typical π 0 event display showing two rings . . . . . . . . . . . . . . . . . . . . . .
5.12 Reconstructed invariant π 0 mass . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.13 Low energy stopping muon reconstructed momentum and Cherenkov angle . . . .
5.14 Low energy stopping muon Pp.e. /Pθ and MC/Data . . . . . . . . . . . . . . . . .
5.15 High energy stopping muon momentum/range . . . . . . . . . . . . . . . . . . . .
5.16 Time variation of decay electron momentum . . . . . . . . . . . . . . . . . . . . .
5.17 Time variation of momentum/range for stopping muons . . . . . . . . . . . . . .
5.18 Summary of energy scale calibrations . . . . . . . . . . . . . . . . . . . . . . . . .

108
109
109
111
112
113
114
115
116
117
118
119

6.1 Event Categorisation into FC, PC and UPMU . . . . . . . . . . . . . . . . . . . .
6.2 Expected energy spectrum for FC, PC and UPMU . . . . . . . . . . . . . . . . .
6.3 FC1 Reduction: PE300 distributions . . . . . . . . . . . . . . . . . . . . . . . . .
6.4 FC1 and FC2 Reduction: NHITA800 distributions . . . . . . . . . . . . . . . . . .
6.5 FC2 Reduction: PEmax /PE300 distributions . . . . . . . . . . . . . . . . . . . . .
6.6 FC3 Reduction: through-going muon SK1 NHITAin and NHITAout distributions .
6.7 FC3 Reduction: through-going muon SK2 NHITAin and NHITAout distributions .
6.8 FC3 Reduction: stopping muon NHITAin distributions . . . . . . . . . . . . . . .
6.9 Illustration of the cable hole muon veto system . . . . . . . . . . . . . . . . . . .
6.10 Reconstructed vertex distributions before and after the veto counter installation .
6.11 FC3 Reduction: Flasher Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.12 Event display of a typical flasher event . . . . . . . . . . . . . . . . . . . . . . . .
6.13 FC3 Reduction: flasher SK1 NMIN100 distributions . . . . . . . . . . . . . . . . .
6.14 FC3 Reduction: flasher SK2 NMIN100 distributions . . . . . . . . . . . . . . . . .
6.15 FC4 Reduction: Estimator r against Nmatch scatter plots . . . . . . . . . . . . . .
6.16 FC5 Reduction: stopping muon NHITAin distributions . . . . . . . . . . . . . . .
6.17 FC5 Reduction: invisible muon NHITACearly distributions . . . . . . . . . . . . .
6.18 FC5 Reduction: accidental coincidence muon PElate distributions . . . . . . . . .
6.19 PC1 Reduction: TWIDA distributions . . . . . . . . . . . . . . . . . . . . . . . .
6.20 PC Second Reduction cluster algorithm . . . . . . . . . . . . . . . . . . . . . . .
6.21 PC2 Reduction: NCLSTA2 against PE200 scatter plots . . . . . . . . . . . . . . .
6.22 PC2 Reduction: NHITACmin distributions . . . . . . . . . . . . . . . . . . . . . .
6.23 PC2 Reduction: NHITAendcap against NHITAside scatter plots . . . . . . . . . . .
6.24 PC2 Reduction: NHITAC2 against PE200 scatter plots . . . . . . . . . . . . . . .

120
121
122
123
124
125
125
126
127
128
128
129
129
130
132
133
134
134
137
138
139
140
140
141

Ring Edge Search P E(θ) distribution and it’s second derivative . . . . . . . . . .
The treatment of Cherenkov radiation in TDC-Fit . . . . . . . . . . . . . . . . .
Illustration of the concept of the Hough Transform . . . . . . . . . . . . . . . . .
An example charge map resulting from the Hough Transform . . . . . . . . . . .
Illustration of Cherenkov radiation from a muon . . . . . . . . . . . . . . . . . . .
PID efficiency for single-ring events . . . . . . . . . . . . . . . . . . . . . . . . . .
A typical Sub-event type Decay Electron . . . . . . . . . . . . . . . . . . . . . . .

153
153
155
156
160
161
163

7.1
7.2
7.3
7.4
7.5
7.6
7.7

7.8

Reconstruction efficiency for single-ring events before and after the ring number
correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
7.9 The relationship of momentum for π 0 and γs for NC1π 0 events . . . . . . . . . . 166
7.10 Flux of Upward going muon events against energy by event type . . . . . . . . . 168
8.1
8.2
8.3
8.4
8.5
8.6

L/E Oscillation Survival Probability . . . . . . . . . . . . . . . . . . . . . . . . . 170
PC Separation, P Eanti /P Eexp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
Separation of overlapping rings in the PC Sample . . . . . . . . . . . . . . . . . . 175
Eν Reconstruction from Evis2 fitting . . . . . . . . . . . . . . . . . . . . . . . . . 177
L/E Analysis Energy Resolution for SK4 . . . . . . . . . . . . . . . . . . . . . . . 177
Eν Resolution against lepton momentum . . . . . . . . . . . . . . . . . . . . . . . 178(a) SK1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178(b) SK2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178(c) SK3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178(d) SK4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
8.7 Flightpath vs Zenith . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
8.8 L/E Angular Resolution for SK4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
8.9 Angular Resolution against energy . . . . . . . . . . . . . . . . . . . . . . . . . . 181(a) SK1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181(b) SK2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181(c) SK3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181(d) SK4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
8.10 L/E Sample Contributions and L/E Contours . . . . . . . . . . . . . . . . . . . . 182
8.11 L/E Resolution Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
8.12 L/E Resolution Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
8.13 L/E (true/rec) Peak and Resolution . . . . . . . . . . . . . . . . . . . . . . . . . 186
8.14 True L/E over Reconstructed L/E Ratio, and Resolution Cut Contour . . . . . . 188
8.15 Neutrino Flight Path Length Variation . . . . . . . . . . . . . . . . . . . . . . . . 192
8.16 Flux Calculation Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
8.17 2-Flavour L/E distributions and L/E over null oscillation, SK 1+2+3+4 . . . . . 199
8.18 2-Flavour L/E Analysis SK 1+2+3+4 Oscillation Allowed Region . . . . . . . . . 200
8.19 2-Flavour L/E Analysis SK 1+2+3+4 ∆m232 and Sin2 2θ23 1D contours . . . . . . 201
8.20 2-Flavour L/E distributions over the null oscillation expectation for each SK period202
8.21 2-Flavour L/E Analysis oscillation parameter allowed region for each SK period . 203(a) SK1 2D Allowed Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203(b) SK2 2D Allowed Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203(c) SK3 2D Allowed Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203(d) SK4 2D Allowed Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
8.22 2-Flavour L/E distributions over the null oscillation expectation for each SK period204
8.23 2-Flavour L/E Analysis comparisons, Decay and Decoherence, other experiments 205
9.1
9.2
9.3
9.4
9.5

L/E at the 2011 MINOS Best Fit Values . . . . . . . . . . . . . . . . . . . . . . .
Muon Neutrino Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Muon Neutrino Flux Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
L/E Resolution Contour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Second Sample Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

208
210
211
212
213

(a) 70-80 % Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(b) 70-90 % Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(c) 70-100 % Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.6 Second Sample Binning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(a) 70-80 % Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(b) 70-90 % Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(c) 70-100 % Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.7 Sensitivity to Second Sample Regions . . . . . . . . . . . . . . . . . . . . . . . . .
9.8 Neutrino Anti-Neutrino L/E Bestfit Distributions . . . . . . . . . . . . . . . . . .
9.9 Neutrino Anti-Neutrino L/E bestfit over null oscillation prediction . . . . . . . .
9.10 2D Allowed Region Contours for Neutrino and Anti-Neutrino . . . . . . . . . . .
9.11 sin2 2θ23 Contours for Neutrino and Anti-Neutrino . . . . . . . . . . . . . . . . .
9.12 ∆m232 Contours for Neutrino and Anti-Neutrino . . . . . . . . . . . . . . . . . . .
9.13 sin2 2θ23 and ∆m232 Contours for Normal Oscillation . . . . . . . . . . . . . . . .
9.14 Estimates of The Systematic Errors . . . . . . . . . . . . . . . . . . . . . . . . . .
9.15 Systematics at Best Fit Neutrino Anti-Neutrino Analysis . . . . . . . . . . . . . .
9.16 2D Allowed Region Contours for Normal Oscillation . . . . . . . . . . . . . . . .
9.17 Neutrino and Anti-Neutrino Allowed Region Comparisons . . . . . . . . . . . . .

213
213
213
214
214
214
214
215
220
221
222
223
223
225
226
227
228
229

10.1 P (νµ → νe ) Distributions for Normal and Inverted Hierarchies . . . . . . . . . . .
(a) P (νµ → νe ) Normal Hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . .
(b) P (ν̄µ → ν̄e ) Normal Hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . .
(c) P (νµ → νe ) Inverted Hierarchy . . . . . . . . . . . . . . . . . . . . . . . . .
(d) P (ν̄µ → ν̄e ) Inverted Hierarchy . . . . . . . . . . . . . . . . . . . . . . . . .
10.2 3-Flavour L/E Bestfit Distributions . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3 3-Flavour L/E Bestfit Over Null Oscillation Expectation . . . . . . . . . . . . . .
10.4 3-Flavour L/E allowed regions against sin2 θ23 . . . . . . . . . . . . . . . . . . . .
10.5 3-Flavour L/E 1D Allowed Regions . . . . . . . . . . . . . . . . . . . . . . . . . .
10.6 L/E 2-Flavour and 3-Flavour allowed regions against sin2 2θ23 . . . . . . . . . . .
10.7 Systematics at Best Fit 3-Flavour L/E Analysis, Normal Hierarchy . . . . . . . .
10.8 Systematics at Best Fit 3-Flavour L/E Analysis, Inverted Hierarchy . . . . . . . .
10.9 3-Flavour L/E Distribution Ratios . . . . . . . . . . . . . . . . . . . . . . . . . .
10.103-Flavour L/E χ2 and Systematics . . . . . . . . . . . . . . . . . . . . . . . . . .
10.113-Flavour L/E 2D Allowed Region Comparisons . . . . . . . . . . . . . . . . . . .

234
234
234
234
234
236
237
237
238
239
241
242
243
244
245

11.1 NSI Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(a) P (νµ → νµ ) Neutrino εµτ = 0.02 . . . . . . . . . . . . . . . . . . . . . . . .
(b) P (ν̄µ → ν̄µ ) Anti-Neutrino εµτ = 0.02 . . . . . . . . . . . . . . . . . . . . .
(c) P (νµ → νµ ) Neutrino, Oscillation only . . . . . . . . . . . . . . . . . . . . .
(d) P (ν̄µ → ν̄µ ) Anti-Neutrino, Oscillation only . . . . . . . . . . . . . . . . . .
(e) P (νµ → νµ ) Neutrino ετ τ − εµµ = 0.08 . . . . . . . . . . . . . . . . . . . . .
(f) P (ν̄µ → ν̄µ ) Anti-Neutrino ετ τ − εµµ = 0.08 . . . . . . . . . . . . . . . . . .
11.2 NSI Bestfit L/E Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3 NSI Bestfit L/E Over No Oscillation Expectation . . . . . . . . . . . . . . . . . .
11.4 Bestfit L/E Compared With Various NSI Parameters . . . . . . . . . . . . . . . .
11.5 NSI and Oscillation 2D Allowed Regions . . . . . . . . . . . . . . . . . . . . . . .

254
254
254
254
254
254
254
256
257
257
258

11.6 NSI 1D Allowed Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.7 Oscillation 1D Allowed Regions, 4D NSI Analysis . . . . . . . . . . . . . . . . . .
11.8 ∆m231 vs εµτ Allowed Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.9 Systematics at Best Fit NSI L/E Analysis . . . . . . . . . . . . . . . . . . . . . .
11.10NSI Comparisons: εµτ × ε0 plane . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.11NSI Comparisons: εµτ × ∆m231 plane . . . . . . . . . . . . . . . . . . . . . . . . .

259
260
260
262
263
265

A.1 SK 1 to 4: Eν Reconstruction from Evis2 fitting . . . . . . . . . . . . . . . . . . .
(a) SK1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(b) SK2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(c) SK3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(d) SK4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.2 L/E Energy Resolution, SK1 and SK2 . . . . . . . . . . . . . . . . . . . . . . . .
A.3 L/E Energy Resolution, SK3 and SK4 . . . . . . . . . . . . . . . . . . . . . . . .
A.4 L/E Angular Resolution, SK1 and SK2 . . . . . . . . . . . . . . . . . . . . . . . .
A.5 L/E Angular Resolution, SK3 and SK4 . . . . . . . . . . . . . . . . . . . . . . . .
A.6 SK 1 to 4: True L/E over Reconstructed L/E Ratio . . . . . . . . . . . . . . . . .
A.7 SK 1 to 4: L/E Resolution Cut by Sample . . . . . . . . . . . . . . . . . . . . . .

273
273
273
273
273
274
275
276
277
278
279

List of Tables
1.1

Global fit of oscillation parameters . . . . . . . . . . . . . . . . . . . . . . . . . .

11

3.1
3.2

Water Purification System components and their purpose. . . . . . . . . . . . . .
Air Purification System components and their purpose. . . . . . . . . . . . . . . .

79
80

4.1

List of the simulated processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.1
6.2
6.3
6.4

FC Reduction: detection efficiency after each step . . . . . . . . . . . . . . . . . .
FC Reduction: Estimated upper limit of the background contamination . . . . .
PC Reduction: detection efficiency after each step . . . . . . . . . . . . . . . . . .
PC Reduction: Estimated upper limit of the background contamination . . . . .

7.1

The vertex and angular resolution of MS-Fit . . . . . . . . . . . . . . . . . . . . . 161

8.1
8.2
8.3
8.4
8.5

Summary of events for the L/E distributions from SK1 to SK4 . . . . . . . . . .
2-Flavour L/E Neutrino Flux Uncertainties . . . . . . . . . . . . . . . . . . . . .
2-Flavour L/E Neutrino Interaction Uncertainties . . . . . . . . . . . . . . . . . .
2-Flavour L/E Neutrino Reconstruction Uncertainties . . . . . . . . . . . . . . .
2-Flavour L/E Neutrino Reduction Uncertainties . . . . . . . . . . . . . . . . . .

187
194
195
197
198

9.1
9.2
9.3
9.4

ν ν̄ L/E Neutrino Flux Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . .
ν ν̄ L/E Neutrino Interaction Uncertainties . . . . . . . . . . . . . . . . . . . . . .
ν ν̄ L/E Neutrino Reconstruction Uncertainties . . . . . . . . . . . . . . . . . . .
ν ν̄ L/E Neutrino Reduction Uncertainties . . . . . . . . . . . . . . . . . . . . . .

217
218
219
220

10.1 Oscillation Parameter Results Summary . . . . . . . . . . . . . . . . . . . . . . .
10.2 3F L/E Neutrino Interaction Uncertainties . . . . . . . . . . . . . . . . . . . . . .
10.3 3F L/E Neutrino Flux Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . .
10.4 3F L/E Neutrino Reconstruction Uncertainties for Normal Hierarchy . . . . . . .
10.5 3F L/E Neutrino Reconstruction Uncertainties for Inverted Hierarchy . . . . . . .
10.6 3F L/E Neutrino Reduction Uncertainties for Normal Hierarchy . . . . . . . . . .
10.7 3F L/E Neutrino Reduction Uncertainties for Inverted Hierarchy . . . . . . . . .

246
247
248
249
250
251
251

11.1 3F Osc+NSI L/E Neutrino Interaction Uncertainties . . . . . . . . . . . . . . . .
11.2 3F Osc+NSI L/E Neutrino Reduction Uncertainties . . . . . . . . . . . . . . . .
11.3 3F Osc+NSI L/E Neutrino Flux Uncertainties . . . . . . . . . . . . . . . . . . . .
11.4 3F Osc+NSI L/E Neutrino Reconstruction Uncertainties . . . . . . . . . . . . . .

267
267
268
269

xvi

135
136
148
149

Chapter 1

Introduction
1.1

Introduction to Neutrinos

The neutrino is one of the more mysterious of the elementary particles. It is the only, currently
known, particle that solely interacts by the Weak Force, and it is the second most abundant
substance in the universe after the photon. This means that a neutrino will very readily transverse
the whole Earth without interacting! Indeed, ten thousand billion neutrinos from the sun passthrough an area the size of your hand every second, [1] and that is not including the many otherneutrino sources!
The neutrino first came into our awareness to explain the continuous energy distribution ofemitted electrons from β-decay. In 1930, Wolfgang Pauli [2] postulated that a ghostly particle,
the neutrino, was carrying off some of the energy the β-electron was expected to have. He wasworried because the particle he had postulated, at that time, would have been very difficult to
detect. It took a further 26 years until the neutrino was finally discovered in 1956 by Cowan
and Reines [3, 4]. Their experiment made use of a highly distinctive delayed-coincidence signal
from inverse β-decay, ν̄e + p → n + e+ , where the positron was detected promptly followed by adelayed γ signal from the neutron capture a few µs later.
Of course there have since been many more experiments, employing massive detectors tobetter catch neutrinos and gradually establishing our knowledge of them. Super-Kamiokande isa modern experiment, weighing at 50 kton, that can detect as many as 20–30 neutrinos per day.
We know that there are 3 flavours of active neutrinos, one for each flavour of lepton, with the νµdiscovered in 1962 [5] and the ντ in 2000 [6]. However the flavour ratios of neutrinos detectedfrom the sun and the atmosphere were not as expected, and so multiple theories were proposedto account for the deficit of the expected neutrino events measured.
After years of experimentation, a theory known as “Neutrino Oscillations” [7, 8] is thought tobe the most likely explanation for the anomaly in flavour ratios observed. This theory proposesthat the flavour of a neutrino detected may change from the flavour at its creation. The probability for the change depends on the flight path length, L (km), and the energy of the neutrino,Eν (GeV). More precisely, how frequently the neutrino flavours cycle between each other, oroscillate, has an L/E dependence. The Super-Kamiokande L/E analysis [9] excluded the otherproposed theories.
In recent years, since 2009, the MINOS experiment has been collecting data with an antineutrino beam produced by a particle accelerator. Surprisingly, when they applied the standard
1

Figure 1.1: Neutrino and Anti-neutrino parameter 90% C.L. allowed regions from MINOS [11].
The initial anti-neutrino result reported in 2011 [10] can be seen as the dashed-red line. Thisregion was distinct from the neutrino parameters (gray shaded area), which formed part of the
motivation of the topics in this thesis. The updated anti-neutrino result reported in 2012 [11]can be seen as the blue solid contour. Clearly, the neutrino and anti-neutrino parameters are
now in agreement. The anti-neutrino search at SK with the Zenith analysis [12] is shown by thedash-dotted green contour.

neutrino oscillation analysis to their initial data in 2010, the oscillation parameters they measured
were considerably different from that of the best fits to their neutrino-beam data [10]. Specifically,a much larger ∆m̄2 was reported in 2011 compared with the neutrino case, which can be seen in
the dashed red contour (2009-2010 data taking period) in Figure 1.1. At this time the ν and ν̄
oscillation parameters measured at MINOS were consistent only at the 2% C.L. These surprising
results were an early inspiration for the analyses in this thesis.
It should be noted that MINOS continued to improve their statistics with a beam optimisedfor ν̄µ , as well as performing additional checks. After running the ν̄µ beam from 2009 to 2011,
MINOS reported on an improved analysis in 2012 that had an allowed region that overlappedwith the neutrino measurement [11], which can be seen in solid blue in Figure 1.1.
However, are the flavour transitions of neutrinos and anti-neutrinos truly governed by thesame mechanism? Until recently, this property was simply assumed. As the foundation of Neutrino Oscillations depends on this holding true, it is crucial to have independent confirmations
that neutrinos and anti-neutrinos oscillate in the same way. Although the MINOS experimenthas shown consistency among the ν and ν̄ oscillation parameters, another confirmation by anexperiment with a different scope of sensitivity is important. This question has also opened thepossibility for alternative explanations, such as sub-dominant Non-Standard Neutrino Interactions (NSI) acting in the background together with Neutrino Oscillations. These scenarios also
must be investigated.
2

1.2

Thesis Motivations

If nature is such that through an oscillation analysis we are able to measure differing parameters
for neutrinos and anti-neutrinos, could this effect be seen at Super-Kamiokande (SK)? Theoretically, Neutrino Oscillations should be CPT symmetric and so the parameters measured for ν andν̄ should be the same. CPT symmetry is a fundamental characteristic of the Standard Model,
however the apparent discrepancies in 2011 lead to considerations that this property should bechecked in neutrinos, rather than assumed.
The data at SK are composed of neutrino and anti-neutrino constituents, however these were
not separated as SK has very little power to distinguish the charge of outgoing leptons. So
a direct check on independent particle and anti-particle oscillation parameters would have to
involve oscillating neutrinos and anti-neutrinos independently in the simulation and compare theoutput with the data. Such an analysis was carried out at SK with distributions for the incoming
neutrino direction with respect to the vertical axis of the detector (a Zenith analysis) [12], which
can be seen in the dash-dotted green line in Figure 1.1. The analyses in this thesis aim toreconstruct the L/E pattern expected for oscillations, and use these distributions to investigate
neutrino oscillations accurately or deviations from pure oscillations. If no differences in the ν andν̄ oscillation patterns were found, this would form another confirmation of the CPT symmetry
in neutrinos within the sensitivity of SK.
The apparent appearance of a different ∆m2 for neutrinos and anti-neutrinos assuming vacuum oscillations could also be explained by oscillations with one ∆m2 together with NeutralCurrent (NC) Non-Standard Neutrino Interactions (NSI) of neutrinos with matter [13]. In an
analysis of oscillations in vacuum, the effect of NSI acting in the background1 would causethe ∆m2 to be measured larger for ν̄ and smaller for ν from the actual ∆m2 . So this was apromising explanation for the results reported at MINOS. With this as an initial motivation, an
NSI-oscillation analysis with the L/E distributions was also carried out for this thesis.

1.2.1

The L/E Analysis with Atmospheric Neutrinos at Super-Kamiokande

Super-Kamiokande (SK) has collected atmospheric neutrino data for over 10 years, covering awide range of energies and baselines. This makes the data ideal for checking a variety of oscillation parameters and for observing potentially unexpected effects; compared to an oscillation
experiment tuned to a particular range of L/E. SK is also the largest pure water Cherenkov
detector in the world, in the GeV range, giving it a respectable neutrino event rate of about
20–30 events per day.
There are two clear figures of merit for performing an L/E analysis of atmospheric neutrino
data at SK. Atmospheric neutrinos traverse great distances through matter, from ∼10 km toover 104 km, with flight paths including the Earth’s mantle and core. This gives atmospheric
neutrino experiments the best sensitivity to matter effects, such as those expected from NC NSI
or from the Mikheyev–Smirnov–Wolfenstein (MSW) effect (Section 1.4.3). In addition, since
the L/E Analysis directly reconstructs the characteristic pattern of neutrino oscillations overthis wide range of distances, it provides the strongest constraint on the atmospheric ∆m2 fromatmospheric neutrinos. This makes it an ideal choice for studies that are highly dependent on
the measurement of ∆m2 .
1

Assuming a negative εµτ NSI term and normal neutrino mass hierarchy3

1.2.2

Neutrino Anti-Neutrino L/E Analysis

The L/E Analysis has the best discrimination of ∆m2 from atmospheric neutrino studies, and so
could potentially have the sensitivity to pick out a secondary ∆m̄2 for anti-neutrinos underlying
the data. Since SK cannot distinguish ν from ν̄, this investigation would involve oscillating ν andν̄ independently, and then fitting an L/E distribution with two ∆m2 , and two sin2 θ. From this,the allowed ranges of these parameters can be found. An equivalent measurement for particle
and anti-particle would confirm the CPT symmetry in neutrino oscillations, while a significant
difference between these parameters would signify the need for new physics.

1.2.3

3-Flavour L/E Analysis

In 2012, the oscillation parameter θ13 , which allows νµ ↔ νe mixing, was found to be non-zeroand not-so-small [14]. So it is better to do analyses with 3-flavour oscillations rather than onlywith the 2-flavour approximation. For this reason, a 3-flavour L/E Analysis was performed forthe first time. All other L/E analyses with neutrino data from any experiment have relied on the
2-flavour approximation. Whether the L/E distributions are still good for measuring oscillation
parameters under the presence of 3-flavour and matter effects will be investigated in this thesis.
The L/E Analysis can provide an independent measurement of the oscillation parametersfrom the Zenith 3-flavour analysis [15], by the measurement of a different variable and by the useof a sub-sample of data. The analyses in this thesis directly measure the distribution of the L/Evariable, a fundamental characteristic of neutrino oscillations. In addition, this measurement ismade with selected samples of neutrinos with a precise resolution in L/E. So an independent
determination of the oscillation parameters can be made from the same source SK data. Furthermore, this result should be more accurate than any previous L/E analysis since it is freefrom the approximations of a 2-flavour approach.

1.2.4

NSI L/E Analysis

The observation of neutrino oscillations and the necessary neutrino masses already require an
extension to the Standard Model, which originally postulated massless neutrinos. So the properties of neutrinos act as a good probe for potential new physics beyond the Standard Model.
Neutral-Current neutrino NSI in matter are a category of possible non-standard effects that canbe investigated with atmospheric neutrinos. A 3-Flavour Oscillation and NSI analysis in theµτ –sector with atmospheric neutrino data was carried out for the first time. All previous measurements of NSI with atmospheric neutrinos in the µτ –sector have made use of 2-flavour hybrid
models.
Atmospheric neutrinos traverse great distances through matter, spanning the entire Earth,including the mantle and core. This provides unparalleled sensitivity to potential matter effects,
such as in this NC matter NSI measurement. Particularly, sensitivity to the εµτ NSI term isdependent on the measurement of ∆m2 , making the L/E Analysis a good choice to study NSI.

1.3

Atmospheric Neutrinos

The Earth is constantly bombarded by a massive flux of primary comic rays, ∼90% protons and∼9% α-particles [17], that interact with nuclei in the atmosphere to produce secondary particles,
mostly pions and some kaons, in hadronic showers. The decay of these mesons and subsequent4

Figure 1.2: Atmospheric neutrino flux ratios on the left, and the zenith distribution of the flux
on the right, calculated by Honda et al. in 2011 [16]. The left figure shows muon to electron
flavour ratio of atmospheric neutrinos, as well as neutrino to anti-neutrino ratios. The red solid
is for the Honda 2011 calculation, while comparisons with other calculations are also shown. The
right plot has been averaged over all azimuthal angles and is for the Kamioka site. The neutrinos
arrive from directly above the detector for cos θ = 1.0, and from directly below at cos θ = −1.0.
The flux is nearly up-down symmetric above around ∼3 GeV.
particles produce atmospheric neutrinos. The main contributor to these neutrinos comes fromthe decay chains of pions:
π + → µ+ + νµ
→ e+ + νe + ν µ ,
π

−

(1.1)

−

→ µ + ν̄µ→ e− + ν̄e + νµ .

(1.2)

As seen in these decay chains, each pion produces two muon neutrinos and one electron neutrino.
So from the atmospheric neutrino flux, a (νµ + ν̄µ )/(νe + ν̄e ) flavour ratio of 2 is expected. For
higher energy cosmic rays, the muons have a greater probability of reaching the ground before
decaying. So for neutrinos above a few GeV, there are less electron neutrinos and the flavour
ratio increases, which can be seen in Figure 1.2. A nice feature of this ratio is that all the flux
calculations essentially agree on its value and variation with energy, leading to a much smaller
uncertainty, of around 3%, in the quantity. While the absolute flux of each calculation agree
within about 10% for neutrinos below ∼10 GeV.
The zenith dependence of the atmospheric neutrino flux for sample energies is also shownin Figure 1.2. Where cos θ = 1.0 corresponds to downward going neutrinos relative to the
Kamioka site; and cos θ = -1.0 to upward going neutrinos. It can be seen that above ∼3 GeV, thedistributions are largely up-down symmetric. There is a geomagnetic cutoff rigidity, below whichprimary comic rays do not make it through the magnetosphere to produce secondary particles
in the atmosphere. The cutoff rigidity varies at different locations in the world. Below ∼3 GeV
at Kamioka, due to this cutoff, the up-down neutrino fluxes are asymmetric.
Atmospheric neutrino flux calculations show that a (νµ + ν̄µ )/(νe + ν̄e ) flavour ratio of 2
or above is expected and that this ratio is a reliable quantity to measure, as well as the fluxes
5being up-down symmetric above a few GeV. However experiments have measured a deficit in thisratio, and in particular a deficit in νµ or ν̄µ arriving from below the detector. This result can be
explained by the theory of Neutrino Oscillations. The atmosphere provides a source of neutrinos
that travel between ∼10 km to ∼13,000 km to reach a detector, and so atmospheric experiments
are in a good position to probe the neutrino flight path over energy (L/E) dependence of neutrino
oscillations.

1.4

Neutrino Oscillations

1.4.1

3-Flavour Oscillations

Many experiments have shown that neutrinos can change from one flavour to another duringtransit, and the theory which best describes this is Neutrino Oscillations. At production, or
after a measurement, a neutrino is in one of three flavour eigenstates: |νe i, |νµ i, |ντ i. Thisflavour eigenstate is defined by the flavour of the lepton associated with the neutrino during
creation or detection. During transit, the neutrino is a superposition of three mass eigenstates:|ν1 i, |ν2 i, |ν3 i. The composition of the mixing is described by the 3 × 3 Unitary Lepton Mixing
Matrix, U , also known as the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) Mixing Matrix. The
two sets of eigenstates are related by the equation:|να i =

3X∗
Uαi
|νi i,(1.3)

i=1

where α = e, µ, or τ label one of the flavour eigenstates, and i = 1, 2, or 3 indicate the mass
eigenstates. An insightful parameterisation of U for vacuum oscillations is:
0 s13 e−iδcp
c12 s12 0 −s12 c12 0 ,
1
0
0
0 1
0
c13



1
0
0
c13



0
U = 0 c23 s23
0 −s23 c23−s13 eiδcp(1.4)

where sij ≡ sin θij and cij ≡ cos θij . U can be seen in Eq. (1.4) as three independent rotations
characterised by the mixing angles θ12 , θ23 , and θ13 . While δcp is a complex phase that allows
for CP violations in the neutrino sector2 . The neutrino in transit will be one of the νi mass
eigenstates, however since we do not know which mass eigenstate was created, we treat the time
evolution as a superposition of each of the mass eigenstates [18].
|να (t)i =X∗ −iEi tUαie|νi (t = 0)i,

(1.5)

i

2

To allow for the case of Majorana neutrinos in which the neutrino is its own antiparticle, another matrix
containing two more complex phases can be multiplied on to U . Since these phases are not of interest in thisthesis, they have not been included. Similarly, the case for more sterile mass eigenstates has not been considered.

6where Ei is an energy eigenvalue of νi . The probability amplitude of detecting a neutrino in theflavour eigenstate |νβ i at time t, from one created as |να i at time t = 0:A(να → νβ )(t) ≡ hνβ |να (t)iX
∗ −iEi t=hνβ |Uαi
e
|νi i

(1.6)
(1.7)

i

=

X X
i

=
=

X

(1.8)

∗
Uαi
Uγi e−iEi t hνβ |νγ i(1.9)

γ=e,µ,τ

X X
i∗ −iEi thνβ |Uαie
Uγi |νγ i

γ=e,µ,τ
∗Uαi
Uβi e−iEi t ,(1.10)iwhere in the line (1.8), U has been used to express the propagated νi back into a superpositionof flavour eigenstates, using the sum with label γ = e, µ or τ . The probability for the flavour
transition is then given by:
P (να → νβ )(t) = |A(να → νβ )(t)|2X∗∗ −i(Ei −Ej )t
=
e
.
Uαi
Uβi Uαj Uβj(1.11)

i,j

Considering natural units (c = } = 1) and using the following approximation:
s


q
m2i
m2i
m2
2
2
Ei = pi + mi = pi 1 + 2 ' pi 1 + 2 = pi + i ,
2pi
pi
2pi
the energy component of the exponential phase in Equation (1.11) becomes:
!


m2j
m2i
− pj +
Ei − Ej = pi +
2pi
2pj
=

∆m2ij
,
2Eν

(1.12)(1.13)(1.14)

where ∆m2ij = m2i − m2j is the mass squared difference between two mass eigenstates, and therelativistic neutrino energy Eν ' |pi | = |pj |, neglecting the mass contribution. Experiments
typically do not measure the flight time of the neutrino, but the flight path between the sourceand detector, Lν . Since Lν ≡ t with a factor of c, Equation (1.11) can be written:!
X∆m2ij Lν∗∗.
(1.15)P (να → νβ )(t) =Uαi Uβi Uαj Uβj exp −i
2Eν
i,jIn the case for anti-neutrinos, assuming that CPT invariance is maintained, the treatment of theflavour transition probability is the same only that the complex conjugate of U would be usedin the anti-neutrino case: P (να → νβ ; U ∗ ) ≡ P (ν̄α → ν̄β ; U ). If U is real, these probabilities areequivalent; and if U is complex, these probabilities differ in general.
7Figure 1.3: Neutrino Oscillations survival probability
typical for atmospheric neutrinos, where the following
parameters were assumed: sin2 2θ = 1.0 and ∆m2 =2.2 × 10−3 eV2 . Since oscillations depend on energy,
three plots have been shown: Eν = 1 GeV in black,Eν = 10 GeV in red, Eν = 100 GeV in green.

1.4.2

2-Flavour Approximation

The rotation in the 12 plane shown in Eq. (1.4) is associated with solar mixing, whereas the 23
plane rotation is associated with atmospheric experiments. In 1999, CHOOZ set an upper limiton θ13 . 12◦ , so it was known that θ13 is small [19]. This means that the middle rotation in U
tends to the identity, and U decouples to two 2 × 2 mixing phenomena to a good approximation.
Since experiments tend to have sensitivity to one of these rotations, as an initial analysis the
mixing between two flavours only was usually considered. These rotations can be written as:  ναcos θij=νβ− sin θijsin θijcos θij

 
νi
,
νj

(1.16)

where α and β are two of e, µ or τ , whereas i and j correspond to the associated mass states.
The time evolution of a neutrino state, |να i for example, can be treated like so:|να (t)i = e−iEi t cos θij |νi i + e−iEj t sin θij |νj i.

(1.17)

Following the same procedure as the 3-flavour case, and the energy approximation in Equation (1.12), the transition probability of a two-flavour oscillation can be shown to be:
P (να → νβ ) = |hνβ |να (t)i|2= sin2 2θij sin2
22

= sin 2θij sin∆m2ji Lν
4Eν!

1.267∆m2ji LνEν(1.18)!
,(1.19)

where ∆m2ji = m2j − m2i is the mass squared difference, Lν is the baseline, Eν is the neutrinoenergy, and making use of the identities sin 2x = 2 sin x cos x and 2 sin2 x = 1−cos 2x. The figure“1.267” in Equation (1.19) allows the use of the units ∆m2ji in eV2 , Lν in km and Eν in GeV. Of
8

course, the survival probability that is normally measured by experiments can be written as:!
2L
1.27∆mνji
P (να → να ) = 1 − sin2 2θij sin2.
(1.20)
Eν
It can be seen in Equation (1.20) that the magnitude of the effect is determined by θij , while
the frequency of the oscillations in L/E by ∆m2ji . Some examples of this survival probability
can be seen in Figure 1.3. The first oscillation maximum, where the survival probability is at aminimum, occurs at:∆m2ji Lνπ=.
(1.21)
Eν
2.534
This relationship can be considered to tune the sensitivity in ∆m2ji of an experiment. Here aresome examples:
CHOOZ

L = 1 kmE = 0.0018 GeV

T2K

L = 295 km E = 0.6 GeV

KamLAND L = 180 km E = 0.0018 GeV−→ ∆m2 ∼ 2.23 × 10−3 eV2 ,−→ ∆m2 ∼ 2.52 × 10−3 eV2 ,−→ ∆m2 ∼ 1.24 × 10−5 eV2 .

CHOOZ (Section 2.3.1) and KamLAND (Section 2.3.2) were reactor experiments, and they had
sensitivity to oscillations above the stated ∆m2 here. T2K (Section 2.4.3) is an accelerator experiment and was tuned to maximise its sensitivity around this ∆m2 . Neutrino Oscillations canessentially be defined by two mass square differences, one at ∆m2 ∼ 10−5 eV2 for the oscillations
driven by solar parameters; and ∆m2 ∼ 10−3 eV2 for oscillations governed by atmospheric parameters and θ13 effects. Since the ∆m2 differ largely in magnitude, oscillation experiments canbe approximated by one 2-flavour 2×2 rotation matrix.

1.4.3

Matter Effects

The above discussion of neutrino oscillations was for propagation in vacuum. However, as neutrinos traverse through matter they undergo coherent forward scattering on electrons, protons
and neutrons. This leads to oscillation with effective mass square differences and mixing angles,that vary from vacuum oscillations. This is known as the MSW effect [20–22], or the matter
effect.
To understand how this effect occurs [18], consider the propagation of neutrinos in matterby the following Schrödinger equation:i∂Ψ(t) = HΨ(t),∂t(1.22)

where Ψα (t) is a component of the above wave function that gives the amplitude for a neutrino
with flavour α at time t. H is the hamiltonian and a 3 × 3 matrix for three flavours. This
hamiltonian has interaction energies that result from neutrino-matter interactions through W
or Z exchange. All neutrino flavours can interact through NC Z exchange, while only electron
neutrinos interact via CC W exchange, as matter has a high density of electrons. For standard
model interactions, NC interactions result in no flavour change and so Z exchange does notcontribute to the oscillation effects observed3 , which depend only on the relative phases of theneutrino eigenstates.
3

non-standard neutrino interactions could allow NC flavour change, see Section 1.5

9

νe

e

W
Figure 1.4: Matter effects that have an influence on
flavour oscillations occur via the W exchange interaction,
shown in this Feynman diagram.
eνeHowever, νe may interact via W exchange, which results in a contribution that does have aneffect on the oscillation transitions. A feynman diagram of the W exchange interaction is shownin Figure 1.4. The electron-electron element of H would gain the interaction energy:√
V = 2GF Ne ,(1.23)
where GF is the Fermi coupling constant, and Ne is the electron number density per unit volume
in the neutrinos flight path. For ν̄e propagation the interaction energy becomes −V .
For an illustration of the effect [18], consider oscillations in matter involving two flavours νeand νµ , then H becomes:∆m2M − cos 2θM sin 2θMH=,(1.24)
sin 2θM
cos 2θM
4Eν
where ∆m2M is the effective mass squared difference, θM is the effective mixing angle, and Eν isthe neutrino energy. These effective parameters are defined by:q(1.25)∆m2M = ∆m2 sin2 2θ + (cos 2θ − xν )2 ,sin2 2θM =sin2 2θ
,sin2 2θ + (cos 2θ − xν )2where xν quantifies the degree of the effect and is defined:√
2 2GF Ne Eν
xν ≡
.
∆m2

(1.26)(1.27)

It can be seen that these effective parameters reduce to their vacuum equivalent, ∆m2 and θ,
when the parameter xν vanishes as Ne tends to 0. For anti-neutrinos xν̄ (≡ −xν ) is used instead.
As matter effects (including the non-standard case) have a different effect on neutrinos and antineutrinos, they can lead to a difference in the oscillation parameters observed for particles andanti-particles. This should not be confused with CP violations, which need to be distinguishedfrom matter effects. Finally, for an idea of the magnitude of the matter effects [18], consider anaccelerator experiment sending neutrinos through the Earth’s crust over a baseline of 1000 km,
with ∆m231 ' 2.4 × 10−3 eV2 , then:Exν '.
(1.28)
12 GeV
So for Eν = 0.6 GeV, the peak T2K beam energy, the effect is small; for Eν = 2 GeV, the peakbeam energy for NOνA, the effect is considerable; and for Eν ∼ 17 GeV, as in the case of OPERA,
the effect would be large. Note that these mentioned experiments have varying baselines, andwere mentioned to provide realistic neutrino energies.
101.4.4

Status of the Oscillation Parameter ResearchThis a brief discussion of the status of the oscillation parameter search, for an comprehensive
review of neutrino oscillation experiments see Chapter 2. Since 1968, solar neutrino experiments [23–26] have measured 30–50% of the flux expected from scrupulously calculated Standard
Solar Models (SSM), which was known as the solar neutrino problem. A similar deficit in muon
atmospheric neutrinos was observed in 1988 by Kamiokande [27, 28] and IMB [29], which was
dubbed the atmospheric anomaly. These measurements were the first indication for the need ofa theory such as Neutrino Oscillations.
The first real evidence for oscillations came from Super-Kamiokande in 1998, from the atmospheric zenith analysis [30, 31], favouring the theory of oscillations over alternatives in the2004 atmospheric L/E analysis [9]. Global analyses of the solar oscillation parameters, ∆m221
and θ12 , from solar neutrino experiments [32] have greatly constrained the allowed regions. The
solar experiment SNO measured the total flux of neutrinos coming from the sun with an NC
reaction [33], which agreed with the SSMs, confirming the flavour changing interpretation of
the deficit. Additionally the reactor experiment, KamLAND, that had sensitivity to solar oscillations, greatly restricted the ∆m221 allowed region [34]. Along with the Super-Kamiokande
atmospheric analyses, neutrino accelerator experiments, such as T2K [35] and MINOS [36], haveconfined the atmospheric oscillation parameters, ∆m232 and θ23 .
The final mixing angle, θ13 , proved to be quite evasive. CHOOZ put an upper limit of around12◦ in 1999 [19]. While in 2011, Double CHOOZ [38], T2K [39] and MINOS [40] gave indicationsof a non-zero θ13 . Finally, in 2012 the parameter was measured by the reactor experiments firstly
by Daya Bay [14], and confirmed shortly afterwards by RENO [41] and Double CHOOZ [42].
The mass hierarchy, or the ordering of the neutrino mass eigenstates, and the degree of CP
violations in neutrinos are still unknown. Accelerator experiments are beginning to confine theallowed values of δcp [37, 43], and attempting to observe differences between the mass hierarchy
assumptions. The resulting parameters from a global fit to the neutrino oscillation data after the
Neutrino 2012 conference [44], is shown Table 1.1. Recent oscillation parameter allowed regions
are shown in Figures 1.5 and 1.6.

Parameter

Value

∆m221
∆m231
sin2 θ12
sin2 θ23
sin2 θ13−5 eV2
7.62+0.19−0.19 × 10
−3 eV22.55+0.06−0.09 × 10
0.320+0.016−0.0170.613+0.022
0.600+0.026−0.040−0.031+0.0026
0.0246+0.0029
0.0250−0.0028−0.0027

Table 1.1: Global fit of oscillation parameters after the Neutrino 2012 conference [44]. The left
figures correspond to normal hierarchy and the right to inverted hierarchy, where these differ.
The errors stated are at the 1 σ level. Note that in this global analysis there was a local minimum
2of sin2 θ23 = 0.427+0.034−0.027 with ∆χ = 0.02 above the global minimum.

11

Figure 1.5: Global solar oscillation parameter allowed regions from 2011 [32] on the left, andrecent atmospheric allowed regions from 2013 [35] on the right. The left figure came from a
combined analysis of the solar experiment data from SNO, Borexino, radiochemical experiments
and SK, as well as from KamLAND. The 95% C.L. regions are shown for ∆m221 and θ12 . The
right figure shows recent results from the accelerator experiments T2K and MINOS, as well as
the zenith and L/E analyses from SK, all at the 90% C.L. The L/E allowed region presented ingreen will be discussed as one of the analyses in this thesis.

Figure 1.6: T2K 2013 allowed regions forsin2 2θ13 for varying δcp [37]. The top panel
is for normal hierarchy and the bottom forinverted. The black solid line gives the best
fit and the orange hatched band displays theaverage θ13 value from the PDG, 2012, whichlargely comes from reactor experiments.

12

νβ

να

Figure 1.7: An example Feynman diagram for a flavourchanging neutral current (FCNC) non-standard neutrino
interaction (NSI) on a down quark. The figure illustrates a non-standard process that would allow neutrinos
to change flavour as they propagate through matter.
d

1.5

d

Non-Standard Neutrino Interactions (NSI)

Non-Standard Neutrino Interactions (NSI) are interactions of neutrinos beyond the Standard
Model that are predicted by various theories. The NSIs of interest in this thesis are NSIs of
neutrinos with matter. In this case the interaction probability amplitudes are dependent on thedensity of matter traversed by the neutrino, and therefore the flavour transition probability isdependent on the flight path of the neutrino through matter. This is not necessarily the case,as there also are predicted neutrino NSI that couple to something other than matter, as well
producing other properties [45].
Generally, NSI can arise even if neutrinos are massless [46–48], although most models expect
neutrinos to have mass [49–56]. For example NSI can be seen in massive neutrinos where theirmass is generated by a see-saw mechanism, naturally resulting in a degree of non-diagonality of
the neutrino states [49, 57]; or with the breaking of R parity in supersymmetric models [50, 57].
In general, these matter NSIs depend on the density of matter on the neutrino flight path, while
not showing the E −1 dependence seen in neutrino oscillations [45].
Early fits considering purely NSI were in agreement with the atmospheric data [58]. However,with higher statistics data and particularly with the addition of the high energy upward-going
muon sample, NSIs appeared to be at best a sub-dominant effect to oscillations [57, 59, 60]. For
this reason, an approach incorporating both neutrino oscillations while allowing for NSI effects
was implemented in this thesis.

1.5.1

Treatment of NSI

The main topic of interest regarding neutrino NSI is in how they can generate flavour transitions
as a sub-dominant effect. Therefore the flavour transition probabilities were calculated with amodel independent approach described by a Hamiltonian including both Neutrino Oscillations
and neutrino matter NSI [57].
These NSI can be classified into two categories: Flavour Changing Neutral Current (FCNC) [61–64] and lepton Non-Universality (NU) [65]. FCNC allows flavour transitions by the neutral current forward scattering of neutrinos in matter and take the form να + f → νβ + f . Here theneutrino converts from flavour α to flavour β by the interaction with a fermion, f . An example
Feynman diagram for an FCNC interaction on a down quark is shown in Figure 1.7. NU allows
the interaction probability amplitude of a neutrino with a fermion to vary depending on theflavour of the neutrino. A common feature of these interactions is that they are independentof the neutrino energy, however as previously stated, they should depend on the density anddistance of the matter traversed.
To illustrate how these matter NSI can be incorporated into an oscillation model, consider
13

the Standard Model weak interaction hamiltonion:
GF
HW = √ (Jµ† Jµ + ρJµN C JµN C ),2

(1.29)

where Jµ and JµN C are the weak current. For neutrino scattering on a fermion, the effective
hamiltonion becomes:
GF
ef f
HW
= √ [f¯γµ (1 − γ5 )να ][ν̄α γ µ (1 − γ5 )f ]2√
1
+ 2GF ρ[ν̄α γµ (1 − γ5 )να ][f¯γ µ (gV − gA γ5 )f ],2(1.30)

where ρ = 1 and L,R label the left- and right-handed fermions. gV and gA are:1
gV = gL + gR = − + 2 sin2 θW ,2
1
gA = gL − gR = − ,2(1.31)(1.32)

with θW as the weak mixing angle, or Weinberg angle. Then applying the Fierz transformation
to the first term of Equation (1.30), separating the neutrino and fermion factors:
GF
ef f= √ [ν̄α γµ (1 − γ5 )να ][f¯γ µ (cV − cA γ5 )f ],HW
2(1.33)

where cV and cA are corrections to the vector- and axial vector- weak charge, respectively, andare defined:gV + 1(α = f ),(1.34)cV =gV(α 6= f ),

gA + 1(α = f ),cA =(1.35)gA(α 6= f ).
As neutrinos propagate through and interact with matter, they gain an interaction energy, V ,from the fermion f , derived from Equation (1.33):√
VSM = 2GF cV Nf ,(1.36)
where Nf is the fermion number density in the matter traversed. In the case α 6= f , theinteraction energy becomes:√
VSM = 2GF cV Nf√= 2GF gV Nf√= 2GF (gL (f ) + gR (f ))Nf .
(1.37)
Equation (1.37) shows the interaction energy gained in a standard model neutrino interaction
with matter. Extending this to allow us to quantify the effect of NSI in a model independent
way, we can redefine V as:√
VN SI = 2GF (εfαβL + εfαβR )Nf ,(1.38)
14

where the εfαβP (P = L,R) parameterise the strength of NSI for an interaction with a left- orright-handed fermion, and correspond to the gP (f ) neutral current coupling in the standard
model. However, with atmospheric neutrinos we cannot distinguish interactions with left- andright-handed fermions in the matter propagated, so we consider εfαβ ≡ εfαβL + εfαβR . Furthermore,
the analysis normalisation is set by assuming interactions only with the down quarks in matter,following the convention set by previous analyses [58]. So we can drop the fermion superscript,
taking f = d, and work with εαβ parameters. NSI cases outside of this assumption can beconsidered by rescaling the εαβ parameters reported in this thesis.
Considering the 2-flavour case in the να –νβ sector, the effective potential can be written as:
H(r) = H0 + Vef f




√
εαα ε∗αβp 0=+ 2GF Nf (~r),
0 pεαβ εββ(1.39)(1.40)

where p is the momentum of the neutrino, and GF is the Fermi coupling constant. The εαβrepresents the amplitude of the FCNC interactions, while the quantity εββ − εαα represents thedifference in the amplitude for να or νβ interactions with matter (NU). Nf (~r is the fermion (dquark) density as a function of position, ~r, along the flight path of the neutrino. The Schrödinger
equation can be written as:
Z L|ψ(t)i = exp − iH(r)dr |ψ(0)i,(1.41)
0where the integration is carried out along the neutrino flight path and L is the distance travelled.
Then the survival probability of να becomes:
r


√
ε2αβ
(εββ − εαα )22
2
2GF Xf εαβ +,(1.42)P (να → να ) = 1 −sin(ε −ε )24ε2αβ + ββ 4 αα
where Xf is the column density of fermions, f , integrated over the neutrino flight path:Z L
Xf =Nf (~r)dr.
(1.43)
0

It can be seen that the survival probability has an oscillitory magnitude that is a functionof the column density of fermions on the neutrino path, but does not depend on the neutrinoenergy. This survival probability can be seen in Figure 1.8.
Since pure NSI alone cannot simultaneously explain the whole energy range of the atmospheric
neutrino data, the effects of both neutrino oscillations and NSI can be included in one effective
hamiltonion:
εee ε∗eµ ε∗eτ
0
0
0
√
1 
0  U † ± VM SW ± 2GF Nf (~r) εeµ εµµ ε∗µτ  ,Hαβ =U 0 ∆m221(1.44)
2E
2
0
0
∆m31
εeτ εµτ ετ τwhere the additional interaction potentials are positive for neutrinos and negative for antineutrinos. The first term is the vacuum oscillation term; the second is from MSW matter
effects on electrons; and the third allows for NSI. In this way, the dominant mechanism for theflavour transitions are expected to come from oscillations, while deviations due to NSI could
possibly be measured.
15

Figure 1.8: NSI survival probability for να → να as
a function of the flight path in log10 (L (km)). The
parameter sets are as follows, εαβ = 1.0 and εββ −εαα = 0.01 in black; εαβ = 1.0 and εββ − εαα = 1.0 in
red; and εαβ = 10.0 and εββ −εαα = 0.1 in green. For
this calculation, the average fermion density, hNf (~r)i,along the flight path of the neutrino was used.

1.5.2

Previous Work with Atmospheric Neutrinos and NSIWith the initial statistics of early atmospheric neutrino data from SK-I, pure NSI could explain the νµ deficit as an alternative to neutrino oscillations [58]. However, with the additionof greater statistics and higher energy upward-going muon samples, pure NSI could no longer
simultaneously explain the full energy range of atmospheric data [57, 59, 60].
Since the atmospheric data could no longer be explained by pure NSI alone, and with thesuccess of the oscillation description at SK, analyses began to include oscillations and NSI together [45, 57]. This was done by including oscillations and µτ –sector matter NSI in a 2-flavour
Hamiltonian. The NSI contribution to the Hamiltonian [57] had the form:


√
0 ε
HNSI = ± 2GF Nf (r),(1.45)
ε ε0
where Nf was the fermion number density and was taken to be the density of down-quarks as
the normalisation for these analyses – the same normalisation used in this thesis. The + (−)
sign holds for neutrinos (anti-neutrinos). This allowed for neutral-current-like interactions withmatter, with the off-diagonal ε coupling for flavour-changing (FC) interactions; and the diagonal
ε0 (≡ ετ τ − εµµ ) as the flavour conserving non-universality (NU) parameter. The mechanism was
similar to the MSW matter interactions.
The first of these analyses was reported in 2002 [57] and made use of the 79 kton yr SK and
MACRO data sets. The best-fit was at ∆m2 = 2.4 × 10−3 eV2 , sin2 2θ = 0.99, ε = −9.1 × 10−3 ,and ε0 = −1.9 × 10−3 . The allowed regions can be seen in the left set of 4 plots in Figure 1.9.
The oscillation mechanism was stable with the introduction of the perturbation from NSI. It was
concluded that FC at the order of 1% could be present, while νµ and ντ interactions were likely
to be universal. The 99% C.L. limits obtained from marginalising over the other parameters
were −0.03 < ε < 0.02 and |ε0 | < 0.07.
Another analysis was published in 2004 [45], that looked at oscillations with sub-dominant
effects from new physics including matter NSI. In the latter case, the treatment was essentiallythe same as the 2002 analysis [57], and used the 1489 day SK and K2K data sets. The best-fit
points were close to the best-fit for pure oscillations, so it was concluded that there was no16

Figure 1.9: NSI allowed regions from the first oscillation+NSI hybrid analyses with atmospheric
neutrinos. In the figures, ε represents the FCNC εµτ parameter and ε0 represents the NU(ετ τ − εµµ ) parameter. The left four figures were reported in 2001 using the 79 kton yr SK
and MACRO data sets [57]. The two oscillation parameters were integrated out, and the shaded
regions refer to 90%, 95%, 99% and 99.7% C.L. with two parameters. While the right four figures
are from 2004 and used the 1489 day SK and K2K data sets [45]. The undisplayed parameters
were marginalised out.
evidence for new physics even as a sub-dominant effect. The allowed regions are shown in theright 4 plots in Figure 1.9. The oscillation mechanism was found to be robust, and the 90% (3σ)
bounds on matter NSI were:(−0.021) − 0.013 6 ε 6 0.010(0.017),
0|ε | 6 0.029(0.052)(1.46)(HNSI real),|ε| 6 0.013(0.021),
0|ε | 6 0.034(0.060)(1.47)(1.48)(HNSI complex).

(1.49)(1.50)

These bounds were well-constrained, however it was called into question whether a 2-flavour
treatment was sufficient for atmospheric neutrinos [13, 66–68]. In a 2004 analysis of atmospheric
neutrinos and νe mixing NSI [66, 67], it was shown that at least ετ τ bounds were significantly
relaxed by the addition of non-zero εeτ . Values of ετ τ up to O(1) were allowed, along the parabolaετ τ = |εeτ |2 /(1+εee ) where the oscillations mimic vacuum oscillations with the same dependence
on the neutrino flight path and energy. This allowed region can be seen in Figure 1.10. So withthe large range of baselines of atmospheric neutrinos, that also traverse the matter densities of the
mantle and core, 3-flavour effects become more significant. In this thesis, a 3-flavour approach to
NSI was conducted. Accelerator baselines at the order of a couple hundred kilometres, like thatof MINOS and T2K, would be less sensitive to these 3-flavour effects and a 2-flavour approachis sufficient [69].
The early results from MINOS running in anti-neutrino mode reported a significantly larger−3 eV2 than the value obtained from their neutrino analysis [10], and also∆m̄2 = 3.36+0.45−0.40 × 10
17

Figure 1.10: Effect of νe mixing with electron
flavour ε NSI in atmospheric neutrinos [66]. Thisis a 2-D cross-section at εee = −0.15, where
the shaded regions represent 95%, 99% and 3.6σ
C.L. for 3 degrees of freedom. This is inverted hierarchy where |εeτ | and |ετ τ | extend up to O(1)at the 95% C.L. For normal hierarchy this extent is up to ∼0.6. The inner contour represents a condition of zero eigenvalue, where NSI
mimics vacuum oscillations along the parabolaετ τ = |εeτ |2 /(1 + εee )allowed a larger deviation for sin2 2θ̄ = 0.86 ± 0.11. In an attempt to explain these results, a few
analyses were carried out in 2010 considering the effect of matter NSIs [13, 70].
The apparent difference in the νµ and ν̄µ data could be explained by the oscillation phasereceiving a contribution from matter effects that retards (advances) the total phase as νµ (ν̄µ )propagate through matter [13]. The effect of this shift in phase would cause an analysis assuming
vacuum oscillations to measure a ∆m2 of lower (higher) value. A phenomenology involving
matter NSI would have one value for ∆m2 in between the vacuum oscillation measurements.
For normal hierarchy, this effect could be produced with a negative value for εµτ ; and a positive
value for the inverted hierarchy case.
In 2010, the MINOS data for νµ and ν̄µ were simultaneously fit by Mann et al. [13] with
−3 eV2 , 0.90 ± 0.05, −(0.12 ±the best-fit at (∆m2 , sin2 2θ23 , R(εµτ )|Ve |) = (2.56+0.27−0.24 × 100.21)|Ve |) [13]. The allowed regions of the separate and combined fits of this analysis can beseen in Figure 1.11. It can be seen in the separate νµ (ν̄µ ) allowed region that an increase in themagnitude of |εµτ | favours a larger (smaller) value for ∆m2 . The ∆m2 converge into a single
value for the combined fit. In the analysis, electron number density was assumed in the Earth’s
crust, and so the MINOS data could be explained with matter NSI at around 12% the strengthof the MSW effect. This result was not taken as evidence for non-zero εµτ , as εµτ = 0 was
within 1 σ, however the NSI fit worked as well as a pure oscillation fit and could fit the νµ andν̄µ data simultaneously with 3 parameters. Considering the much more significant matter effects
in atmospheric neutrinos, the difference in the ∆m2 assuming vacuum oscillations would growwith energy up to ∼20 GeV. However, since νµ and ν̄µ cannot be separated on an event-by-event
basis, this effect would be averaged over.
Another analysis of the MINOS data by Kopp et al. [70] assumed that NC-like matter NSI
were already disfavoured by the 2-flavour atmospheric NSI analyses, but the data could be
explained by CC-like NSI at the production or detection of the neutrinos within NSI bounds at
the time. However, the paper included an initial analysis with NC matter NSI. The analysis also
2m
had an electron number density normalisation, and had the best-fit parameters (εmµτ , ετ τ , sin θ23 ,2−3
2∆m32 ) = (−0.40, −2.16, 0.38, 2.86 × 10 eV ). Where the m superscript denoted matter NSI
to distinguish from the CC case. The allowed regions can be seen in Figure 1.12.
m
The |εmµτ | ∼ 0.4 measured here was considerably larger than the |εµτ | ∼ 0.1 measured byMann et al. [13], which used the same data set and normalisation. The difference was thought
to be due to Mann et al. fitting the oscillation probability to the ratio of observed to expected18

Figure 1.11: Fit of NC-like matter NSI [13] to explain the 2010 MINOS data [10]. The left plotshows the NSI fits to the νµ (contours) and ν̄µ (shaded regions) data separately. With increasing|εµτ |, the ∆m2 increases for neutrinos; while the ∆m2 decreases for anti-neutrinos. The rightplot shows the fit to the combined data, and at the best-fit the ∆m2 converge to one value, in
between the vacuum oscillation values (triangles).
event numbers from MINOS, while this analysis used a modified version of GLoBES to simulate
the energy resolution effects at MINOS, and could reproduce the MINOS results rather well.
m
Despite the large best-fit value for |εmτ τ |, the result was compatible with ετ τ = 0 at the 90% C.L.
Interestingly however, pure standard oscillations was ruled out at the 90% C.L (with the MINOS
data in 2010).
With the significant results from these matter NSI analyses with MINOS data, there wasgood motivation to look for differences in vacuum oscillation parameters between neutrinos andanti-neutrinos and to search for sub-dominant NSI with atmospheric neutrinos at SK – the topic
of this thesis. There has already been an initial NSI search at SK making use of the zenith
distributions, reported in 2011 [73]. The analysis in the µτ –sector used a 2-flavour Hamiltonian,and followed the model used in the first NSI+Oscillation analyses of atmospheric data [45],and used the down-quark number density normalisation. The best-fit was at sin2 2θ23 = 1.00,∆m232 = 2.2 × 10−3 eV2 , ε = 1.0 × 10−3 , ε0 = −2.7 × 10−2 , χ2min = 838.9/746.0 d.o.f. The allowed
regions for the oscillation parameters and for the NSI parameters are shown in Figure 1.13. The
90% C.L. limits for the NSI parameters were:|ε| < 1.1 × 10−2 ,−4.9 × 10−2 < ε0 < 4.9 × 10−2 .
From looking at the NSI allowed regions for each of the samples separately, it was found that theε parameter was mostly constrained by the upward-going muon sample, and the ε0 parameterby the partially-contained and stopping muon samples. A 3-flavour oscillation analysis with
NSI in the eτ –sector was also carried out (separately from the µτ –sector analysis). The NSI
analysis in this thesis was conducted with a 3-flavour Oscillation+NSI model focussed mostly on
the µτ –sector, using high resolution L/E distributions to look for any deviations from standard
oscillations. However, since the L/E value cannot be reconstructed with sufficient resolution for
the upward-going muon samples, they were not included in the analyses in this thesis.
In 2013, the MINOS collaboration looked for NSI in their data [69]. At the MINOS baseline,
the µτ –sector NSI are effectively disjoint from the eτ -sector, so a 2-flavour approach was taken.
The long propagation path of neutrinos in matter gives an enhanced sensitivity to NC-like matter
19

Figure 1.12: NC-like NSI allowed regions in the µτ –sector from a modified GLoBES simulation
fit by Kopp et al. [70] to the 2010 MINOS νµ and ν̄µ data [10]. In each panel the third NSIparameter was marginalised out, as were the oscillation parameters. The best-fit for εmµτ wasconsiderably larger than the other fit to the same data from Mann et al. [13]. While pure standard
oscillations were ruled out at the 90% C.L. The stringent 2-flavour atmospheric limits [57, 71]
and general limits [72] are shown in the gray shaded regions. The left-most panel shows thearg(εmµτ ) parameter marginalised out, while the effect of the parameter is shown in the other twopanels.

Figure 1.13: Allowed regions from a 2-flavour Oscillation+NSI analysis in the µτ –sector with
the Zenith distributions at SK [73]. The undisplayed parameters were integrated out. In theleft panel, the solid contours are for the Oscillation+NSI hybrid model, and the dashed curves
for standard 2-flavour Oscillations. The right panels show the NSI allowed regions, where theparameters were defined as ε ≡ εµτ (FCNC) and ε0 ≡ (ετ τ − εµµ ) (NU). The stars represent thebest-fit points.

20

Figure 1.14: Allowed regions for NC-like NSI from the MINOS collaboration [69]. The 68% and
90% C.L. contours are shown for 2D projections of the (∆m2 , sin2 2θ, εµτ ) parameter space withthe third parameter marginalised out. The left-most panel shows εµτ against ∆m2 ; the middle
is of εµτ against sin2 2θ; and the right shows the oscillation parameter allowed regions for the νµand ν̄µ data combined.

NSI, so the paper focussed on that case. CC-like NSI along the neutrino path would produce anoutgoing charged lepton, which gets absorbed by the Earth’s matter and would not be observed.
On the other hand, NC-like NSI along the neutrino flight path produces a final state neutrino
with another flavour, altering the flavour composition of the beam. Three parameters were fit,
and the best-fit point was at:−3eV2 ,∆m2 = 2.39+0.14−0.11 × 10

sin2 2θ = 1.00+0.00−0.06 ,εµτ = −0.07+0.08−0.08 ,
with the allowed region −0.20 < εµτ < 0.07 (90% C.L.). Note that the analysis assumed theelectron number density in matter, as opposed to down-quark normalisation. So the magnitudeof these εµτ regions are three times larger than for down-quark density normalisation, to describe
the same effect. Figure 1.14 shows the 2D allowed regions with the third parameter marginalisedout. The oscillation parameter allowed regions were in agreement with previously reported
regions by MINOS. Within the errors the fit was consistent with no contribution from flavourchanging NSI.
A summary of the bounds on NSI was published in 2009 [72], which also quoted the ε regions
assuming electron number density, and so are also 3 times larger than ε regions measured in thisthesis to describe the same effect. The 90% C.L. bounds were
4.2 0.33 3.0|εαβ | < 0.33 0.068 0.33 ,3.0 0.33
21
L

L
where thesuperscript denoted interactions in Earth matter and the columns and rows run
through the e, µ and τ flavour. So the εαβ are arranged in the same form as the NSI matrix inEquation (11.3). The constraints on the parameters of interest in this thesis, εµτ and ετ τ , werefairly loosely bound. Whether the tighter constraints from 2-flavour atmospheric analyses holdin a 3-flavour framework will be determined in this thesis.
21

1.6

Outline of Thesis

The aim of this thesis is:
1. To determine if any difference in oscillation parameters can be observed between neutrinos
and anti-neutrinos, assuming no matter effects, with an L/E analysis at SK; or to confirm
the CPT symmetry of neutrino oscillations.
2. To perform the first 3-flavour L/E Analysis with non-zero θ13 , to check that this is still agood method to make oscillation measurements, and to obtain more accurate oscillation
parameters free from 2-flavour approximations.
3. To search for NC matter NSI in the µτ –sector in the atmospheric neutrino data with a
3-flavour hybrid model of Oscillations and NSI, for the first time. To determine the degreeof non-standard neutrino interactions allowed by the data, in a model-independent way.
These searches were initially inspired by the discrepancy in νµ and ν̄µ data around 2011, and theinterest in the literature that NSI could potentially be acting sub-dominantly behind neutrinomass mixing. However, these analyses are further motivated to confirm the CPT symmetry inneutrinos, and for the search for new non-standard physics. In addition, now that we know θ13
is not-so-small, it is better to perform oscillation analyses in 3-flavours. Atmospheric neutrinos
cover a large range of energies and baselines and have an enhanced sensitivity to matter effects,
such as in NSI or the MSW effect. Moreover, the L/E Analysis has the best discrimination of∆m231 from atmospheric neutrinos, making it a good analysis to perform these studies.
To this end, the chapters in this thesis are arranged as follows. Chapter 2 covers a comprehensive overview of over 40 years of experimental evidence for neutrino oscillations, including
the many ways the ghostly neutrino has been detected, and the unravelling of the anomalies toour present understanding. Chapter 3 describes Super-Kamiokande, the detector that collected
the analysis data. Chapter 4 explains details of the simulation, from the determination of theatmospheric neutrino flux, to the interactions and detector simulation. Chapter 5 highlights
some of the calibrations to ensure the energy, time and charge accuracy at SK and the extractionof input values to the simulation. Chapter 6 details how the raw data is reduced to neutrino
events and Chapter 7 illustrates how physical properties are reconstructed from the data.
For the analyses and the main contributions of the author, Chapter 8 explains the details of
the 2-flavour L/E analysis and is relevant to all other analyses in this thesis. Chapter 9 covers
an analysis that searches for the difference in neutrino and anti-neutrino oscillation parametersassuming the vacuum condition, while expanding the available data with an additional resolution
sample. Chapter 10 describes the extension of the L/E Analysis to 3-flavour oscillations, whilethe 3-flavour Oscillation and NSI analysis is covered in Chapter 11. Finally, the conclusions of
this thesis are written in Chapter 12.

22

List of Abbreviations:
ADC: Analogue to Digital ConverterATM: Analogue Timing Module

CC: Charge-Current

FC: Fully-Contained

FCFV : Fully-Contained Fiducial Volume
FV: Fiducial VolumeHE: High Energy. High Energy Trigger

HV: High Voltage

ID: Inner Detector

LE: Low Energy. Low Energy Trigger

L/E: Neutrino Flight Length over Neutrino Energy

MC: Monte Carlo

MSW: Mikheyev–Smirnov–Wolfenstein effect (matter effect)NC: Neutral-Current

NSI: Non-Standard (Neutrino) Interactions

OD: Outer Detector

PC: Partially-Containedp.e.

: Photo-electrons

PMT: Photo-Multiplier Tube

QAC

: Charge to Analogue Converter

QTC: Charge to Time Converter

QE: Quantum Efficiency

SK: Super-Kamiokande

TAC: Time to Analogue Converter

TOF: Time of Flight

UPMU : Upward-Going Muons

23

Chapter 2

Experimental Evidence for
Neutrino Oscillations
The neutrino is difficult to detect, Pauli himself wondered if it could ever be seen. However,
over the years experimentalists have managed to detect the neutrino in the laboratory, amass
statistics of neutrino fluxes, and find some quite surprising results. One such outcome is thatneutrinos are thought to oscillate in such a way that their flavour at detection may differ fromtheir flavour at creation – Neutrino Oscillations. This theory however, requires that neutrinos
have mass while the neutrinos in the Standard Model are massless. There has been a wealthof evidence for Neutrino Oscillations collected over the last 40 or so years, and some of thehighlights have been collected here.

2.1

Solar Neutrino Experiments

The most obvious source of neutrinos is the sun, producing a tremendous flux of neutrinos
during the fusion processes that are the origin of almost all the energy used on Earth today.
Since neutrinos easily pass through matter, they can provide a window to study the interactions
occurring at the core of the sun. There are a number of chains of interactions at the solar core,fusing nuclei into increasingly heavier elements. Interactions like the following:
4p → 4 He + 2e+ + 2νe + γ(2.1)

where the above equation is an average representative reaction of the proton-proton-chain (ppchain), and shows the process of four protons, p, fusing into a Helium nucleus, and emitting
neutrinos and photons. There are actually multiple steps and several possible branches thatthe reaction cycle can follow, accurately calculated and tested in the Standard Solar Model(SSM)[74, 75]. Other branches can lead on to produce 7 Be or 8 B, each also producing neutrinos.
A summary of the energy spectrum of the emitted neutrino fluxes at the Suns core, labelledby the parent isotope it is associated with, can be seen in Figure. 2.1. The three main fusion
reactions emitting neutrinos are [76]:p + p −→ d + e+ + νe

pp neutrinos
7
8

Be neutrinos
B neutrinos

7

−

7

Be + e −→ Li + νe88+

B −→ Be + e + νe
24

Emax = 420 keV
E = 860 keV (mainly)
Emax = 14 MeV

(2.2)
(2.3)
(2.4)

Figure 2.1: Fluxes of solar neutrinos as predicted from the BS05(OP) Standard Solar Model [74]The labels indicate the source reaction of the neutrinos. The relative abundances can be seen.
The pp neutrinos have the largest flux, but their low energies made them difficult to detect andwere measured in radio-chemical experiments. The 7 Be mono-energetic peak was challenging todistinguish and was seen in Borexino. The 8 B neutrinos extend to higher energies and could beseen by Cherenkov detectors such as SK and SNO.
where the d is a deuteron and the reactions have the relative fluxes: φpp : φ7 Be : φ8 B ≈ 1 : 0.08 :0001. The most abundant source of neutrinos is the pp-chain, but the energy of these neutrinos
is in the order of a fraction of an MeV and are difficult to detect. There have been a number ofsolar neutrino experiments, each employing a different detection method and having sensitivity
to a different region of the solar neutrino spectrum.

2.1.1

Homestake Chlorine Experiment

The Chlorine experiment conducted by Raymond Davis et al. began in 1967, and reported thefirst detection of solar neutrinos in 1968 [23, 77–79]. The experiment continued to take data
for around 25 years, improving the statistics and accuracy of its initial findings. The detection
method made use of 615 metric tons of tetrachloroethylene (perchloroethylene), C2 Cl4 , that
captured solar neutrinos by the process 37 Cl(ν, e− )37 Ar. The reaction has an energy threshold of
0.814 MeV, giving it sensitivity to the 7 Be and 8 B solar neutrino fluxes. The SSM predicted that
78% of the relevant flux is from 8 B neutrinos, while 13% originate from 7 Be neutrinos [71]. The
detector was a horizontal cylindrical tank and was built 4400 m.w.e.1 underground, to greatly
reduce the background from cosmic rays, in the Homestake gold mine at Lead, South Dakota.
The detector facility can be seen in Figure 2.2.
Neutrino detection involved removing the 37 Ar from the liquid chlorine and then observing
the 37 Ar decay (35.0 day half-life) in miniature proportional counters. To achieve this eductors
1
m.w.e is metres water equivalent, the depth of water in metres that would give the same reduction in flux of
Cosmic Rays. 2.650 times the depth of standard rock.

25

Figure 2.2: The Chlorine Experiment facility on the left and 37 Ar production rate on theright [78]. The large tank holding the tetrachloroethylene and helium/argon gas can be seen, as
well as the pipes and pumps for circulating and extracting from the gas. The 37 Ar production
rate from the maximum-likelihood analysis can be seen from the data between 1970 and 1994.
The final ‘average’ point is from an analysis combining all the data.
circulated helium from the gas at the top of the tank into the chlorine liquid, which maintained anequilibrium between dissolved argon in the chlorine and gaseous argon in the gas mixture above.
The helium/argon gas was pumped through an argon extraction system that involved 77◦ K
charcoal traps that absorbed the argon. The charcoal was heated to remove the argon/noble
gas mixture, and the other noble gasses were removed with two stages of gas chromatography.
From tests, it was determined that 95% of the argon can be removed from a helium purge over
22 hours [23].
Afterwards the 37 Ar decay events, by electron capture releasing 2.82 keV Auger electrons,
was observed in miniature proportional counters over a period of 250-400 days. Most of the
background was rejected by selecting for the energy and characteristic pulse rise time. The
output of the selection was a time series of events including signal and background. Finally, amaximum-likelihood analysis for the series of events with variables for production rate of 37 Ar in
the detector, and background rate in the counter, was carried out. The resulting 37 Ar production
rate was attributed to the solar neutrino flux. The 37 Ar production rate from 1970 to 1994 isshown in Figure 2.2, with the last point from a combined analysis of all the data [78].
Over the whole data taking period, a consistently smaller flux than predicted by the SSM
was measured [78]. The neutrino capture rate was:RCl = 2.56 ± 0.16 ± 0.16 SNU(2.5)

where 1 Solar Neutrino Unit (SNU) is the equivalent to the neutrino flux producing 10−36 captures per target atom per second. The neutrino capture rate compared with the SSM prediction
was [71]:RCl= 0.3 ± 0.03
(2.6)
SSM
The ratio of νe observed was around 30% of that expected. Since this result could not be
explained by the Standard Model, this was the beginning of the so-called solar neutrino problem.
Davis continued the experiment to verify the result for around 25 years, earning him the shared
Nobel Prize in Physics in 2002 [79].
26

2.1.2

Gallium Experiments

Since 1990, there have been more radiochemical solar neutrino experiments using Galium as
their detection mass, through the 71 Ga(ν, e− )71 Ge reaction. It has a low threshold at 233 keV,which gave these experiments sensitivity down to the pp-neutrinos. Gallium allowed for theobservation of 93% of the solar neutrino flux, and without these experiments, the majority of theflux would not have been measured. The SAGE experiment began in Russia in 1990 with 30 tonsof liquid metallic Gallium (increased to 57 tons since 1991) [25], and the GALLEX experiment
ran in Gran Sasso, Italy, from 1991–1997 with 30.3 tons of gallium in the form of GaCl3 -HCl [80].
The GALLEX experiment was superseded by GNO from 1998–2003. The expected flux of solar
neutrinos from the SSM for gallium was 128+9−7 SNU, of which the pp neutrino component was
69.7 SNU [25].
The concept of these experiments were similar to the Chlorine Experiment. The GALLEX(GALLium EXperiment) experiment made use of 30.3 t of gallium containing 12 t of 71 Ga in
aqueous gallium chloride solution at 101 t. This was contained in a 7 m high 70 m3 tank. Each
exposure began with the addition of ≈1 mg of a stable germanium carrier, and lasted about 3-4
weeks, after which the Ge would be extracted.
The acidity of the solution and the high concentration of chlorine ensured that the germanium
would be in the form of the volatile tetrachloride, GeCl4 . Which could be separated from the nonvolatile GaCl3 by bubbling nitrogen gas through the liquid. The GeCl4 was reabsorbed in water,
and chemically converted to germane, GeH4 , which was then counted in specialised miniaturevery-low-background proportional counters. Tests showed that the extraction efficiency was
greater than 99%.
In 1992, GALLEX reported on the first observation of solar pp-neutrinos, with the full expected flux, and reduced fluxes for 8 B and 7 Be neutrinos [76, 81]. The result in 1999 fromGALLEX I-IV [80] was
RGALLAX = 77.5 ± 6.2(stat.)+4.3−4.7 (syst.) SNU(2.7)

where the errors were at 1σ. The GALLAX data from the solar runs, together with the GNO
data, can be seen in Figure 2.3.
The SAGE (Soviet-American Gallium Experiment) experiment made use of 50 t of gallium inthe form of a liquid metal and was distributed over seven 2 m3 chemical reactors. The runs beganwith the addition of 700 µg of a germanium carrier uniformly distributed over all the reactors,
and lasted 3-4 weeks. The reactors had motors for stirring the liquid metal to thoroughly mix
in the Ge carrier and the extraction reagent during an extraction.
To extract the Ge [82], a reagent containing de-ionised water, HCl and H2 O2 was added to
the Ga, which turns it into an emulsion of droplets. The Ge dissolved in Ga migrated to thesurface of the droplets where it becomes oxidised. This oxide was dissolved with more HCl, andimmediately decanted out.
The Ge was concentrated firstly by evaporating it. More HCl was added to increase thevolatility of the GeCl4 , and an air flow was used to sweep it from the solution, after which it wasabsorbed in 1 l of de-ionized water. The Ge in the swept solution was further concentrated by 3stages of solvent extraction, into 100 ml of low-tritium water. This solution was then synthesised
into germane (GeH4 ), where it was swept by helium and separated using gas chromatography.
The final gas mixture was then inserted into a miniature proportional counter. Since 1997 (SAGE27

Figure 2.3: The gallium solar neutrino rate from SAGE on the left [25], and the combined result
from GALLAX and GNO on the right [24]. In the SAGE data, the triangles represent an analysis
with the L and K peaks, and circles for K peaks alone. The final three points show combined
analysis points for (1) L Peak, (2) K Peak, and (3) all data. The GALLEX and GNO data spana full solar cycle and side reaction contributions have been subtracted. The error bars on both
figures are 1σ statistical.
III), the quantity of the reagents added during an extraction was halved, and two extractions
were performed. The extraction efficiency was ∼80% and improved to over 90% by 1997 [82].
In the counters, 71 Ge decays only by electron capture to the ground state of 71 Ga. These
decays resulted in the emission of Auger electrons observed by the proportional counters, and Xrays that were measured with a reduced efficiency. The signal was a nearly point-like ionisation,
and a fast pulse rise time. These characteristics allowed separation from the background. The
data was put through a selection process, resulting in candidate 71 Ge events with both L- andK- peak regions. These were fitted with a likelihood for a decay curve of a fixed number of 71 Ge
atoms, and two constant backgrounds for each peak. From the fitted parameters the production
rate of Ge per day, and the solar neutrino flux was calculated.
In 2002, after taking 12 years of data, SAGE measured the following capture rate of solar
neutrinos [25]:RSAGE = 70.8+5.3(2.8)−5.2 (stat.) SNU
The systematic errors were not included as they were considerably smaller than the statisticalerror (+3.7−3.2 SNU). Their result was a 6σ deviation from the expectation from the SSM. They
concluded that this was strong evidence for a reduced flux of solar neutrinos from the expectation,and that neutrino matter oscillations in the LMA range best described the data (see Section 2.1.4
for details about the Large Mixing Angle solution). The data can be seen in Figure 2.3.
The GNO (Gallium Neutrino Observatory) experiment [24] also at LNGS, Gran Sasso, wasthe successor to the GALLEX experiment, and ran from 1998 to 2003. The neutrino target was
still 100 t of gallium chloride, containing 30.3 t of gallium. The observations of GNO improved
the quality of data, and combined with GALLAX cover more than a full solar cycle (1991-2003).
The experimental set-up from GALLEX went through a major overhaul and modernisation
for GNO. The extraction procedure was largely the same as that for GALLAX, although theefficiency improved to an average of 95.7% determined from the recovery of the Ge carriers,
where ≈1 mg was added per run. The electronics were upgraded. New preamplifiers for improved
resolution in the pulses; new data acquisition system that had better pulse discrimination; anda new X-ray calibration source that helped reduce systematic errors [83].
28The analysis involved a neural network for the pulse shape selection, replacing the rise-time
selection used in GALLEX, and was followed by a maximum likelihood analysis. The neuralnetwork had a better noise rejection than the previous method, including the recognition andrejection of fast-rising background pulses.
The result from all the GNO data, and GALLAX+GNO was [24]:RGNO = 62.9+6.0−5.9 SNURGALLEX+GNO = 69.3 ± 4.1(stat.) ± 3.6(syst.) SNU(2.9)(2.10)

With the errors at 1σ and with the GNO errors including both systematic and statistical contributions. The rates were after the 4.55 SNU subtraction from side reactions. The systematic error
was dominated by the counting efficiencies. The results from GALLEX and GNO also showedthat there was a constant production rate of 71 Ge over the whole period. The production rate
for the GALLEX and GNO runs can be seen in Figure 2.3.
In gallium, the SSM estimation is that 54% of the flux comes from pp-neutrinos, 26% from7 Be neutrinos and 11% from 8 B neutrinos. The combined rate of solar neutrinos measured bythese Gallium experiments was [24]:RGa = 68.1 ± 3.75 SNU(2.11)

When this was compared with the SSM prediction, the following ratio was observed:
RGa= 0.52 ± 0.03
SSM

(2.12)

So the Gallium experiments gave a confirmation of the solar neutrino problem, and clearlyan explanation was required for the deficit in solar neutrinos observed.

2.1.3

Kamiokande

The Kamioka Nucleon Decay Experiment (KamiokaNDE) was built to search for proton decay,
and took analysis data from 1987–1995 [26]. The Kamioka Observatory is located 2700 m.w.e
underground in the Kamioka mine, Japan. The detector was a cylindrical tank, 16.1 m tall and15.6 m in diameter, filled with 2140 tons of pure water with 948 photomultiplier tubes (PMTs)
facing inwards from the wall – 20% photodetector coverage. This allowed the measurement ofcharged particles by their Cherenkov radiation. Interaction particles travelling faster than the
speed of light in water would emit Cherenkov radiation forming rings of light on the detector
walls. Surrounding the inner volume, there was an optically separated 4π anti-counter equippedwith 123 PMTs to cut backgrounds that originated from outside the detector. Kamiokande also
had a water purification system to maintain a high level of transparency in the water. The
detector setup can be seen in Figure 2.4.
The detector turned out to be an excellent setup for detecting cosmic neutrinos. It was able
to measure neutrinos in real-time, largely preserving the directional information of the neutrino
trajectory. This allowed Kamiokande to perform studies of time variations of the solar neutrino
flux with solar activity, and confirm the origin of the solar neutrinos.
Neutrinos could be detected by the elastic scattering interaction, ν + e− → ν + e− . The
analysis had a threshold of 7.5 MeV, lowered to 7 MeV at the end of 1991, and was sensitive to29

Figure 2.4: The Kamiokande detector on the left [84], and cos θSun distribution on the right [26].
In the left figure, the bulb-like shapes represent where the PMTs were distributed and the
direction they faced. The right figure shows 1036 days of Kamiokande-III data, with cos θSun = 1
being the direction of the sun. The black solid line was the BP5 SSM prediction, and the dottedline the best fit to the data assuming a flat background.
approximately half of the 8 B neutrino spectrum. The scattering angle, θe , of the recoil electron
largely preserved the direction of the incoming neutrino:
θe2 ≤

2me
Te

(2.13)

where Te and me are the kinetic energy and mass of the recoil electron. The angular resolution wasaffected by the multiple scattering of the electron in water and was 28◦ at 10 MeV. Kamiokande
was the first experiment to measure an excess of events2 in the direction from the sun, confirming
the solar origin of the low energy neutrinos. This can be seen in Figure 2.4.
The flux of solar neutrinos measured over 2079 days was [26]:ΦKamiokande = (2.80 ± 0.19 ± 0.33) × 106 cm−2 s−1(2.14)The main systematic error contributions came from uncertainties in the angular resolution, energy
scale and the fiducial volume cut. The ratio of the flux to the SSM prediction was:
ΦKamiokande= 0.492+0.034−0.033 ± 0.058
SSM(2.15)

Over the 8 years of data, Kamiokande did not observe any statistically significant correlation
between the solar neutrino flux and the solar activity and sun spots. Similarly the daytime andnight time flux were not significantly different.
Again, Kamiokande observed a deficit from the flux of solar neutrinos expected, further establishing the solar neutrino problem. It should be noted that the elastic scattering interaction issensitive to both charge-current (CC) and neutral current (NC) interactions (via Z-boson), which
gives it sensitivity to all neutrino flavours. Although only νe interact via the CC W-exchange,and so νe has a 6 times larger contribution than νµ or ντ . The radiochemical experiments, on
the other hand, only interact by CC interactions.
2compared to other directions30

In addition to these measurements, 11 neutrino events from Supernova 1987A were also luckily
captured [85], and there were significant contributions from analyses of atmospheric neutrinos.
These contributions earned Masatoshi Koshiba the shared Nobel Prize in Physics in 2002.

2.1.4

Super-Kamiokande

The Super-Kamiokande (SK) experiment[86] has been running since 1996 and is the successor
to Kamiokande. Most of the principles are the same as those mentioned above, however thedetector now holds ∼50 ktons of ultra-pure water, with a 22.5 kton fiducial volume, viewed by
11,146 inward facing 20-inch PMTs – with over 40% photodetector coverage. The tank is 41.4 m
tall and has a diameter of 39.3 m. The outer detector (OD), anti-counter, now has a completely
separate volume surrounding the inner detector (ID) and houses 1885 8-inch PMTs. SK was thedetector used in the analyses in this thesis and is described in greater detail in Chapter 3.
The chlorine and gallium experiments and Kamiokande had measured a solar neutrino fluxthat was half or less than that which was expected from the Standard Solar Models. This
firmly established the solar neutrino problem. Kamiokande was a real-time detector that could
observe solar neutrinos on an event-by-event basis and confirmed the solar origin of the neutrinos.
Signals for neutrino oscillation could be a distortion of the neutrino energy spectrum; a difference
between daytime and night time fluxes; or seasonal variations on the flux. Kamiokande began
searches for these signals, although did not observe any statistically significant differences. SK
had ∼25 times the target mass as Kamiokande, and continued the search [87].
The dominant background below 6.5 MeV was from the radioactivity of Radon externally andin the water. To reduce this in SK, the detector area was air-tight with fresh air pumped infrom outside the mine, and the rock above the detector was covered with radon-resistant plastic
sheets. In the air above the water-level in the tank, purified air was pumped in to maintainpositive pressure. This reduced the radon concentration to less than 3 mBq/m3 [86]. In addition,
the water purification system completely filters the 50 ktons of water over a period of 2 months.
Further low energy backgrounds were removed by cuts and reconstruction characteristics [87].
Above 6.5 MeV, the main background comes from the decay of muon-induced spallation products,
e.g. 16 N created by the absorption of cosmic muons on 16 O. These events were removed by cutsand likelihoods functions.
The solar neutrino signal was extracted from the cos θsun distribution, fitting a likelihood
function of the signal and background shapes. The signal could be predicted with the known
angular resolution between the incoming solar neutrino and outgoing lepton, and a mostly flat
background w.r.t. the cos θsun direction. Some neutrino oscillation parameter values predicted
an asymmetry in the flux of solar neutrinos that arrive from above the detector (daytime) andthose that pass through the Earth below the detector (night time). This difference would be
due to the matter effects in the Earth’s mantle and core. For vacuum oscillations, the change ofbaseline with the eccentricity of the Earth’s orbit could create a seasonal variation.
In 1999, SK reported an early day night asymmetry analysis for 504 days of data [88], after
already confirming the solar neutrino deficit in 1998. No significant difference was observed,
allowing a region of the oscillation parameter space to be excluded. This region was associatedwith the regeneration of νe solar neutrinos through matter effects in the Earth. In 2001, aprecision measurement of 18,464 observed solar neutrino events over 1,258 days was reported [87].
The flux was 45.1±0.5(stat.)+1.6−1.4 (stat.)% of the BP2000 SSM prediction. A day night asymmetryof 3.3% from the average flux, and 1.3σ away from zero difference, was observed, providing
31

Figure 2.5: In the left figure (a) shows the allowed regions for oscillation parameters from several
experiments (2002) [89]. The shaded region was from the Homestake Chlorine experiment, theregion in the dashed lines from GALLEX/GNO and SAGE. The hatched area is from the chargecurrent contribution from SNO (discussed later). The (b) figure shows the multiple solutions that
result from combining these results. The right figure shows a zoomed in region for the surviving
LMA solution with the addition of SK data (2002) [89] and SNO NC data to the allowed regionsin the left figure. The most confined region with a dashed-dotted contour was a global fit of all
the data. The dashed contour was with the SNO rates removed, and the dotted contour waswith no radio-chemical rates. The shaded region was the result without using the SSM 8 B flux
calculation.
constraints on neutrino parameters. No seasonal variation beyond the effect from the eccentricityof the Earth’s orbit was found.
Early fits to the deficit of solar neutrinos from each of the experiments, resulted in multiple
solutions as seen in Figure 2.5. These solutions occurred where the CC allowed regions overlapped. These were the Large Mixing Angle (LMA) and Small Mixing Angle (SMA) solutionsat large ∆m221 ; and for smaller ∆m221 there were the LOW and Quasi-Vacuum (Quasi-VAC)
solutions, down to the Vacuum (VAC) solutions for ∆m221 below ∼10−10 eV2 .
For 10−8 eV2 < ∆m221 < 10−3 eV2 , the matter density in the sun strongly affects the oscillation
probability. There was an expected resonant enhancement of the oscillations due to the MSW
effect for θ12 < π4 , and a suppression for θ12 > π4 from an anti-resonance effect. This effectcan be seen around the tan2 θ12 = 1 line drawn in the figure. For ∆m221 below 10−8 eV2 , thematter effects are negligible and there would be vacuum oscillations that are symmetrical aroundθ12 = π4 .
In 2002, SK updated their results for 1496 days of data, including two analyses [89]. One
was a “zenith angle spectrum” analysis looking for daily variations and spectrum distortions but
found no significant difference. This result was independent of the absolute solar neutrino fluxand considerably broke the degeneracy of the previous solutions. When combined with the rate
analysis against the SSM prediction, only a part of the LMA and quasi-VAC solutions remained,
both were large mixing solutions. When the SNO data was combined, allowing the interpretationof appearance of other active flavours in the SK measurement, the LMA solution was favoured.
A zoomed in plot of the surviving solution is also shown in Figure 2.5. In 2004, SK performed a32

Figure 2.6: The oscillation parameter 95% C.L. allowed regions [32] are shown for the combined
solar experiments on the left and with the addition of KamLAND data on the right. The solar
allowed regions came from a combined analysis of the SNO, Borexino, radiochemical experimentresults and the SK contribution.
precise day-night analysis of the SK-I data using a maximum likelihood method [90]. A variationof −1.8 ± 1.6(stat.)+1.3−1.2 (syst.)% was observed, where -2.1% was expected from the best-fit LMA
solution. This result was still consistent with no variation, and some more of the oscillation
parameter space could be excluded.
The greater sensitivity of SK compared to Kamiokande has allowed the solar neutrino analysis
threshold to be lowered to 5 MeV. Again, the main flux concerned was from the 8 B neutrinos,
with a much smaller contribution from the hep-neutrinos. The flux calculation from the SK-I
data, and the ratio w.r.t. the SSM was: [71, 86]:
ΦSK = (2.35 ± 0.02 ± 0.08) × 106 cm−2 s−1
ΦSK= 0.413 ± 0.014
SSM(2.16)(2.17)In 2011, SK released updated results including data from the SK-I to SK-III experimental
periods [32]. By constraining the 8 B neutrino flux to the SNO NC flux, the analysis of the energy
spectrum and time variation at SK, for the first time, favoured only the LMA solution at 95%
C.L. A global fit of all the solar results was performed, including results from SNO, Borexino
and the radiochemical experiments. The allowed resulting allowed region is shown in Figure 2.6.
The addition of KamLAND data greatly constrained the ∆m2 of the allowed region, which isalso shown in Figure 2.6. The best-fit parameters were [32]:sin2 θ12 = 0.30+0.02−0.01(tan2 θ12 = 0.42+0.04−0.02 )(All solar)−5∆m221 = 6.2+1.1
eV2−1.9 × 10
2sin θ12 = 0.31 ± 0.01

(2.18)(2.19)

2(tan θ12 = 0.44 ± 0.03)∆m221 = 7.6 ± 0.2 × 10−5 eV2(Solar + KamLAND)(2.20)(2.21)

A three flavour analysis was also performed combining all the solar data and KamLAND data.
The best fit was sin2 θ13 = 0.025+0.018−0.016 and the upper bound at the 95% C.L. reported was
33

sin2 θ13 < 0.059. In addition, the absolute fluxes for the SK-II and SK-III periods were consistent
with the SK-I measurement.
Recently, in 2013, SK published the first indication of the regeneration of solar electron
neutrinos from terrestrial matter effects [91]. The asymmetry measured was [−3.2 ± 1.1(stat.) ±0.5(syst.)]% which deviated from zero by 2.7σ. This was the first direct indication that matter
effects modify the neutrino oscillation probabilities.

2.1.5

Sudbury Neutrino Observatory

The Sudbury Neutrino Observatory (SNO) experiment began in 1999, at the Creighton mine
∼6,000 m.w.e underground near Sudbury, Canada [92]. It was unique in that it had a combinationof interaction channels to make a clean observation of the solar νe flux, as well as the flux fromall active neutrinos originating from the sun. The comparison of these flux measurements could
determine if the observed deficit in the solar νe flux was due to the oscillations to other active
flavours.
SNO achieved this by utilising 1 kt of heavy water, D2 O, as the neutrino target, allowing thefollowing reactions:νe + d −→ p + p + e−(CC)

(2.22)

νx + d −→ p + n + νx(NC)

(2.23)(ES)(2.24)−−

νx + e −→ νx + e

where νx is one of νe , νµ , or ντ , d is a deuteron and p a proton. The charged-current (CC)interaction in Equation (2.22), is sensitive only to νe and produces an electron that can bedetected by Cherenkov radiation. Through this interaction the flux of solar neutrinos arriving
at the Earth with electron flavour could be determined. The neutral-current (NC) interaction in
Equation (2.23), is the disintegration of a deuteron by a neutrino. It has a threshold of 2.2 MeV
and is independent of the neutrino flavour. The outgoing neutron was detected by the absorption
on deuterium producing a 6.25 MeV photon, or absorption on 35 Cl producing photons with a totalenergy of 8.6 MeV. MgCl2 was added to the D2 O for the enhanced neutron capture probability.
These photons Compton scatter, knocking out electrons that can be observed by their Cherenkov
light. Finally, the elastic scattering interaction in Equation (2.24), is the same interaction that
solar neutrinos are detected at Super-Kamiokande and is primarily sensitive to νe , while thecross-section for νµ or ντ is a factor of 61 smaller than for νe . Through the combination of theseinteractions, SNO could make a measurement of the solar neutrino flux and search for oscillations
independent of the SSM predictions.
The detector consisted of a 12 m diameter transparent spherical acrylic vessel that contained
1 kt of ultra pure heavy water. The vessel was supported by 10 rope loops of synthetic fibre,
and was surrounded by a 17.8 m diameter geodesic sphere made from stainless steel struts thatheld the 9,438 inward-facing PMTs, as seen in Figure 2.7. This was all housed in a barrelshaped cavity, 22 m in diameter and 34 m in height, and was filled with purified light water thatprovided shielding and support. The water shield attenuated low energy (< 4 MeV) γ rays fromradioactivity in the PMT array, as well as high energy γ rays from the cavity. In the main
detector volume, the total contribution from all radioactive sources was < 0.2% for low energy
backgrounds and < 0.8% for the high energy sources.
In 2001, SNO released the first results on the ES and CC measurements of the solar 8 B
flux [93]. The ES measurement was consistent with the value measured at SK, however the CC
34

Figure 2.7: The left figure shows the SNO detector [92], and the SNO CC measurement is on
the right [93]. The upper right figure shows the kinetic energy distribution of the extracted
CC events in the data points. The solid histogram was the 8 B flux from the null oscillation
prediction, scaled to the data. No deviation from the 8 B spectral shape was seen. The lower
right plot displays the ratio of the CC events to the prediction. Also shown is the accurate
ES rate from SK. The CC distribution was consistently smaller than the ES flux. The bands
represent the most significant energy dependent systematic errors.
result, that was sensitive only to νe , was considerably smaller. From the discrepancy of these
two results, there was good evidence for flavour transitions in solar neutrinos. In addition, thisresult was inconsistent with the null oscillation hypothesis that all observed solar neutrinos areof electron flavour. The data were taken over a live time of 240.95 days and were divided into
two sets: 70% was used to establish analysis techniques and 30% was kept for a blind test of astatistical bias in the analysis. The analysis of the two data sets were statistically consistent. The
analysis had a high threshold of 6.75 MeV, cutting out most low energy radioactive backgrounds
and most of the neutron capture events.
The data were separated into contributions from CC, ES, and remaining neutron events by
probability density functions (pdf) of effective kinetic energy, the event direction relative tothe solar direction, and from a volume weighted radial variable. From the 1169 fiducial events
above threshold, the maximum likelihood analysis output 975.4 ± 39.7 CC events, 106.1 ± 15.2
ES events and 87.5 ± 24.7 neutron events. The CC spectrum could be extracted by repeating
the signal extraction with the constraint on the shape of the CC pdf removed. The resulting
distributions are shown in Figure 2.7. The 8 B neutrino fluxes assuming standard spectrum shapewere measured to be:
+0.12
φSNO
CC = 1.75 ± 0.07(stat.)−0.11 (syst.)

× 106 cm−2 s−1(2.25)+0.16
φSNO
ES = 2.39 ± 0.34(stat.)−0.14 (syst.)

× 106 cm−2 s−1(2.26)The difference between the 8 B flux derived from the SNO ES measurement and CC measurement was 0.64 ± 0.40 × 106 cm−2 s−1 , which was a 1.6σ difference. The SNO ES rate wasconsistent with the accurately measured rate by SK. The difference between the SK ES rateand SNO CC rate was 0.57 ± 0.17 × 106 cm−2 s−1 , a deviation of 3.3σ or a probability of 0.04%
35Figure 2.8: The original measurement of the 8 B solar neutrino flux from the three reactions at
SNO from 2002 [33] on the left, and the updated result from 2005 [94] on the right. The bands
show the possible flavour compositions of active neutrinos from the measured information andthe ±1σ uncertainty. The dashed band was the SSM prediction, and the NC measurement in
the blue band agrees well with it. The red band shows the CC measurement and the green bandfor the ES result. The intersection of the bands give the flavour composition in φe and φµτ atthe detection point. The 2002 result assumed no distortion in the 8 B spectrum, while the 2005
measurement made no assumptions on the neutrino energy spectrum. The grey band shows the
SK ES measurement.
that the SK measurement could statistically fluctuate down to the SNO CC measurement. This
inconsistency was evidence for non-electron active flavour constituents in the solar neutrino flux.
Since the νe component was determined with the CC measurement, the νµτ component could be
inferred from the ES measurements and therefore the total flux. These were determined to be:φ(νµτ ) = 3.69 ± 1.13

× 106 cm−2 s−1(2.27)φ(νx ) = 5.44 ± 0.99

× 106 cm−2 s−1(2.28)

This inferred total flux was in excellent agreement with the SSM prediction! The additional
information from the νe component of the solar neutrino flux could give a good indication of thetotal observed flux.
In 2002, SNO reported on a direct measurement of the total solar neutrino flux from NC
reactions on deuterium, which could determine the total flux independent of neutrino flavour [33].
The threshold for the reaction in Equation (2.23) was 2.2 MeV, and the analysis threshold wassensitive to the neutrons at 5 MeV. To reduce backgrounds data with vertices in the fiducial
volume of 550 cm from the centre were used. In these conditions the neutron detection efficiency
was measured to be 14.4% through calibrations.
The main backgrounds were from low levels of bismuth and thallium, 214 Bi and 208 Tl, inthe detector materials, from uranium and thorium decay daughter chains. The activity of theseisotopes caused low energy Cherenkov events and the photodisintegration of D2 O, releasing freeneutrons. Several ex situ and in situ studies were carried out to determine the neutron andbackground production rate. The expected background rates were 1.0 ± 0.2 neutrons per day
in the target region, and 1.3+0.4−0.5 neutrons per day from external sources. These rates were36

approximately 12% of the expected signal from NC interactions from the SSM prediction. In
addition, a 250 ms dead time after every event was required to reduce contamination from cosmic
ray and atmospheric neutrino induced neutrons, while the background contribution from subCherenkov threshold muons were expected to be small.
Using a maximum likelihood analysis, the fluxes for each reaction were measured to be:
+0.06
+0.09
φSNO
CC = 1.76−0.05 (stat.)−0.09 (syst.)

× 106 cm−2 s−1(2.29)

+0.24
+0.12
φSNO
ES = 2.39−0.23 (stat.)−0.12 (syst.)

× 106 cm−2 s−1(2.30)−2 −1(2.31)

+0.44
+0.46
φSNO
NC = 5.09−0.43 (stat.)−0.43 (syst.)

6

× 10 cms

The NC flux, that was independent of neutrino flavour, was considerably larger than the CC
flux, that depends on only νe , strongly implying neutrino flavour transitions such as those in
Neutrino Oscillations. From these fluxes, SNO calculated the components from electron (φe )and other active flavours (φµτ ):+0.09
φe = 1.76+0.05−0.05 (stat.)−0.09 (syst.)+0.48
φµτ = 3.41+0.45−0.45 (stat.)−0.45 (syst.)

× 106 cm−2 s−1(2.32)−2 −1(2.33)

6

× 10 cms

This φµτ component was 5.3σ above zero. When combined with the SK ES measurement at thetime, this component became φµτ = 3.45+0.65−0.62 , which was 5.5σ above zero. So this was strong
evidence for flavour transitions of solar neutrinos. The bands for the flavour constituents of theflux are shown in Figure 2.8. The bands came from the unknown flavour composition of eachmeasurement, with the exception of the CC result that was induced by νe regardless of a νµ orντ component, and from the uncertainty in the measurements.
The other remarkable thing about the SNO NC measurement was that it was consistent withthe contemporary SSM prediction for 8 B neutrinos at φSSM = 5.05+1.01−0.81 . Prior to this result, allsolar neutrino experiments were reporting a deficit in the solar neutrino flux. These experiments
were primarily sensitive to νe , and indeed there was a deficit in the electron flavour contribution.
SNO has shown that when all flavours were combined, the total solar flux of active 8 B neutrinos
was in agreement with the theoretical SSM prediction.
Also in 2002, SNO made the first direct measurement of the day-night asymmetry in the νeflux and the total solar neutrino flux [95]. The νe (CC) events had a day minus night asymmetryof 14.0%±6.3%+1.5−1.4 %. If the total flux of active neutrinos was constrained to have no asymmetry,the νe asymmetry reduced to 7.0%±4.9%+1.3−1.2 %. The SNO results constrained a lot of the allowed
oscillation parameter space, and when combined in a global analysis with other solar neutrino
experiments, the LMA solution was strongly favoured.
SNO began the “salt phase” of their experiment in June 2001 when they added 2 metrictons of NaCl to the D2 O volume, which continued through until October 2003. The additionof NaCl increased the neutron capture efficiency by a factor of three, and the capture on 35 Cl
created a distinctive isotropic pattern of γ rays with a higher total energy than the single photon
from neutron capture on deuteron. The NC events could therefore be more accurately separated
statistically, allowing for a more precise measurement of the NC rate. In 2004, SNO released
an improved measurement of the total solar neutrino flux with the added NaCl [96]. Thismeasurement had no assumptions about the energy dependence of the νe survival probability
and was found to be:
φSNO
NC = 5.21 ± 0.27(stat.) ± 0.38(syst.)
37

× 106 cm−2 s−1(2.34)

Maximal mixing was also rejected with a significance of 5.4σ. In 2005, the extracted integral
CC rate was consistent with no day-night asymmetry, which was expected for the best fit LMA
solution [94]. The updated solar neutrino flux flavour composition bands can also be seen in
Figure 2.8. For the oscillation parameters, see the global fits that include SNO data in the SK
solar review, Section 2.1.4.

2.1.6

Borexino

The Borexino experiment began taking data in 2007 with a low background liquid scintillator
detector 3800 m.w.e. underground at Gran Sasso, Italy [97]. The main aim of the experiment
was to make real-time measurements of solar neutrinos below 2 MeV, which accounts for over
99% of the flux. Radiochemical experiments had low thresholds, but the result of their measurements were integrated solar neutrino rates, losing individual event information. Whereas water
Cherenkov detectors could measure in real-time, but were limited to thresholds above 5 MeV
accounting for 0.01% of the total flux, due to low light yields from cherenkov light and naturalradioactive backgrounds.
The real-time measurements prior to Borexino were primarily sensitive to 8 B neutrinos. So
one of Borexino’s first goals was to measure the mono-energetic 0.862 MeV line of the 7 Be solar
neutrino flux [98, 99], which was expected to constitute ≈10% of the total flux. The data at thetime favoured the MSW LMA oscillation solution, which predicted matter enhanced oscillations
at the 8 B neutrino energies that reduced to vacuum oscillations at low energies in the vicinityof 7 Be. The νe survival probability increased from ≈0.33 for high energies up to ≈0.6 aroundthe 7 Be mono-energetic line. So this measurement could provide a good test of the favoured
oscillation solution.
The detection mechanism at Borexino was through the neutrino elastic scattering on electrons
in liquid scintillator that produced sufficient light to observe low energy events. The reaction
was sensitive to all flavours of neutrinos, although primarily to electron neutrinos. The signal
efficiency of the scintillator was ≈50 times greater than water Cerenkov, but no directionality
could be observed and the signal electrons could not be distinguished from natural radioactivity.
So Borexino relied on producing an environment with extremely low radioactivity.
The detector was made up of layers of concentric spherical containers [97] as seen in Figure 2.9. The Inner Vessel (IV) had a radius of 4.25 m and contained 300 tons of scintillator. The
scintillator was pseudocumene (PC) doped with 2,5-diphenyloxazole (PPO) at a concentration of
1.5 g/l. Surrounding this was a 5.5 m radius Outer Vessel (OV) that contained PC and 5.0 g/l ofdimethylphthalate (DMP). The DMP quenches the residual scintillation of the PC. So this layer
acts as a passive shield against radon and other backgrounds outside the detector. Outside thiswas a 6.85 m radius stainless steel sphere, which also contained the PC-DMP buffer fluid and held
2212 8” PMTs. The encompassing tank of radius 9 m and height 16.9 m, contained ultra-purewater and acted as a passive shield and comic muon veto. The main signal was observed fromthe IV, and the outer layers provided ≈5.5 m of m.w.e. passive shielding from external radiation.
Many careful considerations were taken during the preparation, transportation and handlingof detector materials to minimise all sources of radioactivity. A fiducial volume was chosen to
efficiently remove external backgrounds, in this volume the predicted γ background was lessthan 0.5 counts/(d · 100 t) in the 250-800 keV range. Backgrounds from Rn daughters from theU decay chain were removed by detecting the delayed coincidence of 214 Bi–214 Po and vetoing up
to 3 hours prior to the coincidence. The 14 C intrinsic to the scintillator provides a background38

Figure 2.9: The Borexino detector on the left [98], and the 7 Be measurement on the right [98].
The detector geometry is shown, a description of each layer is written in the text. The rightfigure shows the 7 Be analysis in which the peak from α particles from 210 Po decay was statistically subtracted. The remnant contribution was still considered in the fit shown by the pinkline. Background contributions from 85 Kr (blue) and 210 Bi (green) were considered. Finally thedistinctive ‘shoulder’ contribution from 7 Be is shown in red, and the combined fit in black.
from the 156 keV β decay, limiting signal observations to above 200 keV.
In 2008, Borexino reported on its first measurements of the 7 Be neutrinos [98, 99]. The signal
for the mono-energetic 7 Be neutrinos was an electron recoil with a flat energy distribution anda sharp cut-off at 665 keV. This shoulder in the distribution can be seen by following the red
curve in Figure 2.9. There was a peak in the distribution due to α particles from 210 Po decay,
which was statistically subtracted in the figure. After the subtraction, the energy range between
270–800 keV was dominated by 7 Be neutrinos. A small bump still remained, around 400 keV,which was fitted with the fuchsia curve. Another analysis was performed without this statisticalsubtraction, and limited the fit region to above the 210 Po peak (560–800 keV). Both methods
gave consistent results. Other background contributions, such as from the 85 Kr (blue) and 210 Bi(green) β decays were also fitted. The best value for the 0.862 MeV 7 Be neutrinos was:
R7 Be = (47 ± 7 ± 12)counts/ (day · 100 ton)

(2.35)

Where the ratio to the SSM was [71]:
R7 Be= 0.63 ± 0.18
SSM

(2.36)

One of the main uncertainties came from the unknown component of the 85 Kr β decay
spectrum, which was similar in shape to the signal, but could be distinguished in the shoulder
region. The hypothesis that the results were explained by 85 Kr alone was rejected at >5σ level.
The main systematic uncertainty came from the determination of the fiducial mass and wasestimated to be 25%. Due to the good agreement of the fit in the shoulder region, which could
not be explained by any known backgrounds, Borexino concluded that they had detected 7 Be
solar neutrinos.
In 2010, Borexino reported on a measurement of 8 B neutrinos with a 3 MeV threshold [100].
The measurement made use of 8 B neutrino induced ν-e elastic scattering that resulted in a solar396−2 s−1 . The survival probability was found to be 0.29 ± 0.10,
flux of ΦES
8 B = 2.4 ± 0.4 ± 0.1 × 10 cm
which was 1.9σ different from the 7 Be result and consistent with the MSW-LMA solution thatpredicted a varying survival probability. In 2012, Borexino made the first measurement of the
pep solar neutrinos in the 1.0–1.5 MeV range [101]. The solar flux was estimated to be Φpep =(1.6 ± 0.3) × 108 cm−2 s−1 . The absence of the pep signal was disfavoured at 98% C.L. This result
provided the strongest constraint on the CNO solar neutrinos.

2.1.7

Solar Neutrino Summary

The sun is an abundant source of neutrinos. Their relatively low energy makes them tricky todetect and many experiments have employed widely varying detection mechanisms to study them.
The first solar neutrinos were discovered in 1968 by Davis’ Chlorine Experiment that captured
them with a 0.8 MeV reaction threshold that gave it sensitivity to 7 Be and 8 B neutrinos. About
30% of the flux expected from the Standard Solar Model (SSM) was observed, which marked the
beginning of the solar neutrino problem. The Gallium experiments ran between 1990–2003, which
had a very low energy threshold of 233 keV, giving sensitvity down to pp-neutrinos. They had
complicated procedures for extracting and counting the signal Germanium isotope, and observed
about 52% of the flux of the SSM prediction, further evidence for the solar neutrino deficit.
Since 1987 the Kamiokande experiment achieved real-time detection of solar neutrinos, preserving their directional information and confirming their solar origin. The initial analysis threshold of the elastic scattering interaction was ∼7 MeV and was sensitive primarily to the 8 B neutrinos. Similarly, approximately 49% of the expected flux was observed. Since 1996, Kamiokande
was succeeded by Super-Kamiokande (SK). It had ∼25 times the detector mass and detectedsolar neutrinos with a threshold of about ∼5 MeV. The early global fits, around 2000, to solar
neutrino data had many potential solutions for a Neutrino Oscillation explanation of the flux
deficit.
SNO largely helped to solve the puzzle by measuring the electron flavour neutrinos directlywith a CC reaction in 2001, and the total solar flux independent of flavour with a NC interaction
in 2002. A smaller flux of νe was observed compared to other experiments that had some νµ and ντ
in their signal, meanwhile the total flux measured agreed with the SSM! This result showed thatthe deficit was due to flavour transitions from electron neutrinos to muon and/or tau neutrinos.
The constraints from SNO allowed the global analyses to favour the Large Mixing Angle (LMA)
solution. Since 2007 Borexino has been measuring solar neutrinos below 2 MeV in real-time.
The MSW LMA solution predicted that the νe survival probability increased from ≈0.33 at highenergies to ≈0.6 at lower energies. *Borexino confirmed this with precice measurements of the
0.86 MeV 7 Be flux in 2008 and 1.0–1.5 MeV pep neutrinos in 2012.

2.2

Atmospheric Neutrino Experiments

The Earth is constantly bombarded by cosmic rays that interact with molecules in the air toproduce mainly pions and some kaons. These then decay into neutrinos known as Atmospheric
Neutrinos, by chains like those seen in Equations (2.37, 2.38). The first atmospheric neutrinos
were detected in the 1960s in South Africa [102] and India [103].
40π + −→ µ+ + νµ+(2.37a)+µ −→ e + νe + ν̄µπ − −→ µ− + ν̄µ−(2.37b)(2.38a)−µ −→ e + ν̄e + νµ(2.38b)

The pions produced decay to give a νµ and a muon, which can then decay to give a νe andanother νµ . Naively, two atmospheric νµ would be expected for each νe , a 2:1 ratio of νµ to νe .
Since the higher energy muons reach the ground before they can decay in flight, an even larger
ratio could be expected for higher energy events. However, there was some discrepancy in themeasured ratio which lead to the establishment of Neutrino Oscillations as the dominant theoryof neutrinos.
There were various calculations of the atmospheric neutrino flux produced by the bombarding
cosmic rays. Their absolute fluxes agree within about 20%, however the ratio of flavours is moreaccurate at the 5% level. So atmospheric neutrino experiments generally quote their results in aratio like that in Equation (2.39).

R= 

2.2.1

νµ +ν̄µνe +ν̄eνµ +ν̄µνe +ν̄e obs(2.39)

MC

Kamiokande and IMB

Kamiokande started operation from 1983 and went through several upgrades, including an outerdetector veto counter in 1985 (Kamiokande-II) and an electronics upgrade in 1990 (KamiokandeIII) [27, 28]. The detector was able to detect neutrinos by the water cherenkov light emitted by
the lepton products of neutrino interactions. It was originally constructed to search for nucleon
decay, and studied atmospheric neutrinos as a principle background. The experiment was able
to find a significantly smaller νµ flux than expected through this research.
The experiment ran 2700 m.w.e. underground near Kamioka, Gifu Prefecture, Japan, whereSuper-Kamiokande currently runs. The detector was a 4.5 kton cylindrical tank (15.6 m ø ×
16 m height) filled with pure water and surrounded by 1000 inward facing Photo-Multiplier Tubes.
The detector can be seen in Figure 2.4. In 1988, Kamiokande observed (59 ± 7)% muon-likeevents from that expected at less than 1.33 GeV, which could not be explained by systematic
detector effects or uncertainties in the atmospheric neutrino flux [27]. Similarly, in 1994, a flavour
double ratio of 0.57+0.08−0.07 ± 0.07 compared to the expectation was observed for events greater than
1.33 GeV [28].
Some early Zenith distributions can be seen in Figure 2.10. The first results released werefor Sub-Gev events (<1.33 GeV), a deficit can be seen in the left most figure, but there appears
to be no zenith dependence. Kamiokande later carried out an analysis on Multi-GeV events(>1.33 GeV), which have a greater angular correspondence to the neutrino direction (an RMS
of 15-20◦ , compared to ∼60◦ for sub-GeV). The middle figure shows that the deficit seems to be41

Figure 2.10: Kamiokande Zenith Distributions. The left most figure is for sub-GeV events(<1.33 GeV) [27]; the middle for multi-GeV events (>1.33 GeV) [28]; and the right most figureis the zenith distribution of the flavour ratio for Multi-GeV events [28]. The (a) panels arefor electron-like events and the (b) panels for muon-like events. The dashed and dotted lines
correspond to early neutrino oscillation monte-carlo.
coming from upward muon neutrino events that have travelled through the Earth. This zenith
dependence can be more clearly seen in the flavour ratio.
The IMB experiment ran from 1986 to 1991 and utilised a larger 8-kton cherenkov detector
consisting of a cuboid tank (18 m × 17 m × 22.5 m) with 2048 PMTs on the 6 faces, 1570 m.w.e.
underground at Cleveland, Ohio, US [29]. Using two independent analyses, one of particle
identification and one requiring a muon decay signal, IMB reported a deficit of muon neutrinos(or excess of electron neutrinos) from the expectation. In their final data set, the fraction ofmuon-like events was 0.36 ± 0.02(stat.) ± 0.02(syst.), where 0.51 ± 0.01(stat.) ± 0.05(syst.) was
expected [29]. The percentage of events with a muon decay signal was 33 ± 2(stat.)% events
while 43 ± 1(stat.)% were expected [29].
Kamiokande measured about 60% muon-like events compared to the expectation from theStandard Model predictions, and the flavour ratios measured by IMB supported this discrepancy.
This unexpected result became known as the Atmospheric Anomaly, and both experiments suggested that Neutrino Oscillations could be a possible explanation for these results. The later
higher energy Kamiokande analysis also seemed to show a zenith angle correlation of the muonneutrino deficit, further supporting the Neutrino Oscillation picture.

2.2.2

Fréjus and NUSEXThe Fréjus experiment [104] collected data between 1984 and 1988, in an underground laboratory
under the Alps between Italy and France. The detector was a fine grain iron-calorimeter built
to better understand the atmospheric neutrino flux and properties, as well as for backgroundstudies for nucleon decay and neutrino astronomy. It started with 240 tons, gradually increasing
to 900 tons. The detector consisted of 912 flash chambers and iron planes interspersed with 113
planes of geiger tubes. It could reconstruct muon tracks internally (contained), as well as fromthe surrounding rock (upward going stopping muons and horizontal through-going muons).
The fine granularity of the detector allowed it to measure up to 10 TeV muons and search someof the parameter space not covered by Kamiokande and IMB. However, they found no evidence
for neutrino oscillations and their measurement of the νµ rate agreed with the expectation.
The NUSEX experiment ran from 1982 to 1988 [105]. The detector was similar in the designconcept to Fréjus, but was smaller at 150 tons of active mass and ran from 1982 to 1988. It was
42

Figure 2.11: Super-Kamiokande Zenith distributions [31]. The top panels show e-like events, andthe bottom panels show µ-like events. The left 4 panels contain sub-GeV events (<1.33 GeV), andthe right 4 panels contain multi-GeV events (>1.33 GeV). Partially Contained events are µ-likeevents where the muon escapes the inner volume of the detector. The zenith angle dependenceof the muon neutrino deficit can be seen in the higher energy muon events.
located at the Mont Blanc tunnel in France. Similarly to Fréjus they measured no deficit in themuon neutrino rate from the expectation.
The agreement of these two experiments with the Standard Model prediction suggested that
the Atmospheric Anomaly could possibly have been the result of a systematic effect. Later
atmospheric experiments provided more evidence in support of Neutrino Oscillations.

2.2.3

Super-Kamiokande Atmospheric Neutrinos

Super-Kamiokande (SK), described in Chapter 3, is also capable of measuring atmospheric neutrinos. The experiment collected high statistics data with greater accuracy than Kamiokande.
They measured the neutrino distributions relative to their incoming zenith angle, where a cos θof 1 was directly above the detector and -1 corresponded to neutrinos coming from the otherside of the Earth below the detector. In 1998, SK released a measurement that showed the
deficit of muon neutrinos had a zenith angle dependence, and that the observed results could be
explained well by νµ → ντ neutrino oscillations [30, 31, 106–109]. They also showed that νµ → νeoscillations were disfavoured and that these observations could not be explained by any known
uncertainties. This result was key evidence for neutrino oscillations.
Figure 2.11 shows the zenith distributions presented in 1998. The top panels display thezenith distributions for the electron-like events, which are more or less consistent with the no
oscillation prediction in the hashed histograms. However, the zenith dependence of the muon
events can be seen in the lower panels. The neutrino energy of the events increases in the
figures from left to right. This zenith dependence becomes more apparent as the neutrino energy
increases, where there is a greater angular correspondence between the directions of the neutrino
43

Figure 2.12: Super-Kamiokande L/E distributions on the left [9], and atmospheric oscillation
parameter allowed regions from 2010 3-flavour analysis [15]. On the left, the ratio of the data
to the null oscillation prediction as a function of the reconstructed L/E is shown. If there wereno oscillations, points distributed around 1 would be expected. Best fits for 2-flavour νµ → ντoscillations (solid line), neutrino decay (dashed line), and neutrino decoherence (dotted line) are
shown. On the right, the allowed regions are for the normal mass hierarchy, at 68% C.L. in thethin red line, 90% C.L. in the medium blue line, and 99% C.L. in the thick yellow line.
and the lepton product of the interactions. These results show that the deficit in the muon
neutrinos mostly comes from upward going neutrinos that have travelled the order of 10,000 km.
By breaking the events into energy groups, an energy dependence of the deficit can also
be probed. The solid lines in Figure 2.11 represented the best-fit νµ → ντ neutrino oscillation
prediction. SK showed these oscillations to be in good agreement with the data and reported at
the 90% confidence level the oscillation parameters were sin2 2θ > 0.82 and 5 × 10−4 < ∆m2 <6 × 10−3 eV2 .
Prior to the zenith measurement, SK reported on their first measurement of the double
flavour ratio of 0.61 ± 0.03(stat.) ± 0.05(syst.) for the sub-Gev energy range in early 1998 [107],
which was consistent with the Kamiokande, IMB and Soudan-2 experiments. A little later asimilar ratio was published for the multi-GeV range, and some zenith asymmetry of the deficit
was observed [106]. With an updated data set of 535 days, still in 1998, SK reported on the firstevidence for neutrino oscillations visible from the zenith, or flight path, dependence of the deficitof muon neutrinos [30, 31]. This result pointed to neutrino oscillations as a likely explanation
for all the anomalies seen in neutrino experiments. This same zenith dependence was seen in thehigher energy through-going muon events, in which a high energy neutrino interacts in the rocksurrounding the detector and the muon product is observed [110]. In 2000, SK was able to reject
the oscillation to sterile neutrinos at the 99% C.L. while the two-flavour νµ ↔ ντ oscillations
were consistent with all the data [108].
These results were the first evidence that neutrinos oscillate, something that the later Kamiokande
results were also hinting at. Although, there were multiple competing theories that could produce a deficit in the number of muon neutrinos observed with increasing zenith angle. One of44

the characteristic features of Neutrino Oscillations, however, is that it has an L/E dependence:a dependence on the ratio of the neutrino flight length to its energy. In 2004, SK released a
result from an analysis that aimed to reconstruct the oscillatory L/E signature of neutrino oscillations [9], which can be seen in Figure 2.12. The analysis was able to show a significant dip
at L/E = 500 km/GeV in the data, which was reproduced by neutrino oscillations, but not by
neutrino decay and neutrino decoherence. The minimum χ2 best fit for neutrino decay was 3.4σ
larger than the neutrino oscillation best fit; and the best fit for neutrino decoherence was 3.8σ
larger. This was the first evidence that the neutrino survival probability follows the sinusoidal
behaviour of neutrino oscillations.
For more recent analyses, in 2010 there was a report on an investigation into 3-flavour oscillations trying to measure a non-zero θ13 from the data, as well as a search for sub-leading
effects that could determine the quadrant of θ12 , using data from the SK-I to SK-III periods [15].
No evidence for a non-zero θ13 or a preferred θ12 quadrant was found. The data for both normal and inverted hierarchy were consistent, and neither were preferred. The allowed regions forthe atmospheric parameters for normal hierarchy from this analysis can be seen in Figure 2.12.
There has also been a statistical approach with a neural network analysis to observe tau neutrino
appearance in the SK data, published in 2012 [111]. The result was consistent with the expected
number of oscillated τ neutrinos, and excluded the no appearance hypothesis at the 3.8σ level.

2.2.4

Soudan2 and MACROThe Soudan 2 experiment ran from 1989 to 2001 and used a 963 metric ton iron trackingcalorimeter detector that was originally built to study proton decay. The experiment was located2070 m.w.e. underground in northern Minnesota, US. Soudan 2 provided the first independentconfirmation [112] of the Super-Kamiokande (SK) zenith results.
The detector was a honeycomb structure and acted as a time-projection chamber. The
drift tubes were 1.5 cm in diameter and 1 m long, filled with an argon-CO2 gas mixture andcan be seen in Figure 2.13. The tubes were secured in corrugated steel sheets and arranged
into a honeycomb matrix. This geometry was chosen for a more isotropic response. Charged
particles that passed through the drift tubes ionised the gas producing electrons. An electricfield over the tubes caused the electrons to drift at ∼0.6 cm/µs to the ends of the tubes. At
the tube ends there were vertical anode wires that amplify and collect the electrons, and an
induced charge was read from horizontal cathode pads. These two signals matched in time
identify which tube the ionisation occurred in. The coordinate along the tube was given by thedrift time. The track sampling frequency was about once every 3 cm, and tracks and vertices
were reconstructed at ∼0.5 cm resolution. The honeycomb structure made up modules withdimensions of about 1 × 1 × 2.5 m3 , and these modules were built up to construct the 963 ton
detector mass. Surrounding the detector, there were three layers of proportional counters that
operated as a comic ray and external background veto.
Soudan 2 was considerably smaller than SK, and so had a lower event rate, however they wereable to reconstruct clear particle tracks and the event vertices with centimetre resolution. This
enabled them to reconstruct both the lepton and hadronic final state, allowing them to improve
the resolution of the L/E value by 2 to 3 times at low energies. Whereas the recoil proton wasvery difficult to see in SK, so there was a low resolution in L/E for events below ∼1 GeV.
Figure 2.14 shows the L/E distributions reconstructed by Soudan 2 in 2003 [112]. No significant deficit or excess was seen in the νe distributions, however a deficit in the νµ events can be
45

Figure 2.13: The Soudan 2 drift tube on the left [113], and the MACRO detector on theright [114]. In the left figure is a schematic of the main active element in the Soudan 2 detector, a drift tube. An electric field was set up from the centre of the tube to the ends byelectrodes embedded in the circumference of the cylinders. This caused the ionisation electrons
to drift to the ends and collect on the anode wires, and induce a charge on the cathode pads.
A honeycomb matrix of these tubes built up the detector. On the right, the arrangement of thedetection mechanisms in a MACRO supermodule is shown. The dark grey planes were scintillator (3 horizontal, 2 vertical), the many light grey planes were streamer tubes, and there were afew Track Etch detectors shown in black.
seen from around log10 (L/E) > 1.5. Soudan 2 also reproduced zenith distributions similar to
those from SK, and reported an oscillation parameter space that enclosed the SK result, which
can also be seen in Figure 2.14.
The MACRO experiment started taking data during construction from 1989 and recorded
their main dataset between 1994 and 2000 [115]. The detector used layers of scintillator for timing
and streamer tubes for particle tracking, and was located an average of 3800 m.w.e. undergroundat Gran Sasso, Italy. One of the main goals of the experiment was to study atmospheric neutrino
oscillations. The experiment primarily observed upward-going muons that were produced byneutrino interactions in the rock below the detector. Although, they also recorded neutrino
events that originated within the detector as well.
The MACRO detector was made up from six supermodules, lined up side-by-side with total
dimensions 76.5 × 12 × 9.3 m3 , and a total weight of 5.3 kton [114]. A diagram of one of thesupermodules can be seen in Figure 2.13. Each module consisted of three independent detection
mechanisms: liquid scintillation counters, streamer tubes, and nuclear track (track-etch) detectors. The arrangement of these three sub-detectors can be seen in the figure. The scintillator
was a mixture of 96.4% mineral oil, 3.6% pseudocumene, and some PPO and bis-MSB. Each endof the counters were viewed by 8-inch PMTs. The coordinate along the counter could be determined by the time-of-flight, from the relative time difference at each end. The streamer tubes
had 3 × 3 cm2 cross-sectional area, were filled with a 73% He and 27% n-pentane gas mixture andwere read out by anode wires in the centre of the tube. The second coordinate in each plane wasgiven by pick-up strips. The track-etch detectors were mainly for magnetic monopole detection.
The MACRO results were in agreement with νµ → ντ oscillations, and excluded the no46

Figure 2.14: Soudan 2 L/E distributions of their high resolution sample on the left [112], andthe oscillation parameter allowed regions from MACRO and Soudan2 [112]. The upper plot isfor νe events, and the lower for νµ events. The dashed histogram is the Monte Carlo prediction
without oscillations, while the solid histogram is with oscillations. A deficit can be seen fromlog10 (L/E) > 1.5 in the νµ L/E distribution. The dotted line is the contribution expected frominteractions in the surrounding rock, and is included in the Monte Carlo distributions. In theright figure, the Soudan2 90% allowed region is shown in the solid line, MACRO in the dottedline and the contemporary SK result in the dashed line.
oscillation hypothesis at 5σ. They reported a more constrained oscillation parameter spacethan Soudan 2, which also encompassed the SK result as seen in Figure 2.14. Although thelimited dataset of Fréjus and NUSEX did not observe oscillations, Soudan 2 and MACRO had
similar detection methods and reported results in agreement with neutrino oscillations. Thisconfirmation from independent experiments strengthened the evidence for neutrino oscillations,making it very unlikely that the observation was a systematic effect or statistical fluctuation.

2.2.5

Atmospheric Neutrino Summary

Neutrinos are produced by the collision of cosmic rays in the atmosphere across the globe, making
an abundant source of neutrinos with largely varying flight-paths from ∼10 km to ∼10,000 km,
and a spectrum of energies. This made them an ideal probe into the flight-length and energy
dependence predicted by Neutrino Oscillations. In 1988, Kamiokande observed about ∼60% of
the νµ events from the Standard Model expectation. This was supported by the IMB results
observing closer to ∼70% of the expectation. These unexpected measurements were known as
the Atmospheric Anomaly.
The early iron-calorimeter experiments in the 1980’s, Fréjus and NUSEX, measured a νµ rate
in agreement with the expectation. So it was possible the deficits seen in the first Cherenkov
neutrino experiments were due to a systematic effect. However, the first real evidence for Oscillations came from the 50 kton water Cherenkov experiment Super-Kamiokande (SK) in 1998.
47

A clear zenith angle dependence was observed in the νµ events, which showed the deficit grewwith flight-path. There were several competing explanations for this and the 2004 L/E Analysis
at SK demonstrated that Neutrino Oscillations provided the best interpretation, by reproducing
the first dip in the predicted sinusoidal pattern. Around the year ∼2000, the later trackingcalorimeter experiments Soudan 2 and MACRO, which had similar detection methods to Fréjus
and NUSEX, gave an independent confirmation of the Neutrino Oscillation evidence observedat SK. These results could not be explained by any known systematic uncertainty, establishing
firm evidence for Oscillations from atmospheric neutrino experiments.

2.3

Reactor Neutrino Experiments

Nuclear reactors provide a source of ν̄e through the β − decay of the neutron rich fragments fromthe nuclear fission reactions of uranium and plutonium. In order for these fission products to bestable, on average 6 neutrons decay to 6 protons releasing 6 ν̄e per fission. The energy release ineach fission is around 200 MeV, so there are on average about 2 × 1020 ν̄e /sec for each GWthermalof power output. These neutrinos can be used to probe ν̄e → ν̄x oscillations, where x is a speciesof neutrino other than the electron neutrino. Since the energy of neutrinos produced by a reactor
is a few MeV, there is insufficient energy to produce a muon or tau lepton at detection. For
this reason, only the possibility of a deficit of ν̄e flux could be observed and reactor neutrino
experiments were therefore known as disappearance experiments.
Typically reactor neutrinos are detected by the inverse β-decay reaction:
ν̄e + p → e+ + n(2.40)The signature for this reaction is the prompt detection of the e+ from its annihilation with anelectron into two 511 keV γ-rays, followed by the delayed neutron capture. The coincidence of
these two events can be used to reject the majority of the background. It has a reaction thresholdof 1.804 MeV.
There were many reactor experiments that did not observe neutrino oscillations: Gosgen,Krasnoyarsk, Bugey, CHOOZ and Palo Verde. However, they set limits on oscillation parameters,
particularly, CHOOZ provided a stringent limit on θ13 . The later Reactor experiments were thefirst to measure θ13 .

2.3.1

CHOOZ

The CHOOZ experiment [19, 116, 117] took their main data between March 1997 to July 1998
and was located in a villiage of the same name, in France. The detector was located 1 km awayfrom two pressurized water reactors with a combined thermal power of 8.5 GW. These reactors
were just coming online, and only one reactor was on for over 80% of the experiment live time.
However, this situation allowed CHOOZ to understand their neutrino source over a range ofconditions.
The detector was a cylindrical vessel 5.5 m in diameter, 5.5 m deep, and can be seen in
Figure 2.15. The target was a 5-ton plexiglass container filled with hydrogen rich paraffinic
liquid scintillator for the “free protons,” and 0.09% gadolinium for the high neutron capture
cross section and ∼8 MeV γ-ray signal. The target container was observed by 192 8-inch PMTs
with 15% coverage, and outside that there was an optically isolated veto region. These regions
outside the target region were filled with undoped scintillator.
48

Figure 2.15: The CHOOZ detector [116] on the left and the CHOOZ ν̄e ↔ ν̄x oscillation excluded
parameter space [19] on the right. This measurement put stringent limits on νe ↔ νx oscillation.
The hashed region shows the Kamiokande 90% allowed oscillation region. This was excluded,
supporting the νµ ↔ ντ oscillation hypothesis for atmospheric neutrinos.
The neutrino source flavour composition was nearly pure (∼100% ν̄e ), and the intensity of
the flux was known to within 2%. They had a sensitivity down to 10−3 eV2 in ∆m2 , which wasan order of magnitude better than previous reactor experiments. The ratio of the measured ν̄eflux to the expectation was [19]:R = 1.01 ± 2.8%(stat) ± 2.7%(syst)(2.41)and so no evidence for oscillations was observed. Their oscillation analysis allowed them to
exclude some of the parameter space (Figure 2.15), for ν̄e ↔ ν̄x transitions, including the allowed
region observed at Kamiokande. CHOOZ removed the possibility of explaining the AtmosphericAnomaly with νµ ↔ νe oscillations, which supported the νµ ↔ ντ explanation from Kamiokande.

2.3.2

KamLAND

KamLand acquired its primary data between 2002 and 2009 [34, 118–120], and was situated at
the old Kamiokande site, near Kamioka, Japan. The detector was located an average distance of∼180 km (138-214 km) from 26 reactors accounting for over 79% of the ν̄e flux. The remaining
flux originated from one close reactor and many reactors over 295 km away. This considerable ν̄eflux concentrated within a narrow range around ∼180 km, put the detector in an ideal position
to test neutrino oscillations that could explain the solar anomaly.
The detector was a 1 kton sphere of 18 m diameter, containing a 13 m diameter transparent
balloon filled with ultra-pure liquid scintillator. The balloon was suspended in non-scintillating
oil, and there were 1879 PMTs mounted on the inside of the sphere. Outside the sphere there
was a 3.2 kton water cherenkov outer detector, primarily for tagging background events. Electron
anti-neutrinos were detected by the inverse β decay interaction, as in Equation (2.40), with the
2.2 MeV γ-ray from neutron capture on hydrogen ∼200 µs later.
49

Figure 2.16: The KamLAND detector [119] on the left and the measured spectral distortion
L/E [120] on the right. The ratio of the observed ν̄e over the no-oscillation expectation, against
the reconstructed L/E. Best-fit for neutrino oscillations in the solid purple histogram; decay
in the dashed red; and decoherence in the green dot-dashed histogram. The plot assumes all
neutrinos follow an L0 = 180 km flight path.
KamLAND’s first results in early 2003 clearly showed, for the first time, the disappearanceof ν̄e [119]. The ratio of measured ν̄e to the expectation was:R = 0.611 ± 0.085(stat) ± 0.041(syst)(2.42)

This data helped constrain the solar oscillation parameter space, excluding all but the largemixing-angle (LMA-MSW) solution.
In a 2005 analysis [120], with increased statistics, ν̄e disappearance was observed at the
99.998% significance level, 4σ away from the non-oscillation hypothesis. In this analysis they
showed the first evidence for distortion in the energy spectrum of ν̄e , which was in agreementwith oscillations, seen in Figure 2.16.
KamLAND continued to improve the precision of its measurements. In 2008 they released themost precise measurement of ∆m221 and an L/E plot showing almost two cycles of the oscillation
pattern [34], which can be seen in Figure 2.17. In a combined analysis of the KamLAND and−5 eV2solar experiment data, the reported oscillation parameters were ∆m221 = 7.59+0.21−0.21 × 10
and tan2 θ12 = 0.47+0.06−0.05 .
Since 2009, the detector was upgraded to KamLAND-Zen, a neutrino-less double beta decay
experiment with a world-class sensitivity, but still struggling with backgrounds [121].

2.3.3

Daya BayThe Daya Bay experiment started taking data at the end of 2011, and in March 2012 it publishedthe first observation of non-zero θ13 [14], with a significance of 5.2σ. The data were collected
over 55 days, although the result was based on around 10,000 ν̄e candidates in the far detectors,
and about 80,000 in the near detectors. The experiment was located at the reactor complex at
Daya Bay, China.
The Daya Bay experiment was located around six 2.9 GW (thermal) pressurised water nuclear
reactors, arranged into three pairs of nuclear power plants. The ν̄e flux was measured with 650

Figure 2.17: KamLAND 2008 Distributions [34] The left plot shows the updated L/E plot,
displaying nearly two oscillation cycles. L0 = 180 km is the flux-weighted average distance to thereactors. The right plot shows the neutrino oscillation parameter allowed space from KamLAND
and solar neutrino experiments. It can be seen that the KamLAND data puts a large constraint
on ∆m221 .
functionally identical anti-neutrino detectors arranged in two near experimental halls (470 m
and 576 m) and one far at the oscillation maximum (1648 m). The arrangement can be seen in
Figure 2.18. This allowed them to measure the relative differences in the ν̄e flux, independentof correlated uncertainties and minimised uncorrelated reactor uncertainties. The analysis used
the following survival probability:
Psur ≈ 1 − sin2 2θ13 sin2 (1.267∆m231 L/E)

(2.43)

This can be used for reactor based experiments and gives an unambiguous determination of θ13 ,as the disappearance of ν̄e driven by θ12 and ∆m221 is negligible at baselines of around ∼1 km.
The detection method made use of inverse β-decay (Equation (2.40)) and consisted of a 3.1 m
diameter inner acrylic vessel containing 20 tons of gadolinium-doped liquid scintillator as thetarget. Outside this there was a 4 m diameter acrylic vessel holding 20 tons of liquid scintillator.
These were contained in a 5 m diameter stainless steel cylindrical vessel, filled with 37 tons ofmineral oil to shield the inner vessels from radioactivity, and housed 192 PMTs. The detectors
were surrounded by a >2.5 m of high-purity water segmented into two water cherenkov detectors
to remove spallation neutrons and cosmogenic backgrounds.
The experiment observed a deficit in the ν̄e flux in the far experimental hall. The ratio of
the measured flux to the expectation was [14]:R = 0.940 ± 0.011(stat) ± 0.004(syst)(2.44)

sin2 2θ13 = 0.092 ± 0.016(stat) ± 0.005(syst)(2.45)and the best-fit value was:with a χ2 /NDF of 4.26/4 for 4 degrees of freedom. The 6.0% rate deficit in the far hall can also
be seen in Figure 2.18. The experiment also made an energy spectrum measurement where the51

Figure 2.18: The Daya Bay experimental arrangement is shown on the left and the deficit
measurement on the right. In the diagram, the reactors are shown by dots in pairs forming
nuclear power plants (NPP). Six anti-neutrino detectors (AD) are located in three experimental
halls (EH). In the right figure, the ratio of the measured flux over the expected flux assuming
no oscillations is shown. The expected signal was corrected with the best-fit normalisation
parameter. The detectors in EH3 have been displaced slightly for clarity. The best-fit survival
probability is shown by the red curve, and the χ2 against sin2 2θ13 is shown in the top right.
expected spectrum was calculated with the near detectors. The distortion was in agreement withthe best-fit θ13 measurment, further supporting the non-zero θ13 result.

2.3.4

Double CHOOZThe Double CHOOZ is the successor to the CHOOZ experiment and its aim is to improve thesensitivity to the θ13 parameter, reaching down to sin2 2θ13 = 0.03 after three years of operation
with two detectors [122]. A new near detector was planned for construction at the reactor site,
identical to the far detector, allowing a relative normalisation systematic error of 0.6%.
Double CHOOZ also had multiple improvements on their detector design to help reduce someuncertainties. The detector was made up of 4 concentric cylindrical tanks, which can be seenin Figure 2.19. The inner most acrylic vessel contained 10.3 m3 of gadolinium-doped scintillator
as the neutrino target. The next acrylic vessel added a 55 cm layer of liquid scintillator as a γcatcher. Outside this there was a 105 cm thick layer of mineral oil to shield from the radioactivityof the PMTs, and was a major improvement from CHOOZ. All of this was contained in a stainlesssteel vessel housing 390 10-inch PMTs and was referred to as the Inner Detector. Outside themain detector there was an inner veto with a scintillator layer, and some further outer vetoplanes above the detector.
In January 2012, Double CHOOZ released an indication for ν̄e disappearance at a detector∼1 km from two 4.25 GW (thermal) reactors, and a non-zero value for θ13 [38]. In July 2012,
they provided evidence for ν̄e disappearance, and made a measurment on θ13 based on both therate and energy spectrum [42]:sin2 2θ13 = 0.109 ± 0.030(stat) ± 0.025(syst)
52(2.46)

Figure 2.19: Double CHOOZ detector [38] on the left and disappearance measurement on theright [42], with θ13 fit. See the text for the detector description. The right figure shows theprompt energy spectrum; the no-oscillation prediction in the blue dotted line; and the best fit
in the solid red line at sin2 2θ13 = 0.109 and ∆m231 = 2.32 × 10−3 eV2 . The orange bands show
the systematic uncertainties on the best-fit. The background contribution is shown in green, andalso in the inset. Comparisons with the no-oscillation prediction are shown beneath the main
plot.

Their result excluded the no-oscillation hypothesis at 2.9σ. The analysis fit is shown in Figure 2.19. This supported the measurement at Daya Bay published on March 2012, which was
primarily a rate-only analysis.

2.3.5

RENO

RENO is an experiment set up around six 2.8 GW (thermal) reactors in Korea. In April 2012,
they released their observation of ν̄e disappearance, excluding the non-oscillating hypothesis at
4.9σ [41].
RENO had two identical anti-neutrino detectors located at 294 m and 1383 m from the centreof the reactor array. As in the Daya Bay set-up, and the planned set-up for Double CHOOZ,
making a relative measurement from two identical detectors considerably reduces the systematic
errors.
The detector design can be seen in Figure 2.20. The main components were similar to Daya
Bay and Double CHOOZ. The inner most acrylic vessel had 16 t (18.3 m3 ) of ∼0.1% Gadolinium
doped hydrocarbon liquid scintillator. The outer acrylic vessel was a 60 cm thick layer scintillator
γ-catcher. Outside that was a 70 cm thick buffer layer of 65 tons of mineral oil with 354 10-inch
inward facing PMTs. The outer detector was a 1.5 m thick volume of highly purified waterobserved by 67 10-inch PMTs.
The RENO result was based on 229 days of data taking, in which they collected around∼17,000 ν̄e candidates at the far detector and about ∼154,000 at the near detector. The ratio
53

Figure 2.20: The RENO Detector on the left and the θ13 measurement from the rate disappearance on the right. The top plot on the right shows the χ2 distribution as a function ofsin2 2θ13 ; and the lower plot shows the ratio of the measured events to the number expected with
no-oscillations at the two detectors. The smooth curve is from the best fit.
of ν̄e observed to that expected was:R = 0.920 ± 0.009(stat) ± 0.014(syst)(2.47)

sin2 2θ13 = 0.113 ± 0.013(stat) ± 0.019(syst)(2.48)and the best-fit value was:

2.3.6

Reactor Neutrino Summary

Reactor neutrino oscillation experiments make use of the nuclear fission reaction as a neutrino
source. Many early reactor experiment did not observe oscillations, but they set limits on the θ13parameter. Notably in 1999, CHOOZ with a detector ∼1 km from the reactor set the standard
limit on the parameter for many years, and excluded the possibility for νµ ↔ νe oscillations
for atmospheric neutrinos, supporting the νµ ↔ ντ explanation from Kamiokande. In 2003
KamLAND, that had an average baseline of ∼180 km, reported on the first measurement of ν̄edisappearance driven by the solar parameters θ12 and ∆m221 . With their updated results in 2005
and 2008 they were able to greatly constrain the solar ∆m221 in global analyses, as well as showing
nearly two oscillation cycles in L/E.
Around 2011, three reactor experiments were all aiming for a measurement of θ13 . In January
2012, Double CHOOZ was the first to report anti-neutrino disappearance at around ∼1 km froma reactor where the solar parameters are negligible. This result gave an indication for a non-zerovalue for θ13 . Daya Bay released the first measurement of θ13 and evidence for ν̄e -disappearance
driven by the parameter, with 5.2σ significance in March 2012. This result was confirmed
shortly afterwards at RENO, with 4.9σ significance. A little later Double CHOOZ provided
more evidence with an analysis based on both the energy spectrum and rate, but still with one
detector and with a 2.9σ significance.
54

2.4

Accelerator Neutrino Experiments

The first two sources of neutrinos discussed, solar and atmospheric, are a natural flux that canbe investigated. Reactor anti-neutrinos were human-made, so the details of the flux could bebetter understood, however the reactor conditions were outside the control of the experiments.
Accelerator neutrinos, on the other hand, were a directed beam of human-made neutrinos withwell-understood properties that could be studied. The direction, energy, momentum and composition of the neutrino beams were determined by the experiments. So accelerator experiments
could give a controlled test of the natural phenomena observed in other experiments.
Typically the beams were created by accelerating protons up to a few GeV, where the powerof the beam was more important for creating an intense beam of neutrinos, rather than achievingvery high energies. The protons were then collided into a target mass, resulting in a spray ofpions and some kaons (quite like atmospheric neutrinos). The pions were then collected andfocussed by magnets in the direction of the detector. This was followed by a region where thepions could decay in flight, and finally a muon dump to stop unwanted penetrating particles.
The result was a directed beam of neutrinos, typically with a high purity of muon neutrinos. An
example of focusing magnets from K2K can be seen in Figure 2.21, and a beamline from T2K in
Figure 2.24.
The earlier short baseline (few hundred metres) accelerator neutrino experiments did notobserve any oscillations. The first long baseline (few hundred kilometres) experiments aimed tomake a controlled confirmation of the oscillations seen in atmospheric neutrinos, and to constrain
the atmospheric ∆m232 . Following this, their aim was to make a measurement of θ13 , and thenthe mass hierarchy and δcp if θ13 was large enough. Whereas the reactor experiments could makea pure measurement of θ13 ; for accelerator experiments the measurement of θ13 also dependedon δcp , θ23 and the mass hierarchy. This made a measurement of θ13 more difficult, but it alsomeant these experiments could potentially measure the remaining parameters.

2.4.1

K2K

K2K (KEK To Kamioka) was the first accelerator neutrino long-baseline experiment, that sent anearly pure νµ beam 250 km from the Japanese accelerator facility KEK to the Super Kamiokande(SK) detector in Kamioka [123–125]. This baseline and the mean neutrino energy of 1.3 GeV
allowed the atmospheric ∆m2 to be probed by an accelerator based experiment for the firsttime. The main data was taken from June 1999 to November 2004. The stability of the beam
was measured by monitors downstream from the target and beam dump.
A neutrino near detector (ND) was built 300 m from the proton target to verify the stability
and direction of the beam, and to determine the energy spectrum and flux normalisation. The
neutrino flux at SK was estimated by multiplying a far/near ratio to this spectrum. The ND
consisted of a 1 kt water cherenkov detector (1KT) and a fine-grained detector (FGD), which
can be seen in Figure 2.21. The 1KT was used to measure the flux normalisation, as it had thesame detection method as SK most of the systematic uncertainties cancelled. The 1KT had ahigh efficiency for reconstructing the muon momentum below 1 GeV/c, while the FGD had ahigh efficiency above 1 GeV/c. Together, these detection systems measured the energy spectrum.
The FGD consisted of a scintillating fiber and water detector (SciFi), a lead-glass calorimeter(LG) and a muon range detector (MRD). The FGD could reconstruct event tracks, and evenclassify events by using the track reconstructed by the recoil proton when it was visible. The
55

Figure 2.21: The K2K horn magnets on the left and the near detector configuration on theright [125]. The horns each have a 250 kA current that flows through the inside and returns on
the outside, setting up a toroidal magnetic field inside. This collects the pions of one polarity,
π + here, into the forward direction. A second horn further focusses towards the far detector.
The production target was a 66 cm aluminium rod that also served as the inner conductor of thehorn. The near detector configuration after the lead-glass calorimeter was replaced with SciBar,
is shown.
events at SK were selected using GPS timing information to correspond with the beam timing,
which almost entirely cut out any atmospheric neutrino events. For details of the far detector,SK, see Sections 2.1.4,2.2.3 or Chapter 3.
The analysis used a maximum-likelihood method that included both the number of events
observed (normalisation term) and the energy spectrum (shape term). The presence or absenceof neutrino oscillations depended on both of these terms.
In early 2003, K2K reported an indication for neutrino oscillations with disappearance of theνµ accelerator neutrino flux [123]. They estimated that the probability the measurement was due
to statistical fluctuations in the absence of oscillations was less than 1%. In 2005, K2K reported
evidence for neutrino oscillations at the 4σ level [124]. In this second data taking period, there
was no LG which was then replaced with SciBar around half-way through the run. SciBar was
made of scintillator strips read out by wavelength shifting fibres and multianode PMTs, It was
fully active and did not have the aluminium water tanks like SciFi. SK also had half the PMT
coverage (SK-II) during this period. The final results from K2K [125] can be seen in Figure 2.22.

2.4.2

MINOS

The Main Injector Neutrino Oscillation Search (MINOS) experiment is an accelerator experimentthat sends a neutrino beam from Fermilab to a near detector (ND) at 1 km and a far detector(FD) 735 km away in the Soudan mine in Minnesota, US [10, 36, 40, 43]. MINOS collected datafrom 2005.
The neutrinos are produced with the NuMI (Neutrino Main Injector) beam at Fermilab.
120 GeV protons are extracted in 10 µs spills, bent 3.3◦ downwards towards the FD and collidewith a 95.4 cm long target. The positively or negatively charged particles produced are focussed
by two magnetic horns, 10 m apart, depending on the current polarity. These particles thendecay in a 675 m long, 2 m diameter, vacuum decay pipe to produce a νµ or ν̄µ beam.
The MINOS detectors were both steel-scintillator tracking calorimeters with 1.3 T toroidal
magnetic fields. They were made with octagonal planes of steel plates (2.54 cm thick) and planes
56Figure 2.22: K2K reconstructed neutrino energy spectrum on the left and the allowed oscillation
parameter space on the right [125]. In the left plot, the best-fit with neutrino oscillations is shownin the solid red histogram; and the no-oscillation expectation in the dashed blue histogram. The
right plot shows a comparison of the K2K allowed regions against the 2004 SK L/E analysis
allowed region [9].
of scintillating plastic strips (1 cm thick). These strips were arranged 45◦ to the vertical and 90◦
relative to their neighbours. The FD was 5.4 kton and 705 m underground with 484 8 m wideplanes. It triggered with a hit within a 100 µs window centred on the beam spill timing. The ND
was 0.98 kton and 103 m underground with 282 irregular 4×6 m2 planes. It had a ∼105 times
larger data rate than the FD and had a continuous 53.1 MHz sampling during the beam spill.
In 2006, MINOS reported disappearance of νµ when they observed 215 events at the FD
where 336±14 were expected without oscillations [36]. In 2010, MINOS slightly improved theconstraint on θ13 in their νµ → νe oscillation search [126], from the stringent limit set by CHOOZ.
In 2011, after running anti-neutrino mode, MINOS published the first direct measurement of ν̄µdisappearance [10]. Surprisingly, the allowed region for ν̄µ oscillation reported had a considerablylarger ∆m2 and almost excluded the allowed region for νµ oscillation. This result was one of
the motivations for the analyses in this thesis. However, an updated result of the anti-neutrino
analysis showed an allowed region that overlapped with the allowed region for neutrinos, as seenin Figure 1.1.
MINOS was also continuing to look for νe appearance, and their results in July 2011 excluded
the θ13 = 0 hypothesis at the 89% C.L [40]. The flavour composition of the beam was expected
to be 98.7% νµ + ν̄µ and 1.3% νe + ν̄e . MINOS was sensitive to θ13 through νµ → νe oscillations
and they searched for an excess of νe events. The oscillation probability, to leading order, wasgiven by:P (νµ → νe ) ≈ sin2 (θ23 ) sin2 (2θ13 ) sin2 (1.27∆m232 L/E)
(2.49)The probability varies with matter effects and leptonic CP violation [127], so results were presented as a function of δcp and mass hierarchy.
The events were classified by spatial patterns of energy deposition. νe events were morecompact and close to the hadronic shower, typically traversing 6-12 planes. The analysis, firstly,used selection criteria to largely cut out the background events. The main technique to separate
57

Figure 2.23: MINOS results from 2013, combining νµ and ν̄µ beam mode data [43]. The left
figure shows the allowed region for 2 sin2 (2θ13 ) sin2 (θ23 ) with varying δcp , with the upper plot
for normal mass hierarchy and the lower plot for inverted hierarchy. The right figure shows thedifference of the likelihood from the bestfit value, for multiple parameter configurations. The
hash-dotted horizontal lines mark the 68% and 90% C.L. indicating the disfavoured values.
the signal from the background was a nearest-neighbour algorithm, “library event matching”(LEM), where each candidate was compared to 5 × 107 simulated signal and background eventswith a likelihood. The final classification was then made with a neural network. The best-fit+0.071of the analysis was 2 sin2 (θ23 ) sin2 (2θ13 ) = 0.041+0.047−0.031 (0.079−0.053 ) for the normal (inverted)hierarchy and at δcp = 0, which was an improvement on the T2K result at the time. The limit
MINOS reported was 2 sin2 (θ23 ) sin2 (2θ13 ) < 0.12(0.20) at 90% confidence level and δcp = 0.
These results from accelerator experiments were hints that θ13 was large enough to be measured
by the current generation of experiments. Indeed, in 2012 the reactor experiments made the firstmeasurements on θ13 (see from Section 2.3.3).
In 2013 MINOS published an analysis that was a first search for ν̄µ → ν̄e oscillations; andthey extracted parameters using both the νµ and ν̄µ beam mode data [43]. The LEM method
was used again to define 3 parameters that were put into a neural network along with thereconstructed neutrino energy. By assuming δcp = 0, θ23 < π4 and a mass hierarchy, they defined
the new limits: 0.01 < 2 sin2 (2θ13 ) sin2 (θ23 ) < 0.12 at 90% C.L. for normal hierarchy; and0.03 < 2 sin2 (2θ13 ) sin2 (θ23 ) < 0.18 at 90% C.L. for inverted hierarchy. Additionally, by setting
sin2 (2θ13 ) = 0.098 ± 0.013 from reactor data they have began to exclude some of the parameter
space from δcp , θ23 octant, and the mass hierarchy, as seen in Figure 2.23.

2.4.3

T2K

The T2K (Tokai-to-Kamioka) experiment is the successor to K2K and sends a neutrino beam
295 km from the new MW-class proton synchrotron, J-PARC, at Tokai to Super-Kamiokande(SK) as the far detector (FD) [35, 37, 39, 128–131]. Data collection began in January 2010.
T2K made use of an off-axis method, in which the neutrino beam was intentionally directed2.5◦ away from the FD. This created a muon neutrino beam with a narrow-band in energy
and a peak energy at ∼0.6 GeV, which maximised oscillations for νµ -disappearance at 295 km,
58Figure 2.24: T2K beamline on the left and near detector on the right [129]. The neutrino beam
is made by colliding 8 bunches of 30 GeV protons into a graphite target. The production targetand downstream is shown in the left figure. A blown-out view of the ND280 near detector isshown on the right.
and minimised backgrounds from higher energy neutrino interactions. The beam profile wasconstantly monitored and the beam steering was within ±1 mrad. The main physics goals of theexperiment was to make precision measurements of the atmospheric oscillation parameters andto achieve a 20 times better sensitivity to θ13 compared to the CHOOZ limit [129].
At J-PARC, eight bunches of 30 GeV protons were extracted from the main ring per spill andguided by super-conducting magnets to the 90 cm long graphite production target. Three pulsed
250 kA magnetic horns sign selected the outgoing charged pions and kaons and focussed them inthe beam direction. These decayed in flight into neutrinos in the 96 m long decay volume. Thiswas followed by a beam dump and muon monitors. The setup can be seen in Figure 2.24.
The near detector complex (ND280) was located 280 m down-stream from the production
target. It consisted of two detectors. The Interactive Neutrino GRID (INGRID) was on-axis
and monitored the beam direction, intensity and profile. It was made up of 14 7-ton identicalmodules arranged into a vertical column and a horizontal row centred on the beam - forming a“+” sign - and two additional modules off-axis. The detectors were constructed with 9 iron plates
and 11 tracking scintillator planes sandwiched together. Each tracking plane had 24 horizontal
and 24 vertical scintillator bars.
The other near detector was off-axis and was designed to study neutrino interactions and to
calculate beam properties at the FD. The configuration can be seen in Figure 2.24. It consistedof a π 0 detector (P0D), 3 large volume time projection chambers (TPCs), 2 fine-grained tracking
detectors (FGDs), and all surrounded by electromagnetic calorimeters (ECAL). The detector
was encased in the 0.2 T refurbished UA1 magnet, which was also fitted with a side muon range
detector (SMRD).
The main backgrounds to the νµ → νe search were intrinsic νe contamination in the beam
as well as an NC π 0 interaction that could be misidentified as a signal event. Misidentified π 0
events were suppressed by forcing the reconstruction of two rings with the expected light patternassuming two electromagnetic showers, and cutting on the invariant mass Minv < 105 MeV/c2 .
The detector uncertainties came mainly from tracking and PID efficiencies, while the main physics
systematic errors were related to the interaction modelling. Neutrino interaction simulations used59

Figure 2.25: T2K νµ disappearance energy spectrum on the left and allowed parameter regions
on the right compared with other recent results on the right [35]. The bottom panels on the leftshow the ratio of data and best-fit to the no-oscillation prediction for two energy ranges, 0-6 GeV
and 0.3-1.0 GeV. The right plot shows the 90% C.L. allowed regions. The black contour is forthe recent T2K result, while the green contour shows one of the analyses in this thesis.

the NEUT event generator, maintained at SK. These predictions were compared with other event
generators, such as GENIE and NuWro, as well as against experimental data from SciBooNE,
MiniBooNE and K2K.
For the analyses at T2K, comparison of the number of candidate events with the expected
number of events without any oscillations was important. A sample of inclusive νµ CC-interactions
at the ND were compared with the MC prediction at the ND to get a normalisation factor. Thisfactor was applied to the MC at the FD, allowing partial cancellation of the cross-section modelling and flux uncertainties.
In June 2011, T2K reported an indication of νµ → νe appearance with 2.5σ significance,
and the first indication for non-zero θ13 [39]. From about 11 months beam time (Run 1 and2), 6 candidate νe events were observed with an expectation of 1.5 ± 0.3(5.5 ± 1.0) events forsin2 2θ13 = 0(0.1). The 90% confidence interval observed for δcp = 0 and sin2 2θ23 = 1.0 was
0.03(0.04) < sin2 2θ13 < 0.28(0.34) for normal (inverted) neutrino mass hierarchy. The best fit
values were 0.11 (0.14).
In January 2012, T2K submitted the first results for νµ disappearance using the off-axis
technique [130]. Two analysis methods were used on the data from Run 1 and 2. One compared
the number of observed events in bins of reconstructed energy to the expectation using a binned
likelihood-ratio method. The other was a maximum likelihood method with a “normalisation”
term for the observed events, and a “shape” term for the unbinned likelihood for the reconstructed
neutrino energy spectrum. The two analyses were in agreement and the measured parameterallowed regions for two flavour νµ → ντ oscillations were consistent with those reported byprevious experiments.
In 2013, T2K updated their data and analysis to show the first evidence for νe appearance
in a νµ beam, with 3.1σ significance [131]. The data was from Run 1, 2 and 3 and included 11
candidate events over an expectation of 3.3±0.4 background events. Run 3 was taken after T2K
and J-Parc recovered from the 2011 Tohoku earthquake, and approximately doubled the size ofthe overall data set. Over the same data period, a precision oscillation parameter measurement
60Figure 2.26: T2K allowed regions for sin2 2θ13 for varying δcp on the left and a scan of δcp on theright [37]. In the left figure, the top panel is for normal hierarchy and the bottom for inverted.
The black solid line gives the best fit and the orange hatched band displays the average θ13 value
from the PDG, 2012. The right figure shows the likelihood for δcp marginalised over sin2 2θ13 ,sin2 θ23 , and ∆m232 , while θ13 was further constrained by recent results. Solid black is for normal
hierarchy and dotted red for inverted. The values above the cyan lines are disfavoured at the
90% C.L.
was made in a νµ disappearance analysis [35]. 58 events were observed with 205±17(syst.) events
expected in a no-oscillation hypothesis. The survival probability used in the analysis included
the effects from θ13 , and maximal mixing was observed, θ23 ≈ π4 . The allowed regions can beseen in Figure 2.25, where a comparison with the SK L/E analysis is shown and will be describedin this thesis.
At the end of 2013, T2K reported the first observation of electron neutrino appearance in amuon neutrino beam with a significance of 7.2σ over the sin2 2θ13 = 0 hypothesis [37]. For further
background reduction, a new maximum-likelihood method using time and charge probability
density functions for every PMT hit was used. The ratio of the background π 0 likelihood to thesignal νe likelihood and the reconstructed π 0 mass were used to form a 2D cut. This removed
69% more of the π 0 background compared to the previous selection, with only a 2% loss in signal
efficiency. In the fit the unmeasured oscillation parameters were fixed, including δcp and the+0.045
mass hierarchy. The best-fit value measured was sin2 2θ13 = 0.140+0.038−0.032 (0.170−0.037 ) for δcp = 0
and normal (inverted) neutrino mass hierarchy. The allowed region of sin2 2θ13 as a function ofδcp for both mass hierarchy hypotheses can be seen in Figure 2.26.
By using the precisely measured value of θ13 from reactor experiments, T2K also began to
explore CP violation, which can also be seen in Figure 2.26. The combined reactor and T2K
measurements favour δcp = − π2 . The 90% exclusion limits were determined by the FeldmanCousins method.

2.4.4

MiniBooNE

There has been some evidence for anomalies in short-baseline experiments with oscillations occurring at L/Eν ∼ 1 m/MeV. These controversial results began with the evidence for ν̄µ → ν̄e61

oscillations at the ∆m2 ∼ 1 eV2 at the Liquid Scintillator Neutrino Detector (LSND) experiment,with 3.8σ significance [132]. A compatible effect was seen in a re-analysis of reactor anti-neutrino
data with baselines < 100 m, where an average ratio of 0.943 ± 0.023 for the observed to expected
rates was found – the reactor anti-neutrino anomaly [133]. There was further evidence from radioactive source experiments at GALLEX and SAGE, where they observed a deficit of electron
neutrinos from radioactive sources inserted into the detectors – the gallium anomaly [134]. These
anomalies were often interpreted as oscillations at ∆m2new > 1.5 eV2 to a fourth sterile neutrino,
νnew , beyond the standard model.
MiniBooNE is a short baseline experiment designed to either confirm or refute the result
seen at LSND. The neutrinos were produced at the Fermilab Booster neutrino beam, from 8 GeV
protons colliding on to a 71 cm long Beryllium target. The produced particles were focussed witha 174 kA horn, passed through a 60 cm diameter collimater and decayed in a 50 m long air-filled
tunnel.
The detector was 541 m from the target, and was a spherical tank with a radius of 610 cm
filled with 800 tons of pure mineral oil. This allowed for both the production of directional
Cherenkov light and isotropic scintillation light. The tank was optically separated into a 575 cm
radius inner volume with 1280 PMTs, and a 35 cm radius outer volume acting as a veto with 240
PMTs.
The νµ spectrum peaked at ∼600 MeV (∼400 MeV for ν̄µ ) and spanned out to 3000 MeV. CC
νµ and νe events were distinguished by their distinct patterns in Cherenkov and scintillation light.
In addition a decay electron was seen from the muon 80% of the time. The main backgrounds werea single electromagnetic shower from νµ induced NC π 0 events; single photons from radioactive∆ decays; and photons from ν interactions outside the detector penetrating the veto.
In 2007, MiniBooNE released its initial results and found no significant excess above thebackground [135]. The data could exclude two-neutrino appearance-only oscillations as an explanation of the anomaly seen at LSND. However, they did observe a little excess below 475 MeV
that could not be explained by a two-flavour neutrino model. Later on, a 3.0σ excess of these
low-energy electron neutrino events was reported.
In 2010, MiniBooNE searched for ν̄µ → ν̄e , which was the same mode as LSND, and observed
an excess of ν̄e events above 475 MeV [136]. There was only a 0.5% probability that the data
could be explained by the backgrounds. Their results were consistent with ν̄µ → ν̄e oscillationsin the 0.1 to 1 eV2 ∆m2 range, and consistent with LSND.
In 2013, MiniBooNE reported on an improved ν̄µ → ν̄e search, and an updated νµ → νeresult [137]. They improved their understanding of the possible backgrounds, with in situ measurements at both MiniBooNE and SciBooNE (SciBar detector upstream). In anti-neutrinomode, an excess of 78.4 ± 28.5 events, 2.8σ, was observed. The allowed region had some overlap
with the LSND result. The data was consistent with ν̄µ → ν̄e oscillations in the 0.01 to 1 eV2∆m2 range. The updated result in neutrino mode also found an excess of 162 ± 47.8 events,
3.8σ, at low energies. Although the energy distribution was not compatible with a simple 2
neutrino model and the resulting allowed region was marginally compatible with LSND. The
allowed regions are shown in Figure 2.27.
These inconsistencies could perhaps be explained by some unknown systematic uncertainties
or backgrounds, or possibly by models with several sterile neutrinos and allowing CP violating
effects. In any case, MiniBooNE has measured some anomalies from the 3ν oscillations seen inother experiments, and these require explanation.
62

Figure 2.27: MiniBooNE 2013 distributions on the left and the allowed regions on the right [137].
The top panels show anti-neutrino mode and the bottom panels for neutrino mode. The 2ν bestfits are shown in the black histograms on the left figure and by star points on the right. The
shaded regions in the right figure show the LSND allowed regions.

2.4.5

OPERA

The OPERA experiment sends neutrinos from the CNGS νµ beam at CERN to the LNGS
underground laboratory, 730 km away in Gran Sasso, Italy [138–141]. The main goal of the
OPERA experiment was to observe an appearance signal from neutrino oscillations and to prove
unambiguously that νµ → ντ oscillation is the dominant atmospheric transition channel. The
CNGS was a high energy beam optimised for νµ CC interactions, producing neutrinos with anaverage energy of ∼17 GeV. The ν̄µ contamination was 2.1%, and νe and ν̄e less than 1%. OPERA
had their first interactions in 2006, and took their main data from 2008 until the end of 2012.
The OPERA detector was an interesting hybrid of detection technologies, allowing it toachieve sub-micrometric resolution. Firstly there was a veto, which was followed by two identicalsuper modules (SM) each containing 625 tons of 75 000 emulsion/lead ECC (emulsion cloud
chamber) modules, or “bricks.” The SM also consisted of a scintillator target tracker detector(TT) to trigger the read-out and locate interactions in the bricks, and a muon spectrometer.
Each brick had 56 1 mm thick lead plates sandwiched with 57 emulsion films.
A dedicated pair of emulsion films, Changeable Sheets (CS), was glued to the downstream faceof each brick. The CS and TT were used together by an automated system to locate the hit bricks
and extract them. The bricks were exposed to X-Rays and high energy cosmic rays for alignment,
before being developed. These bricks were then scanned by automatic scanning microscopes atlaboratories at LNGS, Europe and Japan. The scan resulted in a 3D reconstruction of theneutrino interaction, as well as micrometric resolution of possible secondary short-lived decay
particles. The bricks and how they relate to the CS and TT can be seen in Figure 2.28.
The τ signal was identified by its characteristic topologies with either one prong (electron,
63

Figure 2.28: Schematic of the OPERA detector elements on the left and the 3rd ντ candidateon the right. How the emulsion/lead ECC bricks, CS and TT reconstruct an example ντ eventis shown in the left figure [138]. The right figure shows the topology of the event reported in
the 2014 analysis [141]. The inset is a zoomed in view of the primary and decay vertex. The τ
decayed in the plastic, where there was a kink for the outgoing muon. p0 was a hadron, and the
EM shower from the γ can be seen from film 41.
muon or hadron) or three prongs. A software algorithm selected bricks with a high probabilityof containing the interaction vertex. Contamination from events outside the brick was estimatedto be 4.5%. The CS of an extracted brick was developed and used to help reconstruct tracks
within the brick. The efficiency of the brick finding procedure was estimated to be 77% for 1-2
brick events and 83% for up to 4 brick events. One characteristic feature of the signal was a kinkin the decay topology.
In 2010, OPERA reported on their first ντ candidate event [139]. The event was muon-less,and survived after a stringent selection procedure. Each of the tracks were carefully identified
to build up the topology. Possible backgrounds were c − c̄ pair production with one charm
particle not being identified; a primary hadron from an unidentified νµ CC-interaction; or νµ NCinteraction with no nuclear fragment associated with the secondary interaction. OPERA carriedout extensive background studies to understand the probability of the background to the proposed
channel, h− (π 0 )ντ , fluctuating to one event. The event was found to have 2.36σ significance of
not being a background event. When all decay modes were considered, the significance bacame
2.01σ. The expected number of νt au events in the 2010 data sample was 0.54 ± 0.13(syst.), and
OPERA claimed the observation of their first ντ CC interaction.
In 2014, OPERA reported evidence for νµ → ντ oscillations at the 3.4σ level after observing
their 3rd ντ candidate in 5 years of data [141]. This included two ντ candidate events in the
hadronic decay channels (one prong in 2009, three-prong in 2011), and a new candidate in themuonic decay channel. The topology of the new event can be seen in Figure 2.28. The main
background for the new candidate was large-angle muon scattering in lead (80%), and charmed
particle decays (20%). Although, the event was in a region of the parameter space free frombackground, considering the τ decay in the low-density low-Z plastic and the 690 MeV/c traverse
momentum. As before, OPERA measured the probability for each of the backgrounds, and had
an expectation of 1.7 ντ events in the analysed data. The absence of a νµ → ντ oscillation signal
was excluded at 3.4σ.
64One of the analyses at OPERA was a measurement of the velocity of neutrinos. It became
global news in 2011 when they reported neutrinos arriving 60.7 ns earlier than light would havetaken to travel the same 730 km. The result had a reported significance of 6σ, and the announcement had followed 6 months of rigorous cross-checking, so it had to be considered seriously.
However, in the following year, two substantial systematic effects were discovered. One was afaulty connection in the fibre optic cable between the GPS receiver and the OPERA master clock
resulting in a 73 ns earlier arrival time; and the other was a clock on an electronic board ticking
faster than the specified 10 MHz, reducing the anomalous result. OPERA has since modified thesystematic uncertainties in their paper to be consistent with the speed of light [140].

2.4.6

Accelerator Neutrino Summary

Accelerator neutrino experiments provided a controlled man-made source of neutrinos to closely
study the natural phenomena seen in other experiments. The early short-baseline experiments(∼100 m) did not observe any indication for oscillations. In 1999, K2K was the first long-baseline
experiment with a 295 km baseline and produced 1.3 GeV neutrinos allowing it to probe theatmospheric oscillations observed. In 2003 they reported on an indication for νµ disappearance,
which reached 4σ significance by 2005.
MINOS followed from 2005, with a 735 km baseline, and began the search for the remaining
oscillation parameters: θ13 , δcp , and the mass hierarchy. In 2010, they slightly improved theconstraint on θ13 from the CHOOZ limit with their νµ → νe search. In 2011, their measured
parameter allowed region for anti-neutrinos differed from the region for neutrinos. This result waspart of the motivation for this thesis. However, later anti-neutrino results were consistent with
the neutrino measurements. T2K was the successor to K2K and had an off-axis beam to narrow
the band in energy of the neutrinos reaching the far detector. In June 2011 they reported on thefirst νµ → νe appearance and non-zero θ13 indication with 2.5σ significance, with the observationof 6 candidate events over a 1.5±0.3 expectation for θ13 = 0. By 2013, T2K measured a non-zeroθ13 at the 7.2σ level. Both MINOS and T2K have been working to confine the parameter spacefor θ13 and δcp , and to determine the mass-hierarchy and the octant of the θ23 parameter.
There has been an anomalous result in accelerator experiments since 2001 when LSND observed ν̄µ → ν̄e oscillations at ∆m2 ∼ 1 eV2 , with 3.8σ significance. The results from MiniBooNE
in 2013 were consistent with LSND. These anomalies have to be explained in some way, perhaps
by the existence of sterile neutrinos. In 2010 OPERA observed the first νµ → ντ appearance
candidate event. By 2014 they had reported on 3 candidate events and evidence for νµ → ντ appearance at the 3.4σ level. OPERA also infamously reported on a 6σ result for faster-than-lightneutrinos when they measured neutrinos arriving at the detector 60.7 ns earlier than light would
take to travel the same path. This turned out to be a due to a systematic error with a faulty
connection of a fibre optic cable.

2.5

Future Neutrino Experiments

Recent experiments have done well to establish a non-zero θ13 , providing allowed regions for
all the mixing angles and mass square differences in a 3 neutrino framework. With θ13 being
reasonably large, current experiments can begin to probe for CP violation in the lepton sector;
to attempt to determine the nature of the neutrino mass hierarchy; and to resolve the octant of
θ23 .
65

Figure 2.29: Possible Neutrino Factory layout on the left and Beta-Beam Facility Layout on theright [143]. There are many challenges accelerating muons, the left figure shows some of theelements such as phase rotation, ionisation cooling and several stages of acceleration. Shown
here is a triangular storage ring with straight sections pointing to detectors. On the right is apotential beta beam set-up using existing facilities at CERN, the beta-beam storage ring pointing
to detectors would need to be built.
Many physicists believe that the small masses in neutrinos are related to physics at theextremely high energy-scale, such as in the See-saw Mechanism, that cannot be probed directly.
There are also theories, such as Leptogenesis, that suggest CP violation in neutrinos could
partially explain why we live in a universe composed of matter, rather than equal quantities
of matter and anti-matter. In addition, large scale water cherenkov detectors can also search
for nucleon decay which is an important measurement for Grand Unified Theories. There isconsiderable interest in the further understanding of the properties of neutrinos.
Perhaps the current generation of experiments will be adequate to make these measurements.
However, the next generation of experiments are being designed and proposed to make precise
and accurate measurements of these evasive parameters, as well as to probe a variety of new
physics. Some of these projects are highlighted here.

2.5.1

Neutrino Factory and Beta Beams

The Neutrino Factory and Beta-Beam experiments both propose new ways of creating neutrino
beams. The Neutrino Factory could produce a high-intensity high-energy neutrino beam byaccelerating muons and allowing them to decay in a storage ring with long straight sections
pointing in a desired direction [142]. This differs from the conventional accelerator neutrino
experiments that produce their main flux from the decay of charged mesons. A potential Neutrino
Factory layout can be seen in Figure 2.29.
Neutrino beams from meson decay have a few limitations. The finite precision that thebeam fluxes can be predicted are a significant source of systematic uncertainties; and the wrongflavour contamination from kaon decays are a nuisance. They are also primarily νµ beams
making νe → ντ oscillation searches difficult. In addition, high energy beams will also have someντ contamination, which could be a problem for νµ → ντ oscillation searches.
The advantage of a beam from muon decays is that the absolute flux can easily and precisely
be calculated. It also has precisely known composition. Stored µ− will produce 50% muon
neutrinos and 50% electron anti-neutrinos. Only one type of neutrino and one type of anti66

neutrino. For µ+ decay, the flavours are reversed. So precise measurements can be made fromνµ , νe , ν̄µ , and ν̄e by selecting the muon charge and monitoring its properties in the accelerator.
There have been discussions of the potential of measuring νe (ν̄e ) → νµ (ν̄µ ) oscillations, known
as the golden channel for making precision measurements [144]. Since there is only one type ofneutrino and anti-neutrino in the beam composition, the signal would come from a “wrong-sign”
muon in a detector with good muon charge identification power. The international design study
for the neutrino factory can be seen in Reference [145].
Beta-Beams could be made by accelerating radioactive ions to high energies, and allowing
them to decay in storage rings with long straight sections pointing to a detector [146]. For
example, this could be achieved by producing 6 He ions, which β-decay:
62+−→ 63 Li3+ + e− + ν̄e
2 He(2.50)

producing a Lorentz boosted collimated ν̄e beam of a single flavour, and with a well-known
energy spectrum and intensity.
It has already been demonstrated at CERN ISOLDE that it is possible to produce ≈ 108
6 He ions per second, and this method could be used for producing a high intensity neutrino
beam. The construction of a storage ring would be similar to the effort involved in building one
for a muon-decay Neutrino Factory. Although, shielding on some accelerator components would
be needed to protect them from the decay electrons and lithium ions. With technologies that
already exist, it could be possible to produce focussed low-energy beams of νe or ν̄e for precision
neutrino physics. A possible Beta-Beam set-up can be seen in Figure 2.29.

2.5.2

Megaton Detectors: Hyper-Kamiokande, DUSEL, and LAGUNAHyper-Kamiokande (HK) will be a next-generation water Cherenkov underground detector at themegaton scale with a design based on the proven technology from Super-Kamiokande (SK) [147].
It will have sensitivity far beyond that of SK over a range of physics searches – proton decay,
atmospheric neutrinos and astronomical neutrinos. It will be located 8 km south of SK, and also
295 km from Tokai and will act as the far detector for T2HK.
The detector will have a 1,750 m.w.e. of rock overburden, and a total (fiducial) mass willbe 20 (25) times larger than SK, at 0.99 (0.56) million metric tons. The design consists of 2
cylindrical tanks 250 m long, 48 m wide and 54 m high, as seen in Figure 2.30. The inner detector
will have 20% photo-cathode coverage with 99,000 20-inch PMTs, and the outer detector will
have 25,000 8-inch PMTs.
HK would have a rich neutrino physics program. The detector would have unprecedented
sensitivity to oscillation parameters. Used with the upgraded J-PARC beamline, T2HK would
be able to obtain 3σ significance for 74% of the δcp parameter space. HK by itself shouldhave the sensitivity to determine the mass hierarchy at 3σ for 46% of the domain of δcp . If
sin2 2θ23 < 0.99, the octant of θ23 could be determined. HK will extend its sensitivity to nucleon
decay, for example for the partial lifetime of the decay mode p → e+ π 0 to 1.3 × 1035 years. HK
can also make contributions for many other topics, such as supernova neutrinos, supernova relicneutrinos, and search for dark matter WIMPs.
DUSEL (Deep Underground Science and Engineering Laboratory) is a new undergroundlaboratory the US is planning to build with 6,200 m.w.e. of rock overburden at the Homestake
Mine at Lead, South Dakota [149]. Several designs for neutrino detectors are being considered.
A megaton scale, modular, multi-purpose water Cherenkov detector is one possibility [149]. The
67

Figure 2.30: Graphic of the Hyper-Kamiokande detector design on the left [147], and the various
designs for the LAGUNA detectors towards the right [148]. HK has two horizontal cylindrical tanks at ∼1 MT. LENA is scintillator detector at 50 kT; MEMPHYS has modular water
cherenkov shafts at ∼600 kT for three; and GLACIER is a liquid argon TPC Cherenkov detectorat ∼100 kT per vessel.

design is an array of 10 individual cylindrical 100 kT water cherenkov detectors with a total massof 1,000 kT and 800 kT fiducial mass, and a photocathode coverage of 14%. Each module would
be 50 m in diameter and 50 m high. The physics topics DUSEL could study would be similar toHK, with similar sensitivities. In addition, the modular design could allow some of the detectors
to be filled with a different target material, giving sensitivity to a lower energy threshold thanwater can achieve. DUSEL are also considering liquid argon detectors for LBNE.
The LAGUNA, or Large Apparatus studying Grand Unification and Neutrino Astrophysics),
is a project in Europe to build a detector of ∼50 kT to ∼650 kT for next-generation nucleon decay
searches and neutrino physics [148]. Many sites around Europe are being considered for extendingor building a new underground laboratory of an appropriate scale and distance from CERN. The
project has three candidates for the detector design: GLACIER, LENA, and MEMPHYS. A
graphic of the designs can be seen in Figure 2.30.
GLACIER (The Giant Liquid Argon Charge Imaging ExpeRiment) is an underground liquidargon TPC detector of around ∼100 kT [150]. This detector design is very interesting, as it canreconstruct high resolution tracks, produces scintillation light, and cherenkov rings. There arealso some studies into magnetising the volume, to provide charge discrimination. LENA (Low
Energy Neutrino Astronomy) is a liquid-scintillator detector of around 50 kT [151]. It would beparticularly sensitive for supernova, solar, geo- and reactor neutrinos. It could search for dark
matter, proton decay and potentially act as a far detector for LAGUNA-LBNO. MEMPHYS(MEgaton Mass PHYSics) is the European design for a water Cherenkov detector at half amegaton, with a modular design of three 215 kT cylindrical tanks [152]. The modular set-up
would allow for 100% up-time for supernova neutrinos, and one module could potentially be
Gadolinium loaded for improved supernova detection. The baseline design assumes 3 modules,
with a total fiducial mass of 440 kT.
68

Figure 2.31: The scale of the NOνA detectors on the left and an overview of the T2HK baselineon the right. The NOνA detectors were made up of hundreds of layers of PVC cells filled withmineral oil and liquid scintillator. The T2HK baseline would run from an upgraded J-PARC in
Tokai to the proposed Hyper-Kamiokande.

2.5.3

Off-Axis Super-beams: NOνA and T2HK

The NOνA, or NuMI Off-Axis νe Appearance, experiment aims to build a new detector that
could achieve a factor of 10 times better sensitivity to the νµ → νe channel than MINOS [153].
The recent results world-wide suggest that NOνA is in a good position to determine the orderingof neutrino masses and measure the first indications of CP violation in neutrinos [154]. NOνA
began taking data with the first modules in 2013, and the construction will be finished in 2014.
The next-generation of accelerator experiments require more intense beams of neutrinos andmore massive detectors. The NuMI beamline is being upgraded from 300 kW to 700 kW of beam
power, to produce a 10 µs beam spill every 1.33 s with 4.9 × 1013 protons per pulse. The far
detector will be a 14 kT, finely segmented 64% active tracking calorimeter. The detector is built
up from PVC cells with (6 × 4) cm cross-section and extending the full 15.6 m height or width ofthe detector. The cells are filled with mineral oil and liquid scintillator. The scintillation lightis transported by wave-length shifting fibres, to 32-channel avalanche photodiodes. The FD is
made up of 928 layers of 384 cells. The ND is 329 tons, located at a 1 km baseline and is builtwith the same components. The ND has 4.1 m width and height, and 206 layers of 96 cells.
The NOνA detectors are being constructed 14 mrad off-axis, with the FD situated in a newlaboratory in Northern Minnesota, 810 km away from Fermilab. The off-axis position provides
a narrow energy-band at the oscillation maximum, peaked at 2 GeV, while also reducing backgrounds. The plan is to run for 6 years, with the time equally dispersed for neutrino mode andanti-neutrino mode. Within the first year NOνA expects to make a 5σ observation of θ13 -driven
νµ → νe oscillations [154]. The analysis tools have been updated and first results could be
released at the end of 2014.
T2HK, or Tokai to Hyper-Kamiokande, is a natural extension of the T2K experiment. The
proposition is to upgrade the J-PARC accelerator to 750 kW of beam power and to use theproposed Hyper-Kamiokande (HK) as the far detector [155]. The main goal of the experiment
would be the discovery of CP asymmetry, where they expect to measure CP violation with morethan 3σ significance over 43 of the domain of δCP , if the systematic errors are suppressed to 5%
and the mass hierarchy is known. The plan is to run for 10 years, 3 years in neutrino mode and7 in anti-neutrino mode. The proposed construction schedule of HK would take 7 years from2016, so the T2HK experiment would begin in 2023.
In summer 2013, the LINAC was upgraded from 181 MeV to provide 400 MeV beam energy.
69

For the 2014 summer shutdown, the peak current will be increased from 30 mA to 50 mA. These
LINAC upgrades should bring the beam power up to 450 kW. There would be problems raising
the beam energy from 30 GeV to 50 GeV, so they propose to double the repetition rate to obtaina 1.3 s cycle of 30 GeV beam energy and 2.0 × 1014 protons per pulse. With this upgrade scheme,it could be possible to reach 750 kW beam power in 2017 [155].
There are other next-generation projects that have been put forward LBNE, the Long Baseline
Neutrino Experiment, to be constructed by 2022 and LAGUNA-LBNO, a Long Baseline Neutrino
Oscillations experiment using one of the LAGUNA detectors.

2.6

Neutrino Experiments Summary

There are many sources of neutrinos, from the fusion reactions in the sun; the collision of cosmic
rays in the atmosphere; nuclear fission in nuclear reactors; and created at accelerator beamlines;
as well as other astronomical sources. The first solar neutrinos were detected in 1968 in theHomestake Chlorine Experiment where 30% of the flux expected by the Standard Solar Models
was observed. This was the beginning of the Solar Neutrino Problem. These findings wereconfirmed between 1992 and 2003 by the gallium experiments, GALLEX, SAGE and GNO.
From 1987 to 1995, Kamiokande used a water Cherenkov detector and measured solar neutrinos
in real-time. They established the solar origin of the neutrinos, while also observing a deficit
from the SSM.
Kamiokande, and a similar experiment IMB, also measured a deficit in the flavour ratio ofmuon to electron in atmospheric neutrinos. They observed about 60% of the muon-like events
expected and this was the beginning of the Atmospheric Anomaly. These two anomalies were
early indications that a theory such as Neutrino Oscillations would be needed to explain theseresults.
Super-Kamiokande (SK) was the successor to Kamiokande, had ∼25 times the target mass,
and ran from 1996. The experiment helped constrain the oscillation parameter space for solar
neutrinos, but there were many potential solutions around the year 2000. SK observed a clear
zenith dependence in the measured atmospheric neutrinos in 1998, which was key evidence for
Neutrino Oscillations. This result showed that the deficit in neutrinos observed was dependent
on their flight-path. In 2004, SK disfavoured alternative explanations for the zenith dependence,with an L/E analysis. In 2001, SNO measured the electron flavour component of the solar
neutrino flux with their CC reaction and determined that a proportion of the flux was of anotherflavour. In 2002, SNO reported on a NC measurement that revealed the total solar neutrino flux
regardless of flavour, and this was consistent with the SSM! This was clear evidence that solar
electron neutrinos were oscillating into muon and tau flavour. The SNO results also allowed
global analyses of the solar neutrino parameters to reduce to the MSW Large-mixing Angle(LMA) solution. From 2008 to 2012, Borexino measured low energy solar neutrinos in real-timeand was also in support of the MSW LMA solution. In the early 2000s, the atmospheric neutrino
results from SK were confirmed by the tracking calorimeter experiments Soudan 2 and MACRO.
In 1999, the reactor experiment CHOOZ set a stringent limit on the remaining mixing angle
to be determined, θ13 . In 2003, KamLAND was the first reactor experiment to measure disappearance of an ν̄e reactor flux. It had an average baseline of ∼180 km from many reactors inJapan and the deficit was driven by the solar neutrino parameters. This allowed their result togreatly confine the solar ∆m221 . They also measured almost two cycles in their L/E analysis.
70

In 2011, three reactor experiments were all aiming for a measurement of θ13 : Double CHOOZ,
Daya Bay and RENO. In January 2012, Double CHOOZ gave the first indication of a non-zeroθ13 . Daya Bay was the first experiment to report on a non-zero θ13 in March 2012, with 5.2σ
significance. Results shortly afterward from RENO and Double CHOOZ were consistent withthe measured value.
Accelerator experiments provide a controlled test of observed phenomena with a precise anddirected man-made neutrino source. K2K was the first long-baseline experiment at 250 km, andin 2003 confirmed the atmospheric parameter allowed regions. Similar support was reported byMINOS in 2006 and T2K in 2012. T2K published the first indication of νµ → νe appearance
in 2011, with 6 candidate events over an expectation of 1.5 ± 0.3. By the end of 2013, T2K
improved the significance of this result to 7.2σ. Both T2K and MINOS have been restricting
the parameter space for the remaining parameters, θ13 and δcp , while trying to determine theneutrino mass hierarchy and octant of θ23 . In 2013, results from MiniBooNE were in agreementwith anomalous oscillations seen at LSND that suggested the existence of sterile neutrinos. Whilein 2010, OPERA observed their first ντ candidate for νµ → ντ appearance. By 2014 they had 3
candidates and observed appearance at the 3.4σ level.
The challenge for future experiments is to determine the neutrino mass hierarchy, δcp , andthe octant of θ23 . A next-generation off-axis beam, NOνA has begun taking data in 2014.
T2HK depends on Hyper-Kamiokande, a megaton water-cherenkov detector, so may not begin
until 2023. The Neutrino Factory proposes to make a precise well-understood neutrino beamby muon decay in a storage ring. While Beta Beams suggest accelerating ions and letting them
decay in a storage ring to create low energy electron neutrino beams. There are many othernext-generation proposals, for the DUSEL underground lab in the US, LAGUNA in Europe, andHyper-Kamiokande in Japan.

71Chapter 3

Super-Kamiokande Detector
Super-Kamiokande is a 50 kton water Cherenkov detector located in Gifu Prefecture, Japan. 1 It
was constructed in a zinc mine under the peak of Mt. Ikenoyama to reduce contamination fromcosmic-ray backgrounds. The mean rock overburden is ∼1000 m (2700 m water equivalent) andacts as a shield, reducing the cosmic ray muon background by 5 orders of magnitude comparedwith that on the surface of the Earth. The detector is also segmented into a main Inner Detector
region and a slimmer Outer Detector region, hereafter referred to as the ID and OD, respectively.
The OD is a principle part of the detector design, allowing the remaining cosmic-ray backgrounds
to be eliminated. The combination of the ID and OD allow Super-Kamiokande (SK) to determine
which interaction vertices originate from within the detector, and which are due to particles
and radiation entering from outside the detector. An illustration of the detector showing its
cylindrical structure, and Inner and Outer detector regions, is given in Figure 3.1.
The main physics aims of SK are to search for the existence of nucleon decay and the researchof neutrinos. The neutrinos reaching SK are from many sources: the core of the sun; producedby cosmic-rays in the atmosphere; created in accelerators and from astrophysical sources suchas supernovae. Solar neutrinos have low energies in the order of a few MeV. While atmospheric
neutrinos arrive at the SK site from the atmosphere surrounding the entire globe and havean energy range from a couple of MeV to multiple TeV. SK has been the Far Detector of thefirst neutrino beamline experiment KEK-to-Kamioka (K2K) [125], which produced accelerator
neutrinos by the collision of accelerated protons on a target and focusing the interaction anddecay products that include neutrinos. At the time of writing, SK is currently acting as the Far
Detector of the neutrino beamline experiment Tokai-to-Kamioka (T2K) [128], the successor toK2K and offering a higher intensity and precise neutrino beam.
The Super-Kamiokande experiment started taking data in April 1996 and continued for five
years of continuous observation until the detector maintenance in July 2001. This data taking
period is referred to as SK-I. While refilling the water after the maintenance, an accident occurredin November 2001 in which more than a half of the PMTs were destroyed. The SK detector was
rebuilt after the accident with half of the original PMT density in the ID and resumed observationfrom October 2002. Continuous data taking continued for three years until October 2005, which isreferred to as the SK-II running period. There was a reconstruction effort to restore the ID PMT
density back to the original design specification, and observation was restarted in June 2006. SK
continued operation until September 2008, which is the SK-III running period. New upgraded1

The geographic coordinates of the site are 36◦ 25’N and 137◦ 18’E, and the altitude above sea level is 370 m

72

Figure 3.1: The Super-Kamiokande Detector [156] is located under Mt. Ikenoyama to reduce
cosmic-ray backgrounds. These backgrounds are further reduced by segmenting the detector into
the Inner Dector (ID) and Outer Detector (OD) regions.
electronics were installed, and data taking was resumed in September 2008. This is known as
the SK-IV running period and is still continuing at the time of writing.

3.1

Cherenkov Radiation

Super-Kamiokande is able to make physical measurements primarily by observing Cherenkov
radiation [157]. When relativistic charged particles have a velocity that exceeds the velocity oflight in the medium, Cherenkov photons are emitted:
v≥

c
n

(3.1)

where v is the velocity of the charged particle, c is the velocity of light, and n is the index ofrefraction of the medium. So there is a velocity threshold before the particle begins to emit
Chrenkov photons, which depends on the index of refraction of the medium. It is useful toconsider the momentum threshold, which is also dependent on the mass of the particle. The
refractive index of water is about 1.34, so the momentum thresholds of Cherenkov radiation forelectrons, muons and charged pions are 0.57, 118 and 156 MeV/c, respectively. The proton ismuch more massive, and has a relatively large momentum threshold of about 1.07 GeV/c.
There is a characteristic angle from the trajectory of the particle to the emission direction of
the Cherenkov photons. The half-cone opening angle, Cherenkov angle θc , can be calculated:cos θc =

1
nβ

(3.2)

where β = vc . For highly relativistic particles, with β ≈ 1, the Cherenkov angle in water is about
42◦ .
73

NUM
4
RUN
9000
SUBRUN
396
EVENT
64753459
DATE **-Jul-15
TIME
13:41:42
TOT PE:
3858.0MAX PE:
24.0
NMHIT : 1456
ANT-PE:
25.4
ANT-MX:
4.2
NMHITA:30

90/00/00:NoYet:NoYet
90/00/00:NoYet:NoYet
90/00/00:NoYet:NoYet
90/00/00:NoYet:NoYet
90/00/00:NoYet:NoYet
90/00/00:NoYet:NoYet
90/00/00:;R= 0:NoYet
R :Z : PHI : GOOD
0.00: 0.00: 0.00:0.000
CANG : RTOT : AMOM : MS

RunMODE:NORMALTRG ID :00000111
T diff.:0.264E+04us
FEVSK :81002803
nOD YK/LW: 2/ 1
SUB EV : 0/ 1
Dec-e: 0( 0/ 0/ 0)*NO KEKGPS RED*

Comnt;

Figure 3.2: A typical neutrino event observed by Super-Kamiokande. The cylindrical structure
has been unrolled, and the clear ring is due to a charged particle emitting Cherenkov photons
until falling below threshold, resulting in the ring image as opposed to a filled circle.
The number of Cherenkov photons, N , emitted as a double derivative with respect to thewavelength, λ, and charged particle tragectory, x, is given by:


d2 N
2πα
1
= 2 1−
(3.3)
dxdλ
λ
(nβ)2
where α is the fine structure constant. The PMTs in SK are sensitive to photons in the wavelength
range of around 300 nm to 600 nm, and the number of Cherenkov photons emitted in this range
is about 340 photons/cm. The Cherenkov wavelength spectrum in water together with the PMT
quantum efficiency is shown in Figure 3.3.
So in a typical event, a charged particle will emit Cherenkov photons at θc ≈ 42◦ while
depositing energy in the detector, until its momentum drops below the Cherenkov threshold.
The result of this is a ring of photons are projected on to the wall of the tank and detectedby the PMTs. In this fashion, SK can detect various charged particles and reconstruct physicalvariables by the analysis of Cherenkov rings. A typical neutrino event is shown in Figure 3.2,
where the ring is created by detecting the lepton product of the neutrino interaction.

3.2

Detector Details

3.2.1

Super-Kamiokande Tank

The Super-Kamiokande detector is 41.4 m high, 39.3 m in diameter and contains 50 kton of highly
purified water. In between the inner and outer sensitive regions, there is a PMT support structurethat holds the inward-facing PMTs of the ID and the outward-facing PMTs of the OD. The nonsensitive regions of the structure are covered with opaque sheets to optically separate the ID andthe OD. The PMT structure is shown in Figure 3.4.
74

Figure 3.3: The relative Cherenkov spectrum inwater as a function of wavelength is shown. The
upper curve gives an idea of the wavelength region the Cherenkov photons are emitted in. The
lower curve shows the percentage quantum efficiency (QE) of the PMTs. The QE peaks inthe same region as the peak of the Cherenkov
photon emission.

Tyvek
8" PMT

Top

Black sheet

Figure 3.4: Technical drawing of the SuperKamiokande PMT Support Structure [156].
A segment of each of the detector regions isshown. The inward facing 20" PMTs of the
Inner Detector, and an outward facing 8" OD
PMT can be seen. The black sheet covers theregion between the PMTs in the ID to optically separate it from the OD. The intermediate space between the OD PMTs is covered
with highly reflective Tyvek to increase lightcollection efficiency.

Barrel20" PMT

Bottom75

The Inner Detector (ID) is 36.2 m high, 33.8 m in diameter and contains 32 kton of water.
It detects photons with 11,146 inward-facing 20-inch PMTs. With the full PMT density, thePMTs are spaced uniformly at intervals of 70 cm giving an effective photocathode coverage ofabout 40% in the ID. In the SK-II period, the remaining 5,182 20-inch PMTs after the accident
were re-distributed such that they were uniformly spaced, with a reduced photocathode coverageof around 20%. To prevent light-leaks between the inner and outer sensitive regions, the space
between the PMTs is covered with an opaque black sheet.
The Outer Detector (OD) is the outer sensitive region surrounding the ID and has a width of
2.2 m along the barrel wall, and is 2.05 m thick at the top and bottom. It acts as an effective vetoto reduce background events in the data, or to tag outgoing charged particles. It detects photons
with 1,885 8-inch PMTs outward-facing from the PMT support structure. The photocathodecoverage is considerably less than that of the ID, so each OD PMT is attached to a 60 cm × 60 cm
wavelength shifting plate to improve the light collection efficiency. This is further improved bycovering the walls with a highly reflective (∼90%) tyvek sheet. The OD is also a 2 m thick wall ofwater that surrounds the ID and serves as a shield to attenuate gamma rays and neutrons fromradioactivity in the rock.
For SK-III, the top and bottom regions of the OD were optically separated from the barrel
regions by running the tyvek sheet directly upward and downward from the barrel region. This
allowed Partially-Contained events, which exit the tank usually through either the top/bottom
or barrel regions, to be distinguished from cosmic-ray muon Clipper events which activate boththe top/bottom and barrel regions. This helped improve the reduction of the backgrounds, as
discussed in Section 6.3.
There is also a 55 cm thick dead region between the ID and OD, which is where the PMT
support structure is situated. This stainless steel structure also houses the signal and HV cablesof all the PMTs and routes them out of the detector.

3.2.2

Photo-Multiplier Tubes (PMTs)

The ID photo-multiplier tubes (PMTs) are Hamamatsu R3600 PMTs and have a diameter of
50 cm (20 inch). The 20-inch PMT was originally developed by Hamamatsu in cooperation
with the Kamiokande collaboration [158]. For Super-Kamiokande, the dynode structure andbleeder circuit were improved to achieve better collection efficiency, energy resolution and timing
response [159]. A technical drawing of the PMT is shown in Figure 3.5.
The photocathode of the PMT is coated with a layer of bialkali (Sb-K-Cs) due to its low
thermionic emission and its high spectral sensitivity to the blue of Cherenkov light [159]. Figure 3.3 shows the close match of the quantum efficiency (QE) and Cherenkov emission spectra.
The QE peaks at 22% at around 360–400 nm, which can be seen more clearly in Figure 3.6. The
collection efficiency at the first dynode averages at 70%, and is uniform across the whole PMT
surface within ±7%. The gain through the venetian blind dynode structure is 107 for the ID
PMTs with a supply high voltage of 1500–2000 V. The single photoelectron signal can be clearly
discerned, which is also shown in Figure 3.6. The transit time spread of the 1 p.e. signal is about
2.2 nsec and the average dark noise rate is around 3 kHz at the 0.25 p.e. threshold.
The geomagnetic field at the detector site is about 450 mG, while a magnetic field above
100 mG affects the photoelectron trajectories in the PMTs, reducing the timing resolution. To
compensate for this, 26 sets of horizontal and vertical Helmholtz coils were set up around theperimeter of the tank reducing the magnetic field in the PMTs to around 50 mG.
76

Figure 3.5: Technical drawing of the Inner Detector PMT [156].

distribution (arbitrary unit)

Quantum efficiency

After the accident in 2001 where one ID PMT at the bottom of the tank imploded, causing a
chain reaction that destroyed around 60% of the ID and OD PMTs, an acrylic cover was designed
for the PMTs. The acrylic is 12 mm thick and UV-transparent, while the side of the PMTs areprotected with a fiberglass casing. There are also holes in the casing to allow water to flow inside.
The PMT acrylic and fiberglass case is shown in Figure 3.7. The transparency of the acrylic canbe seen in Figure 3.8, which is more than 94% above 330 nm for photons of normal incidence.
In the OD there are 1,885 8-inch PMTs, which are mostly Hamamatsu R1408 PMTs andsome are reused from the IMB experiment [160]. The effective photocathode coverage in the OD
is increased by attaching a 60 cm × 60 cm × 1.3 cm wavelength shifter plate, that collects lightthat would not have reached the OD PMT directly otherwise. The light collection efficiency is
increased by 60%. There is a slight decrease in the timing resolution from 13 nsec → 15 nsec for

0.2

0.1

0

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

300

400

500

600
700
Wave length (nm)

0

1

2
3
photo-electron

4

5

Figure 3.6: The ID PMT quantum efficiency [156], showing a peak of 22% at around 360–400 nm
is shown on the left, and the single photoelectron distribution on the right, showing a clear signal
at 1 p.e.
77Figure 3.7: Cover of PMTs since SK-II

1
0.98
0.96
0.94
0.92
0.9
0.88
0.86

300

350

400

450

500

550

Figure 3.8: PMT cover transparency

78

600

Component
1 µm Mesh Filter
Heat Exchanger
Cartridge Polisher
Ultra-Violet Sterilizer
Rn-Free Air Dissolving Tank
Reverse Osmosis Filter
Vacuum Degasifier
Ultra Filter
Membrane Degasifier

PurposeRemoves dust and small particles.
Cools water to reduce PMT dark rate and suppress thegrowth of bacteria. Detector water is kept at 14 ◦ C.
Removes over 99% of heavy ions (Na+ , Cl− , Ca2+ ).
Kills bacteria.
Dissolves Rn-free air into the water, improves efficiency at
the vacuum degasifier stage.
Removes Radium and minute particles of the order of 1 nm
Removes dissolved gases in water, such as oxygen and radon.
Removes small particles of the order of 10 nm.
Removes radon dissolved in water.

Table 3.1: Water Purification System components and their purpose.

the single photoelectron signal, by attaching the waveshifter. However, in the OD, observing theextra photons is of much more importance than better timing resolution.

3.3

Water Purification System

The original source of the Super-Kamiokande water is spring water in the mine. This wateris highly purified by the water system, shown in Figure 3.9 with each component explained inTable 3.1. The SK water is continuously circulated at a rate of about 35 ton/hour through thepurification system to maintain high water transparency and to remove radioactive isotopes suchas Radon (Rn) and Radium (Ra). Such isotopes are significant backgrounds for low energy
analyses.
The resistivity of the water flowing out of the detector for purification is about 11 MΩ·cm andaverages at 18.20 MΩ·cm after purification, approaching the chemical limit. The light attenuation
length is about ∼100 m, and particles larger than 0.2 nm are reduced to 6 particles/cc.

3.4

Air Purification System

To minimise the radon level in the detector water, SK also uses an air purification system. The
seasonal variation of the radon air in the mine is shown in Figure 3.11. The winter levels,
where cool air flows in from outside, are comparably low at ∼100–300 Bq/m3 , however thesummer levels, where air flows from deep within the mine, are ∼2,000–3,000 Bq/m3 . Firstly
fresh air is pumped into the SK dome area at a rate of 10 m3 /minute, resulting in a typical radon
concentration of ∼20–30 mBq/m3 at the SK dome and control room.
From here radon-free air is produced by the air purification system outlined in Figure 3.10,
with the components explained in Table 3.2. This system pumps the purified air into the SK
tank, above the water level at a positive pressure to prevent any Radon present in the dome
air from dissolving in the highly purified water [161]. The resulting radon concentration in theradon-free air is less than 3 mBq/m3 .
79

Figure 3.9: Water System [156].
COMPRESSORCARBONCOLUMNBUFFERTANK

AIR FILTER(0.3mm)AIR DRIERCARBONCOLUMNHEATEXCHANGER

AIR FILTER(0.01mm)
AIR FILTER(0.1mm)

COOLEDCHARCOAL(-40 oC)

SUPER-KAMIOKANDE AIR PURIFICATION SYSTEM

Figure 3.10: Air purification system [156].
Component
Compressor
Air Filter
Buffer Tank
Air Drier
Carbon Columns
Air Filter
Cooled Charcoal

PurposeCompresses the air to 7–8.5 atmospheric pressure.
Removes dusts of size ∼0.3 µm.
Stores the air.
Dries the air and removes CO2 gas to improve the efficiencyof the Carbon Columns.
Removes radon gas using activated charcoal.
Further Removes small dust and particles of ∼0.01 µm.
Removes the remaining Rn gas with charcoal at –40◦ C.

Table 3.2: Air Purification System components and their purpose.

80

2000 Radon monitor readings at SuperK
3500

Mine Air (at sink)
Control Room Air

3000

radon level [Bq/m^3]

2500

2000

1500

1000

500

0
01/01/00

03/01/00

05/01/00

07/01/00

09/01/00

11/01/00

01/01/01

Figure 3.11: Radon levels [156] of the air in the mine is shown by the upper histogram. The
lower histogram at the base shows the radon levels in the SK control room.

3.5

Electronics and Data Aquisition System

3.5.1

Electronics and Data Aquisition: SK-I to SK-III

The electronics for the SK-I to SK-III period make use of custom-built ATM (Analogue-TimingModule) modules of the TKO (TRISTAN KEK Online) standard [162]. The ATM modules
receive the integrated charge signal and arrival timing information of each PMT. A block diagramof the ATM is shown in Figure 3.12. There are two pairs of readouts that are used in succession
to avoid dead time in the data taking. As an example, this allows both a muon and its decay
electron to be recorded. Each of these readouts consist of a QAC (Charge to Analogue Converter)
and a TAC (Time to Analogue Converter).
The PMT signal is first amplified 100 times at the ATM, and then divided into four signalsthat are routed to different components. One is sent to the discriminator, which checks the
PMT signal exceeds the threshold of 0.25 p.e.s . If a sufficient signal is received a 400 nsec timing
window is opened by a gate signal to the QAC, and a start signal is generated and sent to the
TAC. This is for an individual PMT. This hit PMT then generates a HITSUM signal which issent to a global trigger module. This trigger module receives HITSUM signals from each of thehit PMTs, if the HITSUM threshold is exceeded, then all the event information is recorded. At
the ATM, if there is no global trigger within 1.3 µsec then all the information in the QAC andTAC readouts are cleared. Two of the divided PMT signals are sent to each QAC, and the fourthis an output signal of the ATM without the discriminator threshold and is a PMTSUM signal.
In the case a global trigger is issued, a stop signal is sent to the TAC then the readout
information from the QAC and TAC are digitised by an ADC (Analogue to Digital Converter),and stored in the ATM’s internal FIFO memory. The ATM has a 450 pC dynamic range with aresolution of 0.2 pC, and 1.3 µsec dynamic range with a resolution of 0.4 nsec.
The DAQ (Data Aquisition) records the event information from all the PMTs when a global
trigger is issued. A block diagram of the DAQ is illustrated in Figure 3.13. Each ATM caters
to 12 PMTs, with about 20 ATMs housed in each of the 48 TKO crates. Every 16 events, the81to PMTSUM
HITOUTDISCRI.
CURRENT SPLITTER&AMPLIFIER

PMT INPUT

W:200ns

to HITSUM

ONE-SHOT HIT
W:300ns

SELF GATESTART/STOP

TAC-A

TAC-A

CLEAR
GATE

QAC-A

QAC-A

to ADCSTART/STOP

TAC-B

TAC-B

CLEAR
GATE

THRESHOLD

QAC-B

QAC-B

DELAY

GATE
TRIGGERPED_START

Figure 3.12: Input proportion of the ATM

digitised data in the ATM FIFO memory are sent to the SMP (Super Memory Partner) in the
VME. Each SMP looks after the 20 ATMs, and each VME crate houses 6 SMPs. The 8 VME
crates hold the SMPs, which are read out by 8 slave computers and send the data to the online
host computer.

Outer Detector Electronics and DAQ: SK-I to SK-III
The OD DAQ consists of paddle cards that distribute high voltage through coaxial cable to the
OD PMTs, and also handle the PMT signals. A block diagram of the OD DAQ can be seen in
Figure 3.14. The paddle cards receive the PMT signals through a high voltage capacitor andsend them to QTC (Charge to Time Converter) Modules. The OD PMT signals are recorded asa rectangular signal with a width proportional to the charge, and the time given by the leadingedge. The QTC modules have a threshold of 0.25 p.e.s, and generate a HITSUM signal which
is sent to the global trigger module. If a global trigger is issued, a LeCroy 1877 multi-hit TDC
module converts the QTC signal into the charge and timing information. The DPM (Dual Port
Memory) modules in VME crates read out the digitised data in the TDC modules, which is sent
to a slave computer and then the online host computer. The TDC module can record up to
8 QTC pulses with a resolution of 0.5 nsec. The dynamic range is set to 16 µsec starting from
10 µsec before the global trigger timing.
82

20-inch PMT
SCH
ATM
x 20
x 240

ATM
GONG

20-inch PMTSCHATMinterface
Ultra Sparc

x 20
x 240

SMP

ATM

SMP

GONG

online CPU(slave)

SMP
SMP
SMP
SMP

x6

Super Memory Partner

VME
20-inch PMT
SCHUltra Sparc

ATM

online CPU(slave)

x 20
x 240

Ultra Sparc

ATM
GONG

online CPU(host)

Analog Timing Module

online CPU(slave)
TKO

FDDI

Ultra Sparc

Ultra sparc

online CPU(slave)
Ultra Sparc

online CPU(slave)
Ultra Sparc

FDDI

online CPU(slave)
Ultra Sparc

online CPU(slave)

online CPU(slave)interface
Ultra Sparc
SMP
SMP

Ultra Sparc

online CPU(slave)

SMP
SMP
SMP
SMP

Super Memory Partnerinterface

VME
VME

interrupt reg.
TRG

20-inch PMT
SCH
TRIGGERATM

TRIGGER

x 20
x 240

PROCESSORHIT INFORMATION
ATM
GONG

Analog Timing ModuleTKO

PMT x 11200

ATM x ~1000

SMP x 48

online CPU(slave) x 9

Figure 3.13: Inner Detector DAQ

83

Local Hitsum

QTC
Modulex40/quadx10/quad

TDC
x5
4 LSBsHigh Voltage

16 bit event #TDC

FASTBUSPMT

Paddle Card

StruckSupply~470/quadrant

FSCC

AUX

84 pin data ribbon cables

Ethernet FSCC Control

Figure 3.14: Outer Detector DAQ

3.5.2

Hardware Trigger

In the DAQ sections for the ID and OD, HITSUM signals and the global trigger module werementioned. When an ID PMT is hit, with a charge above the 0.25 p.e. threshold, the ATM
generates a rectangular 200 nsec wide 15 mV pulse as a HITSUM signal. The global trigger
module receives the HITSUM signals from hit PMTs in the ID, and sums them up to generate
an overall ID-HITSUM signal. A diagram of the ID Trigger is shown in Figure 3.15.
There are multiple trigger thresholds for the ID-HITSUM within a 200 nsec window: HighEnergy (HE) which is –340 mV (31 hits), Low Energy (LE) which is –320 mV (29 hits) andSuper Low Energy (SLE). The LE trigger corresponds to the signal expected from a 5.7 MeV
electron (with 50% trigger efficiency) and the SLE trigger is equivalent to a 4.6 MeV signal. The
trigger rates are ∼5 Hz for HE and ∼11 Hz for LE, with energy thresholds of 10 MeV and 8 MeV,
respectively.
The trigger for the OD is similar. When an OD PMT exceeds the 0.25 p.e. threshold, theQTC generates a rectangular 200 nsec wide 20 mV pulse as a HITSUM signal. These are summed
to generate the OD-HITSUM signal, with a threshold of 19 hits within a 200 nsec time window.
There are four types of trigger signals in total (HE, LE, SLE and OD) that are fed into thehardware trigger module: TRG. If any one of these trigger signals is received, the TRG module
generates a global trigger signal and the event information is recorded.

3.5.3

Upgraded Electronics and Data Aquisition for SK-IV

For SK4, all of the front-end electronics were upgraded in September 2008 [163]. The newelectronics are based on a quick charge to time converter and a multi-hit time to digital converter.
The new system allows every hit to be processed, and the triggering was changed to be software
based, in place of the previous hardware trigger. This removed the down time from the DAQ,
and made it necessary to transmit large volumes of data, up to 470 MB/s, which was achieved
with Gigabit and 10-Gigabit Ethernet technologies.
In the previous electronics, the bottleneck for the data transfer was the TKO, limited atabout 2 MB/s for 20 ATM boards. This corresponded to a data rate of about 1.4 kHz perchannel, while the PMT dark rate is typically 4.5 kHz. The new electronics can handle much84

Analog signal
from PMT
ATM threshold
400 nsec width

ADC gate
TDC startand stop

stop

start

200 nsec width

HITSUM
Global triggerATM
Sum of HITSUM
Master threshold
Global trigger
CentralHut
Figure 3.15: Inner Detector Trigger

85

Figure 3.16: SK-IV New Charge to Time Converter (QTC), which were custom ApplicationSpecific Integrated Circuits (ASIC) [164].

larger data throughput, allowing the event trigger threshold to be lowered. This is importantfor making precise measurements of the low energy solar neutrinos, by observing the energy
dependence of the Neutrino Oscillations. The new system also allows the measurement of agreater number of events from a neutrino burst from a supernova. From a supernova at thegalactic centre, around 8 thousand events are expected within 10 seconds. The previous DAQ
could handle 60,000 events in this time window, while the new DAQ can process 100 times more.
Another advantage is the software-based trigger, making it possible to trigger data recording
for more complex conditions. An example of where this would be important is distinguishingrelic neutrinos above the atmospheric neutrino background. This can be done by observing aneutron capture by a 2.2 MeV γ emission 200 µs after the initial event. This energy was too small
to trigger an event in the previous system. In the new system, the trigger can be configured tokeep data for some time period after an event, which could be searched.

3.5.4

SK-IV New Electronics

The new electronics are called QBEEs, short for QTC (Charge to Time Converter) Based Electronics with Ethernet. Primarily, these measure the charge and arrival timing from the PMTs
and produce a 6-byte cell per hit and send the data to the online PCs. Each QBEE board
was equipped with eight QTC chips, which were custom Application-Specific Integrated Circuits(ASIC) [164]. Each of these QTCs, shown as a block diagram in Figure 3.16, had three inputs(i.e. 24 channels per QBEE) and three measurement ranges per channel (small, medium, andlarge), to cover a much wider dynamic range than the previous electronics. Previously, charge
saturation would occur at 600 pC, while the new QTC can measure over 2000 pC (∼1000 p.e.)
without saturation. The QTC produces an output pulse whose width is proportional to the input
charge, and leading edge indicates the timing.
The output of the QTC is fed to a multi-hit TDC (Time to Digital Converter), which detects
the leading and falling edges. The charge resolution is about 5% at 1 p.e. and less than 2% above
3 p.e., similar to the previous ATM resolution. The digitised signal from one hit is arranged into
a 6-byte cell containing charge, timing and input channel information. For more details about
the ASIC QTC, see Reference [164].
86

Figure 3.17: SK-IV New Online System [163].

3.5.5

SK-IV New Online System

A block diagram of the new online system can be seen in Figure 3.17. The data from about 550
QBEE boards are transferred as TCP packets to 20 front-end PCs. Data processed from thefront-end PCs are sent via 10-Gigabit Ethernet to 10 Merger PCs. Here, the data are merged
and the software trigger is applied to select candidate events. The triggered events are then sent
to an Organiser PC, which writes merged data onto the disk for offline analysis. The PCs areLinux based, with 2 GB RAM and 3 GHz Intel dual-core CPUs (Xeon 5160).
The software trigger can be used to set conditions for recording data, which can easily be
added or modified to the requirements of physics analysis. The basic trigger was set up to reproduce the previous hardware trigger conditions, and depended on the number of hits exceeding athreshold within a 200 ns time window. The software trigger allows for special triggers, such as
the neutron trigger that looks for an emitted gamma ray from neutron capture; the T2K trigger
for selecting a wide time window around the beam timing; and external triggers for detector
calibrations. For more details about the new online system, see Reference [163].

87

Chapter 4Monte Carlo Simulation
4.1

Overview

The Monte Carlo (MC) method was used to simulate atmospheric neutrino events in SuperKamiokande. To be able to simulate events that are just like the data, there were many considerations. Firstly the neutrino flux at the SK site needs to be calculated by considering the
global data for primary cosmic-rays, their interactions that produce neutrinos and the following
propagation to the SK detector. Once the neutrino flux at SK was estimated, cross-section datacollected by many experiments can be used to predict how the neutrinos interact in the detector
and the resulting kinematics. The third step was to propagate all the interaction products ina simulated detector geometry, and the Cherenkov photons to the PMTs. The simulation also
considered the electronics, such that simulated output similar to the data could be generated.
Since the MC also includes all the true physical parameters, it can be compared with the data
to understand the physical meaning of what was observed.

4.2

Atmospheric Neutrino Flux

There are several models for estimating the neutrino flux from the primary comic-ray flux. They
are referred to here as the Honda flux [16, 165–167] Fluka flux [168], and Bartol flux [169]. The
main flux model adopted for the atmospheric data analyses at Super-Kamiokande is the Honda
flux, and the others were used to estimate the systematic uncertainties on the Honda calculation.
The Honda flux is calculated up to 10 TeV, so the Volkova flux [170] was used above 10 TeV, with
the normalisation adjusted to match the Honda flux continuously.
There are many experiments globally that are trying to determine the flux of primary cosmicrays, which has been summarised in Figure 4.1 together with the fitting used in the Honda flux
model. This data serves as the initial input into the atmospheric neutrino flux calculation. The
primary comic-ray proton flux up to 100 GeV has been precisely measured by the BESS [171] andAMS [172] experiments. The curve was adjusted slightly to agree with emulsion chamber data
above ∼10 TeV [166]. The more recent improvements to the Honda flux have been improvementsin the interaction models [16, 167].
The variation of the solar wind affects the cosmic-ray flux, particularly at the lower energy
range of the spectrum. The flux at the Earth is higher with high solar activity, referred to as
the solar maximum, and lower with low solar activity, the solar minimum. For cosmic-rays of88

Figure 4.1: Measurements of primary cosmic ray proton flux and the model used in the
Honda flux calculation. The data are taken from Webber [173] (crosses), LEAP [174] (upward triangles), MASS1 [175] (open circles), CAPRICE [176] (vertical diamonds), IMAX [177](downward triangles), BESS98 [171] (circles), AMS [172] (squares), Ryan [178] (horizontal diamonds), JACEE [179] (downward open triangles), Ivanenko [180] (upward open triangles), Kawamura [181] (open squares) and Runjob [182] (open diamonds).
around ∼1 GeV, the difference between the solar maximum and minimum is more than a factor of
two. However, the influence decreases to ∼10% at ∼10 GeV. There is also an effect from the geomagnetic field. It deflects cosmic-rays and is characterised by a rigidity (= momentum/charge)
cutoff. Cosmic-rays with momentum lower than the cutoff do not arrive at the Earth. The
cosmic-ray flux above 100 GeV, which is responsible for ≥ 10 GeV neutrinos, is not affected by
the solar activity nor the geomagnetic field.
When the primary cosmic ray protons and nuclei enter that atmosphere and interact with
nuclei in the air, secondary particles are produced, which consist mostly of pions and kaons.
The density structure of the atmosphere is given by the US Standard Atmosphere model [183],
whereas the geomagnetic field model is based on the IGRF2005 model [184].

4.2.1

Hadronic Interaction Models

Two models are used for the hadronic interactions of cosmic-rays with the nuclei in the air. In
the case of the Honda 2006 flux [167] and earlier these models were NUCRIN [185] for primarycosmic-ray energies < 5 GeV and DPMJET-III [186] for > 5 GeV. These models determine how
the meson products are produced, and the atmospheric neutrinos are produced when they decay.
Secondary cosmic-ray muons are also produced in the decay chain, and their flux has beenmeasured by experiments such as BESS [187] and L3+C [188]. The DMPJET-III hadronicmodel was modified using the measurements of BESS and L3+C, and used in the Honda flux
calculation [167]. The ratio of the cosmic-ray muon spectrum after the modification is shown in
Figure 4.2. After the modification, the hadronic interaction model agrees with the data within.10% in the 1–100 GeV muon energy range.
The main differences in the Honda 2011 flux [16] was a modification to the hadronic interac89

1.5

Data/Calculation

Tsukuba (Oct. ’02)
Norikura (Sep. ’99)
L3+C
1.0
DEIS
MUTRON
0.7

0

10

10

1

10

2

10

3

10

4

pµ (GeV/c)

Figure 4.2: Secondary cosmic-ray muon fluxes normalized by the calculation [167]. Data were observed at Tsukuba(open box) and Norikura(open triangle) with the BESS detector [189, 190], andat CERN(open circle) with the L3 detector [188]. For reference, DEIS [191] and MUTRON [192]are plotted. Dashed lines represent the sum of errors in data and calculation. Figure is takenfrom [167].
tion models, which performed better particularly in the low energy region. To better reproduce
the atmospheric muon spectra at lower energies, at sea level and at mountain altitudes, the JAM
hadronic interaction model [193] was used. In addition the HARP experiment greatly improved
our understanding of hadronic interactions at low energies [194], and in particular by colliding
protons on thin N2 and O2 targets [195]. The JAM model was found to agree better with the
HARP data. In the Honda 2011 calculation, JAM was used between 0.2 GeV to 32 GeV, and forenergies above this the modified DPMJET-III model was used as before. Despite these changes,
the zenith dependence and flavour ratios of the atmospheric neutrino flux are very similar to
before. The main difference is a change of the absolute value of the atmospheric neutrino flux
below 1 GeV. A comparison between the low energy HARP data and the JAM and modified
DPMJET-III models can be seen in Figure 4.3.

4.2.2

3-Dimensional Flux Calculation

The calculation of the neutrino flux is treated in a 3-dimensional way. The curvature andpropagation of charged particles in the geomagnetic field as well as 3-dimensional interaction
topologies are considered. Treating the flux in 3-dimensions actually has a significant effect on
the directional flux observed at SK, compared with a simpler 1-dimensional treatment. The twomain features of the 3D calculation are that there is an increase in the neutrino flux in the nearhorizontal direction, and that there is also lower production height of neutrinos, also significant
in the near-horizontal direction.
The increase in the near-horizontal neutrino flux is explained by the increased effective areathat can generate near-horizontal neutrinos, illustrated in Figure 4.4. The lower energy secondary
charged pions and muons, produced from the hadronic interactions of primary cosmic-rays, arealso affected. They have an effect from their transverse momentum and also experience curvature
90Figure 4.3: Comparison of the HARP data on forward pion production of low energy hadronicinteractions [195] with the JAM and modified DPMJET-III model [16]. The HARP data on N2
and O2 targets were combined in a ratio of 78.5% N2 and 21.5% O2 to simulate an air target.

Figure 4.4: A schematic view of the effective areas of primary cosmic rays interacting with air
nuclei for 1-dimensional and 3-dimensional calculations. Arrows written by solid lines show theprimary cosmic rays and dotted lines show the neutrinos. The 3-dimensional calculation gives
larger areas for near-horizontal direction.

91

20

34

(a) vertical

νµ

16
14
12

νe

10
8

30
28
26
24
22
20
18

6
10

(b) horizontal

32
Production height (km)Production height (km)

18

16
-1

1
Eν (GeV)10-11Eν (GeV)Figure 4.5: 50 % accumulation probability of neutrino production height for (a) nearvertical (cosθ > 0.95) and (b) near-horizontal (|cosθ| < 0.05) directions. Thick and thin solid
curves are for νµ and νe by the 3-dimensional calculation, and thick and thin dashed curves arefor νµ and νe by the 1-dimensional calculation, respectively.
in the geomagnetic field. This influences primarily low energy neutrinos at < 1 GeV. Some zenith
plots showing the enhancement of the neutrino flux in the horizontal direction for lower energies
is shown in Figure 4.6, with the earlier 1D Honda flux and the 3D Fluka flux also drawn forcomparison. Due to the stronger than average geomagnetic field above SK, the neutrino flux ispredicted to be up-down asymmetric below a few GeV. However, due to the relatively smaller
angular correlation of the neutrino and lepton directions at low energies, this up-down asymmetryis smeared.
Neutrinos being produced at lower altitudes in the 3D calculation, than in the 1D calculation,
can be explained by the difference in the path length of the primary cosmic-rays. Referring again
to Figure 4.4, and considering direct 1D paths, it can be seen that the path taken through theatmosphere is larger at the horizontal zenith direction than for more vertical zenith directions. In
the 3D calculation, a primary cosmic-ray that has a more vertical incident angle and then curves
to the horizontal direction has a shorter path length than the equivalent horizontal 1D case. This
results in neutrinos reaching lower altitudes in the 3D calculation before interacting. Since thiseffect depends on the curvature of primary cosmic-rays, the influence is most prominent for low
energy neutrinos at < 1 GeV. The simulated neutrino production height for the 3D Honda flux
and the 1D Honda flux are shown in Figure 4.5. Further details of the differences of the 3D and
1D calculations can be found in Reference [165].

4.2.3

The Neutrino Flux at Super-Kamiokande

The resulting neutrino flux observed at the SK site from the Honda 2011 flux calculation [16] isshown in Figure 4.7, where the flux has been averaged over all directions. The Fluka and Bartol
fluxes were also drawn for comparison. The flux of the electron flavour neutrinos drops at afaster rate than the muon flavours with increasing energy. This is explained by the higher energymuons reaching the surface of the Earth before they can decay and produce electron neutrinos.
92

4000

300

Honda flux
Fluka flux
Honda-1D
flux

-2

-1

-1

Flux(m sec sr )0.1-0.3 GeV

1.0-3.2 GeV

–

νµ+νµ
200

2000
100
–

νe+νe
0
-1.0

0

1.0

0
-1.0

0

1.0

0

1.0

1200

0.3-1.0 GeV>3.2 GeV-1-1

Flux(m sec sr )

40

800

-2

20
400

0
-1.0

0

1.0

cosΘ

0
-1.0

cosΘ

Figure 4.6: Zenith angle dependence of the atmospheric neutrino flux. Boxes show the Honda
flux, dots show the Bartol flux and solid lines show the Honda-1D flux. The higher flux is forνµ + ν̄µ and the lower flux for νe + ν̄e in each plot. An enhancement of the neutrino flux is seen
for near-horizontal directions at energies below 1 GeV in 3-dimensional calculations, the Honda
flux and the Fluka flux. On the other hand, no enhancement is seen in the Honda-1D flux.

93Figure 4.7: Predictions of the direction averaged atmospheric neutrino flux (Left) and the flavour
ratio and neutrino anti-neutrino ratios (Right), from Reference [16]. The Honda 2011 flux isshown in solid red [16]; the Honda 2006 calculation in dash-dotted blue [167]; the Bartol flux [169]in dashed navy; and the Fluka flux [168] in dotted green.
The flavour ratio is also shown in Figure 4.7. The (νµ +ν̄µ )/(νe +ν̄e ) ratio is around two up untila few GeV, and then increases with energy.
The neutrino anti-neutrino ratios have also been overlaid in the right plot of Figure 4.7. Thereis a good agreement on these ratios between the flux models. The Bartol flux agrees a little lessat higher energies, but this is likely due to an incomplete treatment of the kaon component of
the atmospheric neutrino and anti-neutrino fluxes, which becomes increasingly relevant at higher
energies. The fact that these ratios are well-understood is important, as the MC was divided
into its neutrino and anti-neutrino constituents in the analysis in Chapter 9.

4.3

Neutrino Interactions

The interactions of neutrinos in the simulation are handled by an interaction generator called
NEUT [196]. The libraries were originally developed for research into proton decay and neutrinos for the Kamiokande experiment, and then modified for Super-Kamiokande. The following
charged current (CC) and neutral current (NC) interactions are simulated:
CC/NC (Quasi-)Elastic Scatteringν + N → l + N0

CC/NC Single Meson Production

ν + N → l + N 0 + meson

CC/NC Single Gamma Production ν + N → l + N 0 + γ
CC/NC Deep Inelastic Scatteringν + N → l + N 0 + hadrons

CC/NC Coherent Pion Production ν + 16 O → l + 16 O + π
94-(a) νµn→µ p

ANL
BNL
SKAT

Serpukhov
GGM (1977)
GGM (1979)

σ (10-38 cm2)

σ (10-38 cm2)

2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

0

2

4

6
8
10
Eν (GeV)

12

14

2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

–

GGM (1977)
GGM (1979)
Serpukhov
SKAT+(b) νµp→µ n

0

2

4

6
8
10
Eν (GeV)12

14

Figure 4.8: Cross-sections for quasi-elastic scattering of (a) νµ and (b) ν̄µ . The solid line is for
NEUT on free nucleons, and the dashed line is for NEUT on bound nuclei. The experimental
data are from ANL [200], Gargamelle [201–203], BNL [204], Serpukhov [205] and SKAT [206].
where N and N 0 are nucleon (proton or neutron) states before and after the interaction and l is
a lepton.
Since the cross section of neutrino-electron elastic scattering is about three orders of magnitude smaller than that of the neutrino-nucleon interactions at a neutrino energy of ∼1 GeV, thisinteraction mode is neglected in the simulation. Since the author did not work on the neutrino
interactions at Super-Kamiokande, a summary will be written here. For a detailed account fromsomeone who did work on them, see Reference [197].

4.3.1

Quasi-Elastic Neutrino-Nucleon Scattering

In NEUT, the CC Quasi-Elastic Scattering (CCQE) on free protons (the hydrogen atoms inwater are essentially free) was based on the Llewellyn Smith Model [198]; while the interactions
on bound nucleons in 16 O, and the nuclear effects, are treated with the Smith and Moniz [199]model. The characteristic of a quasi-elastic interaction is that the nucleon does not break up
from the reaction, and are interactions of the form:
νµ + n → µ− + p+ν̄µ + p → µ + n
(−)

(−)

ν +p→ ν +p

(4.1)
(4.2)
(4.3)

where the n is a neutron, p is a proton, and the interaction in Equation (4.3) is an exampleof NC elastic scattering. A comparison of the cross-sections for quasi-elastic scattering between
experimental data and the NEUT calculation can be seen in Figure 4.8.
The elastic scattering on free protons is based on dipole-type form factors, in which thethe vector mass MV is set to be 0.84 GeV and the axial vector mass MA is set to be 1.21 GeV.
These were set from the experimental measurements by the SciFi [207] and SciBar [208] detectors(K2K), as well as MiniBooNE [209]. This MA value is also used for Single Meson Production.
The effect of a larger MA value is that interactions with larger Q2 and larger scattering angles
are enhanced. The uncertainty in the MA value was taken to be 10%. The axial vector coupling
constant, gA , is set to 1.232 measured by polarised nucleon beta-decay [210, 211]. Other than95

the dipole form factors, mentioned above that are used in NEUT, a few other types of form
factor have been put forward [212–214]. The difference in the cross-section from these models
and NEUT is estimated to be < 10%.
When considering interactions on bound nucleons in 16 O, nuclear effects such as the Pauli
exclusion principle or the Fermi motion of the nucleons should be taken into account. The nuclear effects in NEUT are treated with the Smith and Moniz [199] model. To allow quasi-elastic
scattering to occur, the outgoing momentum of the nucleons in the interactions is required tobe larger than the Fermi surface momentum. NEUT takes the Fermi surface momentum to be
225 MeV, which was determined to reproduce the quasi-elastic peak in electron scattering. The
systematic uncertainties on the neutrino interactions and nuclear effects are estimated by considering other models. Other such models have been proposed, for example, by Nieves et al. [215]and Nakamura et al. [216].
The cross-sections for NC elastic scattering are estimated from the CC cross-sections by thefollowing relations [217, 218]:σ(νp → νp) = 0.153 × σ(νn → e− p)
+

4.3.2

(4.4)

σ(ν̄p → ν̄p) = 0.218 × σ(ν̄p → e n)

(4.5)

σ(νn → νn) = 1.5 × σ(νp → νp)(4.6)σ(ν̄n → ν̄n) = 1.0 × σ(ν̄p → ν̄p)

(4.7)

Single Meson Production

When the hadronic invariant mass (W ) is . 2 GeV/c2 , single meson production via baryon resonances are the dominant hadron production processes. The model used in NEUT is based on theRein and Sehgal model [219, 220]. This model considers single pion production, however it wasadapted for NEUT to encompass the single meson production of η and K mesons, by altering
the decay of the resonances. The meson is produced via intermediate baryon resonances:
ν + N → l +N∗→N 0 + meson(4.8)

where N ∗ is the baryon resonance, and N and N 0 are the nucleon states before and after theinteraction.
Lepton polarisation and mass effects are considered [221, 222], which results in a suppressionof the cross-section at lower Q2 . The angular distribution of the pion in the final state for the∆(1232) resonance is given by a method proposed by Rein [220]. The angular distribution of
the other resonances are taken as isotropic in the resonance rest frame. In the decay of a baryon
resonance, the Pauli blocking effect is considered for interactions off a bound nucleon by requiring
the momentum of the nucleon to be greater than the Fermi surface momentum.
The meson-less decay of the baryon resonance is also considered [223], and 20% of the resonances decay in 16 O with only a lepton and nucleon in the final state. The process occurs in thepresence of other nucleons like so:N ∗ + N → N 0 + N 00
where N ∗ is a baryon resonance and N, N 0 , N 00 are nucleon states.
96

(4.9)

σ/Eν (10-38cm2 GeV-1)

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

GRV94LOGRV94LO (w/ B-Y corr.)
GRV98LO
GRV98LO (w/ B-Y corr.)

CCFR 90
CDHSW 87
GGM-PS 79

0

10

CHARM 88
BNL 80
CRS 8020

30 50

BEBC-WBB 79
IHEP-JINR 96
IHEP-ITEP 79

CCFRR 84
SKAT

100

200

150

250

Eν (GeV)Figure 4.9: Cross-sections for charged current deep inelastic scattering νµ and ν̄µ interactions.
The upper (lower) curves are for νµ (ν̄µ ).

4.3.3

Deep Ineslastic Scattering

The cross-section of CC Deep Inelastic Scattering (DIS) was determined using nucleon structure
functions taken from the parton distribution functions (PDF) calculated by Glück et al. [224].
The PDF in the lower Q2 region was improved by the corrections of Bodek and Yang [225].
Cross-sections of the CC νµ and ν µ interactions are plotted in Figure 4.9.
The treatment of neutral current interactions in NEUT for deep inelastic scattering are estimated from the charge-current case by relations determined by experimental measurements [226,
227]:

0.26
( Eν < 3 GeV )σ(νN → νX)
0.26
+
0.04
(E
/3
−
1)
(
3 GeV ≤ Eν < 6 GeV )(4.10)=νσ(νN → µ− X)
0.30( Eν ≥ 6 GeV )
0.39( Eν < 3 GeV )σ(ν̄N → ν̄X)
0.39
−
0.02
(E
/3
−
1)
(
3 GeV ≤ Eν < 6 GeV )=(4.11)νσ(ν̄N → µ+ X)
0.37( Eν ≥ 6 GeV )Depending on the invariant mass W , the kinematics of the hadronic system is simulated by
one of two methods. For the 1.3 GeV/c2 < W < 2.0 GeV/c2 region, only pions are considered as
outgoing mesons. For the DIS treatment of pions, NEUT requires nπ ≥ 2 to avoid overlappingwith the W region for single pion production. The forward-backward asymmetry in the hadronic
centre of mass system of the pion multiplicity is considered using the results from the BEBC
experiment [228]. For the W > 2.0 GeV/c2 region, the kinematics of the hadronic system arecalculated by the CERN Pythia and Jetset package [229], which treats K, η, ρ and so on, as well
as π.

4.3.4

Coherent Pion Production

Coherent pion production is a neutrino interaction with a nucleus as a whole (oxygen here),
which remains intact as there is only a small momentum transfer to the nucleus. The event is97

characterised by an outgoing pion with the same charge as the incoming weak-current, with boththe outgoing pion and lepton having a strong forward angular correlation with the neutrino. To
simulate this interaction, NEUT makes use of the Rein and Sehgal formalism [230]. There isan upper limit on the cross-section for CC coherent pion production, set by the K2K SciBar
detector [231].

4.3.5

Nuclear Effects

To properly understand the interactions observed, it is necessary to consider nuclear effects.
These are secondary interactions of the outgoing mesons, while they are still inside the nucleus.
All the mesons produced from neutrino interactions inside the 16 O nucleus are tracked fromproduction until they are absorbed or leave the nucleus. NEUT employs a cascade model topropagate the π, K and η produced in interactions in the 16 O nucleus. At Eν > 1 GeV, thecross-sections for pion production from neutrino interactions, as well as pion-nucleon interaction
cross-sections are large. So the secondary interactions of pions are particularly important. The
following interactions are considered: inelastic scattering, charge exchange, and absorption. In
NEUT, the Wood-Saxon density distribution [232] determines the initial production point of thepion in the nucleus:Z1ρ(r) = ρ0(4.12)r−cA
1 + expawhere Z is the atomic number, A is the mass number, ρ0 is the average density of the nucleus,
and a and c are the density parameters. For the 16 O nucleus: ρ0 = 0.48 m3π , a = 0.41 fm, c =2.69 fm, Z = 8 and A = 16 [233]. The mean free path, with momentum and position dependence,of each possible interaction is used to determine which pion interaction takes place [234]. Fermi
motion and Pauli blocking are considered. The energy of the outgoing nucleon must be above
the Fermi surface momentum:

1
3
3 2
pF (r) =π ρ(r).
(4.13)
2
Using the results of a phase shift analysis from π–N scattering experiments [235], the angularand momentum distributions of the outgoing pions are determined. The pion interaction simulation was compared with the experimental data for the following three interactions : π−12 C
scattering, π−16 O scattering, and pion photo-production (γ+12 C → π − +X) [236, 237], as shownin Figure 4.10.
For kaons, cross-section measurements of K ± − N scattering [238–240] were used to develop
the elastic scattering and charge exchange treatment in NEUT. For η mesons, η absorption(ηN → N ∗ → π(π)N ) was considered [241], where the pions are tracked as previously described.
Nucleon–nucleon elastic scattering was also considered, using cross-section measurements [242].
These interactions can lead to the production of delta resonances, which decay to produce pions.
This effect is considered with the isobar production model proposed by Lindenbaum et al. [243].

4.4

Detector Simulation (SKDETSIM)

After the flux of neutrinos expected at the detector has been calculated, and the interaction
generator has predicted how many interact and the interaction topology, the tracking of particles
98

+ 16

cross section for π + O (mb)600
inela+abs+cex
inelastic
absorption
charge exchange

500
400

inela+abs+cex

300
200

inelastic

100

absorption

0

charge exchange

0

100

200

300

400

500

600

+

momentum of π (MeV/c)

Figure 4.10: The cross-sections of π + −16 O scattering as a function of π + momentum. The lines
are the cross sections calculated by NEUT for each interaction mode, and the experimental data
points are taking from [237].
in the detector itself was simulated. The simulation can be summarised as three steps. The firstis the tracking of the outgoing particles through the detector and secondly the generation andpropagation of their Cherenkov photons in water. The third is to simulate the PMT response
as well as the electronics readout. The Super-Kamiokande Detector Simulator is referred to as
SKDETSIM.

4.4.1

Particle Tracking

The detector simulation was developed using the CERN GEANT3 libraries [244]. During the
tracking of the particles through the detector, there are multiple processes that can occur depending on the particle type. A list of the processes considered is presented in Table 4.1. For the
hadronic interactions in water, the CALOR package was used [245], which reproduces the pion
interactions well in the low momentum region (∼1 GeV/c). For the pπ ≤ 500 MeV/c momentum
region, a custom program [246] was used that was based on experimental results from π−16 O
scattering [247] and π − p scattering [248]. The systematic uncertainty in the hadron interactions was estimated from the difference between CALOR, which was used in SKDETSIM, andthe FLUKA model.
For the propagation of charged particles, Cherenkov photons are generated with an opening
angle defined by Equation (3.2), and the number and wavelengths by Equation (3.3). Only
photons in the range of 300–700 nm are generated because this is the sensitive region of thePMTs, as shown in Figure 3.3.

4.4.2

Cherenkov Photon Tracking

Each of the Cherenkov photons produced by the passage of charged particles are tracked in thedetector. The photons can undergo several processes. They can be scattered due to local changes
99

γ

e±

µ±

Hadrons(e+ , e− ) pair production
Compton scattering
Photoelectric effectMultiple scattering
Ionization and δ-rays production
Bremsstrahlung
Annihilation of positron
Generation of Cherenkov radiation
Decay in flight
Multiple scattering
Ionization and δ-rays production
Bremsstrahlung
Direct (e+ , e− ) pair production
Nuclear interactions
Generation of Cherenkov radiation
Decay in flight
Multiple scattering
Ionization and δ-rays production
Hadronic interactions
Generation of Cherenkov radiation

Table 4.1: List of the simulated processes.

in density in the water, or they can be absorbed. The probability of either of these processes
occurring are characterised by parameters that depend on the transparency of the water, andare tuned values. The details on how this calibration is performed are written in Section 5.2.
If the Cherenkov photon is absorbed, tracking of that photon simply stops. Otherwise, there
is a possibility that the photon is scattered. If it scatters, there are two angular distributions inwhich the photon may follow. The primary distribution is symmetric in the forward-backward
direction and corresponds to the type of angular distribution expected from Rayleigh Scattering.
There is also a relatively smaller probability that a very forward angular scattering distribution is taken. This corresponds to the asymmetric component expected from Mie Scattering.
When the Cherenkov photons scatter on tiny dust particles in the water that are close to the sizeof the photon wavelength, Mie scattering may occur. The exact angular distribution taken canvary between a symmetric shape and a considerably forward form, depending on the distributionof small particles present in the tank. Since the exact distribution of small particles in the tankis not known, there is an effective treatment in SKDETSIM to account for the possibility offorward scattering.
The probability of the photon being absorbed greatly depends on the transparency of thewater, and varies with the water quality. A study into the possibility of a time varying MonteCarlo to account for the changes in water quality was carried out and is described in Section 5.2.
At the time of writing, the MC is not varied with time, but instead the momentum calculation
is varied with the water transparency. This variation is determined from the very reliable watertransparency measurement from cosmic-ray muons (see Section 5.1.4).
The probability of these processes occurring is described in Section 5.2. The simulation also100accounts for reflections on the black sheet and on the PMT surfaces. The probability of thisoccurring is also a tuned parameter.

4.4.3

PMT Response

Using the measured quantum efficiency, the PMT photocathode surface is simulated as part of thedetector geometry. To determine the charge distribution, the single p.e. distribution is summed
for each photoelectron emitted and smeared by the timing resolution of the PMT. As well as
this, properties of the electronics are simulated like the timing window and charge threshold.

101Chapter 5

Calibration
5.1

Detector Calibration

5.1.1

Relative Gain Calibration

Uniformity of the high voltage (HV) gain of each PMT is necessary to accurately reconstruct the
momentum of particles without introducing systematic differences due to the particle’s vertex
position of direction. It is the goal of the Relative Gain Calibration to adjust the HV of each
PMT such that the gain is approximately uniform across the whole detector. The remaining
differences in gain can then be corrected for in the software.
This calibration is done using a Xe lamp light source and scintillator ball set up as shown in
Figure 5.1 (left). The Xe lamp light is passed through an ultraviolet (UV) filter and a neutral
density (ND) filter, and then split into four optical fibres. One fibre is routed into the tank andinto the scintillator ball, making the light source for the calibration. Another fibre is used totrigger a Xe lamp calibration event, and the others to monitor the intensity of the light flash.
The scintillator ball is made up of a mixture of acrylic resin, BBOT wavelength shifter andMgO powder diffuser. The BBOT wavelength shifter absorbs the UV light and re-emits photons
with a peak wavelength of 440 nm, which is a typical wavelength for Cherenkov light producedin water.
The high voltage value of each PMT is adjusted to obtain uniform gain by measuring therelative gain Gi of the i-th PMT:Qi
Gi =· l2 · exp
Q0 f (θ) i

 
li
L

(5.1)

where Qi is the charge detected by the i-th PMT, Q0 is a normalisation factor, f (θ) is the PMT
acceptance as a function of the photon incidence angle θ, li is the distance from the light source tothe PMT, and L is the attenuation length. This measurement is performed for several positionsof the scintillator ball, changing the voltage.
After adjusting the HV of the PMTs, a relative gain distribution for all the PMTs like thatshown in Figure 5.1 (right) is obtained. The one sigma spread of the relative gain is about
7 %. This figure shows the remaining gain difference of each PMT, which is corrected for in thesoftware.
102

Number of PMTs in Each Bin

1400

1200

σgain = 7.0%
1000

800

600

400

200

0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Relative PMT Gain

Figure 5.1: Relative Gain Calibration with the Xe lamp. The left figure shows the set-up of
the Xe lamp, Scintillator ball and intensity monitoring. The right figure shows the relative gainof all the PMTs after adjusting the HV to obtain approximately uniform gain. The remaining
difference in the gain is shown, which is corrected for in the software.

5.1.2

Absolute Gain Calibration

The main unit of charge used in the Super-Kamiokande software is the number of photoelectrons
detected. The goal of the Absolute Gain Calibration is to determine the conversion factor fromthe pico-Coulomb (pC) measured by the PMTs to the equivalent number of photoelectrons. Thisis done by measuring the single photoelectron charge distribution of the PMTs.
The calibration source is a polyethylene vessel containing Nickel wires and housing a 252Californium (252 Cf) neutron source, as illustrated in Figure 5.2 (left). The spontaneous fission of
252 Cf produces neutrons, which are captured in the Nickel wires causing the immediate emissionof γ-rays in the 6∼9 MeV region.
The Nickel source is lowered into the tank through one of the calibration portholes. Thisis a low intensity source leading to 50∼80 PMTs hit in the whole tank, so only one photon isreaching each PMT. This gives the single photoelectron charge distributions, to calibrate theconversion factor. A typical charge distribution measured by this calibration is also shown in
Figure 5.2 (right). The sharp peak just above 0 pC is due to emitted photoelectrons that missed
the first dynode and therefore receive less amplification. Whereas, the peak at around ∼2 pC is
the single photoelectron signal. The mean value of 2.055 pC is used as the conversion constantfrom the PMT charge to the number of photoelectrons.

5.1.3

Relative Timing Calibration

The timing of each individual PMT depends on the length of the signal cable from each PMT
and also on the charge detected, due to the timewalk effect of larger signals exceeding thethreshold earlier than weaker signals. Particularly for accurate vertex position reconstruction,
the systematic differences of each PMT timing should be minimised. The goal of the Relative
103

mean

2.055pC

5

10

3

2

Ni wire+water

200mm
Cf

1

0

0

pC

190mm

Figure 5.2: Absolute Gain Calibration with the Ni source is shown on the left. The neutrons
emitted by 252 Cf are captured in the Nickel wires, leading to the emission of γ-rays. The intensity
is low such that the single photoelectron signal can be observed as shown on the right. Here,2.055 pC is taken as the conversion factor from pC→p.e.
Timing Calibration is to measure the timing of each PMT as a function of charge (p.e.), and use
this data to correct the differences.
This calibration is performed with a diffuser ball inserted into the tank as shown in Figure 5.3 (left). Firstly a 337 nm wavelength N2 laser emits a pulse with a time width of 3 nsec.
The light is then shifted to 384 nm by a dye laser module, which is the wavelength the PMTs aremost sensitive to. The laser light is split into two optical fibres, one is used for monitoring andtriggering the Relative Timing Calibration event, and the other is routed into the tank.
This fibre is routed into the diffuser ball, where the light is first diffused by a TiO2 diffuser
tip at the centre, and then further diffused by the encompassing LUXDOX silica gel made of
20 nm glass fragments. The laser light causes the diffuser ball to light up more-or-less uniformly
in all directions.
This measurement is repeated for a range of light intensities using an adjustable attenuation
filter, corresponding to a charge measurement from 1 p.e. to a few hundred p.e. A scatter plot of
the timing against charge, referred to as the TQ-map, for each PMT is made, like the one shownin Figure 5.3 (right). The TQ-map is then used to correct the timing of the ID PMTs.

5.1.4

Water Transparency from Cosmic-Ray Muons

A water transparency measurement is made by using abundant downward going cosmic-ray
muons as the calibration source. The energy deposit is approximately constant at around∼2 MeV/cm and a stable source of Cherenkov photons. The selected sample is of verticallydownward going muons that traverse the full length of the tank. The muon track is reconstructed from the entrance and exit points in the ID. The Cherenkov light is assumed to not
104

980
variable
attenuation
filteroptical fiber
optical fiber

Linear Scale

Log Scale960
950

monitor
PMT
sig.

20’PMT
diffuserball

940

T (nsec)

Super Kamiokande
inner tank970

N 2 laser
generator
λ=384nm930
920

trigger
elec.
sig.

910
900

sig.

LUDOX

diffuser tip(TiO2 )20’PMT

sig.

DAQ
elec.

890
880
1

5

10

100

Q (p.e.)Figure 5.3: Relative Timing Calibration with the diffuser ball. The left figure shows the setup of
the laser and the diffuser ball that disperses the light around the tank. The right figure shows thetiming distribution against charge for a PMT, which is used to adjust the timing information.

undergo scattering, then the detected charge (p.e.) for the i-th PMT is defined as:

Qi = Q0 ·



f (θ)
li
· exp −
li
L

(5.2)

where Q0 is a constant, f (θ) is the PMT acceptance, li is the photon path length to the i-th
PMT and L is the attenuation length. The path length li is calculated from the muon track by
assuming a 42◦ opening angle. An illustration of the measurement is shown in Figure 5.4 (left).
Collecting the charge and length information together for a typical run, log (Q·l/f (θ)) as afunction l is also shown in Figure 5.4 (right), where the attenuation length was estimated to be
95 m.
Cosmic-ray muons are continuously measured as part of the recorded data, so they provide
a non-invasive method of measuring the water transparency continuously over time. The time
variation of the light attenuation length is shown in Figure 5.5. This gives a reliable measurementof the water transparency, which is used to modify the momentum reconstruction in the data.
This measurement is also used to tune the charge scale of the MC. This is done by considering
the intersection of the plot on the right of Figure 5.4 where l → 0. The time variation of thisquantity against elapsed days is shown in Figure 5.6.
The effective observed charge at l ∼ 0 is stable during data taking, as the time variation of
the attenuation length is neligible as l → 0. The number of Cherenkov photons generated by
the passage of charged particles and the acceptance of the PMT is adjusted in the MC to matchthe charge intersection quantity between MC and data. The adjustment to the PMT acceptanceis independent of the photon incident angle. The charge scale is adjusted to within 0.4% of the
data.
105

log(Ql/f( θ))500

RUN 3106

P1
P2

5.987
-0.9487E-04

1000

2000

3000

0.4609E-03
0.4444E-06

400
300
200
0

4000

5000

atteunuation length (m)Figure 5.4: Water Transparency from cosmic-ray muons. The left figure shows the muon passing
vertically downward through the tank as a stable source of Cherenkov photons. The right plotshows the quantity used to determing the water transparency.

140
120
100
80
60
40

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

20
0

0

1000

2000
3000
elapse days from 1996/Jan./1

Figure 5.5: Time variation of the water transparency measured from the continuous source ofcosmic-ray muons. The attenuation length recorded is used to adjust the momentum calculationof the data. The attenuation length is shown as a running average over around 1 week of data
points.

106

Figure 5.6: Time variation of the water transparency measured from the continuous sourceof cosmic-ray muons. The attenuation length
recorded is used to adjust the momentum calculation of the data. The attenuation length isshown as a running average over around 1 weekof data points.

5.2

Water Parameter Tuning

In the Monte Carlo simulation, it is necessary to simulate the processes Cherenkov light undergo
during their propagation (see also Section 4.4.3). The photons may be absorbed or scattered,each of which has a wavelength dependence. The goal of this calibration is to obtain wavelength
dependent parameters that characterise the probability of the Cherenkov photons interacting by
these processes in water. This is often referred to as the water parameter calibration.
The light attenuation length in water is due to the absorption of photons and scattering,where the scattering can take a forward-backward symmetric or asymmetric form. The lightattenuation length can therefore be described as:
L(λ) =

1αabs (λ) + αsym (λ) + αasy (λ)(5.3)

where αabs (λ), αsym (λ), and αasy (λ) are the wavelength (λ) dependent attenuation coefficients forabsorption, symmetric scattering and asymmetric scattering, respectively. The light attenuation
length is also measured by cosmic-ray muon data as described in Section 5.1.4. However, thepower of this calibration is that it can give an indication of how much each process contributesto L(λ), and its wavelength dependence.
The water parameter calibration, therefore, makes use of lasers of several wavelengths. The
setup consists of a 337 nm N2 laser, and system of laser dye modules to create the wavelengthsof 371 nm, 400 nm and 420 nm. The configuration is shown in Figure 5.7.
The laser output is then divided into four optical fibres. Two for monitoring, one to trigger thewater parameter calibration event and the fourth is routed to a light injector that is permanently
fixed to the Inner Detector wall. The light injector gives the laser input an opening angle thatcreates a spot of direct light about ∼10 m in diameter at the bottom of the tank, illuminating
more PMTs. It is preferable to light up more PMTs, as this gives a more accurate measurementof the laser light intensity. Figure 5.7 also shows an event display in which the laser spot on thebottom of the tank can be seen.
For the analysis of the laser data, the tank is divided into six regions: the top and five barrel
regions as shown in Figure 5.8. Each time distribution is made up of the PMT hit time with107

LASER
Optical Fiber

Top

337nm
371nm
400nm
420nm
NUM
2
RUN
8399
EVENT
3919
DATE **-Mar- 1
TIME
5:34:38
TOT PE:
820.0
MAX PE:
19.6
NMHIT :
331
ANT-PE:
13.6
ANT-MX:
2.7
NMHITA:
17

Scattering

RunMODE:NORMALTRG ID :00010111
T diff.: 2.08
us
:0.208E-02ms
FSCC:
2FF90
TDC0: 8899.2
Q thr. :0.0
BAD ch.: masked
SUB EV : 0/ 0

BottomFigure 5.7: Water parameter calibration by laser setup. Multiple wavelengths were preparedsuch that the wavelength dependence of photon scattering and absorption could be studied. The
event display shows the output from the laser firing through the light injector installed at the
top of the tank. A ∼10 m spot can be seen at the bottom of the tank, the charge inside is used
to normalise the analysis distributions. The hits elsewhere are due to scattering, and reflections
for later timing.
the time-of-flight from the centre of the bottom spot subtracted. This causes the distributions
to form a peak on the right due to the reflected light. The distributions are also normalised by
the charge detected in the bottom spot to account for variations in intensity of the laser source.
The first bump in the distribution is due to scattered light. The general shape gives information
about the scattering, while the falling slope is influenced by the absorption of photons. The
division of the barrel into the different regions also gives some information about the angular
distribution of the scattering. The distributions are made for both data and MC.
The simulation uses the SKDETSIM software. The light injector is simulated by generating avirtual particle called a geantino at the position of the light injector. The geantino immediately
bursts into a large number of photons, with a specified opening angle pointing into the tank. A
separate MC is generated for each laser wavelength, and the absorption, symmetric scatteringand asymmetric scattering probabilities are varied. By finding the MC with the set of waterparameters that minimises the χ2 with the laser data, the value of each attenuation coefficient
can be found for one wavelength. This measurement is repeated for each of the other wavelengths,
producing the attenuation coefficient, L−1 , plot shown in Figure 5.9.

5.2.1

Rayleigh and Mie Scattering

The primary process by which photons scatter is by Rayleigh Scattering on the local changes indensity in the water, which has a forward-backward symmetric angular distribution. When thephotons encounter a small dust-like particle in the water comparable in size to the wavelength,108

Figure 5.8: The timing distributions used forthe water parameter analysis. Here the detector has been divided into the top and five barrel
regions. The time-of-flight from the laser spot on
the bottom is subtracted. This causes the reflection peak on the right of the distributions. The
charge in the spot is used to normalise the distributions to account for differences in the laser intensity. The first bump gives information about
the scattering and absorption of photons when
the MC is compared with data. Only the firstbump region is used in the χ2 analysis.

attenuation coefficient(1/m)

Light scattering measurement

10

10

-1

-2
absorption

Rayliegh scattering

10-3
Mie scattering

300

400

500

600

700

wavelength(nm)Figure 5.9: Attenuation Coefficient L−1 from the water parameter calibration. By comparing
the MC and data and minimising the χ2 , absorption and scattering information is extracted foreach wavelength. The attenuation coefficients are summed as shown in Equation (5.3).

109then Mie Scattering has an increased probability of occurring. The angular distribution of Mie
scattering varies from a forward-backward symmetric shape to a considerably forward form, depending on the distribution of small dust-like particles in the detector. Since it is difficult toknow the exact size distribution of small particles, there is not a formal treatment of Mie scattering. Instead, the scattering is approximated by a symmetric contribution and an asymmetricforward-scattering component. In this way, the calibration can measure the primary symmetriccontribution, while allowing for forward-scattering.

5.3

Energy Calibration

The momentum of detected particles is reconstructed from the charge information collected by
the PMTs. It is essential to understand the absolute energy scale as accurately as possible, and to
determine the systematic uncertainty of the energy scale for the atmospheric neutrino analyses.
To study this, four sources are used, listed from lowest to highest energy:• Momentum of decay electrons (∼50 MeV/c)• The invariant mass of π 0 events produced by neutrinos (∼140 MeV/c)• Cherenkov angle of low energy stopping muons (200∼500 MeV/c)• Track range of high energy stopping muons (1∼10 GeV/c)
The data from the above sources is compared with the MC prediction to estimate the accuracyof the absolute energy scale.

5.3.1

Decay Electrons

There are many electons detected by Super-Kamiokande from the decay of stopping cosmic-ray
muons. The decay electrons have a spectrum of energies below ∼53 MeV, known as the Michel
spectrum. This energy spectrum is compared between the observed data and MC in order tocheck the energy scale.
The decay electrons are selected by requiring the following criteria:
1. The time interval from a stopping muon event is 1.5 µsec to 8.0 µsec.
2. The number of hit PMTs in a 50 nsec time window is greater than 60 (30 for SK-II).
3. The goodness of the vertex fit is greater than 0.5 .
4. The vertex position is reconstructed more than 2 m away from the ID wall.
The first criterion selects the timing range where the decay electron search is reliable, while thesecond criterion rejects ∼6 MeV γ-rays from µ− capture on a nucleon.
The momentum spectra of the decay electrons is shown in Figure 5.10. It can be seen that the
tail of the spectra extends up to around ∼70 MeV/c, which is a discrepancy from the expectedMichel spectrum. This is due to µ− capture on the K-Shell (1 shell) of an Oxygen atom, resulting
in a decay electron that is influenced by the electric potential of the oxygen nuclei and the µ−orbital motion [249]. The simulation considers this effect, making use of the measured µ+ /µ−
ratio of 1.37 [250].
110

Events

Events

225
200

500

400

175
150

300

125
100

200
75
50

100

25
0

0

10

20

30

40

50
60
70
80
MOMENTUM (MeV/c)

0

0

10

20

30

40

50
60
70
80
MOMENTUM (MeV/c)

Figure 5.10: Momentum spectra for the decay electrons. The michel spectra is expected to havea momentum distribution up to ∼53 MeV/c, whereas this distribution extends towards ∼70 MeV.
This is due to effects from µ− decaying in orbit of an Oxygen nucleus. The effect is consideredin the MC. The left (right) plot shows the distribution for SK-I (SK-II).
The vertex fitter used is one that was developed for low energy neutrino observations foranalyses of solar and supernova neutrinos, where the maximum goodness is 1. The mean valuesof data agree with the MC prediction within 0.6%, 1.6%, and 0.8% for SK-I, SK-II and SK-III,
respectively.

5.3.2

Neutrino Induced π 0 Events

There are NC neutrino interactions whereby a single π 0 is produced, immediately decaying into
two γ-rays. Checking the combined momentum of the two γ-rays and reconstructing the invariant
mass of the π 0 allows the energy scale to be checked in the few hundred MeV energy region. The
invariant mass of the π 0 can be reconstructed as:q
Mπ0 = 2Pγ1 Pγ2 (1 − cos θ)(5.4)
where Pγ1 and Pγ2 are the momenta of the two γ-rays, and θ is the opening angle between them.
The π 0 events are selected from the atmospheric neutrino data by the following criteria:
1. Two Cherenkov rings are reconstructed and both of them are identified as electron-like.
2. No electrons from muon decay were detected.
3. The vertex position is reconstructed more than 2 m away from the ID wall.
The second criterion rejects events where charged pions were also produced with the π 0 (e.g.
π + π 0 ), or charge current events (i.e. µ± π 0 ). A typical π 0 event, where the two γ-ray rings can
be clearly seen is shown in Figure 5.11. The reconstructed momentum of the selected events
111

NUM
509
RUN
5346
SUBRUN
26
EVENT
1228025
DATE 97-Dec-10
TIME
14:55: 9
TOT PE:
2142.2
MAX PE:
15.4
NMHIT : 1115
ANT-PE:
12.5
ANT-MX:
1.9
NMHITA:17

90/00/00:NoYet:NoYet
90/00/00:NoYet:NoYet
90/00/00:NoYet:NoYet
90/00/00:NoYet:NoYet
90/00/00:NoYet:NoYet
90/00/00:NoYet:NoYet
90/00/00:;R= 0:NoYet
R :Z : PHI : GOOD
0.00: 0.00: 0.00:0.000
CANG : RTOT : AMOM : MS

RunMODE:NORMALTRG ID :00000111
T diff.:0.128E+06us
FEVSK :81002003
nOD YK/LW: 0/ 0
SUB EV : 0/ 0
Dec-e: 0( 0/ 0/ 0)*NO KEKGPS RED*

Comnt;

Figure 5.11: A typical π 0 event display clearly showing two rings from the two γ-ray products.
is shown in Figure 5.12. There is a clear peak around ∼140 MeV/c2 , confirming that the they
originated from the decay of a π 0 .
The actual π 0 mass is ∼135 MeV/c2 , whereas the mean invariant mass reconstructed is∼139 MeV/c2 . This shift to a slightly larger mass can be explained by mainly two reasons.
Firstly, the π 0 is produced with an interaction with an Oxygen nucleus, which is left in an excited state. There are de-excitiation γ-rays that are also included in the interaction that add
a few additional MeV to the event. The second is that the γ-rays from the decay of the π 0
propagate for a short distance before breaking into an electromagnetic shower. This results inthe vertex being reconstructed slightly in the forward direction, and the opening angle between
the γs being reconstructed slightly larger, giving a larger π 0 invariant mass.
The MC considers the de-excitation of the oxygen nucleus, so there is a good agreement
between the data and MC. The peak position of the data is higher than that of the MC by 0.6%,
1.6% and 0.8% for SK-I, SK-II and SK-III, respectively.

5.3.3

Low Energy Stopping Muons

There is a close relationship between the Cherenkov opening angle and momentum, if the momentum is not many times larger than the mass of the particle. So using low energy stoppingmuons (< 400 MeV/c), the momentum can be estimated from the reconstructed opening angle.
By comparing the reconstructed momentum from the observed charge (Pp.e. ) to the momentum
estimation from the Cherenkov opening angle (Pθ ) for data and MC, the systematic uncertainty
in the momentum reconstruction can be estimated. The Cherenkov opening angle expressed as
112

160

70

140

60

120

50
100

40
80

30

60

20

40

10

20
0

0

50

100

150

200

250

0

300

0

50

100

150

200

250

300

Mγγ (MeV/c)Mγγ (MeV/c)

Figure 5.12: Reconstructed invariant π 0 mass from the two detected γ-rays. The π 0 mass is∼135 MeV/c2 , whereas ∼139 MeV is observed. This is due to the additional detection of deexcitation γ-rays from the Oxygen, and the slightly forward reconstructed vertex. The left
(right) plot shows the distribution for SK-I (SK-II).
a function of the momentum is:11
cos θc =
=
nβ
n

s
1+

m2
p2

(5.5)

where θc is the Cherenkov opening angle, n is the index of refraction for water, β = vc , whilem and p are the mass and momentum of the ring. Note that electrons and high energy muons
are too relativistic, and the Cherenkov opening angle converges to a limit as β → 1. Since
their momentum is large compared to their mass. In such cases, the momentum cannot be
estimated accurately from the Cherenkov angle. The low energy stopping muons are selected by
the following criteria:
1. The total charge in the ID is less than 1500 p.e. (< 750 for SK-II)
2. One cluster of hit PMTs in the outer detector.
3. The entrance point is on the top wall.
4. The direction is downward (cos θ > 0.9).
5. One decay electron event is detected.
The first criterion selects low energy muons. The second to fourth criteria selects very vertical,
downward going stopping muons. The fifth criterion improves the purity for muon events.
Figure 5.13 shows the momentum and Cherenkov opening angle distributions for the selected
events. In Figure 5.14 the momentum estimated from the charge is compared with the estimation
113

600

Number of events

Number of events

300
250
200
150
100

400
300
200
100

50
0

500

100

150

200

250

300

350

400

450

0

500

30

32

34

36

38

40

42

44

Opening angle (deg.)

Reconst. momentum (MeV/c)

Figure 5.13: Low energy stopping muon reconstructed momentum and Cherenkov angle (SK-II).
from the Cherenkov opening angle, Pp.e. /Pθ , on the left. The ratio is compared between data
and MC in the right plots, as a function of the expected momentum Pθ . The agreement is within
0.7%, 1.3%, and 2.1% for SK-I, SK-II and SK-III, respectively.

5.3.4High Energy Stopping Muons

The track length of high energy stopping muons is approximately proportional to the muon
momentum. So it can be used to make an independent check of the momentum reconstruction(by observed charge) in the 1 GeV/c to 10 GeV/c momentum range. The following criteria areused to select high energy stopping muons:
1. The entrance point is on the top wall.
2. The direction is downward (cos θ > 0.94).
3. One decay electron event is detected.
4. The reconstructed range of the muon track is greater than 7 m.
The first three criteria select very vertical downward going muons. The fourth criterion selects
high energy events. The range of the muon is determined from the distance between the entrance
point of the stopping muon and the vertex position of the following decay electron. Figure 5.15
shows the mean ratio of momentum/range against the range (left), and the MC/Data comparison
for this ratio (right). A clear momentum loss of ∼2.3 MeV/c per cm can be seen. There is aslight momentum dependence observed, however the data and MC are in agreement within 0.7%,
1.1%, and 2.0% for SK-I, SK-II and SK-III, respectively.

5.3.5Time Variation of Energy Scale

The stability of the energy scale is monitored by the time variation of the decay electron andstopping muon measurements. The mean momentum of decay electrons is watched over time asshown in the top panels of Figure 5.16, with the bottom panels showing the difference between
114

MC/DATA

P(p.e.)/P(θ)

1.4
1.3
1.2

1.05
1.04
1.03
1.02

1.1

1.01

1

1

0.9

0.99
0.98

0.8

0.97

0.7
0.6

0.96
0.95

200 225 250 275 300 325 350 375 400 425 450

expected momentum (MeV/c)

expected momentum (MeV/c)
1.2

MC/DATAP(p.e.)/P(θ)

200 225 250 275 300 325 350 375 400 425 450

1.15
1.1

1.05
1.04
1.03
1.02
1.01
1

1.05

0.99
1

0.98
0.97

0.95

0.96
0.9

0.95

200 225 250 275 300 325 350 375 400 425 450

200 225 250 275 300 325 350 375 400 425 450

expected momentum (MeV/c)

expected momentum (MeV/c)

Figure 5.14: Low energy stopping muon distributions. The ratio of the momentum expectedfrom the charge to the momentum estimated from the Cherenkov angle, Pp.e. /Pθ , is shown on
the left. The MC/Data for the Pp.e. /Pθ ratio is shown on the right. The top (bottom) plotsshow the distributions for SK-I (SK-II).

115

M.C./DATA

Momentum(MeV/c)/Range(cm)

1.05

2.6

1.04

2.5

1.03
1.02

2.4

1.01
1

2.3

0.99
2.2

0.98
0.97

2.1

0.96
0

500

1000

1500

2000

2500

0.95

3000 3500
Range (cm)

0

500

1000

1500

2000

2500

3000 3500
range (cm)0500 1000 1500 2000 2500 3000 3500 4000
range (cm)

1.05

2.6

M.C./DATA

Momentum(MeV/c)/Range(cm)

2

1.04

2.5

1.03
1.02

2.4

1.01
1

2.3

0.99
2.2

0.98
0.97

2.1

0.96
2

0

500

1000

1500

2000

2500

0.95

3000 3500
Range (cm)Figure 5.15: High energy stopping muon distributions of the ratio momentum/range against therange (left), and the MC/Data comparison for this ratio (right). A near constant relationshipcan be seen for momentum and range, ∼2.3 MeV per cm. This ratio slightly increases with range(i.e. momentum), however there is still about a ∼1% agreement between MC and data. The top(bottom) plots show the distributions for SK-I (SK-II).

116

43
42
41
40
39

e momentum/track

Momentum(MeV/c)

44

1.05
1.04
1.03
1.02
1.01
1
0.99
0.98
0.97
0.96
0.95

0

250

500

0

250

500

45
44
43
42
41
40
39

750 1000 1250 1500 1750 2000
elapse dayse momentum/track

Momentum(MeV/c)

45

750 1000 1250 1500 1750 2000
elapse days

1.05
1.04
1.03
1.02
1.01
1
0.99
0.98
0.97
0.96
0.95

0

200

400

600

800
1000
elapse days

0

200

400

600

800
1000
elapse days

Figure 5.16: Time variation of the decay electron momentum average (top) and the difference
between data and MC (bottom). The dotted lines show a 1% variation, and the dashed a 2%
variation. The left (right) plots show the time variation in elapsed days since the beginning of
SK-I (SK-II).
data and MC. For the stability of the energy scale at high energies, the average momentum/rangeof stopping muons is monitored as in the top panels of Figure 5.17, while the lower panels show
the difference between data and MC. The RMS of the variation is 0.88%, 0.55%, and 1.79% for
SK-I, SK-II, and SK-III.

5.3.6

Summary of Energy Scale Calibrations

A summary of the absolute energy scale calibration, showing the contribution from each sample,
can be seen in Figure 5.18. A momentum range from a few MeV to about ∼10 GeV is checkedwhich covers the energy range of FC events and low energy PC events well. A good agreement
can be seen between the data and MC for the full momentum range, and the variation in theenergy scale is estimated to be within 0.74%, 1.60%, and 2.08% for SK-I, SK-II and SK-III,
respectively. This absolute variation of the energy scale is combined with the expected variationof the energy scale over time to give an estimation of the systematic error. The systematic
uncertainty for the energy scale is estimated to be 1.1%, 1.7% and 2.7% for SK-I, SK-II, and
SK-III, respectively.

5.3.7

Uniformity of Energy Scale

The uniformity of the detector is measered using the decay electrons from the cosmic ray muons.
They are good calibration sources to check the detector uniformity, because the vertex is distributed uniformly in the fiducial volume and the momentum distribution is almost uniform inall directions. To take into account the muon polarization, only electrons whose direction isperpendicular to the parent muon direction are used. This condition is -0.25< cos Θe↔µ < 0.25,
where cos Θe↔µ is the opening angle between the electron and muon directions. Figure ?? shows
117

1.05
1.04
1.03
1.02
1.01
1
0.99
0.98
0.97
0.96
0.95

Momentum(MeV/c)
0

250

500

0

250

500

2.5
2.45
2.4
2.35
2.3
2.25
2.2

750 1000 1250 1500 1750 2000
elapse days
µ momentum/track

Momentum(MeV/c)µ momentum/track

2.6
2.55
2.5
2.45
2.4
2.35
2.3
2.25
2.2

750 1000 1250 1500 1750 2000
elapse days

1.05
1.04
1.03
1.02
1.01
1
0.99
0.98
0.97
0.96
0.95

0

200

400

600

800
1000
elapse days

0

200

400

600

800
1000
elapse days

Figure 5.17: Time variation of the stopping muon momentum/range average (top) and thedifference between data and MC (bottom). The dotted lines show a 1% variation, and thedashed a 2% variation. The left (right) plots show the time variation in elapsed days since thebeginning of SK-I (SK-II).
the averaged momentum of decay electrons for the Monte Carlo events normalized by that fordata as a function of the zenith angle of the electrons. From this figure, the detector gain is
uniform within ±0.6 % for SK-I and SK-II.

118

SK-I
(MC - DATA) / DATA (%)

8
6
4
2
0
-2

µ rangeµ angle0π massdecay e

-4
-6
-8
10

2

3

10
10
10
momentum range (MeV/c)
SK-II

4

10

5

(MC - DATA) / DATA (%)

8
6
4
2
0
-2

µ rangeµ angle0π massdecay e

-4
-6
-8
10

2

3

10
10
10
momentum range (MeV/c)

4

10

5

Figure 5.18: Summary of energy scale calibrations. The percentage differences between data andMC are shown for: mean momentum from decay electrons; the peak position of the π 0 invariant
mass;

119

Chapter 6

Data Reduction
6.1

Overview

Super-Kamiokande collects around 106 events per day from it’s High Energy and Low Energy
triggers. However, only around 10 events per day are expected in the atmospheric neutrino data.
Most of the events recorded by the DAQ are expected to be background events originating fromcosmic-rays, environmental radiation and their by-products. Therefore an effective procedure isrequired to select the neutrino events of interest from all the background events.
There are four classifications for the atmospheric neutrino events observed: Fully-Contained(FC ), Partially-Contained (PC ), Upward Stopping Muons, and Upward Through-Going Muons(the last two collectively referred to as UPMU ). The events are classified into these categories
depending on the location of the original neutrino interaction vertex and the containment of theinteraction products, as summerised in Figure 6.1.
The FC and PC events require that the neutrino interaction vertices are inside the Fiducial
Volume of the Inner Detector (ID). The Fiducial Volume (FV) is typically defined to be 2 mfrom the ID wall. In the L/E analysis, a slightly larger region is used to increase the statistics.
However, the 2 m FV is assumed for the figures in this chapter. The FV is selected such that theevents can be reconstructed reliably. When all the interaction products are contained within theID, then the event is classified as FC. If one of the particles exits the ID and deposits energy inFully contained

Partially contained

Upward stoppingmuon

Upward through-going
muon

Figure 6.1: Event categorisation of the atmospheric data into FC, PC and the two UPMU
selections. The dashed lines represent incoming neutrinos, the solid lines interaction products,
and the cylinders the Inner Detector volume.
120

Figure 6.2: The expected energy distributions for each of the event classifications. The upper
figure shows the FC and PC events, and UPMU events are shown in the lower figure.
the Outer Detector (OD), then the event is classified as PC.
There is an UPMU sample to study high energy neutrinos. When energetic neutrinos interact
in the rock beneath the Super-Kamiokande detector, sometimes the muons will enter the ID
volume. Unlike the downward-going muons which consist mostly of cosmic-ray products, thesemuons are likely to have been induced by neutrino interactions. If these neutrino-induced muonsstop in the ID then the event is classified as Upward Stopping Muon; and Upward Through-Going
Muon if the muon passes through the ID.
The mean energies of these event categories are ∼1 GeV for FC, ∼10 GeV for PC and Stopping
UPMU, and ∼100 GeV for Through-Going UPMU, which can be seen in Figure 6.2. The data
reduction procedures are different for each of these categories and are described in this chapter.

6.2

Reduction for Fully-Contained Sample

The FC events are separated from PC events by the activity in the OD. FC events require thatthe number of PMT hits in the highest charge cluster is less than 16 (10 for SKI):• NHITAC < 16 (10 for SKI).
There are 5 reduction steps described here. Note that in the SK-II period, the PMT coveragein the ID was around half of that in the other SK periods, so the numbers for SKII are quoted
separately.

6.2.1

First Reduction

The First Reduction removes a lot of the basic background events and has the following criteria:
121

6

10

5

10

4

10

3

10

2

rejected(a) data

Number of events

Number of events

10

10
1
10

3

10

2

(b) final sample
data
MC

10

6

10

5

10

4

10

3

10

2

rejected(a) data

10
1
10

3

10

2

(b) final sample
data
MC

10

10

1

1
0

50

100

150

200

250

300

350

400

450

500

0

50

100

150

200

250

300

350

400

450

500

PE300 (p.e.)PE300 (p.e.)

Figure 6.3: FC First Reduction: PE300 distributions for (a) raw data and (b) the final sample
data and non-oscillated MC normalised to the data. Distributions are for SK-I (left) and SK-II(right). The PE300 cut criteria arrows mark the rejected events.
(1)

PE300 ≥ 200 p.e.s (> 100 p.e.s for SK-II).
PE300 is the maximum number of total p.e.s observed by the ID PMTs in a
sliding 300 nsec time window.

(2)

NHITA800 ≤ 50 or OD trigger is off.
NHITA800 is the number of hit OD PMTs in a fixed 800 nsec time window from−400 nsec to +400 nsec before and after the trigger timing.

(3)

TDIFF > 100 µsec
TDIFF is a time interval to the previous event.

Criterion (1) removes the low energy background events from environmental or induced radioisotopes. Figure 6.3 shows the PE300 distributions for the raw data and the FC final events. The
200 p.e.s (100 p.e.s for SK-II) corresponds to 22 MeV/c of electron momentum. Note that in theanalyses events with visible energy below 30 MeV are not used. Criterion (2) rejects the cosmic
ray muon events. Figure 6.4 shows the NHITA800 distributions for the raw data, FC atmospheric
neutrino Monte Carlo events and the FC final events. Criterion (3) discards events that could
be contaminated by decay electrons from cosmic ray muons that stopped in the ID prior to theevent. The events within 30 µsec after the selected events by the above criteria are selected as
sub-events to keep any decay electrons. These cuts reduce the data size from 106 events/day to∼3000 (∼2200) events/day (for SK-II).

6.2.2

Second Reduction

In the Second Reduction low energy events and cosmic ray muons are rejected by the following
critera:(1)

NHITA800 ≤ 25 if PEtot < 100,000 p.e.s (50,000 for SK-II) or OD trigger is off.

(2)

PEmax /PE300 < 0.5.
PEmax is the maximum number of p.e.s observed by an ID PMT.
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dataMC(a) data

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NHITA800

Figure 6.4: FC Reduction: NHITA800 distributions for (a) raw data, (b) MC with vertex > 2 mfrom the wall, (c) final sample data and MC normalised to the data. Distributions are for SK-I
(left) and SK-II (right). The NHITA800 cut criteria dotted (solid) arrows mark the rejected
events for FC First (Second) Reduction.

Criterion (1) removes more cosmic ray muons by a tighter threshold than the First Reduction.
Figure 6.4 also shows the stricter cut on NHITA800 by the solid lines. Criterion (2) rejects low
energy and electrical noise events, where most of the recorded charge is from a single PMT.
Figure 6.5 shows the PEmax /PE300 distributions for the data after First Reduction, the FC
atmospheric neutrino MC events and the FC final events. This cut also discards some of theFlasher events. Sometimes a PMT spontaneously starts flashing due to discharges in the dynode
structure, recording a very large charge. Such a background is called a Flasher event. Further
Flasher events are reduced at a later stage. After the FC Second Reduction the event rate is∼200 (∼280) events/day (for SK-II).

6.2.3

Third Reduction

After the First and Second Reduction, the remaining background events are mostly electronicnoise events and cosmic ray muons which have a small number of OD hits.

Through-Going Muon Cut
There are still some through-going muon events after the basic cuts in the previous reduction
steps, which have fewer hits in the OD. To eliminate these events, a through-going muon fitter isapplied if the charge in a single PMT is greater than 230 p.e.s. This fitter reconstructs a track by
selecting an entrance point and an exit point. The entrance point is selected from the earliest hit
PMT together with some neighbouring PMTs and the exit point from the centre of the saturated
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1

PEmax/PE300

Figure 6.5: FC Second Reduction: PEmax /PE300 distributions for (a) data after FC1, (b) FC
MC, (c) final sample data and MC normalised to the data. Distributions are for SK-I (left) and
SK-II (right). The PEmax /PE300 cut criteria arrows mark the rejected events.
ID PMTs. The goodness of the fit is defined as:


X 1
1
(ti − Ti )2
Goodness = X
×
exp −
1
2(1.5 × σi )2
σi2
i
2
σi
i

(6.1)

where ti and σi are the observed hit time of the i-th PMT and its resolution, and Ti is the hit
time expected from the entering time of muon and its track.
There is an additional requirment for the SK-II period, that the number of hit ID PMTs
exceeds 1000. The rejection criteria for through-going muons are:(1)

PEmax > 230 p.e.s (and NHIT > 1000 for SK-II)(2)

Goodness of the Through-Going Muon Fit > 0.75

(3)

NHITAin ≥ 10
or
NHITAout ≥ 10
NHITAin (NHITAout ) is the number of hit OD PMTs located within 8 m fromthe entrance (exit) point in a fixed 800 nsec time window.

Figures 6.6 and 6.7 show the number of hit OD PMTs near the entrance and the exit points
for the data after Second Reduction, the FC atmospheric neutrino MC events and the FC final
events. satisfying the above criteria (1) and (2).
Stopping Muon Cut
To remove stopping muons a Stopping Muon Fitter is applied, which finds the entrance point ina similar way to the through-going case. Then the rejection criteria for stopping muons are:
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Figure 6.6: FC Third Reduction: through-going muon, SK1 NHITAin and NHITAout distributions for (a) data after FC2, (b) FC MC, (c) final sample data and MC normalised to the data.
The NHITAin and NHITAout cut criteria arrows mark the rejected events.

(b-in) MC
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Figure 6.7: FC Third Reduction: through-going muon, SK2 NHITAin and NHITAout distributions for (a) data after FC2, (b) FC MC, (c) final sample data and MC normalised to the data.
The NHITAin and NHITAout cut criteria arrows mark the rejected events.

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Number of hit OD PMTs

Figure 6.8: FC Third Reduction: stopping muon, NHITAin distributions for (a) data after
FC2, (b) FC MC, (c) final sample data and MC normalised to the data. Events shown havegoodness > 0.5. Distributions are for SK-I (left) and SK-II (right). The NHITAin cut criteria
arrows mark the rejected events.
(1)

NHITAin ≥ 10
or
NHITAin ≥ 5
if Goodness of Stopping Muon Fit > 0.5
NHITAin is the number of hit OD PMTs located within 8 m from the entrance
point in a fixed 800 nsec time window.

The direction of the muon is reconstructed to maximize the total number of p.e.s inside a conewith a half opening angle of 42◦ . The goodness definition is same as that of through-going muon
fit given in Equation (6.1). Figure 6.8 shows NHITAin distributions for the data after Second
Reduction, the FC atmospheric neutrino MC events and the FC final events. Events shown havegoodness > 0.5.
Cable Hole Muons
On top of the detector tank, there are twelve cable holes to route in and out the data and HV
supply cables. The passage of four out of the twelve cable holes through the OD form an inactive
region, which could allow cosmic-ray muons to pass into the ID without an OD signal to remove
them. To account for this scenario, a set of veto counters (2 m × 2.5 m plastic scintillation
counters) were installed in April 1997, as depicted in Figure 6.9. The rejection criteria for thecable hole muons are:(1)

One veto counter hit(2)

Lveto < 4 m
Lveto is the distance from the cable hole to the the reconstructed vertex.
126

Figure 6.10 shows the reconstructed vertex distributions for the FC 1-ring µ-like events before
and after the installation of the veto counters. The cable hole muons are rejected well by theveto counters. Criteria (2) is there to save events that may have been rejected by electronic noisein the veto counters.
Cosmic ray µ

Veto Counter
Cable Bundle
from PMT
To
Electronics
Hut

Outer
Detector

Inner
Detector

Figure 6.9: Illustration of the cable hole muon veto system, which shows how cosmic-ray muons
passing through an inactive region of the OD can still be rejected.

Flasher Event Cut
Flasher events usually have a broad hit timing distribution compared with that of the neutrino
events. The flasher events which exhibit this feature are removed by the following cut. The cut
criteria are different for SK-I and are as follows:(1)

NMIN100 ≥ 14
or
NMIN100 ≥ 10
if the number of hit ID PMTs < 800
NMIN100 is the minimum number of hit ID PMTs in a sliding 100 nsec time
window from +300 nsec to +800 nsec after the trigger.

From SK-II onwards, the cut criteria are:(1)

NMIN100 ≥ 20

Figure 6.11 shows the timing distribution and the time window searched for (i) a typical flashing
PMT event and (ii) a typical FC neutrino event. NMIN100 is checked within the region enclosedby the arrows. Figure 6.12 shows a typical event display of a flasher event. An almost identicalevent can be seen ∼20 mins later. Figures 6.13 and 6.14 show the NMIN100 distributions for thedata after Second Reduction, the FC atmospheric neutrino MC events and the FC final events.
127

2000

2000

(i)

1000

1000

500

500
Y(cm)

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-500

-1000

-1000

-1500

-1500

-2000
-2000 -1500 -1000 -500

(ii)

1500

Y(cm)

1500

0
500 1000 1500 2000
X(cm)

-2000
-2000 -1500 -1000 -5000500 1000 1500 2000
X(cm)Figure 6.10: Reconstructed vertex distributions for the FC 1-ring µ-like events (i) before theinstallation of the veto counters and (ii) after the installation. The small dashed circles indicate
the positions of the veto counters. Clusters of muon events can clearly be seen before the vetocounters were installed, and no excess is seen after the installation. The outer solid circle shows
the ID wall and the inner solid circle shows the fiducial volume (2 m from the wall).

30

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(ii)

150
100
50
0
800

1000 1200 1400 1600 1800
Time of ID PMT hits (nsec)

Figure 6.11: A typical timing distribtion for (i) a typical flashing PMT event and (ii) a typicalFC neutrino event. The arrows enclose the timing window in which NMIN100 is checked. A clear
difference in the charge distribution can be seen in this window between (i) and (ii).

128

Figure 6.12: Event display of a typical flasher PMT discharging. The time interval between these
two events is around ∼20 mins.

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Figure 6.13: FC Third Reduction: flasher SK1 NMIN100 distributions for (a) data after FC2, (b)FC MC, (c) final sample data and MC normalised to the data. Distributions are for number of
hit ID PMTs > 800 (left) and < 800 (right). The NMIN100 cut criteria arrows mark the rejected
events.

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Figure 6.14: FC Third Reduction: flasher SK2 NMIN100 distributions for (a) data after FC2,(b) FC MC, (c) final sample data and MC normalised to the data. The NMIN100 cut criteria
arrows mark the rejected events.
Accidental Coincidence Events Cut
Sometimes there is an accidental coincidence of a low energy event which opens the trigger
window with low activity in the OD, and a following cosmic-ray muon event that deposits a lotof charge in the ID. The OD hits in the trigger window from the low energy event pass the earlier
NHITA800 cuts, and the cosmic-ray muon in the same event window passes the PE300 cut. These
accidental coincidence events are removed by the following cuts:(1)

NHITAoff ≥ 20
NHITAoff is the number of hit OD PMTs in a fixed 500 nsec off-timing window
from +400 nsec to +900 nsec after the trigger timing.

(2)

PEoff > 5000 p.e.s (2500 for SK-II)
PEoff is the number of p.e.s observed by ID PMTs in a fixed 500 nsec off-timing
window from +400 nsec to +900 nsec.

Low Energy Events Cut
The remaining low energy events are from the decay of radio-isotopes and electrical noise. Such
events are removed by the following criteria:(1)

NHIT50 < 50 (25 for SK-II)
NHIT50 is the number of hit ID PMTs in a sliding 50 nsec time window.

where NHIT50 is counted after subtracting the time of flight (TOF) of each observed photon
assuming all photons are generated at a point. The vertex is determined as the position at which130the timing residual distribution is peaked. NHIT50 =50 corresponds to visible energy of 9 MeV
and is low enough to not lose efficiency for contained neutrino events with Evis >30 MeV.
After the FC Third Reduction, the event rate is ∼45 (∼21) events/day (for SK-II).

6.2.4

Fourth Reduction

In the Fourth Reduction, further flasher events are removed by using an intelligent patternmatching algorithm. As shown in Figure 6.12, Flasher events tend to repeat with very similar
hit patterns in the detector over the course of hours and days. Repeated events with a similar hit
pattern are very unlikely to be due to neutrino events. So an algorithm that looks for matching
hit patterns is used to reject repeating Flasher events. The algorithm steps are as follows:(1)

Divide the ID wall into 1450 patches of 2 m×2 m square.

(2)

Compute the correlation factor r by comparing the total charge in each patch of
two events, A and B. The correlation is defined as :


A
B
B
1 X QAi − hQ i × Qi − hQ i
r=
(6.2)
N
σA × σBiwhere N is the number of the patches, and hQA(B) i and σA(B) are the averaged
charge and its standard deviation for event A and B, respectively.
(3)

Calculate the distance (DISTmax )between the PMTs with the maximum pulse
heights in the two compared events(4)

If DISTmax < 75 cm, an offset value is added to r :r=r+0.15.

(5)

If r exceeds the threshold(rth ), events A and B are recognized as matched events.
rth is defined as :Br > rth = 0.168 × log10 ((PEA
tot + PEtot )/2.) + 0.130,(6.3)

where PEtot is the total number of p.e.s observed in the ID.
(6)

Repeat the above calculation over 10,000 events around the target event andcount the number of matched events.

(7)

Remove the events with large correlation factor r, or large number of matched
events.

The cut criteria showing which events are rejected is shown in Figure 6.15 for the data after
FC Third Reduction and the FC atmospheric MC.
The event rate after the FC Fourth Reduction is ∼18 events/day.

6.2.5

Fifth Reduction

After the Fourth Reduction, the number of remaining background events are few and these areremoved by criteria specialised for each background event type.
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Maximum Correlation

Maximum Correlation

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Figure 6.15: FC Fourth Reduction: Estimator r against Nmatch scatter plots for (a) data after
FC3, (b) FC MC. Distributions are for SK-I (left) and SK-II (right). The r estimator–Nmatch
cut region arrows mark the rejected events.
Stopping Muon Cut
The remaining stopping muons are rejected by tighter criteria than those in the Third Reduction
stage. Events satisfying the following criteria are removed:(1)

NHITAin ≥ 5
NHITAin is the number of hit OD PMTs located within 8 m from the entrance
point in a sliding 200 nsec time window from −400 nsec to +400 nsec.

This time the entrance position in the OD is estimated by a backward extrapolation fromthe reconstructed vertex determined by TDC-fit (see Section 7.2.3). Figure 6.16 shows the
distributions for NHITAin for the data after FC Fourth Reduction, the FC MC and the final
samples.
Invisible Muon Cut
Invisible muon events are caused by cosmic-ray muons with momenta below Cherenkov threshold
and the subsequent decay electrons being observed as the triggered event. These events arecharacterised by a low energy signal from a decay electron and a signal in the OD before thetrigger timing. Invisible muons are removed by the following cut criteria:(1)

PEtot < 1000 p.e.s (500 for SK-II)
PEtot is the total number of p.e.s observed in the ID.

(2)

NHITACearly > 4
NHITACearly is the maximum number of hit PMTs in an OD hit cluster in a
sliding 200 nsec time window from −8800 nsec to −100 nsec.

(3)

NHITACearly + NHITAC500 > 9
NHITACearly>9

if DISTclust < 500 cm
otherwise
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NhitainFigure 6.16: FC Fifth Reduction: stopping muon NHITAin distributions for (a) data after FC4,(b) FC MC, (c) final sample data and MC normalised to the data. Distributions are for SK-I
(left) and SK-II (right). The NHITAin cut criteria arrows mark the rejected events.
NHITAC500 is the number of hit PMTs in the OD hit cluster in a fixed 500 nsec
time window from −100 nsec to +400 nsec.
DISTclust is the distance between two OD hit clusters, which are used for the
NHITACearly and the NHITAC500 .
Figure 6.17 shows the NHITACearly distributions for the data after FC Fourth Reduction,
the FC MC and the final sample, where the criteria for DISTclust > 500 cm is displayed.
Accidental Coincidence Muon Cut
The remaining accidental coincidence muon events after the Third Reduction are rejected by thefollowing cut criteria:(1)PE500 < 300 p.e.s (150 for SK-II)
PE500 is the total number of p.e.s observed in the ID in a fixed 500 nsec time
window from −100 nsec to +400 nsec.

(2)

PElate ≥ 20 p.e.s
PElate is the maximum number of hit OD PMTs in a 200 nsec sliding time window
from +400 nsec to +1600 nsec.

Figure 6.18 shows the PElate distributions for the data after FC Fourth Reduction, the FC
MC and final samples.
Long-Tail Flasher Cut
This is a stricter version of Flasher Cut than in the FC Third Reduction step. Events satisfying
the following criteria are discarded as Flasher events:
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Nhitaearly

NHITACearly

Figure 6.17: FC Fifth Reduction: invisible muon NHITACearly distributions for (a) data after
FC4, (b) FC MC, (c) final sample data and MC normalised to the data. Distributions are for
SK-I (left) and SK-II (right). The NHITACearly cut criteria arrows mark the rejected events for
DISTclust > 500 cm.

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rejected

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PElatePElateFigure 6.18: FC Fifth Reduction: accidental coincidence muon PElate distributions for (a) data
after FC4, (b) FC MC, (c) final sample data and MC normalised to the data. Distributions arefor SK-I (left) and SK-II (right). The PElate cut criteria arrows mark the rejected events.

134

(1)

NMIN100 > 5 if the Goodness of Point-Fit < 0.4
NMIN100 is the minimum number of the hit ID PMTs in a sliding 100 nsec time
window from +300 nsec to +800 nsec.

From SK-II onwards, in addition to the criteria above, the following extra cuts are applied:(3)

Goodness of point fit < 0.3(4)

NHITMIN100 < 6

See Section 7.2.1 for an explanation of Point-Fit.
After the FC Fifth Reduction, the FC event rate is ∼16 events/day.

6.2.6

FC Reduction Summary

Finally, the Fully-Contained Fiducial Volume (FCFV) neutrino events are selected by applying
the FCFV event cuts:• Vertex of neutrino interactions should be inside the fiducial volume (2 m from the ID PMT
surface).
• The number of hit PMTs in the highest charge OD cluster (NHITAC) should be less than
16 (10 for SK-I).
• Visible energy (Evis ) should be greater than 30 MeV.
The detection efficiencies in each reduction step are estimated by the atmospheric neutrinoMonte Carlo events as shown in Table 6.1, where FCFV cuts were applied using the true vertex
information. The FCFV row shows the detection efficiencies using the reconstructed vertex, andthe final row gives the daily event rate for each SK period.
Fully-Contained
Reduction Step
1st Reduction
2nd Reduction
3rd Reduction
4th Reduction
5th Reduction
FCFV
Event Rate (/day)

SK-I
100.00100.00
99.93
99.29
99.26
99.25
8.26±0.07

Selection Efficiency (%)
SK-II
SK-III
SK-IV
99.97
100.00
100.0
99.92
99.98
99.99
99.78
99.81
99.82
99.38
99.30
99.00
99.30
99.24
98.95
99.95
99.62
99.19
8.28±0.10 8.41±0.13 8.14±0.09

Table 6.1: The detection efficiency after each FC Reduction step for each of the SK running
periods. The efficiencies are shown after the FCFV cuts were applied using the true vertex
information. In the FCFV row, the reconstructed vertex is used, and the final row shows theevent rate per day.
For the FC events, the main sources of the background are cosmic-ray muons, Flasher eventsand neutrons from the rock around the detector. The Fiducial Volume cut largely reduces the
background from these sources. The estimated contamination of the backgrounds is shown in
Table 6.2.
135

Fully-Contained
BackgroundCosmic-ray µ Sub-GeV
Multi-GeV
Flasher PMT Sub-GeV
Multi-GeV

Background Contamination (%)
SK-I SK-II SK-III SK-IV
0.2
0.2
0.3
0.1
0.3
0.2
0.3
0.1
0.1
0.4
0.1
0.1
0.5
0.2
0.2
0.1

Table 6.2: Estimated upper limit of the contamination of each background in the FC events.
Note that the Cosmic-ray µ are mainly a background for the µ-like sample, while the Flasher
PMTs mainly affect the e-like sample.

6.3

Reduction for Partially-Contained Sample

The PC events are separated from FC events by the activity in the OD. PC events require thatthe number of PMT hits in the highest charge cluster is greater than 15 (9 for SKI):• NHITAC > 15 (9 for SKI).
Since the PC events inherently require that there be OD activity, the steps for reducing backgrounds are different than those for FC Reduction. There are five reduction steps described here.
Note that in the SK-II period, the PMT coverage in the ID was around half of that in the otherSK periods, so the numbers for SKII are quoted separately. The reduction stages were modified
from the SK-III period and are presented separately where different.

6.3.1

First Reduction

The PC First Reduction aims to reject through-going cosmic ray muons and low energy events.
The selection criteria for SK-I and SK-II are as follows:(1)

PEtot ≥ 1000 p.e.s (≥ 500 for SK-II).
PEtot is the number of p.e.s observed in the ID.

(2)

TWIDA ≤ 260 nsec (≤ 170 nsec for SK-II).
TWIDA is the width of the hit timing distribution in the OD PMTs.

(3)

NCLSTA ≤ 1 (only for SK-I).
NCLSTA is the number of the hit clusters in the OD.

The PC sample requires that exiting particles, mostly muons, must have a track length in the
ID of at least 2 m, which corresponds to a momentum loss of 500 MeV/c for muons. The PEtot
cut in Criteria (1) of 1000 p.e.s (500 for SK-II) corresponds to 310 MeV/c for muons. Criteria (2)removes through-going muons. Through-going muon events have a broad hit timing distribution
and two hit clusters, around the entrance and exit point in the OD. Figure 6.19 shows the
TWIDA distributions for the raw data, the PC atmospheric MC, and the final samples with MC
normalised to the data. The OD during SK-II has increased reflectivity and improved quantum
efficiency in the OD PMTs, so the TWIDA cut criterion was further tuned for SK-II. The hit
clusters for the NCLSTA variable refers to spatial clusters of neighboring hit PMTs.
From SK-III onwards, the segmentation of the OD was used to develop a more efficientthrough-going muon cut. The PC First Reduction criteria for SK-III onwards:
136

9

10

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5

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10

rejected(a) data

Number of events

Number of events

1010(b) MC

10
1
10

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3

(c) final sample

10

2

data
MC

(a) data

rejected(b) MC

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data
MC

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1

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150

200

250

300

350

400

TWIDA (nsec)

0

50

100

150

200

250

300

350

400

TWIDA (nsec)

Figure 6.19: PC First Reduction: TWIDA distributions for (a) raw data, (b) PC atmospheric
MC with vertex position further than 2 m from the wall, (c) final sample data and MC normalised
to the data. Distributions are for SK-I (left) and SK-II (right). The TWIDA cut criteria arrows
mark the rejected events.
(1)

PEtot ≥ 1000 p.e.s(2)

NHITAtop < 10or
NHITAbottom < 10
NHITAtop (NHITAbottom ) is the number of OD hits in the top (bottom) region.

(3)

NHITAendcap < 25
or
NHITAside < 70
NHITAendcap (NHITAside ) is the number of OD hits in the top and bottom (side)
region.

(4)

ODRmean < 2100 cm
if OD hits < 20 in 500 nsec time window
ODRmean is the average distance between all hit pairs.
N −1 N
1 X X
ODRmean =|~xi − ~xj |.
Npair
i=1 j=i+1

Criteria (2) and (3) remove cosmic-ray muons that pass through two regions in the OD, butpass them if only one region is traversed. Criteria (2) removes the through-going muons whichpass through both the top and bottom OD regions. Criteria (3) discards corner clipping muons,
which pass through the endcap and side regions. Since through-gonig muons are expected todeposit energy in two regions of the OD, the variable ODRmean is expected to be larger than a
PC event with activity mainly in one region.
The event rate after the PC First Reduction is around ∼2 × 104 events/day.

6.3.2

Second Reduction

In the PC Second Reduction, an algorithm that finds clusters of OD hits is used to remove
the remaining through-going and stopping muons. The OD (ID) walls are divided into 11 × 11
137

(21 × 21) patches and the charge observed in each patch is counted. The clusters are formed bylooking at the charge gradient to the neighbouring patches, grouping hits into clusters “centred”
around the patch with the largest charge in the cluster. The algorithm is illustrated in Figure 6.20.
For SK-I, the PC Second Reduction criteria are:
(1)

NCLSTA2 ≤ 1
NCLSTA2 is the number of the OD hit clusters including more than 6 hit PMTs.

(2)

NHITACmin < 7
NHITACmin is the minimum number of hit PMTs among top (or bottom) andside regions in the OD hit cluster.

(3)

PE200 > 1000 p.e.s
if NCLSTA2 = 1
PE200 is the number of the observed p.e.s within 200 cm from the highest charge
PMT in the ID hit cluster closest to the OD hit cluster.

more charge

Cluster 1

less charge

Cluster 2

Cluster 3

Figure 6.20: A schematic view of the algorithm to find hit clusters in the PC Second Reduction.
The circles represent the charge observed in each patch, where the size of the circle is proportional
to the number of p.e.s . The arrows represent the vector charge gradient, which point from apatch to the neighbouring patch with the highest charge.
For SK-II, the PC Second Reduction criteria are:
(1)

NCLSTA2(2) ≤ 1
NCLSTA2(2) is the number of the 2nd OD hit clusters including more than 10
hit PMTs.

(2)

NHITAendcap < 20 or NHITAendcap < MAX(NHITAside )(exp(5.8 − 0.023 × NHITAside )if NHITAside < 75
MAX(NHITAside ) =exp(4.675 − 0.008 × NHITAside ) if NHITAside ≥ 75
NHITAendcap is the number of OD hit PMTs in the top and bottom region.
NHITAside is the number of hit OD PMTs in the side region.
138

(a) data(c) final sample (data)(b) MC(d) final sample (MC)NCLSTA2≥2
rejected

1

0

NCLSTA2

≥2

1

0
0

1000 2000 3000 4000 5000 6000 7000 0
PE200 (p.e.)1000 2000 3000 4000 5000 6000 7000
PE200 (p.e.)

Figure 6.21: PC Second Reduction: NCLSTA2 against PE200 scatter plots for (a) data after
PC1, (b) PC MC after PC1, (c) final sample data and MC normalised to the data. Distributions
are for SK-I. The NCLSTA2–PE200 cut region arrows mark the rejected events.
(3)

NHITAC2 < 12 + 0.085 × PE200
NHITAC2 is the number of the OD hit PMTs in the 2nd cluster.

Criteria (2) for both SK-I and SK-II reject corner clipping muons, which leave hit PMTs inboth the top (or bottom) and side regions in the OD. Criteria (3) for SK-I and SK-II remove
stopping muons, using the number of p.e.s in the ID hit cluster located closest to the OD hit
cluster.
The event rate after the PC Second Reduction is ∼8000 events/day.

6.3.3

Third Reduction

In the PC Third Reduction, Flasher events and cosmic-ray stopping muons are discarded. The
Flasher events are removed by the broad timing distribution expected from a Flasher event,which is the same method as that in the FC Third Reduction.
Flasher events are rejected by requiring the following selection criteria:(1)

NMIN100 ≤ 14
or
NMIN100 ≤ 9
if the number of hit ID PMTs < 800 (400 for SK-II)
NMIN100 is the minimum number of hit ID PMTs in a sliding 100 nsec time
window from +300 nsec to +800 nsec.

The stopping muon events are removed by the number of hit OD PMTs near the entrance
position. The entrance position is estimated by a backward extrapolation from the reconstructedvertex and direction of Point-Fit (see Section 7.2.1). Point-Fit is a simple fitter that assumes apoint source for the detected photons. Stopping muons are rejected by requiring the selection
criteria, for both SK-I and SK-II:
139

Number of events
0

20

40

60

80

100

120

140

0

20

40

60

80

160

180

120

140

160

180

10

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(a) data

rejected(b) MC

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1
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2

data
MC

1
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10

20

30

40

50

60

NHITACmin

200
180
160
140
120
100
80
60
40
20
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200
180
160
140
120
100
80
60
40
20
0

200

(b) MC100610(a) data

rejected

10

NHITAendcap

200
180
160
140
120
100
80
60
40
20
0

rejected

8

NHITAendcap

NHITAendcap

200
180
160
140
120
100
80
60
40
20
0

NHITAendcap

Figure 6.22:
PC Second Reduction:
NHITACmin distributions for (a) data
after PC1, (b) PC MC after PC1, (c) final
sample data and MC normalised to the data.
Distributions are for SK-I. The NHITACmincut criteria arrows mark the rejected events.

10

200

NHITAside

rejected

0

20

40

60

80

(a) final sample (data)

100

120

rejected

0

20

40

60

80

140

160

180

200

(b) final sample (MC)

100

120

140

160

180

200

NHITAside

Figure 6.23: PC Second Reduction: NHITAendcap against NHITAside scatter plots for (a) data
after PC1, (b) PC MC after PC1, (c) final sample data and MC normalised to the data. Distributions are for SK-II. The NHITAendcap –NHITAside cut region arrows mark the rejected events.

140

140

(a) data
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80
60

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20

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500

750

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1000

1250

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1750

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80
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rejected(b) final sample
(MC)

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rejected

120(a) final sample(data)

100

40

0rejected

120

100

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NHITAC2140

rejected

120

0

0

250

500

PE200 (p.e.s)

750

1000

1250

1500

1750

2000

PE200 (p.e.s)

Figure 6.24: PC Second Reduction: NHITAC2 against PE200 scatter plots for (a) data after
PC1, (b) PC MC after PC1, (c) final sample data and MC normalised to the data. Distributions
are for SK-II. The NHITAC2–PE200 cut region arrows mark the rejected events.
(2)

NHITAin ≤ 10
NHITAin is the number of hit OD PMTs located within 8 m from the entrance
point in a fixed 500 nsec time window.

For SK-III onwards, Criteria (2) is applied at the PC Fifth Reduction stage.
The event rate after the PC Third Reduction is ∼100 events/day.

6.3.4

Fourth Reduction

In the PC Fourth Reduction, cosmic-ray muons that passed the PC Third Reduction due torelatively small OD activity are removed at this stage.
In this reduction stage, two types of event reconstruction fitters are used: Point-Fit and athrough-going muon fitter. Point-Fit is described in Section 7.2.1. The through-going muon
fitter is also used in the FC Third Reduction. This fitter determines the entrance point as the
position of the earliest hit cluster in the ID. For proper PC events, which are initiated inside theID, the through-going muon fitter cannot determine the entrance point correctly and tends tohave worse goodness when compared with through-going muons. However, the vertex positionand direction can be reasonably well estimated by Point-Fit for both PC events and cosmic-ray
muons. The difference in these characteristics are used to reject cosmic-ray muons.
The selection criteria in the PC Fourth Reduction are as follows for SK-I and SK-II:(1)d~pfit · d~PMT > −0.8d~pfit is the reconstructed direction by Point-Fit, and d~PMT is the direction fromthe reconstructed vertex to the earliest saturated PMT.

(2)

DCORN > 150 cm
DCORN is the distance from the reconstructed vertex from Point-Fit to thenearest edge of the ID.
141

(3)

TLMU < 30 m
if Goodness of Through-Going Muon Fit > 0.85
TLMU is the track length of a muon estimated from the entrance and the exit
points by the through going muon fit.

Criteria (1) rejects cosmic-ray stopping muons which have their entrance point in the opposite
direction to the reconstructed direction by Point-Fit. For PC events, the difference in these
directions should be less anti-parallel than for cosmic-ray stopping muons. Criteria (2) attempts
to remove corner clipping muons. Criteria (3) discards through-going muons which have a longtrack length.
SK-III Onwards Cuts
For SK-III onwards a reconstruction fitter is used to classify each event as one of Stopping Muon,
Through-Going Muon, Multiple Muon or Corner Clipping Muons. 97% of the background eventsare classified as either Stopping Muon or Through-Going Muon. On the other hand, 96% of thePC events are categorised as one of the other muon types. There are five possible selection
criteria based on the fitter:(1)

anglemuon < 90◦anglemuon is the angle between the fitted direction and the vector between the
Point-Fit vertex and the centre of the highest charge OD cluster.

(2)

dotprodmuon > -0.8
dotprodmuon is a similar variable to anglemuon but using the earliest saturated
ID PMT instead of the centre of the highest charge OD cluster.

(3)

lengthmuon < 17.5 m
lengthmuon is the length of the fitted muon track. Long track lengths are likely
to be produced by cosmic-ray muons.

(4)goodnessmuon < 0.52
goodnessmuon is the goodness from the muon fitter.

(5)

cornermuon ≥ 3 m
cornermuon is the distance between the fitted entrance point and the corner of
the tank.

The number of these criteria that are required to pass this reduction step depend on the classification of the event from the fitter. In the case the event is classified as Through-Going Muon,then it has to pass 4 of the 5 criteria before being passed. Similarly, if the event is categorised as
Stopping Muon, then it has to pass 4 of the 5 criteria including the dodprodmuon criterion withan additional requirement. The event must have goodnessmuon < 0.5 or ehitmuon < 10, where
ehitmuon is the number of OD hits located within 8 m from the fitted entrance point in a fixed
500 nsec time window. For the other classifications, 2 of the above 5 criteria have to be passed.
As a final step, low energy background events are removed by requiring the selection:(1)

PEtot ≥ 2900 p.e.s
PEtot is the total number of p.e.s observed in the ID.

PEtot of ∼3000 p.e.s corresponds to a muon momentum of 500 MeV/c, which is safe for PC events
as exiting muons must have at least a momentum of 700 MeV/c to reach the OD.
The event rate after the PC Fourth Reduction is ∼20 events/day.
1426.3.5

Fifth Reduction

In the PC Fifth Reduction stage more elaborate selection criteria specialised for each backgroundsource are applied to remove the remaining non-neutrino events to produce the final PC sample.
The selections for the SK-I to SK-II period and the SK-III to SK-IV period are different, andhave been presented separately here.

SK-I and SK-II Reduction Selections
Low Energy Event Cut
The following selection criteria reject the remaining low energy background events:(1)

PEtot ≥ 3000 p.e.s (1500 for SK-II)
PEtot is the total number of p.e.s observed in the ID.

Again, this requirement corresponds to a muon momentum of 500 MeV/c, where at least a
momentum of 700 MeV/c is necessary to reach the OD. This is a safe cut that will remove a lotof the low energy backgrounds.
Through-Going Muon Cut: OD Clusters
At this stage, there are three sets of selection criteria that attempt to remove the remaining
through-going muons by considering particular properties of these background events. The first
set of cuts make use of the OD hit cluster information obtained by the algorithm in the PC Second
Reduction (see Section 6.3.2). Events which seem to have two OD clusters, from the expected
entrance and exit points of a through-going muon, are removed. The following selection criteria
reject through-going muons:(1)

DISTclust ≤ 20 m
DISTclust is the distance between the highest charge OD hit cluster and thesecond highest one.

(2)

PEAC2nd < 10 p.e.s
PEAC2nd is the number of p.e.s detected in the second highest charge OD hit
cluster.

(3)

NCLSTA5 < 2
NCLST5 is the number of OD hit clusters which contain more than 9 hit PMTs.

In criterion (3), the same clustering algorithm as that in the PC Second Reduction is used,
but with different parameters. The OD wall is divided into 6 × 6 instead of 11 × 11 in the PC
Fifth Reduction to avoid the boundary effect of the patches.
Through-Going Muon Cut: Geji
The second set of through-going muon cuts aims to remove “Geji” muon events. Some cosmic-ray
muon events enter at the edge of the top of the detector, pass parallel to and along the ID wall,and exit through the edge at the bottom. The result is a “centipede”-like track (geji in Japanese)143along the ID wall. Since the light collection efficiency around the edge of the OD is relatively
not so good and since the reconstruction tools do not perform so well at the detector boundaries,
these events pass the prior reduction steps. The reduction selection criteria that rejects these
events is:(1)

NHITAtop < 7 and NHITAbottom < 7
NHITAtop (NHITAbottom ) is the maximum number of hit OD PMTs in a 8 m
radius sphere centered at the top (bottom) edge.

(2)

PEAtop < 10 p.e.s and PEAbottom < 10 p.e.s
PEAtop (PEAbottom ) is the number of p.e.s in OD detected in the same sphereas that for the NHITAtop (NHITAbottom )(3)

TDIFFA × c/40 m < 0.75
or
TDIFFA × c/40 m > 1.5
TDIFFA is a time interval between the averaged hit timing in the top and thebottom spheres.

Criterion (3) checks that the difference in the timing between the top and bottom is notconsistent with a muon passing vertically through the detector.
Through-Going Muon Cut: Fitting
The third set of through-going muon cuts make use of precise reconstruction tools to check thenumber of hit OD PMTs near the entrance and exit points. The vertex position and the ring
direction are reconstructed by a Cherenkov ring pattern based fitter: MS-fit (see Section 7.5).
The entrance and the exit points on the detector wall are estimated by extrapolating the fitted
information. The selection criteria that remove the remaining through-going muons are:
(1)

NHITAin < 5andNHITAout < 5
NHITAin (NHITAout ) is the number of hit OD PMTs within 8 m from the entrance (exit) point.

(2)

TDIFFA × c/TRACK < 0.75
or
TDIFFA × c/TRACK > 1.5
TRACK is distance between the entrance and exit point estimated using thevertex position and the ring direction reconstructed by MS-fit.

Stopping Muon Cut: Fitted Entrance Point
Three sets of selection criteria are are applied to reject stopping muons. In the first cut, thereconstructed vertex position and direction using MS-Fit (Section 7.5) are extrapolated backwards to determine the entrance point. The number of hit OD PMTs near the entrance point
are counted and used to reject muons that entered through the OD. The first set of reduction
criteria that remove stopping muons are:
(1)

NHITAin < 10
NHITAin is the number of OD hit PMTs within 8 m from the reconstructedentrance.
144

Stopping Muon Cut: Fitted Entrance Angle
In the second stopping muon cut, the Cherenkov ring is reconstructed by two different fitters,
TDC-Fit (Section 7.2.3) and MS-Fit (Section 7.5). Then the angle between the reconstructed
direction and the OD hit cluster are compared. In the case of a stopping muon event, the opening
angle is expected to be large (ideally anti-parallel). However, for a PC event, the angle should besmaller (ideally parallel). The reduction selection that discards stopping muons by comparing
this angle are:(1)

ΘTDC-Fit ≤ 90◦ or ΘMS-Fit ≤ 90◦
ΘTDC-Fit (ΘMS-Fit ) is the opening angle between the direction to the OD hit
cluster and the ring direction reconstructed by TDC-fit (MS-fit).

Stopping Muon Cut: Stopping Muon Fit
In the third stopping muon cut, the vertex and direction are estimated by Stopping Muon Fit.
This fitter estimates the entrance position as the earliest hit cluster in the ID. For PC events,
the fitting is not so successful as the vertex is typically not on the wall. However, if the Goodnessof the Stopping Muon Fit is > 0, then the charge inside a 42◦ cone facing the entrance position
is checked. Less than 60% of the charge should be within this cone. Then the selection criteria
for rejecting stopping muons are:(1)

If the Goodness of Stopping Muon Fit > 0(2)

PEcone /PEtot < 0.6
PEcone is the number of p.e.s observed by ID PMTs located inside a 42◦ cone
facing the entrance position.
PEtot is the total number of p.e.s observed in the ID.

(3)

NHITAin ≤ 6
NHITAin is the number of hit OD PMTs within 8 m from the entrance position.

Cable Hole Muon Cut
There are four veto scintillation counters placed over the cable holes on the top of the detector.
Unlike the FC event selection, a PC event may activate those veto counters. So an additional
requirement on the reconstructed direction from TDC-Fit (Section 7.2.3) relative to the direction
from the reconstructed vertex to the veto counter is required. If these are anti-parallel, then theevent is more likely to be a cosmic-ray muon in origin. If there is a veto counter hit, then the
reduction selection requires that the directions are not anti-parallel for a PC event:(1)

If there is one veto counter hit.

(2)d~ring · d~veto-vertex > −0.8d~ring is the reconstructed ring direction by TDC-fit, and d~veto-vertex is the direction from the reconstructed vertex to the hit veto counter.
145

Corner Clipping Muon Cut: Evis Track
Most of the corner clipping muon events are rejected in the PC Second and Third Reduction
steps. In this reduction step, the remaining corner clipping muon events are removed. The
corner clipping muon events have a small hit cluter in the ID, so the vertices are occasionally
mis-reconstructed inside the ID. As a result, the track length from the vertex to the exit point
reconstructed by MS-fit (Section 7.5) is large, while the track length which is estimated by thevisible energy using the energy loss of muons ∼2 MeV/cm is small. This cut is applied in SK-II.
The reduction selection criteria that rejects corner clipping events are the following:(1)

TRACKEvis > TRACK − 1500
if TRACK > 15 m
TRACKEvis is the estimated track length from the visible energy and an energy
loss of ∼2 MeV/cm
TRACK is the track length from the vertex to the exit point estimated from thevertex point and the direction by MS-fit.

When the PC event is properly reconstructed TRACKEvis ≈ TRACK, so Criterion (1) should
hold. In the case of a corner clipping event, TRACK may be reconstructed large and be longer
than TRACKEvis , even after subtracting 15 m.

SK-III and SK-IV Reduction Selections
The PC Reduction for SK-III onwards was modified to improve the efficiency. For the following
reduction steps there are two types of cut: Hard Cuts and Soft Cuts. PC events are required topass all of the Hard Cuts, while they may fail the Soft Cuts once only.
The Hard cuts are:(1)

Through-Going Muon Cut: OD Clusters(2)

Through-Going Muon Cut: Geji

(3)

Stopping Muon Cut: Fitted Entrance Angle(4)

Cable Hole Muon Cut(5)

Corner Clipping Muon Cut: Edge DistanceCriteria (1)–(4) have already been described for the SK-I and SK-II case.
Corner Clipping Muon Cut: Edge DistanceThis is the same criteria that was applied in the PC Fourth Reduction for SK-I and SK-II. Thiscut avoids clipping muon events by requiring the edge is not too close to the edge of the ID. The
reduction selection criteria are:(1)

DCORN > 150 cm
DCORN is the distance from the reconstructed vertex from Point-Fit to thenearest edge of the ID.
146

PC Fifth Reduction Soft Cuts
The soft cuts are:(1)

Through-Going Muon Cut: Fitting(2)

Through-Going Muon Cut: OD Cluster Hits

(3)

Stopping Muon Cut: Fitted Entrance Point(4)

Stopping Muon Cut: Stopping Muon Fit(5)

Stopping Muon Cut: Fitted Entrance Strict(6)

Stopping Muon Cut: Point-Fit Angle(7)

Corner Clipping Muon Cut: Evis Track(8)

Decay Electron Cut

Criteria (1), (3), (4), and (7) have been previously described in the reductions for SK-I and
SK-II.
Through-Going Muon Cut: OD Cluster Hits
The number of hit OD PMTs in the first and second highest charge clusters are used to remove
the remaining through-going muons. The reduction selection critera are:
(1)

NCLSTA5(1) < 10
and
NCLSTA5(2) < 17
NCLSTA5(1) (NCLSTA5(2)) is the number of hit OD PMTs in the first (second)highest charge cluster.

Stopping Muon Cut: Fitted Entrance Strict
The criteria used here to reject stopping muons considers the number of hit OD PMTs within
8 m of the reconstructed entry point. This criteria was applied during the PC Third Reduction
for SK-I and SK-II, but was moved to PC Fifth Reduction for SK-III and SK-IV. In addition
the cut is more strict, however it now requires that the vertices fitted by Point-Fit and MS-Fit
agree within 15 m. The reduction selection criteria for further removing stopping muons are:
(1)

NHITAin ≤ 6
if |P~Point-Fit − P~MS-Fit | < 15 m
NHITAin is the number of hit OD PMTs within 8 m from the entrance position.
P~Point-Fit (P~MS-Fit ) is the reconstructed vertex position from Point-Fit (MS-Fit)

Stopping Muon Cut: Point-Fit Angle
This reduction step to reject stopping muons uses a selection criteria that was present in the PC
Fourth Reduction. However, since this cut was not necessarily required to pass the PC Fourth
Reduction, it is applied again in the Fifth Reduction.
(1)

anglemuon < 90◦anglemuon is the angle between the fitted direction and the vector between the
Point-Fit vertex and the centre of the highest charge OD cluster.
147

Partially-Contained
Reduction Step
1st Reduction
2nd Reduction
3rd Reduction
4th Reduction
5th Reduction
PCFV
Event Rate (/day)

SK-I
98.98
96.74
95.69
89.86
88.66
80.98
0.66±0.02

Selection Efficiency (%)
SK-II
SK-III
SK-IV
98.58
99.09
99.63
93.43
98.52
98.73
92.32
98.51
98.68
84.60
97.87
97.42
82.63
96.61
96.15
74.80
88.80
86.30
0.65±0.04 0.62±0.03 0.66±0.02

Table 6.3: The detection efficiency after each PC Reduction step for each of the SK running
periods. The efficiencies are shown after the PCFV cuts were applied using the true vertex
information. In the PCFV row, the reconstructed vertex is used, and the final row shows theevent rate per day.
Decay Electron Cut
High energy neutrinos interactions produce pions via DIS interactions. The pions then decay
into muons, which further decay into electrons that are tagged as decay electrons associated with
the neutrino event. However, high energy cosmic-ray muons decay in flight and no decay electron
is observed. The reduction selection that removes these cosmic-ray muon events:
(1)

6.3.6

Ndecaye ≥ 1
if Evis > 25 GeV
Ndecaye is the number of tagged decay electrons.

PC Reduction Summary

Summary Finally, the Partially-Contained sample used in analyses are selected by applying these
Fiducial Volume (PCFV) cuts:• Vertex of neutrino interactions should be inside the fiducial volume (2 m from the ID PMT
surface).
• The number of hit PMTs in the highest charge OD cluster (NHITAC) should be larger
than 15 (9 for SK-I).
• Visible energy (Evis ) should be greater than 350 MeV.
(The total observed charge in the ID > 3000 p.e.s (1500 p.e. for SK-II))The detection efficiency after each PC Reduction stage is estimated with the atmospheric neutrinoMonte Carlo, which is summarised in Table 6.3. The final detection efficiency is estimated to be
81%, 74.8%, 88.6% and 1000% for SK-I, SK-II, SK-III and SK-IV, respectively
In the final PC sample, the background events mainly consist of cosmic-ray muons. The final
PC sample is checked by eye-scanning the events in an event display. There are few background
events, most of which are removed by the fiducial volume cut. However, the presence of background events outside the fiducial volume can lead to some contamination. From the eye-scanningand extrapolation of the background distribution outside the fiducial volume, the systematics for148

Partially-Contained
BackgroundCosmic-ray µ

Background Contamination (%)
SK-I SK-II SK-III SK-IV
0.2
0.7
1.8
0.6

Table 6.4: Estimated upper limit of the contamination of cosmic-ray µ in the PC events.
the PC Reduction were estimated. The cosmic-ray muon background contamination in the PC
sample is summarised in Table 6.4
The uncertainties of the PC Reduction efficiencies were estimated from the difference between
data and Monte Carlo for the cut variables. The uncertainties are estimated to be 2.4%, 4.8%,
0.5% and 1000% for SK-I, SK-II, SK-III and SK-IV, respectively. Due to the improvements of
the reduction efficiency for SK-III and SK-IV, there is better agreement between data and MC
and a resulting smaller uncertainty.

149

Chapter 7

Event Reconstruction
7.1

Overview

The extraction of physical properties from the events follows a flow of reconstruction steps:
1. Vertex Reconstruction
The position of the vertex and a direction is initially fitted assuming a point-like source,then the ring edge is found. Following this the position and direction reconstruction are
improved by considering a line source.
2. Ring Counting
Ring candidates are searched for and tested successively by a likelihood and evaluation
functions to determine whether they are real or not. Those that are identified determine
the number of reconstructed rings for the event.
3. Particle Identification
Reconstructed rings are identified to be showering (e-like) or non-showering (µ-like) byconsidering electron and muon charge pattern distributions and their opening angles.
4. Precise Vertex Fitting (MS-fit)
The previous vertex fitting is time-based and improves the reconstruction in the direction
parallel to the particle direction. The PID and pattern information is used by MS-fit tofurther improve the vertex and direction reconstruction.
5. Momentum Reconstruction
The momentum is reconstructed by considering the charge in a cone with a half angleof 70◦ . Corrections for relative gain between data and MC, scattered photons and lightattenuation in water are made.
6. Decay Electron Search
The muon detection efficiency can be increased by searching for decay electrons from muon
decay. These are found and classified into a few different event types.
7. Ring Number Correction
Some mis-identified rings are found and removed from the final reconstruction.
150

8. π 0 Fitting
An algorithm is used to find π 0 events that have been mis-reconstructed as single-ring e-likeevents.

7.2

Vertex Reconstruction

The vertex reconstruction follows a 3-step process:• Point-Fit gives an initial estimation of the vertex position and ring direction assuming apoint-source.
• Ring Edge Search locates the opening angle of the ring and improves the direction
reconstruction.
• TDC-Fit further improves the position and direction by considering a line-source for thephotons and accounts for scattering.

7.2.1

Point-Fit

The first step is to find an approximate vertex with a rough fitting assuming a point source. A~ is assumed and the time of flight from A~ to the PMT is subtracted from the hit
point vertex, A,
0timing of each PMT, ti for the i-th PMT, to give the timing residual:ti = t0i −




n ~
~
|Pi − A|c

(7.1)

where P~i is the position of the i-th PMT, and c/n is the speed of light in water. Then to find
~ the goodness-of-fit is determined by the estimator Gp :the best vertex match for A
1 X
Gp =
exp
N
i

 !

ti − t0 2− 0.5
1.5 × σ(7.2)

where N is the number of hit PMTs, t0 is a free parameter and represents the event timing at
the best-fit , ti is the timing residual (Equation 7.1), σ is the typical timing resolution of a PMT(2.5 ns), and the factor 1.5 is there to improve the performance of the fit. By maximising Gp , a~ with timing t0 .
best-fit vertex can be found for A~ 0 , is used to estimate the particle direction.
At this stage the vertex found from point fit, A~ 0 to each PMT is weighted by the charge at each PMT, qi , and summed to
The direction from A
give an estimated direction:!
X~i − A
~0
P
d~0 =
qi
(7.3)
~ 0|
|P~i − Ai

7.2.2

Ring Edge Search~ 0 , and particle direction, d0 , are used to find theIn this stage, the Point Fit estimated vertex, A
outer edge of the Cherenkov ring and improve the reconstruction of the ring direction.
151

1. A distribution of charge as a function of the opening angle, θ, from the assumed ring
direction to the hit PMT is made: P E(θ). The initial ring direction is taken to be thePoint Fit estimation, d0 . The charge in this distribution is corrected for the effects fromthe water transparency and the PMT acceptance.
2. The outer edge of the Cherenkov ring is found by finding the roots of the second derivativeof the P E(θ) distribution, and satisfying the following conditions:•d2 P E(θ)
=0
dθ2
θedge

• θedge > θpeak , where θpeak is the angle at the peak of the P E(θ) distribution andtherefore θedge is taken to be the angle of the outer edge of the ring.
• If there are multiple θedge candidates, the one closest to θpeak is chosen.
A typical P E(θ) distribution and it’s second derivative is shown in Figure 7.1.
3. The ring direction and θedge are simultaneously optimised to maximise the estimatorQ(θedge ):R θedge

P E(θ)dθQ(θedge ) = 0
×sin θedgedP E(θ)dθ
θ=θedge!2
× exp(θedge − θexp )2
−
2σθ2

!
(7.4)

where θexp is the opening angle of the Cherenkov ring expected from the charge distribution
P E(θ), with angular resolution σθ . The ring direction is adjusted around d0 and the steps
are reitterated until Q(θedge ) is maximised.
The numerator in the first factor of Equation (7.4) favours more charge inside the ring, while
the denominator enhances Q(θedge ) for narrower rings. The middle factor favours sharper rings
and the last factor enhances the estimator if it is close to the expectation.

7.2.3

TDC-FitUsing the Point-Fit vertex and direction, and the ring information as a starting point, TDCFit improves the vertex and ring direction reconstruction. This more precise fitter considers
the Cherenkov light as a line-source, estimating the emission point of the Cherenkov for PMTs
hit within the ring. For PMTs hit outside the ring the scattering of Cherenkov photons is also
considered, and the source point is taken as the assumed vertex. The time residual, the estimatedtiming of the vertex, is calculated as:(~ i − O|~ − n × |P~i − X~ i|t0i − 1c × |X
Hits inside the Cherenkov ring
c
ti =
(7.5)
n
0
~
~
ti − c × |Pi − O|
Hits outside the Cherenkov ring
~ i is the estimated point of emission for the Cherenkov photon, O~ is the assumed vertex,
where Xn is the refractive index of water and P~i is the position of the i-th PMT. From Equation (7.5) it
can be seen that the particle is assumed to travel at the speed of light, c, and that the Cherenkov
photons take a direct path to the PMT. This scenario is illustrated in Figure 7.2. The mean of
the ti is calculated to give t0 , which is an estimation of the vertex timing.
152number of p.e.

150
100
50

second derivarive

0

0

20

40

60

80

10

0

-10

0

20

40

60

80
degreeFigure 7.1: A typical P E(θ) distribution isshown in the upper figure, and the second derivative in the lower figure. The roots of the second
derivative are used to find the outer edge of thering, θedge .

PMTs

Opening
angle θ c

ΟXitrack lengthli

Cherenkov
cone

Figure 7.2: Illustration of Cherenkov radiation.
The Cherenkov photons are assumed to originate from the track of a charged particle, andare emitted at point Xi to the i-th PMT. The
track length from the vertex is given by li .

153

The estimator used to improve the vertex and ring direction is divided into three parts: GI
for PMTs within the ring, GO1 for PMTs outside the ring with an early timing residual (ti ≤ t0 ),and GO2 for PMTs outside the ring with a late timing residual (ti > t0 ). The GI estimator of
the fitting is defined as:
 !

X 1
1
ti − t0 2
GI =exp − ·
(7.6)
2
1.5 · σσi2iwhere σi is the timing resolution of the i-th PMT and depends on the detected charge, whereas
σ is the mean timing resolution for all PMT hits. So GI gives an estimation of how well the time
residual values, ti , from Equation (7.5) converge to a single vertex time, t0 .
For the PMT hits outside the ring, the estimator depends on the difference between ti andt0 and are defined like so:

GO 1 =

X 1
i

GO2 =

σi2

X 1
i

σi2

1
− ·
2

exp

!
 !
ti − t0 2
×2−1
1.5 · σ#!
 !
1ti − t0 2− ·, Gscatt (ti , t0 ) × 2 − 1
2
1.5 · σ"
max exp(for ti ≤ t0 ) (7.7)(for ti > t0 ) (7.8)

where,
Rq
Gscatt (ti , t0 ) =
× exp
1.52

1
− ·
2



ti − t0
1.5 · σ

2 !





Rq
ti − t0
+ 1−
×
exp
−
1.52
60nsec

(7.9)

and,
X
Rq =

qi

θ<θc +3.0

X(7.10)qi

θ<70◦

If ti ≤ t0 then this suggests the PMT hit timing is earlier on average than direct light from theassumed vertex, and the estimator GO1 in Equation (7.7) estimates how well these hits converge
to t0 . The “×2−1” factor acts as a penalty for a poor fitting. If ti > t0 then it is possible that thephotons arriving at the PMT were scattered, and the estimator GO2 in Equation (7.8) considers
this effect. It takes the maximum estimation of either direct light, which is the exponential term,
or after considering a degree of scattering, Gscatt (ti , t0 ) in Equation (7.9).
Gscatt has two terms which depend on the fraction of charge within the ring, Rq in Equation (7.10). The first term is favoured if there is more charge in the ring, and is an estimator for
direct light. The second term is stronger with increasing charge outside the ring, and it favours
the hits which are less scattered.
Then the three estimators are brought together for all the PMT hits to give the total estimator
for the TDC-Fit:
GT =GI + (GO1 or GO2 )
X 1
i

154

σi2

(7.11)

The vertex position and ring direction which maximises GT are taken as the fitted result
from TDC-Fit.

7.3

Ring Counting

After the vertex, direction and ring edge of the first Cherenkov ring is found, an algorithm is run
to search for possible candidates for other rings.

7.3.1

Ring Candidate Search

The algorithm for searching for other ring candidates applies a method known as a Hough
Transform [251]. This technique uses a spherical coordinate system about the reconstructedvertex and draws rings centred on each PMT hit as illustrated in Figure 7.3. These virtual rings
assume a 42◦ cone opening angle, and a circular charge distribution depending on the hit PMT
charge is generated. Summing these ring charge distributions on the (Θ, Φ) plane gives peaks
where the virtual rings overlap, which correspond to the ring direction of candidate rings. A
typical charge distribution is shown in Figure 7.4 and a candidate ring is illustrated as the shaded
region in Figure 7.3.
42 deg. ringhit PMT(possible center)

centerCherenkov ring

(most probable)

Figure 7.3: Illustration of the Hough Transform. Virtual circles representing Cherenkov rings
are drawn around each PMT hit. Points of considerable overlap are likely to be the centre of areal Cherenkov ring, shown as the shaded region.

7.3.2

Ring Candidate Test

The possible candidates for rings are tested by a likelihood function which depends on all therings already found. When N rings have been identified, and candidates are being tested for the(N +1)-th ring, the likelihood that the candidate is a real ring is given by:
155

600
400

Figure 7.4: The virtual circles from the 200
Hough Transform are actually circular charge
distributions, and this is a typical Hough0
Transform charge distribution. This charge
150
is summed on a (Θ, Φ) plane, the peaks areΘ ( 100
where there is considerable charge overlap
degr
and represent directions that are likely to beee 50s)
the centre of real Cherenkov rings.

LN +1 =

Xlog(prob(qiobs ,

300

200
100
00

Φ

)

ees
(degr

exp
n=1 αn · qi,n ))PN +1(7.12)iwhere the probability function is defined as:(qiobs − qiexp )2
1
√(for qiexp > 20 p.e.)
exp
−

2
2σ
exp
2π
prob(qiobs , qi ) =Single p.e. and Poisson convolution (for qiexp < 20 p.e.)(7.13)The likelihood in Equation (7.12) compares the charge observed, qiobs , on the i-th PMTagainst the summed charge expected on the PMT from the N identified rings and the (N +1)-thexp
ring being tested, α ·qi,nfor the n-th ring. The likelihood is maximised by varying the αn factors(1 ≤ n ≤ N + 1), with a lower momentum constraint.
The probability function in Equation (7.13) measures how closely the expected charge fromall the considered rings matches the observed charge by a Gaussian distribution when qiexp >20 p.e. In the case qiexp < 20 p.e. the probability is determined from the convolution of a single
photoelectron distribution and a Poisson distribution.
If none of the (N +1)-th ring candidates satisfy LN +1 ≥ LN , the algorithm is finalised andthe number of rings is determined to be N . In the case LN +1 ≥ LN , the (N +1)-th ring is put
through six evaluation functions:
F1 : The difference LN +1 − LN is corrected by considering the total charge of theevent. If the difference is larger, the candidate is more likely to be a real ring.
F2 : The average of the expected charge near the edge of the (N +1)-th Cherenkov
ring, Qedge , excluding the charge expected from the other N rings is calculated.
If Qedge is larger, the candidate is more likely to be real.
156

F3 : The average of the expected charge outside the (N +1)-th ring, Qout , excluding
the charge expected from the other N rings is calculated. If the difference
Qedge -Qout is larger, the candidate is more likely to be real.
F4 : A residual charge distribution, excluding the residual charge
expected from the~ = P q res · êi is calculated,other N rings is made. Using this, a vector sum V
i iwhere qires is the residual charge for the i-th PMT and êi is the unit vector~ | is larger, the candidatefrom the reconstructed vertex to the i-th PMT. If |V
is more likely to be real.
F5 : The difference in charge between the peak of a candidate ring and the averageof inside and outside the ring. If this difference is larger, the candidate is more
likely to be real.
F6 : The azimuthal symmetry about the direction of the ring is considered, wherea single-ring event is expected to be more symmetric than a multi-ring event.
This evaluation function is only used to separated single and multi ring events.
The weighted sum, F , of these evaluation functions are computed, which constitutes the test
for the (N +1)-th ring candidate. If F ≥ 0, then the (N +1)-th ring is identified as true and thealgorithm is repeated to search for the next ring. On the other hand if F < 0 for all remaining
ring candidates, then the number of rings is determined to be N .

7.4

Particle Identification

After the rings have been identified, a particle identification (PID) algorithm is run. In SuperKamiokande effectively two types of charge distribution are observed from Cherenkov rings:
showering type (e-like), and a non-Showering type (µ-like). The showering events have a diffuse
ring due to an electromagnetic shower or multiple scattering and are most likely to be produced
by an electron or gamma photon. The non-showering events have a more sharp ring edge andare most likely produced by muons or charged pions.

7.4.1

Estimation of the Particle TypeTo estimate the particle type a likelihood is defined assuming a lepton, Ll (l = e or µ), as thearchetype for the showering and non-showering type events. For the n-th ring, the likelihood isdefined as:


Y
X
exp,l
exp,l Ll (n) =prob qiobs , qi,n+qi,n(7.14)
0n0 6=ni
θi,n <1.5θc,n

The product in the likelihood in Equation (7.14) is taken over each PMT which is within 1.5
times the estimated Cherenkov opening angle for the n-th ring, θc,n . The probability functionis the same as that defined in Equation (7.13), which determines how well the expected charge,
qiexp , reproduces the charge observed, qiobs , in the i-th PMT. Here, the expected charge of theexp,ln-th ring is calculated assuming a lepton type l (l = e or µ), qi,n, and adding the charge from all157

the other rings where n0 6= n. The details of these lepton charge distributions is described in thefollowing sections. Ll is maximised by varying the direction of the n-th ring and its Cherenkov
opening angle, while all the other rings are kept fixed.
The probability of a ring being identified as like a particular lepton considers both the charge
pattern and opening angle of the ring for single-ring events. For multi-ring events, only thepattern is considered. For the probability contribution from the charge distribution of the ring,
the above likelihood is taken in the form of a χ2 :
χ2l (n) = −2 ln Ll (n) − constant(7.15)

Then the pattern probability is defined as:
1 χ2l (n) − χ2min
Plpattern (n) = exp −
2
σL2 l

2 !
(7.16)

where the above mentioned minimum χ2 is taken as the minimum of the electron or muon
assumption,
χ2min = min[χ2e , χ2µ ]. σLl characterises how well the likelihood was calculated and is√
2NP M T , where NP M T is the number of PMTs that were used in the calculation of Ll .
The probability from the Cherenkov opening angle, generally for the n-th ring is:

2 

exp,l
θ
−
θ
n
 1 c,nPlangle (n) = exp −(7.17)22σθwhere θc,n is the reconstructed Cherenkov opening angle of the n-th ring, θnexp,l is the expected
opening angle for particle type l (l = e or µ), σθ is the resolution of the estimated Cherenkov
angle.
Then the total probability of the ring being identified as lepton l is the product of the patternand opening angle probabilities:
Pltotal (n) = Plpattern (n) × Plangle (n)(7.18)

noting that Plangle (n) is not used for multi-ring events due to insufficient reconstruction of theCherenkov opening angle when many rings are involved. The ring is identified as µ-like when
the total probability Pµtotal (n) > Petotal (n) and vice-versa for the e-like case.

7.4.2

Expected Charge Distribution: Electrons

The expected charge distributions, in photoelectrons (p.e.), produced by an electron were prepared in advance by using a Monte Carlo simulation. The expected charge distribution, Qexpe (pe , θ),is generated assuming a hypothetical spherical surface of radius Rsph = 16.9 m (same as the radius of the Inner Detector), with the charge collected on a 50 cm diameter circular area which
represents a PMT surface (equivalent to the 20 inch ∼ 50 cm diameter PMT). Qexpe (pe , θ) is
calculated as a function of electron momentum, pe (MeV/c), and the opening angle, θ, from theelectron direction.
The expected charge on the i-th PMT due to the n-th ring is calculated using Qexpe (pe , θ)like so: sph 1.5
 r 
R
i
e
qi,n = αn,e × Qe (pe , θi,n ) ×× exp −× f (Θi )(7.19)ri
λ158

whereeqi,n: Expected charge (p.e.) for the i-th PMT due to the n-th ring

αn,e : Normalisation factor
θi,n : Opening angle between the n-th ring direction and the
direction from the vertex to the i-th PMT
ri : Distance from the vertex to the i-th PMT
Θi : Angle of photon arriving direction relative tothe i-th PMT’s facing direction (acceptance angle)
f (Θi ) : Correction function of the PMT acceptanceλ : Attenuation length of light in water (see Section 5.1.4)

The factor (Rsph /ri )1.5 takes into account the ri dependence of the intensity of the Cherenkov
light, in effect translating the expected charge in a sphere to the SK geometry. The index 1.5
was tuned by a Monte Carlo study.

7.4.3

Expected Charge Distribution: Muons

The expected charge distribution, in photoelectrons (p.e.), produced by a muon is calculated
analytically by the following equation:
µ
qi,n
=

sin2 θi,n + qiknock
αn,µ ×dθ
ri sin θi,n + ri dx
|x=xi

!

 r 
i
× f (Θi )× exp −
λ(7.20)

where
µ
qi,n
: Expected charge (p.e.) for the i-th PMT due to the n-th ring

αn,µ
knock
qi

: Normalisation factor: Observed charge (p.e.) for the i-th PMT due to the knock-on electrons

x : Distance from the vertex along the muon trajectoryxi : Distance from the vertex to the Cherenkov emission point
for the i-th PMT
ri , θi,n , λ, Θi , f (Θ) : Same quantities as in Equation (7.19)
The numerator sin2 θ in Equation (7.20) is proportional to the number of emitted Cherenkovdθ
photons in the direction θi,n (compare Equation (3.3)). The denominator ri (sin θ + ri dx) comesfrom the area that Cherenkov photons are emitted towards, which can be readily understoodfrom Figure 7.5 showing a schematic view of Cherenkov photon emission. When the muon loses
energy, the Cherenkov opening angle also changes and therefore the area subtended by the angle
does as well. The charge contribution from knock-on electrons is qiknock and was estimated by aMonte Carlo simulation.
159

dx
⋅sin
θ

r⋅d
θ
dθ

r
Figure 7.5: lllustration of Cherenkov radiationfrom a muon. Due to the energy loss in water, the Cherenkov opening angle, θ, changes as
the muon momentum changes. When the muon
propagates by dx, Cherenkov photons are emitted into the region dx · sin θ + r · dθ.

7.4.4

θ
dx

muon

Expected Charge Distribution: Scattered Light

The scattered light that hits the PMTs can be distinguished from direct photons by using thetiming information. The time residual, t0i , for the i-th PMT as defined in Equation (7.5) is used
to determine the “off timing” by the criteria:
tpeak − 30nsec < t0i < tpeak + 2σi + 5nsec
tpeak + 2σi + 5nsec < t0i(Direct photons)(7.21)(Scattered photons)(7.22)

where tpeak is the time at the peak position of the time residual distribution, and σi is themeasured timing resolution as a function of observed charge (p.e.).
The charge due to the scattered photons, qiscatt , is estimated from the “Off timing” PMTs andadded to the expected charge for the i-th PMT:
qiexp,l = qil,direct + qiscatt(7.23)

where the label l denotes the lepton type and is either e or µ.

7.4.5

Performance of PID

The PID efficiency for single-ring events is shown in Figure 7.6.

7.5

Precise Vertex Reconstruction (MS-Fit)

The vertex fitter described earlier in Section 7.2.3, TDC-Fit, is a timing based fitter and mainly
improves the vertex reconstruction along the direction parallel to the particle direction. MS-Fit
improves the vertex reconstruction by including the ring pattern information. At this stage, thevertex is refitted with MS-Fit adjusting the vertex position and particle direction using the PID
likelihood, while simultaneously using TDC-Fit to modify the vertex in the direction parallel tothe particle direction using the timing information. This is an iterative process and continues
until the changes in the vertex position and the particle direction is less than 5 cm and 0.5◦ ,respectively. The vertex resolution for single-ring events is approximately 30 cm.
160

PID efficiency

1
0.8
0.6

(a) νe Q.E.

0.4(b) νµ Q.E.
> 5 GeV/c> 5 GeV/c

0.2
0
10

2

10

3

10

4

Lepton Momentum (MeV/c)

10

2

10

3

10

4

Lepton Momentum (MeV/c)

Figure 7.6: The PID efficiency for CC quasi-elastic single-ring events as a function of charged
lepton momentum for (a) νe and (b) νµ .

MS-FitEvent type
FC Sub-GeV
Single-Ring e-likeµ-likeFC Multi-GeV
Single-Ring e-likeµ-likeMulti-Ring µ-like
PC

SK-I

Resolution (cm)
SK-II SK-III SK-IV

Angular Resolution (◦ )SK-I SK-II SK-III SK-IV

31.2
23.8

35.6
30.3

31.1
23.9

31.5
23.5

3.1
1.9

3.2
2.1

3.0
1.9

3.0
1.9

33.5
24.8
67.4
53.6

34.3
26.9
111.8
62.8

33.4
25.1
73.8
52.6

33.5
23.7
87.8
67.0

1.2
0.8

1.3
1.0

1.2
0.8

1.2
0.8

1.0

1.2

0.9

0.9

Table 7.1: The vertex and angular resolution of MS-Fit. The resolution is defined as the width
where 68 % of the events are included.

161

7.6

Decay Electron Search

The selection efficiency of µ-like events can be improved by requiring the detection of the electron
produced by the muon decay (decay electron). After the primary event, the presence of a decay
electron is searched for. The electrons found in this way are classified depending on the following
criteria:• Sub-event type
Decay electrons observed as a separate event (sub-event) as shown in Figure 7.7.
• Primary-event type
Decay electrons observed in the same event as the primary event.
• Split type
Decay electrons that appear at the end of the primary event timing window, and therefore
their energy deposition is split between the primary event and the sub-event.
For the Sub-event type decay electrons, the following criteria are required:
1. The time interval from a primary event(∆t) is < 30 µsec.
2. The total number of hit PMTs is greater than 50 (25 for SK-II).
3. The goodness of vertex fit is greater than 0.5.
4. The number of hit PMTs in a 50 nsec time window is greater than 30 (15 for SK-II).
5. The total number of photoelectrons is less than 2000 (1000 for SK-II).
6. The number of hit PMTs in a 50 nsec time window (N50 ) is greater than 60 (30 for SK-II).
noting that the numbers are different for the SK-II period due to the difference in the phototube
coverage.
For the Primary-event type decay electrons, another peak is searched for in the primary event
time window after the initial main event. Further required conditions are:7. The number of hit PMTs in a 30 nsec time window is greater than 40 (20 for SK-II). (forthe Primary-event type and Split type)
8. 0.1 µsec < ∆t < 0.8 µsec or 1.2 µsec < ∆t < 30 µsec. (for all types)
N50 =60 (30 for SK-II) corresponds to about ∼11 MeV of electron energy. Criterion (1.) rejects
the gamma emission from a µ− capture on 16 O nuclei. The decey electron reconstruction is not
so efficient around 1 µsec and Criterion (8.) rejects the decays around this timing. The efficiency
of detecting these electrons are 80 % and 63 % for µ+ and µ− , respectively.
162

1990/00/00:NoYet:NoYet
1990/00/00:;R= 0:NoYet
R :Z : PHI : GOOD
0.00: 0.00: 0.00:0.000
CANG : RTOT : AMOM : MS

NUM
10
RUN
21511
SUBRUN
1
EVENT
20423
DATE 2003-Jan-16
TIME
21: 8:19
TOT PE:
1353.9MAX PE:
28.3
NMHIT :
672
ANT-PE:
298.4
ANT-MX:
57.9
NMHITA:
112

NUM
10
RUN
21511
SUBRUN
1
EVENT
20425
DATE 2003-Jan-16
TIME
21: 8:19
TOT PE:
155.6
MAX PE:
5.3
NMHIT :
127
ANT-PE:
314.3
ANT-MX:
57.9
NMHITA:109

1990/00/00:NoYet:NoYet
RunMODE:NORMAL
0:NoYetTRG ID :00001011 1990/00/00:;R=R :Z : PHI : GOOD
T diff.:0.231E+05us0.00:
0.00: 0.00:0.000
FEVSK :81002801 CANG : RTOT : AMOM : MSnOD YK/LW:59/66
SUB EV : 0/ 3
Dcye: 0( 0/ 0/ 0/ 0)

RunMODE:NORMALTRG ID :00000011
T diff.: 3.48
us
FEVSK :81002801
nOD YK/LW: 3/ 2
SUB EV : 2/ 3
Dcye: 0( 0/ 0/ 0/ 0)

Comnt;

Comnt;

Figure 7.7: An event display of a Sub-event type decay electron where the time difference between
the two events is ∼3.5 µsec. The primary event (cosmic ray stopping muon) is shown on the left
and the following decay electron event on the right.

7.7

Momentum Reconstruction

The momentum is estimated by considering the relationship between the number of observed
Cherenkov photons inside a cone with a half opening angle of 70◦ and the momentum of theparticles. In the reconstruction, the momentum for individual rings is determined by separating
the observed charge (p.e.) in each hit PMT into the charge contribution from each ring. The
separation of the observed charge is carried out based on the expected charge distribution fromeach ring as a function of the opening angle θ and uniform in azimuthal angle φ. The observed
charge (p.e.) in the i-th PMT from the n-th ring is estimated as:
exp
qi,n
obs
qi,n
= qiobs × P exp
n0 qi,n0(7.24)

obs is the fractional charge from
where qiobs is the total observed charge in the i-th PMT, and qi,nexp
the n-th ring in the i-th PMT. The expected charge is qi,n , and the sum of n0 is over all theidentified rings.
To calculate the charge inside the 70◦ cone, the charge in each PMT is corrected for the lightattenuation in water and the acceptance of the PMT as follows: r  cos Θ 
X
X

GMC 
i
i
obs
 α×
 (7.25)
RT OTn =
q
×
exp
×
−
S
i
i,n

Gdata Lf (Θi )◦◦
θi,n <70−50nsec<ti <250nsec

θi,n <70

whereα: Normalisation factor
163

Gdata , GMC: Relative PMT gain parameter for data and Monte Carlo simulation

θi,n: Opening angle between the n-th ring direction and the i-th PMT
direction

Θi: Angle of photon arriving direction relative to the i-th PMT facing
directionti: TOF subtracted hit timing of the i-th PMT position

L: Light attenuation length in waterri: Distance from the vertex position to the i-th PMTf (Θ): Correction function for the PMT acceptanceSi: Expected charge (p.e.) for the i-th PMT from scattered photons

To reject the effect from muon decay electrons, the summation is restricted to be within thetime window from -50 nsec to +250 nsec around the peak of the TOF subtracted hit timing
distribution. The time variation of observed data is corrected for the attenuation length inwater, L, by the continuous measurement of cosmic ray through-going muons (Section 5.1.4).
The momentum resolution is approximately given by:(
2.6
0.6 + √p(GeV/c)% for electron
σp =(7.26)
0.7√
1.7 + p(GeV/c) % for muon
The resolution is defined as the 1 σ width of the Gaussian fit.

7.8

Ring Number Correction

Some of the rings are mis-identifited and have low momentum or are overlapped with otherreconstructed rings. So at this stage rings that match either of the following 2 sets of criteria
are deemed to be mis-identified and are removed from the reconstruction.
The procedure compares the i-th reconstructed ring with the j-th identified ring, where i 6= j.
The first set of criteria remove the i-th ring if it is of low momentum and overlapping with alarger momentum ring. The second set removes the i-th ring if it has a very low momentum
fraction.
(1-1) pi < pj ; i 6= jwhere pi is the momentum of i-th ring.
(1-2) θij < 30◦
where θij is the angle between the i-th and j-th ring directions.
(1-3) pi cos θij < 60 MeV/c
where pi cos θij is the momentum of i-th ring projected perpendicularly to j-th
ring.
or
164

Figure 7.8: Reconstruction efficiency for single-ring events as a function of momentum before
and after the ring number correction. The left figure shows the νe events and νµ on the right.
The efficiency before the ring number correction is shown by the white points, and the final result
by the black points.
(2-1) pi < 40 MeV/c(2-2) pi /ptot < 0.05
where ptot is the total momentum for all rings.
After removing poorly reconstructed rings by one of either of the two sets of criteria above, theresulting ring counting efficiency for single-ring events can be seen in Figure 7.8

7.9

π 0 Reconstruction

A π 0 produced from an interaction immediately decays into two γs, which can be reconstructed
as a π 0 by the algorithm described in Section 5.3.2. However, if one of the γ from π 0 decay is
missed by the reconstruction, the event may sometimes be classified as a single-ring e-like event.
This mis-identification may occur for the following reasons:
1. The energies of the two γs are considerably asymmetric and the energy of the second γ(Eγ2 ) is too small to be reconstructed as a ring.
2. The rings of the two γs overlap and are reconstructed as one ring when the opening angle
between the two γs is small.
The relationship between the momentum of the second γ and the opening angle between thetwo γs is shown in Figure 7.9. Figure 7.9 also shows the momentum distribution of the expected
NC single π 0 events, with the hatched region showing events mis-identified as single-ring e-like.
The mis-identified events amount to 33% of the NC single π 0 events.
The π 0 fitter assumes the existence of two overlapping rings and compares the observed
charge distribution with the expected charge distribution of two γs. The direction of the more
165

3500

500

3000

400
350

2500

Number of events

Momentum of second γ (MeV/c)

(b)

(a)

450

300
250
200
150

2000
1500
1000

100

500
50
0

0

20

40

60

80

100

120

140

160

0

180

0

200

400

600

800

1000 1200

1400

0Angle between two γs (degree)

True π Momentum (MeV/c)

Figure 7.9: (a) A scatter plot of the true momentum of the second gamma against the true
opening angle between two gammas for NC1π 0 events in the atmospheric Monte Carlo events.
(b) Distribution of the true π 0 momentum for NC1π 0 events in the atmospheric Monte Carloevents. The hatched region shows the NC1π 0 mis-identified as single-ring e-like events.
energetic γ is fixed in the direction from the precise vertex reconstruction, and the second γ issearched for by varying its direction and the energy shared between the two γs until a best-fit
combination is found. The two gamma-rays are assumed to be emitted from one vertex.
The expected charge distribution for the i-th PMT is calculated as follows:ri
11 − q×
qiexp,π = αe × Qexp
× f (Θi ) + qiscatt(7.27)π (θi , pγ , ri ) ×
exp(r
/L)
2
2
i
r +R
i

P MT

where
αe

: Normalisation factor

Qexpπ: Expected photon distribution from a gamma-ray as a function of θi ,pγ and ri

θi: Opening angle between the i-th PMT direction and the ring directionpγ: Initial gamma-ray momentumri: Distance from the vertex to the i-th PMT

RP M T: Radius of a PMT∼25 cm

L: Light attenuation length in waterf (Θi )

: Correction for the PMT acceptance as a function of the photon incidence angle Θiqiscatt: Expected charge for the i-th PMT from scattered photons
166

Qexp is the expected photon distribution as a function of opening angle θi from the ring direction
to the i-th PMT, the initial gamma-ray
pγ and the distance from the vertex to theq momentum
22i-th PMT ri . The factor 1 − ri / ri + RP M T corrects for the solid angle, and exp(-ri /L)
accounts for the light attenuation. Note that the charge contribution from scattered hits isalready included in Equation (7.27), and qiscatt is calculated in the same way as that describedin Section 7.4.4.
The best-fit configuration of two gamma-rays is determined by a log likelihood method. The
likelihood is defined as:



X
Lπ =
log prob qiobs , qiexp,π(7.28)
i=1

The expected charge from a pion, qiexp,π , is optimized by changing the direction and momentum
fraction of the second gamma to achieve the maximum likelihood value for Lπ . The probability
function, prob(qiobs , qiexp ), is the same as that defined in Equation (7.13) and checks how well
qiexp,π matches qiobs .
After the first fit, the π 0 reconstruction process is re-iterated with finer stepping size toimprove the fitting performance for events where two rings are almost overlapped. Such events
give a narrow Lπ distribution. The maximum likelihood between the first and second fit is chosen
as the fitting result.
The invariant mass of the π 0 , Minv , is defined as:q(7.29)
Minv = 2Eγ1 Eγ2 (1 − cos θγγ ).

7.10

Upward-Going Muons

The aim of the upward-going muon samples is to analyse neutrino events with very high energy.
In the L/E analysis, it is difficult to determine the resolution of these events and they are notincluded in the present analysis. Figure 7.10 shows the flux of these upward-going muon events
against the energy of the primary neutrino.

167

0.7

ShoweringNon-showeringdΦ/dlogEυ( 10-13 cm-2 s-1 sr-1)

0.6

Stopping

0.5
0.4
0.3
0.2

Figure 7.10: Flux of Upward going muon events
against energy by event type. The upward muon
events are shown by the dotted line for upward
stopping, the dashed line for non-showering andthe solid line for showering events.

168

0.1
0
1

10

2

10
10
Eυ (GeV)

3

10

4

10

5

Chapter 8

An Update of the 2-Flavour L/E
Analysis
Neutrino Oscillations is a theory where the probability of a neutrino changing from one flavour
to another depends on the ratio of the distance travelled by the neutrino and its energy, Lν /Eν .
The expected oscillatory pattern can be observed by plotting the variable Lν /Eν , reproducing
the characteristics of Neutrino Oscillations. The L/E survival probability for 2-flavour νµ → ντoscillations is shown in Figure 8.1, for the typical parameters sin2 2θ23 = 1.0 and ∆m232 =2.5 × 10−3 eV2 . The oscillation parameters were defined as in the following 2-flavour survival
probability:
1.267∆m232 Lν
P (νµ → νµ ) = 1 − sin2 2θ23 sin2
.
(8.1)
Eν
The analysis in this chapter aims to reconstruct primarily the first dip, at the region where
the oscillation probability reaches its first maximum point. As it can be seen from the figure, thesubsequent oscillation maxima are closely spaced together in log10 (L/E). So it is very challenging
to achieve a high enough resolution to resolve the second or third oscillation maxima. The ∆m232
parameter characterises the range of L/E values that the deficit in νµ occurs; and sin2 2θ23
characterises the magnitude of the deficit. The increased sensitivity to ∆m232 of this analysis
comes from reconstructing the position of the dip. While sensitivity to sin2 2θ23 is achievedby the relative difference of the distribution for no oscillations at low L/E and the averaged
oscillations at high L/E, as well as from the depth of the dip.
The L/E Analysis reconstructs the main characteristics of Neutrino Oscillations by selecting
a precision sample of data with a good resolution in L/E. This gives it an improved sensitivity tothe oscillation parameters, and in particular this analysis has the most discriminating sensitivity
to ∆m232 from atmospheric neutrinos. Relying on this advantage, any discrepancy between
the ∆m232 for neutrinos and anti-neutrinos was investigated to confirm the CPT symmetry in
Neutrino Oscillations in Chapter 9. The strength of the L/E analysis with 3-flavour and matter
effects was considered in Chapter 10, while the deviation from pure 3-flavour oscillations in theprescence of Non-Standard Neutrino Interaction (NSI) effects was studied in Chapter 11.
Each of the analyses share the Lν /Eν distributions. How these variables are extracted andtheir treatment are described in this chapter, along with the standard 2-flavour L/E analysis.
The L/E data quality, the likelihood analysis method, and the systematic uncertainties are alsopresented in detail here. So the explanations that follow are relevant to all the analyses presented
169

Figure 8.1: L/E survival probability for 2flavour νµ → ντ oscillations for sin2 2θ23 = 1.0
and ∆m232 = 2.5 × 10−3 eV2 . The first oscillation
dip is seen at L/E ∼500 km/GeV. The subsequent oscillation maxima are closely spaced together in log10 (L/E)
in Chapters 9, 10, and 11.

8.1

Dataset and Treatment

For the analyses in this thesis a common dataset was used and divided into samples of interest.
The data collected span over 10 years and 4 setups of the detector: SK1 (1489.2 days), SK2(798.6 days), SK3 (518.084 days) and SK4 (1096.7 days). This data has gone through a processof reduction to remove background events, fully described in Chapter 6. Then the following event
samples were collected: Fully-Contained (FC) Single-Ring, FC Multi-Ring, Partially-Contained(PC) Stopping and PC Through-Going. These samples span over a wide range of energies.
Depending on the energy, the event topology differs and so the treatment of the data are different
for each sample. The details of this selection procedure are written here.

8.1.1

Fully-Contained: Single-Ring (FC1R) and Multi-Ring (FCmR)The Fully-Contained samples were chosen to select events that remain within the Fiducial Volumeof the Inner Detector. Typical energies of FC events may range from some 10s of MeV to around
100 GeV. Firstly, to select the FC events, the following cuts were made:• Reconstructed vertex is within the Fiducial Volume. Distance from the vertex to the top
and bottom ID wall should be > 1.5 m, and to the barrel ID wall should be > 1 m.
• Number of hit PMTs in any cluster in the OD should be < 10 (16) for SK1 (SK2-4).
• Visible energy (Evis ) should be > 30 MeV.
The Fiducial Volume defined for FC events used in the L/E distributions was larger than in
the Zenith angle analysis or the PC samples. This was done to increases statistics, which werereduced by requiring a high resolution in Lν /Eν (Section 8.3).
Then to select the Single-Ring (FC1R) events, which span up to ∼10 GeV, the following
additional cuts were made:• The event should be identified as having only one ring.
170• The ring should be reconstructed as µ-like and have momentum pµ > 200 MeV/c.
To select the FC Multi-Ring (FCmR) sample, the FC cuts were applied, followed by thefollowing selections:• The number of rings identified should be > 1.
• The most energetic ring should be reconstructed as µ-like and have momentum pµ > 600 MeV/cand Evis > 600 MeV.

8.1.2

Partially-Contained: PC Stopping And PC Through-Going

The Partially-Contained (PC) sample consisted of events with an outgoing muon that left theInner Detector (ID) and penetrated into the Outer Detector (OD). There was a special treatment
for the energy reconstruction by considering the muon track length and the dE/dx for the muonenergy deposited. The details are described in Section 8.2.1. The cuts made to select the PC
sample were:• Vertex is within the Fiducial Volume. Distance from the vertex to the ID wall should be > 2 m.
• Number of hit PMTs in any cluster in the OD should be > 10 (16) for SK1 (SK2-4).
• Visible energy (Evis ) should be > 350 MeV.
• The total charge deposited in the ID should be > 3000 (1500) photoelectrons for SK1 andSK3-4 (SK2).
• The reconstructed neutrino energy (Evis2 ) should be < 50 GeV.
The last cut is related to the resolution cut, discussed in Section 8.3.4; and Evis2 is the totalenergy reconstructed for all outgoing charged particles, explained in Section 8.2.1. Then the
PC sample was further divided into PC Stopping and PC Through. PC Stopping consisted ofevents where the outgoing muon stopped in the OD, while in the PC Through sample the muonspassed through the OD and exited the detector. The details of how the events were separated
into these two categories is described in Section 8.1.3. To select the PC Stopping events, thefollowing additional cuts were made:• Must be classified as stopping in the OD. P Eanti < P Eexp /1.5.
• The most energetic ring or the second most energetic should be reconstructed as µ-like.
The µ-like cut was made to increase the purity of muon events in this sample. This was notnecessary for the PC Through sample which retained a high purity of muons. To select PC
Through events, the PC cuts were applied then the following PC separation selection was made:• Must be classified as passing through the OD. P Eanti > P Eexp /1.5.
1718.1.3

PC Stopping and Through-Going Separation

The PC sample was separated into two categories. Events in which the muon was expected
to have stopped in the OD were classified as PC Stopping and those which passed through as
PC Through-Going. In the case of PC Stopping, the energy of the muon can be reconstructedaccurately by considering the path it took and calculating the energy loss from dE/dx. On theother hand, the energy calculated in the PC Through sample was a lower limit for the muonenergy.
The separation was made by considering the charge deposited in the OD (P Eanti ) and comparing it with the expected charge deposition in the OD of a muon exiting the OD (P Eexp ).
Specifically, the charge measured was the maximum number of photoelectrons in a sliding 500 ns
window from −400 ns to +500 ns from the event time. If the ratio of these charges, P Eanti /P Eexp ,was less than 0.67 (or 1/1.5) then the event was classified as PC Stopping and PC Through-Goingotherwise. A ratio close to 1 would be expected for a PC Through event.
The expected charge, P Eexp , was a tuned value since the charge distribution in the OD
depends on the structural geometry and the make-up of old and new PMTs in the OD. A MC
sample was used to measure the charge deposited in the OD by the method described above,and this value was normalised by the calculated path through the OD for an exiting muon. This
formed a distribution of charge/cm and had a peak at the charge/cm for a PC Through event.
These peak values were measured for several regions of the OD: Top, Bottom, Barrel, and Edge.
So in the analysis, the path through the OD for a PC event could be calculated and multipliedby the tuned peak values to form the variable P Eexp . Then the P Eanti /P Eexp ratio was used toclassify the event. The distributions of P Eanti /P Eexp (or Qobs /Qexp ) and the cut position are
shown in the left of Figure 8.2.
It was found that due to the charge response in the OD, a significant number of PC Throughevents were leaking into the PC Stopping sample – enough to cause an asymmetry in a zenith
distribution of the PC samples. The effect was that the peak of the P Eanti /P Eexp (or Qobs /Qexp )distributions pulled away from a ratio of 1 with a variation in the Z coordinate or radius. Thiswas corrected for by fitting the peak of Qobs /Qexp against the Z position or radius, and thencalculating a correction factor to make the peak of Qobs /Qexp = 1 at all positions. In the rightof Figure 8.2, the top plot shows the asymmetry of Qobs /Qexp which was fitted; and the bottom
plot shows a heat map of the number of events after applying the correction. A greater numberof events is shown by red and fewer events by blue.
The systematics for the PC separation were calculated with the Qobs /Qexp distributions. The
peak was fitted for data and for MC, then the MC distribution was shifted to make the data andMC peaks match. The percentage of PC Stopping events that changed category was taken as
the estimation for the uncertainty in the PC separation.

8.2

Reconstruction of L/E

Neutrino Oscillations have an L/E dependence so it is vital that both the flight path and theenergy of the neutrino are reconstructed as accurately as possible. If the resolution is not
sufficient, the L/E dependence will average out. This averaging effect can also be seen at large
values of L/E, after the first oscillation maximum, where the oscillations become very rapid and
it is experimentally difficult to resolve the L/E pattern, as seen in Figure 8.1.
172

Figure 8.2: The Qobs /Qexp (≡ P Eanti /P Eexp ) of PC events penetrating the Barrel section isshown on the left, and figures illustrating the PC charge asymmetry correction on the right.
Left: Qexp is the expected charge for a PC Through event, so for a Qobs /Qexp ratio of 1, themuon likely exited the OD. This ratio was used to classify the PC events as shown by the cut
arrows. The black points are data and the black histogram is the non-oscillated MC, the blue
histogram is oscillated MC, and the red hatched histogram are the true stopping events estimatedfrom the MC. Right: the asymmetry of the Qobs /Qexp with the Z coordinate along the Barrel
region in the top figure; and a heat map of Qobs /Qexp after applying the correction to the ratio.
A greater number of events is shown by red and fewer events by blue. After the correction, theratio is flat w.r.t. the Z coordinate, shown by the flat red band.

173

8.2.1

Neutrino Energy

The neutrino energy calculation was based on the total energy reconstructed for all outgoing
charged particles, Evis2 . Since various event topologies were expected, the precise energy reconstruction depended on the sample. In particular for PC events, as some of the energy wasdeposited outside of the ID.
FC Single-Ring
Since the event is fully contained in the Fiducial Volume and only 1 ring was found, the energy
reconstruction is rather straight forward. The total visible energy is taken to be the energy ofthe outgoing muon:
Evis2 = Eµ .
(8.2)
FC Multi-Ring
In the case of FC Multi-Ring events, the energy reconstruction assumes that the most energetic
ring was formed by the outgoing muon from a CC interaction. The remaining rings are assumed
to be pions if the ring is identified as µ-like and electrons if the ring is e-like. Then the totalenergy is the summation over all of these rings:
Evis2 = Eµ +n
X(Eπi or Eei ).

(8.3)

i=2

Careful treatment is required for the µ-like rings expected to be formed from pions. In thestandard reconstruction, the energy of such rings assumes that the source charged particle isa muon, and the Cherenkov energy threshold for the muon is used. So to reconstruct the pionenergy, the assumed muon energy is adjusted by considering the difference in the energy thresholdfor Cherenkov emission.
Whether the particle is a pion or a muon, the visible energy in the ring does not include theenergy before the threshold, like so:Ering = Eµ − Eµth ,orEring = Eπ − Eπth ,(8.4)

where the energy thresholds are defined to be:Eµth = 160 MeV,

(8.5)

Eπth = 212 MeV.

(8.6)

Furthermore, assuming the dE/dx in water is constant and also independent of the energy andthe type of particle, then the number of emitted Cherenkov photons is simply proportional to
the track length. Then the energy deposited by a muon or pion, estimated from the same ring,is equal:Ering = Eµ − Eµth = Eπ − Eπth .
(8.7)
Then the energy of the pion can be reconstructed:
Eπ = (Eπ − Eπth ) + Eπth
= (Eµ − Eµth ) + Eπth
= Eµ + (Eπth − Eµth ).
174

(8.8)

Figure 8.3: Distribution of the momentum of the most energetic muon ring against the track
length in the ID, Pµ /Dinner . The histograms have been divided into Quasi-Elastic (QE) interactions (hatched) and non-QE interactions using the Atmospheric MC PC Single-Ring sample.
Events to the right of the arrow are separated into a muon ring and an electron ring.
Therefore Eπ is assumed to be,
Eπ = Eµ + 52 MeV.

(8.9)

PC Sample
The PC sample consists of events in which the muon escapes the ID and penetrates into the OD.
However, the energy for such events can still be accurately reconstructed by considering the pathof the muon and the energy deposition, dE/dx, in the OD and the dead region between the ID
and OD. Therefore,
Evis2 = EID + Edead + EOD .
(8.10)For PC events, the incoming neutrinos are quite energetic (∼10 GeV), so there is a significant
possibility of collinear outgoing particles being mis-reconstructed as a single ring in the ID.
Therefore a correction is made to the number of rings before the energy in the ID, EID , iscalculated.
These mis-reconstructed rings can be distinguished by comparing the momentum of the ring,Pµ , against the track length in the ID, Dinner , as shown in Figure 8.3. A Quasi-Elastic event
is a clean single ring CC event in which the proton was below Cherenkov threshold, while a
Non-Quasi-Elastic interaction should have multiple rings. A wide tail can be seen in Figure 8.3,
which is due to near-parallel pions and/or other particles causing the reconstructed momentum
to be considerably larger than what would be expected for a muon (∼2.4 MeV/cm).
This is corrected for by separating the muon rings with suspected overlapping particles. If themost energetic muon ring has Pµ /Dinner > 3.0 MeV/cm, then the ring is separated into a muonring with momentum estimated from the track length in the ID and an electron ring. The muon
momentum is calculated as Dinner × dE/dx, where dE/dx = 2.4 MeV/cm. Then the expectedcharge from the muon is subtracted from the overlapped ring, and the remaining charge is used
to reconstruct the energy of the electron. Following this separation, the energy in the ID, EID ,can be calculated with improved resolution in the same way as the FC sample.
175

Edead is the energy deposited by the muon in the dead region that is situated between the
ID and OD. The energy is estimated by calculating the track length of the muon through thedead region and applying a dE/dx of 2.4 MeV/cm:
Edead = Ddead × dE/dx,(8.11)

where Ddead is the track length of the muon in the dead region estimated from the vertex andring direction.
EOD is the energy deposited by the muon in the OD. The track length, DOD , is estimatedin a similar way to that of the Edead case, however a modification is made depending on the PC
classification. For OD Through events the track length is taken to be the full length of the muontrack through the OD, whereas in the OD Stopping sample the track is assumed to be half thislength. Then with a dE/dx = 2.4 MeV/cm, EOD is estimated as:
EOD = 0.5× DOD × dE/dx
EOD =DOD × dE/dx(for OD Stopping events),(for OD Through events).

(8.12)

Reconstructing Eν
Once the total energy of all the charged particles, Evis2 , has been reconstructed, the neutrinoenergy is estimated via a fitted polynomial expression as a function of log(Evis2 ). The relationshipbetween the neutrino energy, Eν , and Evis2 was determined by fitting the following polynomialwith the Monte Carlo:Eνrec = Evis2 × (a + bx + cx2 + dx3 ),(8.13)
where x = log(Evis2 ). The energy fitting for each of the event samples from SK4 can be seen in
Figure 8.4. The energy resolution of the events in each sample can be seen in Figures 8.5, while
the energy resolution against the observed energy is shown in Figure 8.6. These distributions
were made from the MC simulation, showing the expected energy reconstruction resolution forthe data, which was reconstructed in the same way. Details of the 70% resolution cut used inthese figures can be found in Section 8.3.

8.2.2

Neutrino Flight Path Length

To estimate the neutrino flight path length, first the neutrino direction is reconstructed and thezenith angle calculated. The flight length can then be determined from the neutrino incident
zenith angle. The details of the direction reconstruction varies for each event sample.
FC Single-Ring
In the FC Single-Ring sample, the neutrino direction is assumed to be simply the reconstructed
direction of the muon. Then the zenith angle of the neutrino is taken to be:cos Θrecν = cos Θµ ,(8.14)

where cos Θrec
ν and cos Θµ are the cosine of the zenith angle of the reconstructed neutrino andmuon, respectively.
176

Figure 8.4: Eν Reconstruction from fitting Evis2
to the true neutrino energy estimated by theMonte Carlo. The fitting for each of the event
samples is shown, and the PC samples were further divided into single-ring and multi-ring. The
fitted result for each SK period is shown in Appendix A.

70000

SK 4 FC 1-Ring Sub-GeV

60000

σ = 24%2500

SK 4 FC Multi-Ring Sub-GeV

3000

σ = 30%SK 4 PC Stopping
σ = 19%

2000

2500

50000
2000

40000

1500

30000
Number of Events

1000

1000

20000

500

500

10000
0

1500

-1

-0.5

0

0.5

22500

SK 4 FC 1-Ring Multi-GeV

20000

σ = 14%

1

0

-1

-0.5

0

0.5

1

5000

5000

4000

4000

10000

3000

3000

7500

2000

2000

1000

1000

12500

5000
2500
0

-1

-0.5

0

0.5

1

0

-1

-0.5
0
0.5
(Erec-Etrue)/Etrue

-0.5

0

0.5

1

SK 4 PC Through-Going6000

σ = 24%

15000-1

SK 4 FC Multi-Ring Multi-GeV

6000

17500

0

1

0

σ = 44%

-1

-0.5

0

0.5

1

Figure 8.5: L/E Analysis energy resolution for SK4. The difference of the reconstructed andtrue neutrino energy with respect to the true neutrino energy, estimated by the MC, is shown.
Events to the right of zero have been reconstructed with an energy larger than the true energy
and vice versa for the left. The black solid histogram is for events satisfying the 70 % resolution
cut, and the green shaded area corresponds to 68% of the events. The energy resolution for each
SK period is shown in Appendix A.

177

(a) SK1(b) SK2(c) SK3(d) SK4

Figure 8.6: Eν Resolution against the lepton momentum (or Evis2 for FC multi-ring), estimatedwith the MC. Each of the event samples are shown by the markers shown in the legend. The energy resolution for the FC samples are quite similar between each SK period, while the resolution
for the PC samples vary slightly.

178

Figure 8.7: Neutrino flightpath through the
Earth vs the incoming zenith angle of a neutrino. The flightpath changes rapidly for horizontal events. For upward-going events, theflightpath is so long that uncertainties of production in the atmosphere become negligible.
FC Multi-Ring
For the FC Multi-Ring sample, the neutrino direction is reconstructed as the momentum weighted
sum of the directions of all the identified rings. The zenith angle of the neutrino is taken to be:cos Θrecν = cos Θsum ,nXd~sum = pµ · d~1 +(pπ or pe ) · d~i ,(8.15)(8.16)

i=2

where the first ring is the muon ring with direction d1 and momentum pµ . The other rings aretaken to be pions or electrons with momentum pπ or pe , respectively. di is the direction of thei-th ring, with n rings in total. Then d~sum is their momentum weighted direction summed over
all identified rings. From this the summed zenith angle is calculated, Θsum , giving the zenith
angle of the neutrino, Θrecν .
PC Samples
The reconstruction of the neutrino direction for the PC samples is similar to that of the FC
samples. However, for the Multi-Ring events there is an additional weighting factor applied tothe most energetic muon ring in the direction calculation. This factor, α, is taken as 2.0 for OD
Stopping events and 4.0 for OD Through events. This weighting was optimised to get the best
performance for the reconstruction. The zenith angle of the neutrino is calculated like so:cos Θrecν = cos Θµcos Θrecν = cos Θsum
d~sum = α · pµ · d~1 +nX(for Single-Ring),(8.17)(for Multi-Ring),(8.18)(pπ or pe ) · d~i .

i=2

The angular resolution for several samples is shown in Figures 8.8.
179

(8.19)

45000SK 4 FC 1-Ring Sub-GeV

40000

SK 4 FC Multi-Ring Sub-GeV

4000

σ = 61.0°

3500

30000

3000

25000

2500

20000

2000

15000

1500

10000

1000

400

5000

500

200

0

0

Number of Events

35000

0

16000

30

60

SK 4 FC 1-Ring Multi-GeV

800
600

30

60

90

120 150 180

SK 4 FC Multi-Ring Multi-GeV

5000

σ = 18.5°σ = 29.0°

12000

0

16000

10

20

30

40

50

60

SK 4 PC Through-Going14000

σ = 10.7°

10000

8000

3000

8000

6000

2000

6000

4000

4000
1000

2000
0

0

12000

4000

10000

σ = 19.7°

1200
1000

90 120 150 180
0
6000

14000

SK 4 PC Stopping

1400

σ = 45.0°

0

15

30

45

60

75

90

0

2000
0
15 30 45 60 75 90
Angular Resolution (degrees)

0

0

10

20

30

40

50

60

Figure 8.8: Angular resolution in SK4 for the L/E samples, estimated with the MC. The distribution of angles w.r.t. the true neutrino direction is shown. The black solid histogram is forevents satisfying the 70 % resolution cut, and the green shaded region corresponds to 68% of
events. The angular resolution for each SK period is shown in Appendix A.

180

(a) SK1(b) SK2(c) SK3(d) SK4

Figure 8.9: Angular Resolution against the lepton momentum, estimated with the MC. Each of
the event samples are shown by the markers shown in the legend.

Reconstructing Lν

The flight path length of the neutrino is estimated from the reconstructed zenith angle of theneutrino. For upward-going neutrinos traversing much of the Earth (∼ 104 km), there is a close
relationship between the path length and zenith angle. For horizontal events, the variation of
the path length with the zenith angle is rapid (Figure 8.7) and the resolution is expected to belower than for vertical events. Then for downward-going neutrinos, and for horizontal events
as well, the uncertainty in the neutrino production height in the atmosphere (∼10 km) is moresignificant than for upward going neutrinos. The calculation of the flight path length from thezenith angle is based on the Honda flux calculation, and the relationship between these variables
can be seen in Figure 8.7. The angular resolution against observed energy is shown in Figure 8.9.
181

Figure 8.10: The L/E distributions satisfying the 70% resolution cut and oscillated with sometypical parameters (sin2 2θ23 , ∆m232 ) = (1.0, 2.4 × 10−3 eV2 ) are shown on the left. The contributions from each of the samples is shown and labelled in the legend. The dashed histogram shows
the L/E distributions without oscillations. The two bump structure can be understood by studying the L/E contours on the (cos θ, Eν ) plane in the right plot. Just above L/E = 102 km/GeV,
the resolution requirements cut out a lot of horizontal events, which can be seen in the right plotof Figure 8.14.

8.3

L/E Event Resolution Cut

As well as improving the reconstruction of Lν and Eν as described in the previous sections, theresolution of the quantity Lν /Eν was estimated. This was calculated with respect to the neutrino
recenergy and incident zenith angle, for a grid of points in the (cos Θrec
ν ,Eν ) parameter space. A
resolution map was created to cut out events which were likely to have poor resolution and cause
the L/E pattern to become averaged out.
In Figure 8.10, an L/E distribution can be seen on the left, separated by sample, and theL/E contours with respect to neutrino energy and incident zenith angle on the right. The L/E
distributions have had a resolution cut applied to them, keeping events that were expected tohave a relatively good resolution in L/E. This requirement cuts out many of the low energy
events or horizontal-going events.
On the right plot of Figure 8.10, in the region of L/E between 102 –103 km/GeV a narrow
band of mostly horizontal events or low energy events are traced. This accounts for the reduced
statistics in the L/E plot in this region after the resolution cut was applied. The majority of
selected events were either downward-going, L/E between 10–102 km/GeV, or upward-going,
L/E between 103 –104 km/GeV, with more statistics for lower energy events. This explains thecharacteristic two-bump structure of the basic L/E distribution.

8.3.1

Estimation of the L/E Resolution

The L/E resolution was estimated using a MC sample equivalent to 100 years of data. The
resolution was mostly limited by the reconstruction of the neutrino flight path length. This wasdependent on the angular correlation between the incoming neutrino and the measured outgoing
lepton direction. For higher energy events, there is a strong correlation between these directions,
182

z2

z1

“Reconstructed” νdirection“True” νdirection

ɸ
dθ
ω

y2
x2
y1

Figure 8.11: The estimation of the L/E resolution considers the angle dθ between the reconstructed and true neutrino direction, which iswhere the uncertainty in the flight path comesfrom. The “reconstructed” direction, ω, was assumed in a loop, and the “true” direction was assumed to be at an angle dθ. Here direction refers
to the zenith angle, the incoming neutrino direction. The true directions were rotated through
2π in φ w.r.t. the reconstructed direction, sampling 30 steps. Then these true directions wererotated into detector coordinates, assuming ω
was in the ZY-plane of the detector.

of about 10◦ –20◦ . However, for lower energy events the reconstructed lepton direction can differ
considerably from the incoming neutrino direction – upwards of 30◦ below Eν ∼ 1 GeV. See
Figure 8.9 for the calculated angular correlation against energy.
To calculate the effect of this on the L/E resolution, a representative sample of 300–1000
events of a particular neutrino energy range was collected. For each of these events, the angle
between the reconstructed lepton direction and the true neutrino direction, dθ, was calculated.
This gave a distribution of events with a characteristic dθ for a particular neutrino energy.
In the estimation, a particular reconstructed neutrino zenith direction, ω, was assumed asshown in Figure 8.11. Then the “true” neutrino direction was assumed to be at an angle dθfrom this reconstructed direction, and rotated through 2π in φ with respect to the reconstructed
direction1 as depicted in Figure 8.11. In this φ rotation, 30 steps were sampled, giving 9000–30000
test events for a given reconstructed Eνrec and cos Θrec
ν .
The “reconstructed” direction was assumed to be at a zenith angle, ω, in the ZY-plane of thedetector, and the “true” neutrino directions were rotated about the x-axis to match the ZY axesin the detector coordinates — a rotation from the first set of coordinates (green) to the second
set of coordinates (black) in Figure 8.11. Then from this assumed “true” neutrino direction, theflight path length was determined, and the quantity (L/E)true /(L/E)rec. was calculated.

8.3.2

Definition of the L/E Resolution

recThis process was repeated for each point on the (cos Θrec
ν ,Eν ) parameter space, producing a(L/E)true /(L/E)rec. distribution at each point. From the distributions, two quantities were
calculated: the peak of the distribution, and the estimated resolution from that peak. The peak
was defined as the ratio of the (L/E)true /(L/E)rec. distribution containing the most test eventsin a ±25% window from that ratio. Once the peak was found, the resolution was defined as theplus-or-minus-percentage of the peak value that contained 68% of the test events. For example,
if the peak was found at a ratio of 1.0, and 68% of the test events were within a ratio of 0.7–1.3(1.0±30%), then the resolution was defined as 30%.
1

In this estimation of the L/E resolution, the neutrino flux was assumed to be uniform in φ. This estimation
produced a value to cut by depending on the reconstructed Eν and cos Θ of the event, and the geometry of the
Earth. The L/E resolution estimated here was only used for data screening. Since the data and simulation was
treated in the same way in the analysis, it introduced no bias in the results.

183

rec
A sample of (L/E)true /(L/E)rec. distributions from the (cos Θrec
ν ,Eν ) parameter space for
SK1 FC1R MC events can be seen in Figure 8.12. The numbers in the top right of each panel
show the neutrino energy and zenith angle in black and calculated resolution in red. The reddashed lines represent the calculated “peak” of these asymmetrical ratio distributions. While
the grey shaded regions represent the range of 68% of the test events. Generally, the higher
energy events had better resolution due to the stronger angular correlation; and the upward- or
downward-going events had better resolution than the horizontal events. For the low energy andhorizontal events there was no clear “peak,” as the test events were distributed over a wide range
range of ratios. In such cases, the shaded region representing 68% of events can extend muchbeyond the apparent peak.
Figure 8.13 shows heatmaps of the peaks on the left and the resolution on the right, overrecthe (cos Θrec
ν ,Eν ) parameter space. The third dimension is represented by the colour gradient.
The main deficit of νµ was in upward-going samples, so it was more important to have a higher
resolution in L/E in the upward-going direction. In the L/E analysis, the estimation on the L/E
resolution was taken for cos Θ ≤ 0, and reflected at cos Θ = 0 for cos Θ > 0.

8.3.3

L/E Shift

rec
rec
rec
The Lν /Lrecν and Eν /Eν had asymmetric distributions, resulting in a shift of (Lν /Eν )/(Lν /Eν )from unity. This shift was different for each sample and the effect of it was a reduction in resolution and a smearing of the L/E information. This shift was corrected for and the L/E variable
was reconstructed as:
rec 0
rec
rec
(Lrec
(8.20)
ν /Eν ) = δL/E · (Lν /Eν ),where δL/E was taken as the L/E (true/rec.) peak during the L/E resolution calculation, shown in
the left of Figure 8.13, after reflecting the upward-going values to the downward-going direction.
Provided the event has sufficient resolution, the shift in δL/E should bring the reconstructed L/E
closer to the true value.

8.3.4

L/E Resolution Cut

Since the resolution calculation considers a large sample of possible origins of a reconstructedrecevent, it gives an estimation of the reliability of the L/E value at a given point of (cos Θrec
ν ,Eν ).
A larger resolution means that a larger range of true L/E values could produce that event. Whilea smaller resolution means that the reconstructed L/E will be closer to the true value, despite
the uncertainty in the topology of the event. The selection of the L/E resolution to be within
70% was determined by a study to maximise the sensitivity to distinguish neutrino oscillations
from other hypotheses in 2004 [9]. Or in other words, the 70% resolution requirement gives agood balance between improved L/E shape and retaining statistics.
Distributions for the true L/E over reconstructed L/E ratios are shown in the left of Figure 8.14 for each event sample. These distributions were made from the analysis MC, and thetruth information refers to the source of the simulated event. The solid black histogram has had
the <70% resolution cut applied, and has a tighter distribution than without the resolution cut
in dashed blue. The resolution contours for each of the event samples in the (cos θ,Eν ) plane
can be seen on the right of Figure 8.14. A summary of the events of each of the samples after
the <70% resolution cut was applied is shown in Table 8.1. Due to low statistics for high energy
events, the resolution for energetic neutrinos (Evis2 > 10 GeV for FC Single-Ring, Evis2 > 50
184

1600

6.7 GeV
-0.75
17.0%

1400
1200
1000
800

6.7 GeV
-0.25
36.0%

600
500
400

6.7 GeV
0.25
48.0%

500
400

0

200

100

100

0

0

0.5

1

1.5

2

2.5

3.0 GeV
-0.75
21.0%

2500
2000

0.5

1

900

1.5

2

2.5

200

3.0 GeV
-0.25
54.0%

800
700
600

0

0

0.5

1

1.5

2

2.5

3.0 GeV
0.25
104.0%

1000
800
600

400
1000

400

300
200

500

200

100
0

0

0

0.5

1

1.5

2

2.5

1.3 GeV
-0.75
35.0%

1800
1600
1400
1200
1000

0

0

0.5

1

1.5

2

2.5

1.3 GeV
-0.25
81.0%

3000
2500
2000

0

0.5

1

1.5

2

1200
1000

3500

1.5

2

0.6 GeV
-0.75
70.0%

3000
2500

0

2.5

1

5000

1500

3000

1000

2000

500

1000

1.5

2

2.5

0.6 GeV
-0.25
98.0%

6000

4000

0

0.5

7000

2000

0

0

0.5

1

1.5

2

2.5

2

2.5

3.0 GeV
0.75
24.0%

0

0.5

1

1.5

1800

2

2.5

1.3 GeV
0.75
44.0%

1600
1400
1200

400
200

0

1

1.5

600
200

0

0.5

1

800

400

500

0

0.5

1000

600

1000

400
200

0
2200
2000
1800
1600
1400
1200
1000
800
600
400
200
0

2.5

1.3 GeV0.25
529.0%

1400

800
1500

800
600

1000

400

0

0

500

1500

1200

600

200

400
200

6.7 GeV
0.75
17.0%

1400

800

300

300

600

600

0

0.5

1

1.5

2

0

0

0.5

1

2

2.5

0.6 GeV0.25
6057.0%

2000
1800
1600
1400
1200
1000
800
600
400
200
0

2.5

1.5

0

0.5

1

1.5

1400

2

2.5

0.6 GeV0.75
217.0%

1200
1000
800
600
400
200
0

0

0.5

1

1.5

2

2.5

0

0.5

1

1.5

2

2.5

SK1 FC1R (L/E)true/(L/E)rec.

Figure 8.12: The L/E resolution calculation was performed by producing (L/E)true /(L/E)rec.
rec
distributions for each point in (cos Θrec
ν ,Eν ). These plots are samples from this grid, withthe reconstructed neutrino energy and assumed zenith written in the top right of each panel.
The “peak” was found for these asymmetrical distributions, shown by the red dashed line. The
resolution was then calculated from the peak, as the region containing 68% of test events, shownby the grey shaded area. The resolution value is written in red.

185

Figure 8.13: Histograms over the (cos Θ,Eν ) parameter space where the value of the plottedparameter is shown by colour, with the scale shown to the right of each plot. The left plot shows
the L/E (true/rec) peaks and the right plot shows the L/E (true/rec) resolution. The peak isclose to one for high energy downward-going and upward-going events, and the resolution is highfor these events. For horizontal and low-energy events, the L/E ratios are dispersed over a wide
range of values and there is no real peak. The resolution is poor in these regions. On the rightplot, the contours from the outside inward show 60%, 70%, 80% and 90% resolution. In theanalysis, the left half of these plots are reflected in cos Θ = 0 to the downward direction.
GeV for FC Multi-Ring and PC events) cannot be determined and these events were cut fromthe analysis (Section 8.1).

8.4

Likelihood Analysis and χ2 Minimisation

The data were analysed by a binned maximum likelihood technique over 43 L/E bins fromlog10 (L/E) = 0.0 to 4.3. Since the statistics were limited, a likelihood based on Poisson Statistics
was used. The bin by bin correlations of systematics from independent sources were consideredby using the “Pull Method,” [252] which is equivalent to the Covariance Method. The likelihood
is then defined:
Y exp(−N exp )(N exp )Niobs Y
j
exp
obs
i
i
L(N , N ) =×
exp,(8.21)
σjsys
Niobs !
ijwhere Niobs is the number of events observed in the data in the i-th bin, and Niexp is the numberof events in the i-th bin expected from the Monte Carlo simulation. The Niexp are adjusted toaccount for the systematic uncertainties. The 1 sigma value of the j-th systematic is represented
by σjsys . While the fji is the fractional change in the i-th bin from a 1 sigma variation of the j-thsystematic error, and includes information of the bin-by-bin correlations. Finally j are the Pull
Terms that adjusts how much the j-th systematic is applied to the MC, and are minimised toreduce the systematic correction. The Niexp are also adjusted for the particular parameter set of
the theory being tested. The Pull Terms modify the MC like so:


nsys
X
exp
exp
i
Ni → Ni
1+fj · j ,(8.22)
j=1

186

SampleData
Resolution< 70%
SK1 (1489.2 days)
FC
Single-Ring
1568
Multi-Ring
421
PC
Stopping
87
Through-Going459
SK2 (798.6 days)
FC
Single-Ring
865
Multi-Ring233
PC
Stopping
42
Through-Going212
SK3 (518.084 days)
FC
Single-Ring
556
Multi-Ring160
PC
Stopping
36
Through-Going200
SK4 (1096.7 days)
FC
Single-Ring
1196
Multi-Ring
362
PC
Stopping
71
Through-Going424

MC< 70% (νµ CC)

2215.5612.0(98.7 %)
(94.0 %)

120.3
684.9

(93.6 %)
(99.2 %)

1206.3323.0(98.6 %)
(93.6 %)

58.2
312.2

(93.8 %)
(99.2 %)

772.3
223.7

(98.6 %)
(94.3 %)

45.8
275.2

(94.0 %)
(99.0 %)

1627.7488.9(98.4 %)
(94.3 %)

98.3
577.7

(93.7 %)
(99.1 %)

Table 8.1: Summary of events from SK1 to SK4. The left column gives the sample name. The
middle columns show the data numbers for events that were expected to have been reconstructedwithin 70% of the true L/E. The right columns show the same for MC, with an additional column
displaying the percentage of νµ CC events.

187

Figure 8.14: log10 of the ratio of True L/E over Reconstructed L/E on the left and the resolution
cut contours on the right. On the left, the ratio is shown for each of the four event samples forSK4. The solid black histogram has had the <70% resolution cut applied, while the dashed blue
has not. The high resolution samples have a tighter distribution on this L/E true/reconstructeddouble ratio. On the right, the resolution contours on the (cos θ,Eν ) plane can be seen for eachsample for SK4. The distributions for SK1 to SK3 can be found in Appendix A.

where the sum over the systematic errors combines the contributions of each uncertainty to thei-th bin.
The log likelihood is then defined as a χ2 :L(N exp , N obs )χ ≡ −2 ln
L(N obs , N obs )




nsys
nbin
X
X
exp
i
obs
obs
=2
Ni
1+
fj · j − Ni + Ni ln


2

i=1
nsys
X

+

j=1

j=1

j
σjsys

Niobs
P
i
Niexp (1 + nsys
j=1 fj · j )



2
.

(8.23)The term on the right is like a penalty term. If a large adjustment is made to the MC from thesystematics, the χ2 is increased accordingly.
To minimise over all theoretical parameters along with the systematic parameters would
be very computationally intensive. Instead, the pull terms, , are minimised first. The χ2 isdifferentiated with respect to each of the systematic pull terms, with the minimum occurring
when the derivative is equal to zero:
∂χ2
= 0.
∂k
188

(8.24)

Considering the derivative of Equation (8.23) with respect to an arbitrary k :
nbin
X  exp
Niexp fki
∂χ2
k
i
obs
P
=2
Ni fk − Ni
+ 2 sys 2
exp
nsys i∂k(σNi (1 + j=1 fj · j )
k )
i=1

nbin
X  exp
Niobs fkiki
Pnsys i+ sys 2 ,=Ni fk −(σk )1 + j=1 fj · ji=1

(8.25)(8.26)

where the natural log was expanded to consider only the terms dependent on  and the common
factor of 2 was removed by demanding the derivative to be equal to zero. This can then be
rewritten as a condition for the epsilon minimum:k(σksys )2

=

nbin
X
i=1


Niobs
exp
P
− Ni
· fki .
i·
1 + nsysfj
j=1 j(8.27)

This gives nsys (k = 1 → nsys) linear equations, which canrearranged into a matrix equation
P be Pnsys
and solved simultaneously. This is done first by adding nbin
Niobs fji fki j to both sides ofijthe equation:nsys
X  nbinXj

Niobs fji fki +iδjk(σksys )2· j =

nbin
X

Niobs



1
1+

i=1

Pnsys

i
j=1 fj · j

+

nsys
X

fji · j



− Niexp



· fki

j=1

(8.28)
=

nbin
X

Niobs




exp
2
3
1 + S − S + · · · − Ni
· fki ,

(8.29)

i=1

where S =Pnsysi
j=1 fj · j and the binomial series expansion was used:(1 + S)−1 = 1 − S + S 2 − S 3 + S 4 − S 5 + · · ·(8.30)

The above expansion was used on the first fji term on the RHS of Equation (8.28), cancelling out
the second fji term on the RHS. Up to the fifth order, O(S 5 ), was considered in the expansion.
Equation (8.29) can be more simply expressed in the matrix equation:nsys
X

Mjk · j = Vk ,(8.31)jwhere,
 nbinXδjk−→ Mjk ,(σksys )2
i



nbin
X
exp
obs
2
3
Ni
1 + S − S + · · · − Ni
· fki −→ Vk .
Niobs fji fki +(8.32)(8.33)

i=1

Therefore Equation (8.31) is solved simultaneously for each k, to obtain the minimised k . Thiswas done iteratively, initially inputing j = 0 for (j = 1 → nsys) into the Vk . In the next steps
the minimum k were inserted into Vk , for 10 iterations.
189

The Pull Method has several advantages over the Covariance Method, especially when thenumber of bins, N , is greater than the number of systematics, K. The problem of minimisingthe χ2 is reduced from inverting an N × N matrix to inverting a K × K one. In order to further
improve computational efficiency the χ2 was first minimised over the Pull Terms, . Minimising
Equation 8.23 is equivalent to solving nsys linear equations:
nsys
X
j=1

8.5



nbin
nbin
X exp
X
δjk
exp
i
2
obs
i
i
· fki .
+
(1 + (fj · j ) + · · · )Ni − Ni
Ni · fj · fk · k =(σjsys )2i=1(8.34)

i=1

Systematic Uncertainties

The systematic uncertainties used for analysis with the L/E distributions were a subset of the
Zenith Analysis systematic errors. The set of uncertainties have been presented in categories
relating to the errors in the neutrino flux simulation, neutrino interactions, event reconstruction,and reduction of the raw data and Monte-Carlo. The systematic uncertainties are summarisedin Tables 8.2, 8.3, 8.4, and 8.5. The tables show the effect of a 1σ variation of the systematic
uncertainty on the relevant distributions, the bestfit , and the pull column is (j /σj )2 from
Equation (8.23).

8.5.1

Neutrino Flux Uncertainties

The systematic uncertainties in the neutrino flux were estimated by comparing several calculations of the neutrino flux at the Super-Kamiokande Detector. The Honda Flux [16, 165–167] wascompared with the Fluka [168, 253] Flux and Bartol Flux [169, 254]. A Super-Kamiokande paperthat discusses the flux uncertainties in detail can be found in Reference [109]. The atmospheric
neutrino flux uncertainties have been presented in Table 8.2.
Absolute Normalisation (1)The absolute normalisation depends on the uncertainty in the primary cosmic ray flux and on the
differences between the hadronic interaction models used in the flux calculations. In this analysis,
the shape of the L/E pattern was the main concern so this parameter was left essentially free, bysetting it to 1000%. However, the uncertainty in the normalisation is thought to be around 10%,and the best fit in the L/E analyses in this thesis tends to only require about 6% normalisation(despite being free).
Anti-Neutrino/Neutrino Ratio - ν̄µ /νµ (3)The L/E distributions were filled from (νµ + ν̄µ ) samples, so only the uncertainty in the ν̄µ /νµratio was considered. The error in this ratio mainly came from the uncertainty in the π + /π − ratio
produced from primary cosmic rays, especially below ∼5 GeV. The uncertainty was estimatedfrom the differences in the hadronic interaction model predictions between each of the flux
calculations [167–169]. At higher energies, above ∼10 GeV, the uncertainties in the K + /K −ratio become more significant. The uncertainty was common for each SK period and was taken
to be 6%.
190

Upward/Downward Going Neutrino Ratio (4)At low neutrino energies of less than a few GeV, the fluxes of neutrinos are up/down asymmetric.
This is due to the rigidity cutoff with the low energy primary cosmic rays being deflected in thegeomagnetic field. However, for energies above a few GeV the upward and downward fluxes arefor the most part symmetric. This effect can be seen in Figure 1.2. The low energy Zenith
distributions at Super-Kamiokande were not substantially affected, as there is a relatively poor
angular correlation for low energy neutrino interactions. The uncertainty in the up/down ratio
was estimated by comparing the predicted ratios of several flux models. Uncertainties between
0.2% to 0.5% were calculated for several samples as summarised in Table 8.2, which were used
to estimate their effect on the analysis distributions.
Horizontal/Vertical Neutrino Ratio (5)For low energy neutrinos, there is an enhancement of the expected number of horizontal neutrinos, which was implemented in the 3D flux models. This is a characteristic of the 3D natureof the hadronic showers of primary cosmic rays. However this was not seen so clearly in theneutrino zenith distributions due to the relatively weak angular correlation between the incoming neutrino and outgoing lepton below 1 GeV. At larger energies the predicted K/π ratio of
hadronic interactions in the atmosphere becomes more significant. So the uncertainty in thehorizontal/vertical neutrino ratio was estimated by comparing the 3D calculations of each of theflux models. Uncertainties between 0.1% to 2.3% were estimated for several constituent samples
as shown in Table 8.2, which were applied to determine their effect on the analysis distributions.
Neutrino Flight Path Length (6)The neutrino flight path length can be calculated accurately considering the geometry of the
Earth. The main uncertainty came from the neutrino production height in the atmosphere. Thiswas significant for downward-going and horizontal neutrinos where the difference in path length
was considerable. In the case of upward-going events, the uncertainty in production height was
negligible compared to the path length through the Earth, leading to an uncertainty in the path
length of the order of 1%. To estimate this systematic error, the density of the atmosphericstructure in the flux simulation was altered by 10% and the difference in the resulting path
length calculations were used. Figure 8.15 shows the neutrino flight length against the zenith
angle for two different atmospheric densities. The systematic error was taken to be 10%.
Energy Spectrum (7)At energies relevant to Super-Kamiokande, the spectrum of primary cosmic rays protons, accounting for about ∼75% of the flux, could be fit to a function with a form proportional to E γ ,where γ is the spectral index and was found to be −2.74 [109, 255]. This value was a good fit
below 100 GeV, above 100 GeV the spectrum was better fit by γ = −2.71. The primary comic
ray helium nuclei, ∼15% of the flux, could be fit with a spectral index of either −2.64 or −2.74.
The uncertainty in the energy spectrum was estimated by calculating the flux weighted averageof the spectrum index uncertainties [109]. The systematic error was taken to be 5% for all SK
configurations.
191

path length (km)

4

10

3

10

2

10

1

-0.2

0

0.2

0.4

0.6

0.8

1

Zenith angle (cosΘ)
Flux ratio

Figure 8.15: Neutrino flight path length against
the zenith angle for doward-going and horizontal events. The solid curve shows the standard
calculation, and the dashed line was for an atmospheric density structure that was compressed by
10%. The difference tends to zero for upwardgoing events.

10

1.4
1.3
1.2
1.1
1
0.9

Figure 8.16: The ratio of other flux calculations
w.r.t. the Honda flux. The Bartol (solid black)and FLUKA (dashed red) fluxes normalised by
the Honda flux are shown.

0.8
0.7
0.6 −1
10

1

10

102

103
Eν(GeV)

FC Muli-GeV (8) and PC (2) Sample Normalisation
Although there are already considerations for the systematic uncertainties in the absolute normalisation and for the energy spectrum, as written above, there are further energy-dependent
difference between the flux predictions. The differences in the flux of two other calculations
with respect to the Honda flux can be seen in Figure 8.16. This can lead to variations within
the sample distributions. To consider this effect, the uncertainty in the sample-by-sample relative normalisation for multi-GeV events was taken to be 5%. Both the FC Multi-GeV and PC
samples contain Multi-GeV events, so the systematics on these two samples are highly correlated. Another systematic error on the ratio of these two samples was considered to accountfor further differences. Uncertainties between 0.02% to 0.9% were taken over the SK periods asshown in Table 8.2. The FC Sub-GeV samples were not altered by these uncertainties to avoid
double-counting with the other flux errors.
Solar Activity (26)The solar activity varies over an 11 year cycle, affecting the primary cosmic rays and therefore
the atmospheric neutrino flux. To account for the modulation of the solar activity, the change of192

the neutrino flux in a ±1 year period was taken to estimate the uncertainty. This was calculated
as 20%, 50%, 20% and 10% for SK1, SK2, SK3 and SK4, respectively. Where these values were
the plus or minus percentage shift between high and low solar activity fluxes, to determine thechange in the L/E distributions due to the solar activity uncertainty.

8.5.2

Neutrino Interaction Uncertainties

There have been many experiments measuring the cross-sections of neutrino interactions, andseveral theoretical models based on those results. Where the statistics were abundant, there
was generally a good agreement between the theoretical models. However, there can be larger
discrepancies when the statistics are limited. A summary of the systematic uncertainties for theneutrino interactions are listed in Table 8.3.
Quasi-Elastic Total Cross-Section (9)A simulation based on the Smith and Monitz model was used for the CCQE Total Cross-Section
for a bound nucleon. To estimate the uncertainty, this model was compared with the Nievesmodel [215]. The systematic error was taken to be 10%.
Single Meson Production (Total Cross-Section) (10)To estimate the single meson production cross-section, Pions were used due to their large statistics. This same systematic was then applied to the single meson production of other particles
such as η and K. Various experimental results for the νµ p → µ− pπ + interaction were compared against the Monte-Carlo to estimate the systematic error. The uncertainty was set to be
20%. The interactions for charged pion production were better understood than for π 0 , so there
another uncertainty was estimated for that.
Single Meson Production (π 0 /π ± ) (27)There was more discrepancy between the interaction models for π 0 due to the poor experimental
statistics available. By comparing the Hernandez model and the Rein and Sehgal model forthe νµ n → µ− pπ 0 interaction against the related interactions with an outgoing charged pion, a
relative systematic uncertainty of 40% was set for π 0 /π ± .
DIS (Total Cross-Section) (11)The agreement between the models and experimental results for the total cross-section for Deep
Inelastic Scattering interactions, above a few tens of GeV, was considered to estimate this systematic error. An uncertainty of 5% was set.
Coherent π Production (12)Generally the Coherent π Production systematic was estimated by comparing the Rein and Sehgal
model against experimental results. For NC interactions there was about a 50% agreement,
however not much was known about the CC νµ interaction so an uncertainty of 100% was set.
193

ID

Descriptionσ%Systematic Errors in Neutrino Flux
1 Normalisation–
3 ν̄µ /νµ Ratio
6.0
4 Up/Down Ratio
Single-Ring < 400 MeV
0.3
Single-Ring > 400 MeV
0.5
Single-Ring Multi-GeV
0.2
Multi-Ring Sub-GeV
0.2
Multi-Ring Multi-GeV
0.2
PC
0.2
5 Horizontal/Vertical
Single-Ring < 400 MeV
0.1
Single-Ring > 400 MeV
1.9
Single-Ring Multi-GeV
2.3
Multi-Ring Sub-GeV
1.3
Multi-Ring Multi-GeV
1.5
PC
1.7
6 Neutrino Flight Path Length
10.0
7 Energy Spectrum
5.0
8 Sample Normalisation Multi-GeV
SK1
5.0
SK2
5.0
SK3
5.0
SK4
5.0
2 FC Multi-GeV µ / PC
SK1
0.6
SK2
0.5
SK3
0.9
SK4
0.02
26 Solar Activity
SK1
20.0
SK2
50.0
SK3
20.0
SK4
10.0

Systematics
Best Fit %
Pull
6.41
0.66
0.01

0.00000.01230.0001-0.02

0.00030.26-0.98

0.00070.0382

-3.74
-1.61
-0.72
2.28

0.56040.10410.02080.2076

0.00
0.00
0.04
0.00

0.00000.00000.00160.0000

0.13
0.04
0.16
-0.48

0.00000.00000.00010.0023

Table 8.2: Neutrino Flux Uncertainties

194

ID

Description

Systematic Errors in Neutrino Interactions
9 QE Scattering Total Cross-Section
10 Single Meson Production (Total Cross-Section)
11 DIS (Total Cross-Section)
12 Coherent π Production
13 NC/CC Ratio
21 Nuclear Effect
24 Axial-Vector Mass (MA ) [QE and Single Meson]
25 DIS (Bodek Correction)
27 Single-Meson Production (π 0 /π ± )σ%Systematics
Best Fit %

Pull

10.0
20.0
5.0
100.0
20.0
30.0
20.0
20.0
40.0

5.10
-15.63-0.51-33.23
2.51
1.02
4.28
17.19
8.66

0.26030.61080.01020.11040.01570.00120.04580.7386
0.0469

Table 8.3: Neutrino Interaction Uncertainties
NC/CC Ratio (13)As the NC interactions are less well understood, the NC/CC Ratio systematic uncertainty was
taken to be 20%.
Nuclear Effect (21)
Nuclear effects in 16 O nuclei affect the angular correlation between the incident neutrino and theemitted mesons in an interaction. The uncertainty concerns the mean free path of hadrons produced in a neutrino interaction. A 30% systematic error was taken by considering the agreementof the model used and experimental results.
Axial-Vector Mass (MA ) for QE and Single Meson Production (24)There is a dependence on the axial-vector mass, MA , for quasi-elastic scattering and single meson
production. In the simulation MA = 1.21 GeV/c was set. There is about a 10% uncertainty on
the value of MA . The systematic uncertainty on interactions due to this was estimated bycomparing the Q2 spectrum for MA = 1.11 GeV/c and MA = 1.21 GeV/c. The systematic error
was taken to be 20% in the analysis.

8.5.3

Event Reconstruction Uncertainties

There are many algorithms that reconstruct physical variables from raw charge and timing dataon the PMTs, or likelihoods that separate events into different classifications. In these uncertainties, the performance of these likelihoods or the accuracy of the algorithm outputs are estimated.
Ring Counting (16)The algorithm for finding and separating the rings in an event are described in Section 7.3.
A likelihood function divides the events into Single-Ring and Multi-Ring classifications. This
classification can be affected by several uncertainties such as water transparency and the detector
calibrations. The systematic error was estimated by comparing the data and Monte Carlo ring195

counting likelihood distributions. This was done for multiple samples and the 1σ estimations
have been listed in Table 8.4.
Particle Identification - Single-Ring (17) and Multi-Ring (18)
The particle identification (PID) into e-like and µ-like events is determined by a likelihood
algorithm as described in Section 7.4. The uncertainty in the PID was estimated by comparing
these likelihood distributions between the data and Monte Carlo. The classification of the PID
algorithm determines the number of e-like and µ-like events, and so determines the µ-like events
available to the L/E Analysis. The systematic error was calculated for multiple samples, whichhave been presented in Table 8.4.
Energy Calibration (19)The energy reconstruction and uncertainty are studied in the energy scale calibration, whichprimarily uses four sources to cover a wide energy range, as explained in Section 5.3. The
systematic uncertainty in the absolute energy scale for SK1, SK2, SK3, and SK4 were 1.1%,
1.7%, 2.7%, and 2.3%, respectively.
Up-Down Asymmetry of the Energy Calibration (20)The up/down asymmetry of the energy calibration was determined by comparing the decay
electron events from cosmic ray muons. These decay electrons have a nearly uniform momentum
in all directions and have vertices distributed throughout the fiducial volume of the inner detector,
making them an excellent calibration source. The uncertainties for SK1, SK2, SK3, and SK4
were estimated to be 0.6%, 0.6%, 1.3%, and 0.3%, respectively.
PC Stopping and PC Through-Going Categorisation (23, 28, 29)
PC events were separated into stopping and through-going classifications, by comparing thecharge deposited in the OD with the expected charge of a muon that completely passed throughthe OD. Since the OD has a non-uniform response, due to the geometry and distribution of oldand new PMTs, the charge expectation was a tuned value. The PC systematics were estimatedby comparing the data and MC, shifting the MC distribution to match the data more closely
and then observing the percentage of PC Stopping events that change category. Further details
are written in Section 8.1.3. The PC separation uncertainties have been listed in Table 8.4

8.5.4

Reduction Uncertainties

The basic data trigger is rather simple and further steps are required to reduce the raw data to
the events that are interesting for analysis, and remove as many background events as possible.
These uncertainties estimate the efficiency of these reduction processes.
FC Reduction (14)The reduction for fully-contained (FC) events was carried out as described in Section 6.2. The
systematic uncertainty in the FC reduction was estimated by comparing the reduction distributions between data and Monte Carlo for variations in the cut values. The uncertainties for SK1,
196

ID

Description

SK1
σ % Best Fit %Systematic Errors from Reconstruction
16 Ring Counting-0.30
Single-Ring < 400 MeV 0.7
Single-Ring > 400 MeV 0.7
Single-Ring Multi-GeV
1.7
Multi-Ring Sub-GeV-4.5
Multi-Ring Multi-GeV-4.1
17 Particle ID Single-Ring-0.02
Sub-GeV-0.1
Multi-GeV
0.2
18 Particle ID Multi-Ring
1.33
Sub-GeV-3.9
Multi-GeV-2.9
19 Energy Calibration
1.1
-0.12
20 Up-Down Asym of E Calib. 0.6
0.13
23 PC Stop/Through Top
46.1
2.08
28 PC Stop/Through Bottom
22.7
-3.29
29 PC Stop/Through Barrel
7.01
0.07
ID Description
SK3
16 Ring Counting-0.73
Single-Ring < 400 MeV 3.0
Single-Ring > 400 MeV 0.6
Single-Ring Multi-GeV
1.0
Multi-Ring Sub-GeV-2.6
Multi-Ring Multi-GeV-2.1
17 Particle ID Single-Ring
0.01
Sub-GeV-0.4
Multi-GeV-0.5
18 Particle ID Multi-Ring
0.39
Sub-GeV
3.1
Multi-GeV
4.5
19 Energy Calibration
2.7
0.92
20 Up-Down Asym of E Calib. 1.3
0.25
23 PC Stop/Through Top
86.6
-4.47
28 PC Stop/Through Bottom
12.1
0.80
29 PC Stop/Through Barrel
28.7
5.07

Pull

σ%

0.0009

Pull

-0.35

0.0012-0.03

0.00080.60

0.0036

-0.39
0.06
-0.37
-0.40
0.19
SK4
-0.14

0.05290.01110.00040.00090.0004-0.10

0.01010.99

0.0098

0.77
-0.02
2.19
-0.16
0.07

0.11130.00280.00260.00020.0001

2.3
0.7
1.7
-8.2
-0.8
0.0006-0.4-0.1
0.01770.01180.04350.00200.02100.0001

2.2
-3.4
1.7
0.6
19.37
12.9
9.44

0.00530.0002

-3.0
0.6
-1.2
-2.3
2.4
0.0000-0.4-0.4
0.00150.11610.03600.00270.00440.0312

2.2
6.8
2.3
0.3
43.3
11.6
7.4

Table 8.4: Neutrino Reconstruction Uncertainties

197

SK2
Best Fit %ID

Descriptionσ%Systematic Errors from Reduction
14 FC Reduction
0.2
15 PC Reduction
2.4
22 Non-ν Background (µ-like)Cosmic-Ray FC Sub-GeV
0.1
Cosmic-Ray FC Multi-GeV 0.1
Cosmic-Ray PC
0.2
ID Description
14 FC Reduction
0.8
15 PC Reduction
0.5
22 Non-ν Background (µ-like)Cosmic-Ray FC Sub-GeV
0.1
Cosmic-Ray FC Multi-GeV 0.2
Cosmic-Ray PC
1.8

SK1Best Fit %

Pull

σ%

-0.01
-0.68
0.00

0.00190.07970.0000

0.2
4.8

SK2
Best Fit %

Pull

0.00
-1.11
0.11

0.00000.05310.0116

SK4
0.02
0.09
-0.29

0.00660.00760.0828

0.1
0.1
0.7
SK3
-0.01
0.01
0.25

0.00030.0007
0.0613

0.3
1.0
0.1
0.8
4.9

Table 8.5: Neutrino Reduction Uncertainties
SK2, SK3 and SK4 were 0.2%, 0.2%, 0.8% and 0.3%, respectively, and have been summarised inTable 8.5.
PC Reduction (15)The reduction for partially-contained (PC) events is explained in Section 6.3. These uncertainties
were estimated by considering the performance of the algorithm that determines the number ofhits in charge clusters in the outer detector, which calculates the variable called N HIT AC. FC
and PC events are separated by this variable. The effect of varying the cut values on the data
and Monte Carlo distributions were also considered. The uncertainties for SK1, SK2, SK3 andSK4 were 2.4%, 4.8%, 0.5% and 1.0%, respectively, and have been summarised in Table 8.5.
Non-Neutrino Background (22)The non-neutrino background was estimated during the reduction procedure, and mainly comesfrom from cosmic ray muons for the µ-like samples that are relevant to the L/E Analysis. The
uncertainties were estimated for FC Sub-GeV and Multi-GeV samples, as well as PC events, andhave been presented in Table 8.5.

8.6

2-Flavour Oscillation L/E Results

The previous sections have described how the L/E variable was reconstructed as accurately as
possible, and how the L/E resolution was estimated for a given neutrino energy and zenith
angle. For the analysis, the high resolution events were kept – events with an L/E value that was
expected to be within 70% of the true L/E. These precise L/E distributions were compared with198

Figure 8.17: The 2-flavour L/E distributions on the left and the L/E distributions normalised by
the null oscillation Monte Carlo on the right, for the data from SK1 to SK4. The combined data
are shown by the points, and the bestfit oscillated MC in the red solid histogram. The bestfit
point was at (sin2 2θ23 , ∆m232 ) = (1.0, 2.5 × 10−3 eV2 ). The MC without oscillations is shown insolid black, which has been used to normalise the data and bestfit MC in the right plot.
2-flavour oscillated Monte Carlo through a likelihood analysis, while minimising the contribution
from the systematic uncertainties.
The resulting L/E distributions are shown in the left of Figure 8.17. The data consists of thecontributions from SK1 (1489.2 days), SK2 (798.6 days), SK3 (518.1 days) and SK4 (1096.7 days),while the Monte Carlo consists of 500 years of simulated data for each detector configuration andeach of the fully-contained and partially contained event samples. The combined data is shown inthe data points, while the bestfit oscillated Monte Carlo accounting for systematic uncertainties
is plotted in the solid red histogram. Also shown for comparison is the Monte Carlo without
any oscillations in the solid black histogram. It is clear from this plot that the data cannot be
explained without some kind of perturbation such as that introduced by neutrino oscillations.
Meanwhile, the oscillated Monte Carlo provides an excellent fit to the data.
In the right of Figure 8.17 the data and bestfit Monte Carlo have been normalised by the nonoscillated Monte Carlo. This brings out some of the more subtle features. A flat distribution
around 1 would mean that the data could be described without oscillations. For low valuesof L/E, the data and MC are distributed around 1, then drop down into a dip at aroundL/E ∼ (500 ∼ 700) km/GeV. This deficit corresponds to the first point where the oscillation
probability is at a maximum. For the oscillated MC the bestfit parameters were:χ2 = 171.7/169 d.o.f,(8.35)

sin2 2θ23 = 1.000 ± 0.047,(8.36)∆m232 = (2.50 ± 0.27) × 10−3 eV2 .

(8.37)The bestfit systematic errors are summarised in Tables 8.2, 8.3, 8.4, and 8.5. The effect ofa 1σ variation of the systematic uncertainty on the relevant distributions is shown in the σ%column. The fraction of the systematic error applied to the bestfit distributions, the  term in
199

-2

2

2

∆m32(eV )10

Figure 8.18: 2-Flavour L/E Analysis Oscillation
allowed region for the combined dataset from
SK1 to SK4. The L/E analysis has more power
to constrain the ∆m232 compared to the Zenith
Analysis, as it reconstructs the position of theoscillation dip with a high L/E resolution sample. The bestfit point was at (sin2 2θ23 , ∆m232 ) =(1.0, 2.5 × 10−3 eV2 )99% C.L.
90% C.L.
68% C.L.
10

-3

0.7

0.8

0.9

1

2

sin 2θ23

Equation (8.34), is expressed as a percentage in the Bestfit % column. The value in the pullcolumn is (j /σj )2 and can be thought of as the contribution of the j-th systematic error to what
has been called the penalty term in Equation (8.23). So a large contribution from the systematic
uncertainty will result in a poor fit in the χ2 minimisation.
The contours showing the 2 dimensional allowed regions of the oscillation parameters can beseen in Figure 8.18. Since the global minimum was slightly in the non-physical region (sin2 2θ23
= 1.005), the ∆χ2 for 68%, 90% and 99% C.L. in the physical region were calculated to be ∆χ2of 2.358, 4.681, and 9.255, respectively, from the global minimum.
The modification to the ∆χ2 for a best fit that falls in a non-physical region followed themethod described in the 1996 PDG [256]. In this calculation, the ∆χ2 between the minimum inthe non-physical region and the physical region was used to calculate the number of events thatfall in the physical region. For example, if 40% of events fall in the physical region, then the
90% C.L. in the physical region would correspond to the 96% C.L. over the entire range, which
is where the effective physical 90% C.L. boundary would be drawn.
The 1 dimensional contours were drawn as slices through the minimum point, and so theabove calculated ∆χ2 C.L.s for 2 d.o.f. were used. The 1D slices are shown in Figure 8.19. The
1σ (68% C.L.) errors expressed in the bestfit above were calculated from these 1D contours. The
90% C.L. allowed regions were:−3

2.129 × 10

8.7

eV

2

sin2 2θ23 > 0.931,(8.38)< ∆m232 < 2.918 × 10−3 eV2 .

(8.39)

Discussion

The 2-flavour oscillation simulation provides an excellent explanation of the data observed at
Super-Kamiokande. The 90% allowed regions presented here are more confined than those previously published with the data from the SK1 period [9]: sin2 2θ23 > 0.90 and 1.9 × 10−3 eV2 <
200

20

20

17.5

17.5
99% C.L.
90% C.L.
68% C.L.

7.5

2

2

10

12.5
10

2

χ - χ min

12.5

15

2

χ - χ min

15

99% C.L.
90% C.L.
68% C.L.

7.5
5

5

2.5

2.5

0

0
10

-3

2

2

10

-2

0.70.750.80.850.90.95 1 1.051.11.15
2

sin 2θ23∆m32(eV )Figure 8.19: 2-Flavour L/E Analysis 1D contours for ∆m232 and Sin2 2θ23 for the combined
dataset from SK1 to SK4. These were taken as slices through the minimum bestfit point at(sin2 2θ23 , ∆m232 ) = (1.0, 2.5 × 10−3 eV2 )∆m232 < 3.0 × 10−3 eV2 . The allowed regions from the SK1 analysis are shown on the leftof Figure 8.20. The data in this analysis had 3903 days of data, over double the live timecompared to the first result with the SK1 data (1489 days). The result presented here is also
in agreement with a later analysis including SK1 and SK2 data [257]: sin2 2θ23 > 0.94 and1.85 × 10−3 eV2 < ∆m232 < 2.65 × 10−3 eV2 . The allowed regions from the SK1+2 analysis areshown on the right of Figure 8.20. However, the bestfit ∆m232 from this analysis (2.5 × 10−3 eV2 )was slightly higher than these previous analyses (2.4 × 10−3 eV2 for SK1, 2.2 × 10−3 eV2 forSK1+2), and the 90% C.L. constraint on sin2 2θ23 was slightly more relaxed compared to theSK1+2 result. By comparing the 2D oscillation allowed regions from each SK period alone, as
in Figure 8.21, it can be seen that the SK3 and SK4 data relax the constraint on sin2 2θ23 andthe SK4 data favours a larger ∆m232 . The results from each SK period, and each analysis, were
statistically consistent with one another.
The L/E distributions over the null oscillation prediction for each SK period are shown in
Figure 8.22. It seems that for SK3 and SK4 the data at large L/E have more of a tendency to
be at 0.5 or slightly above, which could result in a little more relaxed constraint on sin2 2θ23 .
As for ∆m232 , an argument could be made that in the SK4 data the shoulder region before thedip was more strongly pronounced, such that smaller ∆m232 were more suppressed. The bestfit
for ∆m232 for SK4 was 2.8 × 10−3 eV2 , compared to the ∆m232 of (2.2 ∼ 2.4) × 10−3 eV2 for SK1,SK2 or SK3. In this analysis, the atmospheric flux was updated to the Honda 2011 flux [16], andthe SK1, SK2 and SK3 data and MC were reprocessed with the latest reconstruction software
in 2012.
Traditionally, the L/E analysis provided evidence to favour Neutrino Oscillations to explain
the νµ deficit in the data, and excluded the competing theories of Neutrino Decay and Neutrino
Decoherence. The left plot in Figure 8.23 shows the L/E distributions over the null oscillation
prediction for SK1+2+3+4, together with the three MC bestfits for each of the theories. As
previously shown by the L/E Analysis [9, 257], these competing theories cannot reproduce the201

Figure 8.20: Previously published Standard 2-Flavour L/E results for SK1 in 2004 [9] (left) andSK1 and SK2 in 2009 [257] (right). In these plots sin2 2θ ≡ sin2 2θ23 and ∆m2 ≡ ∆m232 . The
90% C.L. allowed region for sin2 2θ23 was [0.9,1.0] for the SK1 analysis, [0.94,1.0] for SK1+2,
and [0.93,1.0] from this thesis. The 90% C.L. allowed region for ∆m232 (×10−3 eV2 ) was [1.9,3.0]
for SK1, [1.85,2.65] for SK1+2, and [2.13,2.92] from this thesis.
dip in the data, which is a feature of neutrino oscillations. The minimum χ2 for Neutrino
Decay and Decoherence were 187.8/169 d.o.f and 194.8/169 d.o.f, respectively. The bestfit forneutrino oscillations had a minimum χ2 of 171.1/169 d.o.f, and the ∆χ2 for Neutrino Decay andDecoherence from the Oscillation minimum was 16.7 and 23.7, respectively. Taking the squareroot of the ∆χ2 between Oscillations and Decay or Decoherence gave a 4.0 σ and 4.8 σ difference,
respectively, from the oscillation best fit. This is in agreement with past L/E Analysis results,
which already excluded these other hypotheses.
Also shown in Figure 8.23, the right plot compares the 2-flavour L/E results presented herewith the SK Zenith Analysis, MINOS, and T2K results from 2013 [35]. The 2-flavour L/E
Analysis still provides a nice constraint on ∆m232 from the SK data, while the SK atmospheric
neutrino analyses constrain the value of sin2 2θ23 more than the accelerator experiments, for data
up until 20132 .
Further improved sensitivity to oscillation parameters could be achieved with a more accurate
L/E resolution. The L/E analysis with atmospheric neutrinos is limited by the angular correlationof the outgoing lepton direction to the incoming neutrino direction and therefore the inferred νflight path length. A well-defined path length could be achieved in an accelerator experiment,
which could give a better measurement of ∆m232 . Experiments like T2K and MINOS constrain∆m232 very well.
Improved sensitivity to θ23 could be achieved with a very precisely known ν flux. This can bea challenge even for human-made fluxes such as in accelerator experiments producing neutrinos
from meson decay, as there is always some uncertainty in the contamination of the beam. The
main advantage of a Neutrino Factory is the precisely understood ν beam, produced from the2

T2K now provides a better constraint on the atmospheric parameters [258]202

(a) SK1 2D Allowed Region(b) SK2 2D Allowed Region(c) SK3 2D Allowed Region(d) SK4 2D Allowed Region

Figure 8.21: 2-Flavour L/E Analysis oscillation parameter allowed region for each SK period.
The SK1 to SK3 data and MC were reprocessed with the latest reconstruction software in 2012.
In addition the atmospheric flux was modified to Honda 2011 [16]. So these allowed regions differ
from those previously published [9, 257]. The constraint on sin2 2θ23 was relaxed by the SK3and SK4 data, while the SK4 data favours larger ∆m232 . In these plots sin2 2θ ≡ sin2 2θ23 and∆m2 ≡ ∆m232 .

203

Figure 8.22: 2-Flavour L/E distributions over the null oscillation expectation for each SK period.
The dip region is slightly less emphasised in the SK3 and SK4 data leading to a less constrained
sin2 2θ23 . Perhaps the more pronounced shoulder region before the dip in SK4 suppresses smaller∆m232 values more strongly.

204

Figure 8.23: Comparison of the 2-Flavour L/E oscillation with Neutrino Decay and Decoherence
on the left, and with other experiments on the right [35]. The left plot shows the L/E distributions
normalised by the null oscillation prediction. The red curve shows the bestfit for Oscillations,the green for Decoherence and blue for Decay. The right figure compares the 2-Flavour L/E
analysis with other experiments measuring the atmospheric oscillation parameters.
decay in flight of accelerated muons.
However, these accelerator experiments are tuned to a specific value of L/E, typically at theoscillation maximum probability, to achieve this sensitivity. One advantage of an experiment
like SK is that it probes a large range of baselines and energies, so it can potentially discoverunexpected effects outside of tuned sensitivity.
The L/E analysis makes use of the same source of atmospheric data as the Zenith analysis,
however it is able to provide a more constrained region for ∆m232 . Since the L/E Analysis isspecialised to have sensitivity to oscillations, it could be strong for distinguishing sub-dominant
effects beyond oscillations.

205

Chapter 9

Analysis of Neutrino and Anti-Neutrino
Oscillation
Neutrino Oscillations is a Lorentz invariant theory, and so should also be CPT invariant. Decadesof experiments on the oscillation of flavour ratios of neutrinos during transit have been carried out(Chapter 2), and Neutrino Oscillations has been found to be the dominant explanation of almost
all observations. However, up until fairly recently, there had not been comprehensive studies ofneutrino and anti-neutrino oscillations separately. In recent years, accelerator experiments havebeen running in anti-neutrino mode and have been able to directly measure the transitions ofanti-muon neutrinos.
Some of the early results [10] implied that the oscillation parameter ∆m̄232 , measured with
anti-neutrinos, could be significantly larger than the equivalent measurement for neutrinos,∆m232 . Although the tension from these results were later relaxed [11], these early results inspired
more interest in the question of whether neutrino flavour transitions truly are CPT symmetric.
Neutrinos only interact weakly, with very small cross-sections, and so their precise propertiesare difficult to measure. Since neutrinos appear to Oscillate, this implies that neutrinos havemass, yet up until now none of the mass eigenstates of the neutrino have been measured. The
fact that neutrinos should have mass at all was a surprise, as the Standard Model originally
postulated them to be massless. So the properties of neutrinos are somewhat elusive and theunknowns make them mysterious, as compared with the other known elementary particles. Soeven properties such as the fundamental CPT symmetry should be checked in neutrinos. If
neutrinos were found to break CPT symmetry, this would certainly require explanation andwould point towards new physics.
There has been considerable interest in CPT violation in neutrinos [259]. For example, this
could arise if the mass of the neutrino and anti-neutrino differ, mν̄ 6= mν ; or could be allowed by
CPT violating terms in the general relativistically Extended Standard Model (SME) [260, 261].
The observation of Neutrino Oscillations is an amplification of the effect of small parameters,such as neutrino masses, through their interference. In the same way, even a small degree of CPT
asymmetry may be visible through Neutrino Oscillation measurements. In this analysis however,rather than attempting to directly measure the strength of CPT violating terms; indications of
CPT violation could be observed in neutrino oscillations from a measurement of independent
parameters for neutrino and anti-neutrino, if the effect is considerable.
The aim of the Neutrino Anti-Neutrino L/E Analysis was either to confirm the CPT symmetry206

of neutrinos at the sensitivity of SK, or to measure a discrepancy between neutrinos and antineutrinos. This was done by allowing the oscillation parameters for neutrinos and anti-neutrinos
to vary independently in the MC. The ν and ν̄ components were then combined before comparing
against the data, which were not separated into ν and ν̄. The data were not separated because
SK has little power for charge discrimination, as it is a massive water Cherenkov detector, andis not magnetised.
However, the flux of neutrinos has been accurately calculated and constantly improved forover 20 years with any new cosmic ray flux measurements or updated hadronic interaction models.
Although the error of the absolute flux is around ∼10%, the ν/ν̄ ratio is accurately known towithin a few percent. So the relative ν and ν̄ composition is well understood. These contributions
can be oscillated with independent neutrino and anti-neutrino parameters in the MC, to findthe parameter set that best matches the data. If there is a significant difference between theoscillations of ν and ν̄, this could be observed as a distortion of the L/E distributions that canbe described best by separate ν and ν̄ parameters.

9.1

Analysis Principles

9.1.1

Reconstruction of L/E

The reconstruction in this analysis is mostly similar to that described in the Standard L/E
Analysis (Section 8.2). Here, a brief summary of the process follows. Reconstructing the L/Evariable as accurately as possible is important to be able to observe the L/E pattern expected for
Neutrino Oscillations. Firstly the visible energy was reconstructed (Section 8.2.1), taking somefurther considerations. These included the reconstruction of the pion energy in multi-ring events;
or making use of dE/dx for a reconstructed muon path in PC events and separating rings thatwere expected to be overlapping. After all these additional considerations, the visible energy
may still differ from the true neutrino energy. To account for this, a Monte Carlo sample wasused to estimate the neutrino energy for a given visible energy and event classification. This gave
a fitted correction factor as a function of the visible energy. Furthermore, careful momentum
weighting was used to try to reconstruct the neutrino direction and therefore the flight path moreaccurately (Section 8.2.2).
The accuracy of the L/E reconstruction was also estimated depending on the reconstructedenergy and zenith of the neutrino, such that cuts could be made on the expected L/E resolution(Section 8.3). There was also an offset of (L/E)true /(L/E)rec. from unity during the resolution
estimation, which was also accounted for (Section 8.3.3).

9.1.2

Neutrino and Anti-Neutrino Oscillations

Since SK does not have much power to distinguish e− from e+ or µ− from µ+ , it is also very
difficult to separate ν and ν̄ on an event-by-event basis. Instead, a discrepancy could be looked
for statistically by treating the ν and ν̄ independently in the MC simulation. Since the expected
observable difference between the parameters was already small, a simple 2-flavour νµ → ντ
207

600

1.8
2011 MINOS ν BF only
2011 MINOS ν BF only
2011 MINOS mixed BF
Non-Oscillated MC2011 MINOS ν BF only

1.6

2011 MINOS ν BF only
2011 MINOS mixed BF

1.4

400

Number of Events

Number of Events

500

300

200

1.2
1
0.8
0.6
0.4

100
0.2
0

1

10

102

L/E (km/GeV)

103

0

104

1

10

102

L/E (km/GeV)

103

104

Figure 9.1: Reconstructed L/E plots from the MC assuming the best fit values from the 2011
MINOS best fit values [10], during the period of tension between neutrino and anti-neutrino
measurements. The L/E distributions are shown in the left, and have been normalised by thenon-oscillated MC on the right. The blue shows the ν and ν̄ oscillated with the 2011 MINOS best
fit for neutrino oscillation [262]: ∆m232 = ∆m̄232 = 2.32 × 10−3 eV2 and sin2 2θ23 = sin2 2θ̄23 =1.00. The red shows the ν and ν̄ oscillated with the 2011 MINOS best fit for anti-neutrino
oscillation [10]: ∆m232 = ∆m̄232 = 3.36 × 10−3 eV2 and sin2 2θ23 = sin2 2θ̄23 = 0.86. While in
the green MC, the ν events were oscillated with ∆m232 = 2.32 × 10−3 eV2 and sin2 2θ23 = 1.00;
and the ν̄ events were oscillated with ∆m̄232 = 3.36 × 10−3 eV2 and sin2 2θ̄23 = 0.86. These plotsshow how the L/E distributions can be used to distinguish between these kinds of oscillation
scenarios.

208

oscillation scenario was assumed with the survival probabilities below:
P(νµ → νµ ) = 1 − sin2 2θ23 sin2

 1.27∆m2 L 

32,E 1.27∆m̄2 L 
32P(ν̄µ → ν̄µ ) = 1 − sin2 2θ̄23 sin2
.
E

(9.1)
(9.2)

What this means is that the ν and ν̄ were separated in the MC into two samples and thenoscillated independently, keeping track of the oscillation parameters: θ23 and ∆m232 for neutrinoand θ̄23 and ∆m̄232 for anti-neutrino. Then at the analysis stage a (∆m232 , θ23 , ∆m̄232 , θ̄23 ) parameter set was chosen, and the corresponding ν and ν̄ samples were combined to give the L/E
distribution to be compared against the data, which were not separated into ν and ν̄.
Figure 9.1 shows how the L/E distributions can be built by oscillating ν and ν̄ independently
in the MC. The distributions were made from the reconstructed L/E values. The example
parameter sets were based on the best fit results for ν and ν̄ reported by MINOS in 2011 [10, 262],
during the period of tension between these measurements. The blue histogram shows ν and ν̄both oscillated at the ν best fit, which is close to the SK best fit; and the red histogram showsν and ν̄ both oscillated at the ν̄ best fit. Mixed parameters can be seen in the green histogram,in which the ν events were oscillated at the ν best fit, and the ν̄ events at the ν̄ best fit.
It can be seen that the first slope occurs earlier in the red plot at around L/E∼102 km/GeV,
due to the larger ∆m̄232 , while the magnitude of the oscillations decreases due to the smallersin2 2θ̄23 . This is most prominent where the oscillations begin to average at L/E greater than∼103 km/GeV. If the ν̄ truly oscillate independently with parameters such as those reported byMINOS in 2011, the data should be better fitted by the green curve rather than the blue. The
MC can be divided into ν and ν̄ reliably, due to the accurate calculation of the neutrino flux andthe well-understood ν̄/ν ratio.

9.1.3

Neutrino and Anti-Neutrino Flux Ratio

Although there is some discrepancy of the absolute flux between different calculations of theatmospheric neutrino flux at about the ∼10% level, the agreement of the anti-neutrino to neutrino
ratio is much better. The flux of neutrinos is created by primary cosmic rays bombarding the
air molecules in the atmosphere, creating pions and some kaons. These follow decay chains,producing neutrinos at each step.
π + → µ+ + νµ
→ e+ + νe + ν µ ,
π

−

(9.3)

−

→ µ + ν̄µ→ e− + ν̄e + νµ .

(9.4)

Kaons also become more relevant towards higher energies. Kaons can decay into muons andneutrinos, or various combinations of pions. These muons and pions can then further decay to
produce neutrinos. In pion decay, most of the energy goes to the muon; while in kaon decay theenergy is distributed quite evenly over the neutrino and muon, making them more significant at
higher energies and the dominant source of neutrinos above 100 GeV.
The meson production by cosmic rays and the subsequent decay chains are the origin of theneutrino anti-neutrino ratios. The uncertainty in this ratio comes from the uncertainty of the209ν µ Honda 2011
ν µ Honda 2011ν µ Fluka 2003φν × E2ν (m-2 sec-1 sr-1 GeV)

102

ν µ Fluka 2003
ν µ Bartol 2003
ν µ Bartol 2003

10

1
10

-1

1

10

10

2

Eν (GeV)Figure 9.2: The flux of νµ and ν̄µ from various flux calculations. Shown in black is Honda
2011 [16]; Fluka 2003 [168] in blue; and Bartol 2003 [169] in red. The νµ flux is shown by solidlines, and the ν̄µ by dashed lines. The fluxes are similar, and have been multiplied by Eν2 tocompress the vertical scale and make the differences more clear. Generally, compared to Honda,
the Bartol flux seems slightly larger; and the Fluka flux seems slightly smaller. Except that the
Bartol ν̄µ flux falls more rapidly than the other two calculations.
π + /π − ratio from pion production by primary cosmic rays. Above ∼10 GeV, the uncertainty
in the K + /K − ratio also becomes more significant. It can also be seen from Equations (9.3)
and (9.4), that for a pion decay one νµ and one ν̄µ are produced regardless of the π + /π − ratio.
So a ν̄µ /νµ ratio of approximately 1 is expected for any given pion. Therefore the predicted ratio
is very accurate. This does not hold quite true for kaons, because the energies of the neutrinos
from Kaon decay are significantly higher than those of neutrinos produced by the subsequentmuon decay. Figure 9.2 shows the νµ and ν̄µ fluxes for the Honda 2011 [16], Fluka 2003 [168, 253]
and Bartol 2003 [169, 254] calculations. These flux values were very similar, so the fluxes havebeen multiplied by the square of the neutrino energy, Eν2 , to shrink the vertical range of the plot
and make their differences more apparent.
The neutrino and anti-neutrino fluxes are very similar up to a few GeV, after which the ν̄µ /νµratio decreases, which can be seen more clearly in the left of Figure 9.3. There appears to be agood agreement between the Honda and Fluka fluxes, while the ratio decreases more rapidly in
the Bartol flux. This is likely due to an incomplete treatment of the kaon component of the fluxin the Bartol 2003 calculation, which becomes increasingly significant towards higher energies.
In the right of Figure 9.3, the ν̄µ /νµ ratios for Fluka and Bartol have been normalised by the
Honda 2011 calculation. It can be seen that these fluxes agree very well, within 3% over mostof the energy range, indicated by the inner grey dashed horizontal lines. At most these differ by
210

1.1

1.2
Honda 2011

1.05

Fluka 2003 / Honda 2011

Fluka 2003

Bartol 2003 / Honda 2011

1.15

Bartol 20031(νµ / νµ) Double Ratios

1.1
0.95

νµ / νµ

0.9
0.85
0.8
0.75

1.05

1

0.95

0.9
0.7
0.85

0.65
0.6
10

-1

1

10

10

2

0.8

10-1

1

Eν (GeV)

10102Eν (GeV)Figure 9.3: The ν̄µ /νµ ratio of each of the flux models in the left plot, and these ratios normalised
to the Honda 2011 ratio on the right for comparison. The flux of ν̄µ is quite similar to νµ up
until a few GeV, then gradually falls more rapidly. All the fluxes agree well, except that the
Bartol ratio falls more rapidly above 10 GeV. This was likely due to an incomplete treatment of
the kaon component of the flux, which is more relevant at higher energies. The right plot showsthat the fluxes agree within 3% for most of the energy range (inner grey dashed horizontal lines).
While at most within about 5% (outer grey dashed lines).
about 5%, marked by the outer grey dashed lines, with the exception being the Bartol flux athigher energies due to the differences in kaon treatment. In the L/E analysis, only events up to
50 GeV were included, as it is very difficult to determine the L/E resolution for energies above
this. For the systematic uncertainty on the ν̄µ /νµ ratio, a conservative estimate of 6% was taken.
Since the ν̄µ /νµ ratio was well-known, the relative composition of neutrinos and anti-neutrinosin the MC was reliable. So these particle and anti-particle contributions can be oscillated independently to find the combined L/E shape that best matches the data.

9.2

Sensitivity Study for ν and ν̄ Parameters

As a first step of the analysis a sensitivity study was carried out. This was done to determine
how accurately the oscillation parameters could be determined by the L/E distributions, whenthe ν and ν̄ are oscillated independently. It was also used to consider extensions to the Standard
L/E Analysis (Chapter 8).
Although the ν and ν̄ fluxes are quite similar, with the ν flux being about ∼10% higher, theevent rate of ν̄ at SK is much less than for ν, and is about 20-30% of the total events. This
meant that the ν̄ distributions would have considerably less influence on the final L/E analysis
distribution than the ν contribution. So less sensitivity was expected for ν̄.

9.2.1

Increasing Statistics with a Second Resolution Sample

In the Standard L/E Analysis, there was a requirement for events to have a high resolution
in L/E (See Section 8.3). Quantitatively, the requirement was that the resolution of the L/Evariable was within 70% of the expected true value. This <70% resolution cut discarded a lotof the more horizontal events and lower energy events, which can be seen in the contour plot in211

Figure 9.4: L/E resolution contour on the(cos θ, Eν ) plane. The coloured lines show theregion of this parameter space that was cut by
these resolution requirements. It can be seenthat typically low energy and horizontal eventswere cut by imposing a higher L/E resolution
condition. The precise sample selected by theStandard L/E Analysis is shown in red. By considering a second resolution sample outside of
70% the statistics could be increased. The blackline shows the contour for maximum oscillation
probability, and the dashed black lines for halfmaximum oscillation probability.
Figure 9.4.
Of course, to reconstruct the L/E pattern precisely this cut was necessary. However, toincrease the available statistics for the ν̄ parameter sensitivity, the addition of a second resolution
sample was considered. What this means is that the precise sample (<70% resolution) analysis
bins were kept, and additional bins were added for a sample made up from a resolution rangethat was previously discarded.
The ranges considered were 70-80%, 70-90% and 70-100% resolution. L/E distributions ofthese second resolution samples can be seen in Figure 9.5. The blue and green distributions
correspond to the precise sample, and the orange and red distributions correspond to each of theresolution ranges mentioned above. It can be seen that the second samples each cover a range
in L/E previously discarded by the Standard L/E Analysis. The resolution of these samples are
relatively poor compared to the precise sample, however the additional information from theseL/E distributions could help constrain the allowed region for ν̄.

9.2.2

Analysis Bins

The treatment of the data and MC binning from each of the SK periods was different from thatof the Standard L/E Analysis. In the Standard case, the L/E distributions from each of the
SK periods had separate analysis bins. For this analysis, the bins from each SK period weremerged. This gave increased statistics for each bin, and reduced the risk of unstable effects due
to low-statistics bins.
As with the Standard analysis, the precise L/E distributions were divided into 43 bins in
log10 (L/E). Previously there were 43 bins for each of SK1 to SK4, or 172 bins in total. Analysing
the χ2 with merged bins compressed the 172 bins into 43 bins, or one combined L/E distribution.
As mentioned in Section 9.2.1, additional bins were added for a second resolution sample.
Since these secondary samples cover a different region in L/E, it is not a good idea to naively
use the same 43 L/E bins as the precise sample. So the average L/E distribution of ∼400 MC
212

70-80% Total

70% Total

450

70% Res Anti-nu
400

70-90% Total

70-80% Anti-nu

Number of Events

70% Res Anti-nu
400

Number of Events

Number of Events

70% Total

450

70% Res Anti-nu
400

70-100% Anti-nu

350

350

300

300

300

250

250

250

200

200

200

150

150

150

100

100

100

50

50

50

0

0

0.5

1

1.5

2

2.5

3

3.5

70-100% Total

70-90% Anti-nu

350

070% Total

450

4

0

0.5

1

1.5

2

2.5

3

3.5

L/E (GeV/km)

4

0

0

0.5

1

1.5

2

2.5

L/E (GeV/km)

3

3.5

4
L/E (GeV/km)

L/E

L/E

L/E

(a) 70-80 % Sample(b) 70-90 % Sample(c) 70-100 % Sample

Figure 9.5: Second Sample Distributions. The L/E Analysis estimates the accuracy of the reconstruction of the L/E variable. The Standard L/E Analysis, discussed in Chapter 8, requiredthat L/E be reconstructed within 70% of the true value. This corresponds to the blue hatchedhistogram in each of the plots above. The orange hatched histograms show the L/E distributions of possible second samples in an extended resolution range. The green and red hatchedhistograms show the anti-neutrino contribution for the precise sample and second resolution
sample, respectively.
samples of differing oscillation parameters was taken to give an idea of the region in L/E coveredby the second sample. The L/E bin where all bins beyond had 5+ or 6+ events per bin was
found. These positions are marked by lines in Figure 9.6, where the blue line marks where allfollowing bins have 5 or more events, and the green line for 6 or more events per bin.
To compare the addition of each of these second samples to the analysis, the same binningwas used for all the second samples. Since the most strict requirement was from the 70-80%
sample, where bins 11 to 43 had 6 or more events, these bins were chosen for the second sample.
In other words, the L/E bins 11 to 43 were used in the χ2 analysis for the second resolution
sample. These bins for SK1 to SK4 were, of course, also merged giving 33 additional analysis
bins. So the total number of analysis bins used was 43 + 33 = 76.

9.2.3

Sensitivity with the Second Resolution Sample

After merging the L/E bins for SK1 to SK4, then introducing more bins for a second resolution sample, the sensitivity study was carried out. To scan the relevant parameter space 21
parameters were taken for each of ∆m232 , θ23 , ∆m̄232 , and θ̄23 . Where the bar denotes the oscillation parameters applied to the anti-neutrinos in the MC. This gave a parameter space of about
200,000 points (214 ).
The MC was fitted against dummy data with the following input oscillation parameters:(∆m232 , θ23 , ∆m̄232 , θ̄23 ) = (2.1 × 10−3 eV 2 , 1.0, 2.1 × 10−3 eV 2 , 1.0). The resulting sensitivity considering the addition of each of the 70-80%, 70-90%, and 70-100% resolution samples can be seenin Figure 9.7. For each of these possibilities, the sensitivity to ν remained the same, but the
70-90% resolution sample marked by the solid line gave the best sensitivity for ν̄. So the 70-90%
second resolution sample was chosen for the final analysis.
It can be seen that generally there was reduced sensitivity for ν̄ compared to that of ν,213

70-80 Total
70-80% Anti-nu
100

Number of Events

Number of Events

Number of Events

120

70-90% Total

225

70-90% Anti-nu
200

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400

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300

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150
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150

40
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100
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25

0

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L/E (GeV/km)

4

0

0

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1

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2

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L/E (GeV/km)

3

3.5

4
L/E (GeV/km)

L/E

L/E

L/E

(a) 70-80 % Sample(b) 70-90 % Sample(c) 70-100 % Sample

Figure 9.6: Second Sample Binning. The average L/E distribution for ∼400 MC sets with varying
oscillation parameters was used to estimate the bins covered by the second resolution sample.
The blue line in each plot above marks where all following bins have more than 5 events per bin,and the green for 6+ events. The 70-80% sample provided the most constrained binning, so bins
11-43 were used for the second sample.
however the addition of the second sample gave a reasonable improvement considering the low
statistics available.
These allowed regions were drawn by marginalising over the parameters not under consideration. For example, for the neutrino allowed region all the anti-neutrino parameters were
marginalised over. For one particular ν parameter set, this was done by scanning over all the ν̄parameter sets and finding the minimum χ2 . Then the minimum χ2 for the next ν parameter
set was found, and so on to build up the 2D χ2 distribution. The corresponding procedure wasalso done for drawing the anti-neutrino allowed region. This practice was also used for the final
analysis. The confidence levels (C.L.) used were for a distribution of two variables and taken tobe (68%, 90%, 99%) = (2.3, 4.61, 9.21).
The allowed regions for the ν̄ parameters in this treatment should be larger, as any applied
effect by the ν̄ oscillation can be more easily counteracted by the ν oscillation. However, in the
global analysis of all the parameters, the best fit could come from a particular combination ofneutrino and anti-neutrino parameters. These parameters need not be the same if anti-neutrinos
do indeed appear to oscillate differently from neutrinos.

9.3

Likelihood Analysis and χ2 Minimisation

As with the Standard L/E Analysis, a binned maximum likelihood based on Poisson statistics wasused. The “Pull Method” [252] was implemented for minimising the systematics and considering
the bin-by-bin correlations of the independent errors. The differences in the χ2 treatment now
include additional bins from the Second Sample discussed in Section 9.2.1, the bin merging briefly
mentioned in Section 9.2.2, and some other small changes. A much more full description of thebasic concepts of the likelihood was discussed for the Standard L/E Analysis in Section 8.4.
The treatment of the χ2 was a little complicated, as the systematics needed to be considered
for each SK period individually. The systematics for the Second Sample were also considered214

SK-I+II+III+IV, νµ
∆m232

∆m232

SK-I+II+III+IV, νµ10-2

99% C.L.
90% C.L.
68% C.L.

10-3
0.7

0.75

0.8

0.85

0.9

10-2

99% C.L.
90% C.L.
68% C.L.

10-3
0.7

0.95
1
Sin22θ23

0.75

0.8

0.85

0.9

0.95
1
Sin22θ23

Figure 9.7: Sensitivity for each of the Second Sample Regions. The solid curves show the ∆χ2after adding the 70–90 % resolution sample to the 70 % precise resolution sample. The dashedcurves show the ∆χ2 after adding 70–80 % sample, and the dotted curves after adding the 70–
100 % sample. Very little difference can be observed in the neutrino case, however the addition of
the 70–90 % sample has consistently better sensitivity than the other samples for anti-neutrino.
in the same way. This allowed the systematics to be properly minimised before the χ2 for theparameter set was calculated. At the χ2 calculation stage, the bins for SK1 to SK4 were merged,
after applying the appropriate adjustment from the systematics.
As a reminder the definition of the χ2 was:L(N exp , N obs )χ ≡ −2 ln
L(N obs , N obs )


nsys
nbin
X  exp 
X
i
obs
obs
=2
Ni
1+
fj · j − Ni + Ni ln


2

i=1
nsys
X

+

j=1

j=1

j
σjsys

Niobs
P
exp
i
Ni (1 + nsys
j=1 fj · j )



2
.

(9.5)

In the Standard L/E Analysis nbin corresponded to the 43 L/E analysis bins for each SK period,
side-by-side, giving 172 bins in total. For the ν ν̄ L/E Analysis, the first 172 bins were for thePrecise Sample (L/E resolution < 70%). Following this were the 33 Second Sample L/E bins
for each SK period, giving an additional 132 analysis bins, or 172 + 132 = 304 bins in total. As
mentioned in the previous section, the Second Sample chosen had the 70-90% resolution range.
So the systematics were minimised over the 304 bins, so that errors specific to each SK period
could be considered. However, as mentioned, the bins were merged at the χ2 calculation stage.
The 172 Precise Samples bins were merged into 43, and the 132 Second Sample bins were merged
into 33 bins, giving 43+33 = 76 bins in the final χ2 . Note that this treatment slightly differs fromthe definition written in Equation (9.5), as there was summing in each sample over bins with the215same L/E value. In this way, the systematics could be considered properly while improving thestability of the χ2 for low statistics bins.

9.4

Neutrino Anti-Neutrino Analysis

The sensitivity study was helpful to determine the structure of the analysis. However, since
there were 4 free Oscillation parameters compared to only 2 in the Standard L/E Analysis, theparameter space quickly grows very large. In the sensitivity study there were 21 values for eachparameter leading to a parameter space of about 200,000 (214 ) points, since every parameter
combination was considered.
In the Standard analysis, a 201 × 201 parameter space could be scanned comfortably on
one CPU. In order to scan a more reasonable number of ν and ν̄ parameters, the analysis was
extended to work over many CPUs. In the final analysis 41 points for each parameter wereconsidered, giving 2.8 million sets, and was scanned over 100∼200 CPUs.
The parameter space consisted of 41 points in each of sin2 2θ23 and sin2 2θ̄23 from 0.7 to 1.2,
in which values above 1.0 were non-physical. For ∆m232 and ∆m̄232 , 41 points each from 10−2 to10−3 were scanned with logarithmically spaced intervals. Since it was computationally intensive
to scan a more fine grid with 4 free parameters, the 1D χ2 for each parameter were fitted witha spline function to calculate the confidence intervals and central value. In these distributions,the other parameters were marginalised out over the physical region of the parameters.

9.5

Systematic Uncertainties

The systematic uncertainties used for analysis with the L/E distributions were a subset of the
Zenith Analysis systematic errors. The set of uncertainties have been presented in categories
relating to the errors in the neutrino flux simulation, neutrino interactions, event reconstruction,and reduction of the raw data and Monte Carlo. Since the Neutrino Anti-Neutrino L/E Analysis
made use of an additional resolution sample, the bin-by-bin contribution of those systematics
had to be estimated for the new samples. This corresponded to extending the fji in Equation 9.5
to account for the new bins introduced by the second resolution sample. Then the systematics
were minimised simultaneously over the two resolution samples.
The uncertainties summarised here have the same definitions and discussion as the Standard
L/E Analysis. Refer to Section 8.5 for further details. The systematic errors from the best-fit
MC from this analysis are summarised in Tables 9.1, 9.2, 9.3, and 9.4. These uncertainties havealso been displayed graphically in Figure 9.14 for the estimates; and in Figure 9.15 for the best fit
values. Further discussion of the systematics was written together with the results in Section 9.6.

9.6

Neutrino Anti-Neutrino Results

As written above, the analysis was expanded to include two resolution samples. A precise sample
with a resolution within 70%, in which the L/E variable was expected to have been reconstructedwithin 70% of the true L/E value; and a second resolution sample with an expected resolution
between 70%–90%.
216

ID

Descriptionσ%Systematic Errors in Neutrino Flux
1 Normalisation–
3 ν̄µ /νµ Ratio
6.0
4 Up/Down Ratio
Single-Ring < 400 MeV
0.3
Single-Ring > 400 MeV
0.5
Single-Ring Multi-GeV
0.2
Multi-Ring Sub-GeV
0.2
Multi-Ring Multi-GeV
0.2
PC
0.2
5 Horizontal/Vertical
Single-Ring < 400 MeV
0.1
Single-Ring > 400 MeV
1.9
Single-Ring Multi-GeV
2.3
Multi-Ring Sub-GeV
1.3
Multi-Ring Multi-GeV
1.5
PC
1.7
6 Neutrino Flight Path Length
10.0
7 Energy Spectrum
5.0
8 Sample Normalisation Multi-GeV
SK1
5.0
SK2
5.0
SK3
5.0
SK4
5.0
2 FC Multi-GeV µ / PC
SK1
0.6
SK2
0.5
SK3
0.9
SK4
0.02
26 Solar Activity
SK1
20.0
SK2
50.0
SK3
20.0
SK4
10.0

Systematics
 % (σ)Pull

6.51
0.39 (0.065)
0.00 (0.000)

0.00000.00430.00000.01 (0.010)

0.00010.92 (0.092)-0.89 (-0.178)

0.0086
0.0315-3.54 (-0.708)-2.54 (-0.508)-1.35 (-0.270)
0.94 (0.188)

0.5016
0.25830.07290.03550.02 (0.033)
0.00 (0.000)
0.05 (0.056)
0.00 (0.000)

0.00090.00000.00290.00002.00 (0.100)-3.25 (-0.065)-0.25 (-0.013)-0.34 (-0.034)

0.01000.00420.00020.0012

Table 9.1: Neutrino Flux Uncertainties

217

ID

Description

Systematic Errors in Neutrino Interactions
9 QE Scattering Total Cross-Section
10 Single Meson Production (Total Cross-Section)
11 DIS (Total Cross-Section)
12 Coherent π Production
13 NC/CC Ratio
21 Nuclear Effect
24 Axial-Vector Mass (MA ) [QE and Single Meson]
25 DIS (Bodek Correction)
27 Single-Meson Production (π 0 /π ± )σ%Systematics
 % (σ)

Pull

10.0
20.0
5.0
100.0
20.0
30.0
20.0
20.0
40.0

5.17 (0.517)-15.60 (-0.780)-0.46 (-0.092)-22.88 (-0.229)
0.19 (0.010)
0.37 (0.012)
3.85 (0.193)
16.55 (0.828)
11.08 (0.277)

0.2676
0.60840.00850.05240.00010.00010.03710.68450.0767

Table 9.2: Neutrino Interaction Uncertainties

9.6.1

Neutrino Anti-Neutrino L/E Analysis Best Fit

After minimising the χ2 for 2.8 million parameter sets, the best fit MC L/E distributions together
with the L/E data can be seen in Figure 9.8. In the left plot, the precise sample is represented
by the black points for data and the red histograms for MC; and the second sample by the blue
points for data and the magenta histograms for MC. The solid histograms show the best fit,
while the dashed histograms have not been oscillated. The precise sample looks very similar
to the Standard L/E Analysis case; while the L/E values were more smeared in the 70%–90%
resolution distributions, leading to some deficit prior to the first oscillation peak. The data wasreconstructed in the same way, so this effect can be seen in the data too. The plot on the rightshows the same information, but shows the contribution from the anti-neutrino distributions in
the green (precise sample) and red (second sample) hatched histograms.
The bestfit in the physical region corresponded to the following parameters:
χ2 = 69.02/71 d.o.f,(9.6)

sin 2θ23 = 1.00 ± 0.04,−3∆m232 = 2.43+0.32
eV2 ,−0.21 × 10

(9.7)

2

2

sin 2θ̄23 = 1.00 ± 0.08,
−3∆m̄232 = 2.44+0.56
eV2 ,−0.33 × 10

(9.8)
(9.9)
(9.10)

where the errors represent 1σ (68% C.L. for 1 d.o.f.), and the bar denotes the oscillation parameters applied to the anti-neutrino simulated distributions. The central values stated here
were calculated from the minimum of a well-fitted spline function to the ∆χ2 in 1D with allother parameters marginalised out. Considering only the scanned points, the best fit was foundat: (sin2 2θ23 , ∆m232 , sin2 2θ̄23 , ∆m̄232 ) = (1.00, 2.37 × 10−3 eV2 , 1.00, 2.51 × 10−3 eV2 ). The
best fit distributions were plotted from this parameter set. In these analyses, any reference to“marginalising out” a parameter means that the parameter was allowed to be free and the valuethat had the minimum contribution to the χ2 was used.
The bestfit L/E distributions after normalising by the null oscillation prediction to reveal the
L/E oscillation pattern can be seen in Figure 9.9. Again the precise sample looks very similar to
218

ID

DescriptionSK1σ% % (σ)Systematic Errors from Reconstruction
16 Ring Counting-2.42 (-0.242)
Single-Ring < 400 MeV 0.7
Single-Ring > 400 MeV 0.7
Single-Ring Multi-GeV
1.7
Multi-Ring Sub-GeV-4.5
Multi-Ring Multi-GeV-4.1
17 Particle ID Single-Ring-0.02 (-0.020)
Sub-GeV-0.1
Multi-GeV
0.2
18 Particle ID Multi-Ring
0.28 (0.028)
Sub-GeV-3.9
Multi-GeV-2.9
19 Energy Calibration
1.1 -0.38 (-0.345)
20 Up-Down Asym of E Calib. 0.6
0.05 (0.083)
23 PC Stop/Through Top
46.1 1.85 (0.040)
28 PC Stop/Through Bottom
22.7 -3.23 (-0.142)
29 PC Stop/Through Barrel
7.01 0.30 (0.043)
ID Description
SK3
16 Ring Counting
1.50 (0.150)
Single-Ring < 400 MeV 3.0
Single-Ring > 400 MeV 0.6
Single-Ring Multi-GeV
1.0
Multi-Ring Sub-GeV-2.6
Multi-Ring Multi-GeV-2.1
17 Particle ID Single-Ring
0.00 (0.000)
Sub-GeV-0.4
Multi-GeV-0.5
18 Particle ID Multi-Ring
0.32 (0.032)
Sub-GeV
3.1
Multi-GeV
4.5
19 Energy Calibration
2.7
0.91 (0.337)
20 Up-Down Asym of E Calib. 1.3
0.10 (0.077)
23 PC Stop/Through Top
86.6 -4.31 (-0.050)
28 PC Stop/Through Bottom
12.1 0.78 (0.064)
29 PC Stop/Through Barrel
28.7 8.48 (0.295)

Pull

σ%

0.0587Pull-0.50 (-0.050)

0.0025-0.09 (-0.090)

0.00770.52 (0.052)

0.0027-0.34 (-0.200)
0.09 (0.150)-0.30 (-0.015)-0.32 (-0.025)-0.85 (-0.090)
SK4
3.66 (0.366)

0.04020.02130.00020.00060.0081-0.09 (-0.090)

0.0075-0.95 (-0.095)

0.00900.04 (0.017)-0.05 (-0.167)
2.18 (0.050)-0.19 (-0.016)
0.14 (0.019)

0.00030.02580.00250.00030.0003

2.3
0.7
1.7
-8.2
-0.8
0.0004-0.4-0.1
0.00080.11950.00720.00160.02030.0018

2.2
-3.4
1.7
0.6
19.37
12.9
9.44

0.02260.1341

-3.0
0.6
-1.2
-2.3
2.4
0.0000-0.4-0.4
0.00100.11480.00600.00250.00420.0873

2.2
6.8
2.3
0.3
43.3
11.6
7.4

Table 9.3: Neutrino Reconstruction Uncertainties

219

SK2
 % (σ)ID

Descriptionσ%Systematic Errors from Reduction
14 FC Reduction
0.2
15 PC Reduction
2.4
22 Non-ν Background (µ-like)Cosmic-Ray FC Sub-GeV
0.1
Cosmic-Ray FC Multi-GeV 0.1
Cosmic-Ray PC
0.2
ID Description
14 FC Reduction
0.8
15 PC Reduction
0.5
22 Non-ν Background (µ-like)Cosmic-Ray FC Sub-GeV
0.1
Cosmic-Ray FC Multi-GeV 0.2
Cosmic-Ray PC
1.8SK1 % (σ)Pull

σ%-0.01 (-0.050)-0.15 (-0.063)
0.00 (0.000)

0.00310.00380.0000

0.2
4.8

SK2
 % (σ)Pull

0.00 (0.000)-2.63 (-0.548)
0.11 (0.110)

0.00050.30000.0123SK40.02 (0.067)
0.06 (0.060)-0.21 (-0.210)

0.00360.00360.0455

0.1
0.1
0.7
SK3
-0.01 (-0.013)
0.01 (0.020)
0.29 (0.290)

0.00010.00070.0843

0.3
1.0
0.1
0.8
4.9

Table 9.4: Neutrino Reduction Uncertainties

Figure 9.8: Neutrino Anti-Neutrino L/E bestfit distributions. The left plot shows precise sample
with black points for data, red solid histogram for the bestfit MC and dashed red for MC
without oscillations; and the second sample with blue points for data, magenta solid histogram
for the bestfit, and dashed magenta for MC without oscillations. The right plot shows thesame information but showing the contribution from anti-neutrinos in the green hatched (precise
sample) and red hatched (second sample) histograms.

220

Figure 9.9: Neutrino Anti-Neutrino L/E bestfit and data normalised by the non-oscillated Monte
Carlo, for the precise sample on the left and the second resolution sample on the right. The left
plot was similar to the Standard L/E case. The right plot shows more smearing of the deficit,
but the lowest point in the MC seems to correspond to the oscillation maximum. There wasgood agreement between data and MC and a clear departure from unity, even for the second
sample.

the Standard L/E Analysis case. As expected, the L/E shape was smeared in the second sample
distribution. However, it looks like the lowest point in the bestfit distribution corresponds tothe position with maximum oscillation probability. The smearing introduces some deficit before
the oscillation maximum, while the averaging of oscillations at high L/E values looks similar tothe precise sample case. There was a good agreement between the MC and data in the second
sample, and disagreement with the non-oscillated case, suggesting that these distributions should
contribute to constrain the oscillation parameters.
The 2D allowed regions can be seen in Figure 9.10. Here the ∆χ2 was drawn from theglobal minimum in the physical region and the 68%, 90% and 99% C.L. correspond to a ∆χ2 of
2.30, 4.61, and 9.21, respectively. In the ν plot, the ν̄ parameters were marginalised out, andvice versa for the ν̄ plot. Due to the smaller statistics of ν̄ events, the ν̄ allowed region was
significantly larger than for ν. Since ν and ν̄ cannot be separated on an event-by-event basis,the ν and ν̄ parameters were measured in a combined distribution. So the larger statistics of theν events gave it more power to define the overall distribution. The ν and ν̄ allowed regions wereoverlapping, and no significant discrepancy between the two sets of oscillation parameters could
be seen.
The 1D allowed regions for sin2 2θ23 for neutrinos and anti-neutrinos are shown in Figure 9.11.
The same is shown for ∆m232 in Figure 9.12. In these 1D distributions the 68%, 90% and 99%
C.L. correspond to a ∆χ2 of 1.00, 2.71, and 6.63, respectively, and the undisplayed parameters
were marginalised out. The 1σ (68% C.L.) errors above were calculated from a spline function221νµ Oscillation

10-2

νµ Oscillation

99% C.L.

99% C.L.

90% C.L.

90% C.L.

68% C.L.

68% C.L.

2∆m232 (eV2)∆ m32 (eV2)

10-2

10-3
0.7

0.75

0.8

0.85
Sin22θ23

0.9

0.95

10-3
0.7

1

0.75

0.8

0.85
Sin22θ23

0.9

0.95

1

Figure 9.10: 2D allowed region contours for the neutrino oscillation parameters on the left andthe anti-neutrino parameters on the right. These allowed regions were overlapping, and nodiscrepancy between these parameters could be seen. The best fit points found from the 1D
distributions are shown by the stars.
smoothly fitted to the ∆χ2 points. Additionally, the 90% C.L. allowed regions were:−3

sin2 2θ23 > 0.93,(9.11)

sin2 2θ̄23 > 0.86,(9.12)

2.07 × 10 eV < ∆m232 < 2.96 × 10−3 eV2 ,1.89 × 10−3 eV2 < ∆m̄232 < 3.63 × 10−3 eV2 .
2(9.13)(9.14)The vertical coloured lines in Figures 9.11 and 9.12 indicate the positions where the 1D ∆χ2fitted functions cross the 68% and 90% confidence levels. These positions were taken as theconfidence intervals of the oscillation parameters represented by the data. The minimum of thefitted curve was also found to give a more representative central value for the parameter, and isindicated by the light blue downward triangle in the Figures. These central values were quoted as
the best fit above, and are marked by the stars in Figure 9.10. Although the scanned parameter
space had limited granularity, the measured points in the 1D ∆χ2 seemed to follow a smooth
simple line and so could be interpolated accurately.

9.6.2

Analysis With Forced Normal Oscillations

In addition to the ν and ν̄ independent oscillation analysis, another analysis with the ν and ν̄parameters forced to be the same was carried out. This allowed the ν and ν̄ oscillation parameters
to be compared with the normal oscillation case, in the same analysis setting. The stability of
the analysis after the addition of the second resolution sample and the merging of the bins in theχ2 , can be seen by comparing with the Standard 2F L/E Analysis. The best fit was found at:
χ2 = 69.08/73 d.o.f,(9.15)

sin 2θ23 = 1.00 ± 0.03,−3∆m232 = 2.43+0.21
eV2 .
−0.15 × 10

(9.16)

2

222(9.17)

νµ Oscillation ∆χ220νµ Oscillation ∆χ2

20

99% C.L.
18

90% C.L.

16

68% C.L.

14

12

12
min

14

χ2-χ2

min

18

90% C.L.

16

χ2-χ2

99% C.L.

108

68% C.L.

10
8

6

6

4

4

2

2

0

0

0.7

0.75

0.8

0.85
sin22θ23

0.9

0.95

1

0.7

0.75

0.8

0.85
sin22θ23

0.9

0.95

1

Figure 9.11: sin2 2θ contours for neutrinos on the left and anti-neutrinos on the right. The
undisplayed parameters were marginalised out. The points show the scanned parameters, whichwere smoothly fitted with a spline function as shown by the curve. The fitted curve was used to
calculate the 1D 68% and 90% C.L. allowed regions, as well as the minimum point indicated by
the light blue triangle.

νµ Oscillation ∆χ220νµ Oscillation ∆χ2

20

99% C.L.
18

90% C.L.

16

68% C.L.

14

12

12
min

14

χ2-χ2

min

16

χ2-χ2

99% C.L.
18

90% C.L.

108

68% C.L.

10
8

6

6

4

4

2

2

0

0
-3

10

∆m232 (eV2)

-3

10-2

10

2

10-2

∆ m32

Figure 9.12: ∆m232 contours for neutrinos on the left and anti-neutrinos on the right. The
undisplayed parameters were marginalised out. The points show the scanned parameters, whichwere smoothly fitted with a spline function as shown by the curve. The fitted curve was used to
calculate the 1D 68% and 90% C.L. allowed regions, as well as the minimum point indicated by
the light blue triangle.

223

The 90% C.L. allowed regions were found to be:−3

2.19 × 10eV2

sin2 2θ23 > 0.95,(9.18)< ∆m232 < 2.79 × 10−3 eV2 .

(9.19)

The 1σ and 90% C.L. regions were calculated from the 1D plots shown in Figure 9.13. The
undisplayed parameter was marginalised out. The 2D allowed regions from normal oscillations
is shown in the left of Figure 9.16. The allowed regions from this analysis specified in Equations (9.18) and (9.19) seem to be a little more confined than that of the Standard 2-flavour L/E
Analysis, written in Equations (8.38) and (8.39). However, this was mostly due to the different
procedure used to define the 90% C.L. allowed ranges. In the standard analysis case, a crosssection at the best fit point was used to define the 90% C.L. range for 2 DOF. It can be seen in
Figure 9.16 that the 2D allowed regions were very similar, which is discussed in Section 9.7.1.

9.6.3

Systematic Uncertainties

The contribution from the systematic errors were minimised during the χ2 fitting, and the best
fit values are summarised in Tables 9.1, 9.2, 9.3, and 9.4. The 1σ estimates for each of thesystematic uncertainties has been displayed graphically in Figure 9.14; and the best-fit values
relative to these 1σ estimates in Figure 9.15. Although the normalisation was essentially free inthis analysis, as the shape is the main concern with the L/E distributions, the estimate of 10%
has been shown. Relative to this, 0.65 σ of the normalisation systematic was used in the best fit.
If a penalty was included from this term, the χ2 would increase by 0.42, which would still resultin an excellent fit between MC and data.
The contributions from most systematics were below 0.5 σ, while all of them were below
1 σ. This perhaps suggests that the estimated magnitudes of the systematic uncertainties wereconservative, and that the pulls from these errors were not used too much in the best fitting
distribution. A very relevant uncertainty to this analysis was the ν̄µ /νµ ratio, conservatively
estimated at 6%, and its contribution at the best fit was only 0.39% (0.065 σ). So the difference
between ν and ν̄ was largely defined by the ν and ν̄ oscillation parameters rather than thesystematic uncertainty. The larger contributions came from a few systematics such as the MultiGeV normalisation, or interaction uncertainties like the quasi-elastic total cross-section, single
meson production total cross-section, or DIS Bodek correction. Although, the 1 σ estimates forthese were not too large at 5%, 10%, 20% and 20%, respectively.

9.6.4

Analysis Summary

No discrepancy between ν and ν̄ oscillation parameters was found. Despite allowing the additional freedom for ν and ν̄ parameters to oscillate independently, the best fit for these parameters
was essentially identical to the normal oscillation case. The allowed regions between particle andanti-particle oscillation parameters were overlapping, with the ν̄ regions being larger due to thelower statistical significance of the anti-neutrino events to influence the distributions. It has
been shown that, within the sensitivity of this analysis, neutrino oscillations appears to be CPT
symmetric.
224

20

Normal Oscillation ∆χ2Normal Oscillation ∆χ2

20

99% C.L.
18

14

12

12
min

14

10
8

68% C.L.

10
8

6

6

4

4

2

2

0
0.7

90% C.L.

16

68% C.L.

χ2-χ2

χ2-χ2

min

16

99% C.L.
18

90% C.L.

0
0.75

0.8

0.85
sin22θ23

0.9

0.95

1

-3

10

∆m232 (eV2)

10-2

Figure 9.13: sin2 2θ23 and ∆m232 1D Contours for Normal Oscillation, in which the ν and ν̄parameters were forced to be the same. The undisplayed parameter was marginalised out. The
points show the scanned parameters, which were smoothly fitted with a spline function as shownby the curve. The fitted curve was used to calculate the 1D 68% and 90% C.L. allowed regions,as well as the minimum point indicated by the light blue triangle.

9.7

Discussion

9.7.1

Comparisons Between The L/E Analyses

Comparisons of the various L/E Analyses are shown in the right plot of Figure 9.16. It can beseen that the ν, ν̄ and normal oscillation parameters all trace out overlapping regions. The best
fit points were all nearly identical, as indicated by the overlapping green and black stars, andred circle. The Standard 2F Analysis is depicted in the grey hatched region, which covers almost
the same area as the normal oscillation analysis here. It looks like the forced normal analysis
in this chapter has a slightly better constraint on sin2 2θ23 from the additional statistics of thesecond sample, providing a better distinction between the non-oscillated and average oscillated
bins. This corresponds to the difference in the event numbers observed at low and high L/E,
which largely accounts for the sensitivity of the L/E analysis to the sin2 2θ23 parameter.
The shared region between the normal oscillation and standard analysis suggest that the analysis is still stable with the modifications introduced in this chapter. While the second resolution
sample gave increased sensitivity to sub-dominant effects such as in independent anti-neutrino
oscillations or non-standard interactions. The bin merging in the χ2 also increases the stability
in lower statistics bins. At the L/E values near the position of maximum oscillation probability,
the statistics were reduced due to the precise resolution requirements. So the improved stabilityof the χ2 in these regions was preferable.

9.7.2

Fit in The Non-Physical Region

There was a very slightly better fit in the non-physical region with a ∆χ2 = 68.60 and thefollowing parameters: (sin2 2θ23 , ∆m232 , sin2 2θ̄23 , ∆m̄232 ) = (0.91, 2.51 × 10−3 eV2 , 1.20, 2.37 ×
10−3 eV2 ). This may be an indication that the ν̄ parameters could have slightly more influenceon the L/E distributions if the ν̄ events had a little more statistical significance. However, it
is likely that this effect was simply a fitting artefact from the best fit point being located in
225Normalisation

Common
SK1
SK2
SK3
SK4

FC Multi-GeV µ / PCν µ/ ν µ Ratio
Up / Down Ratio
Horizontal / Verticalν Flight Path Length
Energy Spectrum
Normalisation Multi-GeV
QE Scattering Total x-sec
Single Meson Production (x-sec)
DIS
Coherent π Production
NC / CC Ratio
FC Reduction
PC Reduction
Ring Counting
PID Single-Ring
PID Multi-Ring
Energy Calibration
Up / Down Asym of E. Calib.
Nuclear EffectNon-ν Background (µ-like)
PC Stop / Through (Top)
Axial-Vector Mass (M )A

DIS (Bodek Correction)
Solar Activity
Single Meson Production (π0 / π± )PC Stop / Through (Bottom)
PC Stop / Through (Barrel)

20

40
60
80
1σ Systematic Estimates (%)

Figure 9.14: Estimates of the systematic errors

226

100

Normalisation

Common
SK1
SK2
SK3
SK4

FC Multi-GeV µ / PCν µ/ ν µ Ratio
Up / Down Ratio
Horizontal / Verticalν Flight Path Length
Energy Spectrum
Normalisation Multi-GeV
QE Scattering Total x-sec
Single Meson Production (x-sec)
DIS
Coherent π Production
NC / CC Ratio
FC Reduction
PC Reduction
Ring Counting
PID Single-Ring
PID Multi-Ring
Energy Calibration
Up / Down Asym of E. Calib.
Nuclear EffectNon-ν Background (µ-like)
PC Stop / Through (Top)
Axial-Vector Mass (M )A

DIS (Bodek Correction)
Solar Activity
Single Meson Production (π0 / π± )PC Stop / Through (Bottom)
PC Stop / Through (Barrel)

-1.0

-0.8

-0.6

-0.4 -0.2
0.0 0.2
0.4
Systematics at Best Fit (σ)

0.6

0.8

Figure 9.15: Systematics at Best Fit Neutrino Anti-Neutrino Analysis

227

1.0

10-2

Normal Oscillation

0.006
90% C.L. Allowed Regions
ν L/E
ν Zenithν L/E
Normal Oscillation L/EStandard 2F L/E

0.004

2

∆m232 (eV2)∆ m232 or ∆ m32 (eV2)

99% C.L.
90% C.L.
68% C.L.

10-3
0.7

0.75

0.8

0.85
Sin22θ23

0.9

0.95

0.002

0
0.75

1

0.8

0.85
0.9
Sin2 2θ23 or Sin2 2θ23

0.95

1

Figure 9.16: 2D allowed region contours for Normal Oscillation on the left, and comparisons of
the various L/E analyses in linear scale on the right. In the Normal Oscillation, the ν and ν̄parameters were forced to be the same. The 90% C.L. from this analysis is shown by the dashdotted red line in the right plot. The region is very similar to the Standard 2F L/E analysis in
shaded grey. The L/E and Zenith ν̄ have comparable regions, with the Zenith result shifted tolower ∆m̄232 . All parameter regions were in agreement with each other.
between the scanned parameter points, at ∆m232 = 2.43 × 10−3 eV2 . In this case, the additional
power of the ν̄ distributions in the non-physical region could be selected to define a better fitting
total ν + ν̄ distribution.

9.7.3

Comparisons With Other Analyses

A similar ν̄ allowed region was obtained from an anti-neutrino Zenith analysis [12], although it
was shifted to smaller ∆m2 . This can also be seen in the right plot of Figure 9.16. The sin2 2θ̄23
was more confined in the Zenith analysis, perhaps due to greater statistics giving more power todistinguish between the non-oscillated and averaged oscillation bins. Generally the L/E analysis
has more sensitivity than the Zenith analysis to the ∆m2 parameter for normal oscillations, due
to locating the poisition of the “dip” from the first point of maximum oscillation probability.
However, when considering the ν̄ parameters between these analyses, the range extended by∆m̄232 was quite comparable. Both the Zenith and L/E analyses showed neutrinos and antineutrino allowed regions that were in agreement, and no evidence for a discrepancy between
particle and anti-particle oscillation parameters was observed.
In Figure 9.17, comparisons of the anti-neutrino measurements from SK and MINOS canbe seen. The original discrepancy between the ν̄ data (dotted red) and ν data (shaded grey)reported by MINOS in 2011 [10] is also shown for comparison. Their best fit for ν̄ at that timeis indicated with the red filled circle, which was excluded at the 90% C.L. by the L/E analysis ingreen, as well as the Zenith analysis [12] in dashed blue. An updated result from the MINOS ν̄data in 2012 [11] is traced by the dash-dotted grey contour, which also excludes the earlier best
fit point at the 90% C.L. However, the SK analyses gave better constraints on sin2 2θ̄23 , with the
SK data up until 2012. The L/E analysis essentially measures the same oscillation parametersfor ν and ν̄, indicated by the overlapping filled and hollow stars. These best fits were close to theν best fit from MINOS, shown by the open circle, while the Zenith analysis preferred a slightly
smaller value for ∆m̄232 .
228

0.0082∆ m232 or ∆ m32 (eV2)0.006

90% C.L. Allowed Regions
ν SK L/E
ν SK Zenithν MINOS 2009-2010
ν MINOS 2009-2011
ν SK L/E
ν MINOS 2005-2010

0.004

0.002

0
0.5

0.6

0.7
0.8
Sin2 2θ23 or Sin2 2θ23

0.9

1Figure 9.17: Comparisons of various neutrino and anti-neutrino oscillation parameter measurements at the 90% C.L. and in linear scale. The dotted red contour shows the first ν̄ result fromMINOS [10] in 2011, which was distinct from the ν data in the shaded grey region. An updated
result from MINOS in 2012 [11] is shown in the dash-dotted grey contour. Results from SK withdata until around 2012 are also shown. The L/E ν̄ contour is drawn in the green solid line, andjust excluded the MINOS 2011 best fit. Similarly for the Zenith measurement [12] in dashedblue. In addition, the L/E neutrino region has been overlaid (marginalised over ν̄) in the shaded
green contour. The ν and ν̄ best fits from L/E were overlapping (filled and open stars), andclose to the MINOS ν best fit (open circle). All measurements for ν and ν̄ were in agreementfrom 2012.

229

9.7.4The Result and Prospects

All of these results suggest that neutrino oscillations are CPT symmetric, when considering
simple 2 flavour vacuum oscillations. Otherwise, any CPT asymmetry present must be verysmall. The initial surprise on the discrepancy in simple oscillation parameters between neutrino
and anti-neutrino has subsided. However, there is still a possibility for a degree of CP-asymmetry
in neutrino oscillations. This effect is commonly characterised by the δcp parameter in full 3flavour oscillations. As the determination of this parameter requires a very accurate measurementof ν and ν̄ oscillations, it is at the limit of the sensitivity of present day experiments.
Accelerator experiments such as T2K [37] and MINOS [43] are beginning to confine theallowed region of δcp . T2K, MINOS+ and NOνA [154] will continue to improve these constraints.
Meanwhile Hyper-Kamiokande [147], the proposed successor to Super-Kamiokande, will aim tofurther improve neutrino measurements with much greater statistics.
So there is still considerable interest to make neutrino and anti-neutrino measurements independently. Further sensitivity could be achieved by having much better charge discriminationof outgoing leptons, and therefore the power to distinguish neutrino from anti-neutrino. The
India-based Neutrino Observatory, INO [263], has a part of its design to have a magnetised atmospheric neutrino detector on a 50 kT scale. There is also interest to make measurements of
CPT violation in neutrinos at INO [264]. In addition, designs for Neutrino Factories [147] havethe advantage of a very precisely understood neutrino flux from the decay of accelerated muonsand a detector with excellent charge discrimination.

230

Chapter 10

3-Flavour L/E Analysis
It has been common in neutrino oscillation experiments to approximate the neutrino flavour
transition probabilities with 2-flavour models. Since the two mass square differences (∆m221 ∼7.6 × 10−5 eV2 and ∆m231 ∼ 2.5 × 10−3 eV2 ) differ by nearly 2 orders of magnitude, and theparameter that allows cross-mixing between these 2-flavour schemes (θ13 ) was considered to besmall, these approximations were very accurate.
However, since 2012 it has been shown that θ13 is non-zero, and not so small. This was
first indicated by T2K [39], and unambiguously observed by the reactor experiment Daya Bay
in March 2012 [14], with a significance of 5.2 σ, and shortly afterwards confirmed by the reactor
experiments RENO [41] and Double CHOOZ [42]. The reactor experiments made their measurement by observing the disappearance of νµ (into νe ) at a distance of around ∼1 km. In addition,in 2013 the accelerator experiment T2K made the first observation of νe appearance from a νµbeam, with 7.2 σ significance [37].
One of the principle characteristics of neutrino oscillations is that the transition probability
depends on the quantity L/E. However, up until now there has not been a 3-flavour oscillation
analysis considering this characteristic. The analysis presented in this chapter is the first attempt
to study the L/E dependence with 3-flavour oscillations. It has been shown that in 2-flavours,
the L/E analysis provides strong constraints on the oscillation parameters, with the most confining measurement of ∆m231 from atmospheric neutrinos. This L/E characteristic has also beensuccessful at excluding competing theories for the flavour transitions observed in atmospheric
neutrinos.
The aim of this analysis was to confirm that the L/E analysis still provides a meaningfulmeasurement under the presence of 3-flavour effects, such as transitions through matter andνµ ↔ νe mixing through oscillations. As this is the first 3-flavour L/E analysis, measurementson the neutrino mass hierarchy and the octant of θ23 were made. However, the sensitivity tothese parameters was thought to be small, especially since this analysis does not have tuned
sensitivity to them. For the same reason, the influence of δcp was not considered. However,
the transition probabilities were updated to include 3-flavour oscillations and matter effects. To
achieve this, the non-scanned oscillation parameters were set at the best fit from global solar
oscillation parameter results [32], and from measurements of θ13 in early 2012 [14, 41]. At thistime, θ13 was found to be around sin2 θ13 = 0.025.
In recent years, with measurements on the last remaining oscillation parameter θ13 , we havebeen closing in on the final formulation of neutrino oscillations. So it is preferable to carry out
analyses with the more accurate 3-flavour probabilities. Through doing this, more significant
231measurements on the atmospheric parameters θ23 and ∆m231 can be obtained from the L/E
analysis – free from any concerns from making 2-flavour approximations.
Super-Kamiokande (SK) has an impressive data set of atmospheric neutrinos spanning over
10 years, and a wide range of neutrino energies and baselines. This makes analyses at SK
ideal for scanning the whole parameter space of atmospheric oscillation parameters. Not only
can these analyses make precision and significant measurements of these parameters, but they
can potentially observe additional effects that may not be visible in an experiment with tuned
sensitivity. In addition, the updates in this analysis also serve as a precursor to considering
the effects of Non-Standard Neutrino Interactions (NSI) in a 3-flavour framework, which will be
discussed in Chapter 11.

10.1

Moving from 2-Flavours to 3-Flavours

Since ∆m231 and ∆m221 differ by about 1.5 orders of magnitude, Neutrino Oscillations can be
decoupled into two 2-flavour regimes to a good approximation. For the analysis of atmospheric
neutrinos, ∆m231 (≈ ∆m232 ) was most significant and could accurately describe the data with
νµ → ντ oscillaitons [30, 31]. Whereas ∆m221 could be probed with solar and reactor neutrinos [32]. As greater statistics were collected and early evidence for non-zero θ13 emerged, someanalyses were carried out considering 3 flavours, such as the 3-flavour zenith analysis at SK [15].
This is the first L/E analysis to take into account 3-flavour effects.

10.1.1Treatment of Oscillation Probabilities

The νµ deficit in the Standard L/E Analysis discussed in Chapter 8 was described well by the
2-flavour probability:2
2
2 1.27∆m32 Lν
P (νµ → ντ ) = sin 2θ23 sin,(10.1)
Eν
while in this analysis, the neutrino oscillations were treated with the following Hamiltonian:


0
0
0
1
†
0  UkβUαj 0 ∆m221± VM SW ,(10.2)
Hαβ =
2E
2
0
0
∆m31
where U is the Unitary Lepton Mixing Matrix:




1
0
0
c13 0 s13
c12 s12 0
1 0  −s12 c12 0 ,U = 0 c23 s23   0
0 −s23 c23−s13 0 c13
0
0 1(10.3)

and where cij ≡ cos θij and sij ≡ sin θij . Notice that the CP-term, e±iδcp , has been omitted inthis analysis. Matter effects were included in Equation (10.2), in the VM SW term:

VM SW =

√




1 0 0
2GF Ne (~r) 0 0 0 ,
0 0 0
232

(10.4)

where GF = 1.166 × 10−5 GeV−2 is the Fermi constant, and Ne the ambient electron number
density calculated along the neutrino path from production to the detector.
In the L/E distributions, νµ events were selected and form the bulk of the data analysed.
The Standard L/E Analysis considered only the 2-flavour survival probability P (νµ → νµ ). Now
that 3-flavour effects were being considered, mixing between νµ and νe must also be considered.
Oscillation probabilities were also applied to the few remaining νe events for completeness. The
probabilities were modified as follows:Φ(νe ),
Φ(νµ )
Φ(νµ )P (νe ) = P (νe → νe ) + P (νµ → νe ),Φ(νe )

P (νµ ) = P (νµ → νµ ) + P (νe → νµ )(10.5)(10.6)

where Φ(νµ ) and Φ(νe ) are the flux of νµ and νe , respectively, taken from looking up flux tables
on an event-by-event basis.
To calculate the probabilities, the hamiltonian in Equation (10.2) was diagonalised to get theeffective eigenvalues Ĥ = diag(E1 , E2 , E3 ) and used to form an evolution matrix for constantmatter density:j(t, t0 ) =Sβα

3X
0 ∗ 0 −iEi L(Uαi) Uβi e(α, β = e, µ, τ ),(10.7)

i=1

where L is equivalent to t − t0 and U 0 is the Lepton Mixing Matrix in matter. These amplitudes,j, were calculated for the j-th region of constant matter density through the earth. In theSβαsimulation, the interior of the Earth was divided into five regions of constant matter density
using the PREM model [160]. The trajectory of the neutrino through these matter regions wasjdetermined and Sβα
was calculated for each region traversed. Then the probability for the α → βjas follows:transition was computed from the Sβα
2
1 2
J
| ,
Sβα
Pαβ = |Sβα
· · · Sβα(10.8)

for J regions of constant matter density traversed.

10.1.2

Flavour Flux Re-Weighting

In order to consider the mixing between νµ and νe , the events in the MC needed to be re-weighted
to the corresponding flavour. This was done so that all of the reconstruction in the MC was forthe neutrino of the flavour observed, but the origin of the neutrino could be assumed to be of adifferent flavour.
Event-by-event, the expected flux for νµ and νe were checked at the particular neutrino energy
and direction of the event, and whether it was a neutrino or anti-neutrino. These flux weightvariables Φ(νµ ) and Φ(νe ) were found by looking up flux tables, provided by the Honda 2011
flux calculation [16]. The ratio of the fluxes could be used to re-weight the νµ flux to the νe flux
Φ(νe )
with the ratio Φ(ν, or vice-versa.
µ)
233Osc only: Numu to Nue, Normal, antiNu

0

0.6

-0.1
0.5

-0.2
-0.3

Zenith (cosθ)

Zenith (cosθ)Osc only: Numu to Nue, Normal, Nu

0

0.5

-0.1

-0.2

0.4

-0.3

0.4
-0.4

-0.4

-0.5
-0.6

-0.6
0.2

-0.7
-0.8

-0.8

0.1

0.1

-0.9
1

10

102
Energy (GeV)

-1 -1
10

0

(a) P (νµ → νe ) Normal Hierarchy

1

10

102
Energy (GeV)0(b) P (ν̄µ → ν̄e ) Normal Hierarchy
Osc only: Numu to Nue, Inverted, antiNu

0
0.5

-0.1

-0.2

Zenith (cosθ)Osc only: Numu to Nue, Inverted, Nu
Zenith (cosθ)

0.2

-0.7

-0.9
-1 -1
10

0.3

-0.5

0.3

0
0.6
-0.1

-0.2

0.4

-0.3
-0.4

0.5

-0.3
0.4

-0.4
0.3

-0.5

-0.5

-0.6

0.2

-0.7

0.2

-0.7

-0.8

0.1

-0.9
-1 -1
10

0.3

-0.6

-0.8
0.1
-0.9

1

10

102
Energy (GeV)

-1 -1
10

0

(c) P (νµ → νe ) Inverted Hierarchy

1

10

102
Energy (GeV)0(d) P (ν̄µ → ν̄e ) Inverted Hierarchy

Figure 10.1: Oscillation probabilties for νµ → νe and ν̄µ → ν̄e for Normal and Inverted Hierarchies. The oscillation parameters were taken as global fit values from an analysis after Neutrino
2012: ∆m221 = 7.62 × 10−5 eV 2 , ∆m231 = 2.55 × 10−3 eV 2 , sin2 θ23 = 0.613, sin2 θ12 = 0.320, andsin2 θ13 = 0.0246. The effect of the resonance term can be seen at ∼5 GeV for neutrinos in theNormal Hierarchy (a), and anti-neutrinos for the Inverted Hierarchy (d)10.2

Neutrino Mass Hierarchy

One of the remaining pieces of the Neutrino Oscillation model to be determined is the ordering of
the neutrino mass hierarchy. Whether the neutrino masses are arranged naturally in increasing
mass as ν1 , ν2 , ν3 in the Normal Hierarchy or as ν3 , ν1 , ν2 in the Inverted Hierarchy.
The Inverted Hierarchy was simulated to a good approximation by changing the sign of the∆m231 parameter in the Unitary Lepton Matrix written in Equation (10.4). The possibility ofmeasuring the mass hierarchy comes from a resonance term in the interactions with matter,causing an enhancement in the P (νµ → νe ) probability for neutrinos in the Normal Hierarchy
case, and for anti-neutrinos in the Inverted Hierarchy case. The resonance occurs for upwardgoing events at around ∼5 GeV, and can be seen in Figure 10.1. Since the mass hierarchy is
unknown, the analysis was carried out for each case and the results compared.
234

10.3

Analysis Method

The analysis method used a χ2 based on Poisson Statistics and is very similar to the Standard L/E
Analysis (Chapter 8) with all the improvements made in the Neutrino Anti-Neutrino Analysis(Chapter 9.1). Specifically, the improvements were the merging of the bins from SK1 to SK4
during the χ2 calculation to have increased stability for bins with few events; and the additionof a Second Resolution Sample. For details, refer to Section 9.2. This analysis was built on topof the previous analyses described, so the verifications and the details written earlier about theanalysis techniques apply here also. The main addition to the analysis in this chapter was amodification of the probabilities to take into account 3-flavour oscillations and matter effects.
The parameter space scanned over was 201 points in ∆m231 from 1 × 10−3 eV2 to 1 × 10−2 eV2with logarithmically spaced intervals, and 51 linearly spaced points in sin2 θ23 from 0.2 to 0.8.
To include the effects from the other flavours the remaining parameters were set as ∆m221 =7.66×10−5 eV2 , sin2 θ12 = 0.304, and sin2 θ13 = 0.025 for both hierarchies. These parameters wereset by global solar oscillation parameter results [32], and sin2 θ13 by the earliest measurements
in 2012 from Daya Bay [14] and RENO [41].

10.4

Results

The atmospheric oscillation parameters in the MC were varied over the parameter space, andfor each parameter set the L/E distributions were calculated and compared with data. In thisanalysis, the L/E MC distributions were determined with 3-flavour oscillation probabilities.

10.4.13-Flavour Results

After scanning the whole parameter space, the points in (∆m231 , sin2 θ23 ) with the minimum
χ2 were (2.60 × 10−3 eV2 , 0.55) for the Normal Mass Hierarchy and (2.45 × 10−3 eV2 , 0.54) for
the Inverted case. The minimum χ2 for the Normal Hierarchy was 69.56/73 while the Invertedcase was 69.44/73. So a very good fit of the Monte-Carlo to the data was found for both mass
hierarchies, with neither scenario being favoured over the other.
The bestfit L/E distributions have been drawn in Figure 10.2, for both hierarchies. As
suggested by the minimum χ2 values, a very good fit can be seen for both cases. So the histogramsfor both hierarchies appear to be overlapping in the figure, after the systematics have beenminimised in their contribution to get the best fit to the data. As expected, there was a large
discrepancy between the bestfit MC and the MC assuming no oscillations (Null Oscillation MC).
The bestfit L/E distributions in Figure 10.2 were divided through by the null oscillation
expectation, giving the distiribtions in Figure 10.3. This gave an idea of the relative change of the
distributions caused by oscillations for a particular value of L/E. The largest deficit was seen ataround ∼500 km/GeV where you would expect maximum oscillation for ∆m232 ≈ 2.5 × 10−3 eV2 .
Even with sin2 θ13 = 0.025, the oscillation minimum “dip” was seen as clearly as in the 2-flavour
case, suggesting that the L/E Analysis is a good method to study the oscillations in detail. For
large L/E, the oscillations are very rapid and there is an average reduction in the flux. Again,
very little difference was seen in the distributions between the mass hierarchies. For comparison,
the corresponding distribution in the 2-Flavour Standard L/E Analysis was also drawn.
The analysis was carried out by inputting sin2 θ23 , as opposed to the double angle variable2
sin 2θ23 that is often used in 2-flavour models, to potentially observe any difference in the θ23
235

600

600

Precise L/E Data

2nd Sample Data

Normal Bestfit

Normal Bestfit

Null Osc. MC

500

Null Osc. MC

500

Inverted Bestfit

Inverted Bestfit

Number of events

400

Number of events

400

300

300

200

200

100

100

0

2

1

3

10
10
10
L/E (km/GeV) (SK 1+2+3+4)

10

0

4

1

2

3

10
10
10
L/E (km/GeV) (SK 1+2+3+4)

10

4

Figure 10.2: L/E bestfit distributions for the 3-Flavour L/E Analysis, for the precise resolution
sample on the left and second resolution sample on the right. The red solid histogram shows
the bestfit for Normal Hierarchy and the blue dashed histogram for the Inverted case. The reddashed histogram shows the MC L/E distribution with no oscillations applied. Since a very good
fit was obtained for both hierarchies, very little discrepancy can be seen between them and thehistograms were essentially overlapping.

octants. The resulting allowed region can be seen in Figure 10.4, for ∆m231 against sin2 θ23 . The
first immediate observation was that the contours were considerably symmetric across sin2 θ23 =0.5 and that neither octant was significantly favoured over the other. The best fit points forboth hierarchies were for sin2 θ23 > 0.5. However, looking at the 1-dimensional contours drawn
by integrating out the other parameter in Figure 10.5, it can be seen that the χ2 distribution
was very flat about sin2 θ23 = 0.5. This suggests that any value close to sin2 θ23 = 0.5 could havebeen favoured by this analysis.
In the 2D parameter space (Figure 10.4), it appears that the contours for the Normal Hierarchy were shifted very slightly up and to the right compared to the Inverted case. Here the∆χ2 was drawn from the global minimum and the 68%, 90% and 99% C.L. correspond to a∆χ2 of 2.30, 4.61, and 9.21, respectively. This effect can be seen more clearly in the 1D plots
in Figure 10.5. In these 1D distributions the 68%, 90% and 99% C.L. correspond to a ∆χ2 of
1.00, 2.71, and 6.63, respectively. The 1σ errors (68% C.L.) and 90% C.L. allowed regions were
calculated from the 1D distributions. The bestfit parameters with 1σ errors were:

+0.22
2
−3
sin2 θ23 = 0.55+0.07
eV2−0.13 , and ∆m31 = (2.60−0.21 ) × 10

Normal Hierarchy,(10.9)−3∆m231 = (2.45+0.25
eV2−0.16 ) × 10

Inverted Hierarchy,(10.10)2sinθ23 = 0.54+0.07−0.12 ,

and

236

1.8

1.8
Precise L/E Data

1.6

2nd Sample L/E Data

1.6

Normal Bestfit

Data/Prediction (null oscillation)

Data/Prediction (null oscillation)

Normal Bestfit
Inverted Bestfit

1.4

2 Flavour

1.2
1
0.8
0.6
0.4
0.2

Inverted Bestfit

1.4
1.2
1
0.8
0.6
0.4
0.2

0
1

2

3

10
10
10
L/E (km/GeV) (SK 1+2+3+4)

10

0

4

1

2

3

10
10
10
L/E (km/GeV) (SK 1+2+3+4)

10

4

Figure 10.3: L/E bestfit MC over the null oscillation expectation for the 3-Flavour L/E Analysis. The left plot shows the precise resolution sample with the red solid histogram for Normal
Hierarchy and the blue dashed histogram for the Inverted case. As expected from the L/E distributions in Figure 10.2, these two histograms were essentially overlapping here as well. Also
drawn for comparison in the green dashed histogram is the bestfit from the 2-Flavour Standard
L/E Analysis (see Chapter 8). A small difference can be observed between the 2-Flavour and
3-Flavour case. The right plot shows the second resolutions sample, where very little difference
can be observed between the two mass hierarchies.

∆ m231 (eV2)

0.005

0.004

68% C.L.
90% C.L.
99% C.L.
NH Min.
IH Min.

Normal Hierarchy
Inverted Hierarchy

0.003

0.002

0.001

0
0.2

0.4

0.6

0.8
Sin2θ23

Figure 10.4: Oscillation parameter allowed regions for the 3-Flavour L/E Analysis, this timecompared against the single-angle parametersin2 θ23 . The solid contours show the result fromthe 3-Flavour Normal Neutrino Mass Hierarchy
assumption, and the dashed lines for the Inverted case. The star shows the Normal Hierarchy minimum point, and the circle the Inverted
Hierarchy minimum. All the contours look quite
symmetric about sin2 θ23 = 0.5. The Normal
Hierarchy contours appear to be very slightly
shifted up and to the right compared to the Inverted Hierarchy case.

237

18
16

min

20

99% C.L.
90% C.L.
68% C.L.

χ2-χ2

χ2-χ2

min

20
Normal Hierarchy

18

Inverted Hierarchy

16

14

14

12

12

10

10

8

8

6

6

4

4

2

2

0
0

99% C.L.
90% C.L.
68% C.L.

Normal Hierarchy
Inverted Hierarchy

0
0.001

0.002

0.003

0.004
0.005
∆ m231 (eV2)

0.2

0.4

0.6

0.8
sin2θ23

Figure 10.5: 1-Dimentional allowed regions from the 3-Flavour L/E Analysis. In each of theseplots the other parameter was integrated out. The solid histogram is for the Normal Hierarchy
case, and the dashed histogram for Inverted. The confidence levels have been marked by thehorizontal lines. For both parameters, slightly larger values were favoured for Normal Hierarchy
compared to the Inverted case.
and the 90% C.L. allowed ranges were:0.396 < sin2 θ23 < 0.644, and 2.275 × 10−3 eV2 < ∆m231 < 2.995 × 10−3 eV2Normal,(10.11)0.390 < sin2 θ23 < 0.631, and 2.182 × 10−3 eV2 < ∆m231 < 2.842 × 10−3 eV2Inverted.
(10.12)

These allowed ranges were calculated with a smooth spline function that fitted the scanned ∆χ2points well. In this way, the 68% and 90% C.L. ranges were more accurately calculated from theinterpolation.
The data were fit equally well by both the Normal and Inverted Mass Hierarchy hypotheses.
So there was no evidence to favour one over the other. The only difference observed was that in
the Normal Hierarchy case, slightly larger ∆m231 or sin2 θ23 were preferred by the best fit. For
sin2 θ23 , no significant asymmetry was seen around 0.5 to show any preference for a particularoctant. However, for both hierarchies the 90% C.L. extends slightly further above 0.5 than it
does below 0.5. Since the main purpose of this analysis was to determine if an L/E analysis is
still meaningful in a 3-flavour formalism, this issue was not investigated in detail.

10.4.2Comparison With The 2-Flavour L/E Results

The official result from the L/E analysis was determined from a 2-flavour analysis, as describedin Chapter 8. Any L/E analysis performed by a neutrino oscillation experiment has been done in
2-flavours. The results in this chapter are the first attempt to make an L/E measurement with
3-flavour oscillations and matter effects. Here, the 3-flavour results are compared with thosefrom the 2-flavour analysis.
238∆ m231 (eV2)

0.005

0.004

68% C.L.
90% C.L.
99% C.L.
L/E 2 Flavour
NH Min.
IH Min.

Normal Hierarchy
Inverted Hierarchy

0.003

0.002

0.001

0
0.8

0.85

0.9

0.95

1
Sin2 2θ23

Figure 10.6: Oscillation parameter allowed regions against the double angle parameter,sin2 2θ23 , to compare the allowed parameter
space between the 3-Flavour L/E Analysis andthe 2-Flavour Standard L/E Analysis. The solid
contours show the result from the 3-Flavour Normal Neutrino Mass Hierarchy assumption, andthe dashed lines for the Inverted case. The
3 gray shaded regions show the equivalent allowed regions from the Standard L/E Analysis.
The star shows the Normal Hierarchy minimum
point, and the circle the Inverted Hierarchy minimum.

Analyses in 2-flavours generally characterise oscillations with the double angle parameter,sin2 2θ23 . For comparison, the points in the 3-flavour analysis between sin2 θ23 = 0.2 → 0.5
were converted to the double angle equivalent. The allowed regions in this parameter space areshown in Figure 10.6. Looking at these allowed regions, the Inverted Hierarchy contours seemedvery similar to those of the 2-Flavour analysis. Whereas slightly larger ∆m2 was favoured in
the Normal Hierarchy case, and the shift of sin2 θ23 to larger values resulted in the apparently
more constrained region in sin2 2θ23 shown here. The minimum points in (∆m231 , sin2 2θ23 ) were(2.60 × 10−3 eV 2 , 0.991) for Normal Hierarchy and (2.45 × 10−3 eV 2 , 0.995) for the Inverted case.
The Standard L/E Analysis does not include some modifications present in this analysis, suchas the merging of the analysis bins during the χ2 calculation, or the second resolution sample.
However, in the Neutrino Anti-Neutrino Analysis, an analysis assuming normal oscillation was
carried out. In that analysis the neutrino and anti-neutrino parameters were forced to be thesame, and the analysis included these modifications. It was shown in Figure 9.16 that theallowed regions for normal oscillation were almost identical to the Standard L/E regions. As
these modifications were made to increase sensitivity to sub-dominant effects.
The comparable allowed regions to the 2-Flavour Standard L/E Analysis support this analysis
and indicate that the extension to 3-Flavours was successfully implemented. The differencessuggest that more accurate Oscillation parameters and allowed regions can be obtained with theadded information from 3-flavour effects, omitted in the 2-flavour assumption.

10.4.33-Flavour L/E Systematic Uncertainties

The 1 σ estimates of the systematic uncertainties were the same for all L/E analyses and can beseen in Figure 9.14, and the minimised systematics at the best fit for this analysis can be seen in
Figure 10.7 for Normal Hierarchy, and Figure 10.8 for Inverted Hierarchy. For both hierarchies,
the best fit systematic errors were all below 1 σ, with most being below ∼0.4 σ. This suggeststhat the estimates of the systematics were quite conservative. Despite the slight over-estimationof the errors, the contribution of the systematics to the fitting was minimal.
The shape of these systematic pull distributions was very similar for each hierarchy, and239

also for the Neutrino Anti-Neutrino Analysis (Figure 9.15), with only slight differences in the2magnitude of each uncertainty. The
 contribution to the χ from the systematics for the Neutrino
 2
1Anti-Neutrino Analysis was Σ σ = 3.84 and 3.60 from the Standard 2F Analysis; while for the
3-Flavour Normal and Inverted Hierarchy these contributions were 3.31 and 3.54, respectively.
So the 3-Flavour analyses were slightly less dependent on the uncertainties to achieve their best
fit, with the 3F Normal Hierarchy fit requiring the smallest contribution from systematics.
Details of the bestfit systematic uncertainties can be seen in Table 10.3 for the neutrino flux;
Table 10.2 for neutrino interactions; Table 10.4 and 10.5 for the reconstruction in normal andinverted hierarchy, respectively; and Table 10.6 and 10.7 for the reduction in normal and inverted
hierarchy, respectively.
Since a very good fit was achieved for these 3-flavour analyses, with small contributions fromthe systematic uncertainties, the 3-Flavour L/E distributions appear to be an excellent representation of the data. So the 3-Flavour L/E Analysis is great for investigating the atmospheric
neutrino oscillation parameters, with the least assumptions of all the analyses in this thesis.

10.5

Discussion

10.5.1Comparisons Between L/E Analyses

In order to take a closer look at the difference between each of the L/E analyses, ratios of
the L/E distributions between the analyses and Normal and Inverted Hierarchies are shown in
Figure 10.9. The left plot in the figure was drawn for the precise resolution sample, and forthe second resolution sample on the right. In the ratio plots, the blue line depicts the ratio of
Inverted/Normal hierarchy for the precise sample and magenta for the second resolution sample.
Very little difference could be observed in these ratios.
A small difference of about ∼2% between the hierarchies was seen in the precise samples ataround 1,200∼4,000 km/GeV. This was the region of the resonance coming from the sign of the∆m2 and the MSW matter effect, which can be seen in Figure 10.1. The resonance was expected
to occur at a zenith of -0.49 to -0.65 equivalent to path lengths of about 6,200∼8,500 km, andenergies of around 4.2∼5.2 GeV. This corresponds to an L/E range of about 1,200∼2,050 km/GeV.
There was also some resonance expected at a zenith of around -0.9, corresponding to an L/Erange of about 2,400∼4,800 km/GeV. Although small, some difference was seen in the L/E bins
corresponding to the matter resonance. Unfortunately, there was no way to determine whether
this difference occurred for neutrinos or anti-neutrinos, which would be necessary to clearly
distinguish the mass hierarchy. At SK the neutrino events in the data are more statisticallysignificant than anti-neutrinos, so a larger difference from the resonance would be expectedin the case of Normal Hierarchy (in which the resonance occurs for neutrinos) than Inverted(resonance in anti-neutrinos). However, the difference in the Normal and Inverted fits were notsignificant enough to favour one over the other. A very small variation was seen in the second
resolution sample, but it was less than a ∼1% effect.
Also shown in the left of Figure 10.9 is the L/E ratio between the 3-Flavour and 2-Flavour
analyses. There appeared to be a distinctive peak at around 2000 km/GeV. This could be due tothe averaging treatment in the Standard 2-Flavour analysis. For an oscillation phase greater than
2π, the oscillation phase sin2 (1.27L/E) term was replaced with 0.5. This averaging treatment
1

The sum here is over all the systematic uncertainties.

240

Normalisation

Common
SK1
SK2
SK3
SK4

FC Multi-GeV µ / PCν µ/ ν µ Ratio
Up / Down Ratio
Horizontal / Verticalν Flight Path Length
Energy Spectrum
Normalisation Multi-GeV
QE Scattering Total x-sec
Single Meson Production (x-sec)
DIS
Coherent π Production
NC / CC Ratio
FC Reduction
PC Reduction
Ring Counting
PID Single-Ring
PID Multi-Ring
Energy Calibration
Up / Down Asym of E. Calib.
Nuclear EffectNon-ν Background (µ-like)
PC Stop / Through (Top)
Axial-Vector Mass (M )A

DIS (Bodek Correction)
Solar Activity
Single Meson Production (π0 / π± )PC Stop / Through (Bottom)
PC Stop / Through (Barrel)

-1.0

-0.8

-0.6

-0.4 -0.2
0.0 0.2
0.4
Systematics at Best Fit (σ)

0.6

0.8

1.0

Figure 10.7: Systematics at Best Fit 3-Flavour L/E Analysis, Normal Hierarchy241Normalisation
FC Multi-GeV µ / PC

Inverted Hierarchyν µ/ ν µ Ratio
Up / Down Ratio
Horizontal / Verticalν Flight Path Length

Common
SK1
SK2
SK3
SK4

Energy Spectrum
Normalisation Multi-GeV
QE Scattering Total x-sec
Single Meson Production (x-sec)
DIS
Coherent π Production
NC / CC Ratio
FC Reduction
PC Reduction
Ring Counting
PID Single-Ring
PID Multi-Ring
Energy Calibration
Up / Down Asym of E. Calib.
Nuclear EffectNon-ν Background (µ-like)
PC Stop / Through (Top)
Axial-Vector Mass (M )A

DIS (Bodek Correction)
Solar Activity
Single Meson Production (π0 / π± )PC Stop / Through (Bottom)
PC Stop / Through (Barrel)

-1.0

-0.8

-0.6

-0.4 -0.2
0.0 0.2
0.4
Systematics at Best Fit (σ)

0.6

0.8

1.0

Figure 10.8: Systematics at Best Fit 3-Flavour L/E Analysis, Inverted Hierarchy

242

Data/Prediction (null oscillation)

Data/Prediction (null oscillation)

1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

Precise L/E Data
Normal Bestfit
Inverted Bestfit
2 Flavour

1

1.1
1.075
1.05
1.025
1
0.975
0.95
0.925
0.9

2

3

10
10
10
L/E (km/GeV) (SK 1+2+3+4)

10

4

2F / 3F Normal

10

10

2

10

3

10

2nd Sample L/E Data
Normal Bestfit
Inverted Bestfit

1

1.02
1.015
1.01
1.005
1
0.995
0.99
0.985
0.98

Inverted / Normal

1

1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

4

2

3

10
10
10
L/E (km/GeV) (SK 1+2+3+4)

10

4

Inverted / Normal

1

10

10

2

10

3

10

4

Figure 10.9: The top plots show the L/E distributions for the precise resolution sample on theleft, and the second resolution sample on the right. The corresponding lower plots show someratios of the distributions.
would take effect at an L/E just over 2,000 km/GeV. This could cause a more abrupt discrepancy
in the relevant bin, as there was no explicit averaging treatment in the 3-Flavour L/E Analysis.
The statistics in the MC were too high to explain the difference. There was 500 years of simulated
data per SK period, or 2000 years in total, which was normalised to around 10 years live time. Sothe√∼150 events in the bin in question, when converted back to the MC statistics, corresponds
to√
1/ 150 × 200 = 0.6%, or less than 1 event. However, in the data itself a fluctuation of 1/ 150 ∼8% could be expected, and these MC sets were fitted against this same data. Considering thedata in the L/E distribution in Figure 10.2, considerable fluctuation can be seen around L/E
1,000∼4,000 km/GeV. In this particular case, the Standard MC was found to be above the data
point in the L/E∼2,000 km/GeV bin, and the 3F Normal Hierarchy MC was below it. This isvisible with a close inspection of Figure 10.9 at the relevant bin, and clearly both fitted values areallowed by the data point. Otherwise, some difference was expected from properly considering
3-flavour and matter effects. With fewer assumptions, the 3-flavour simulation should be a better
representation of neutrino oscillations.
A comparison of the bin-by-bin χ2 can be seen in Figure 10.10. These distributions were very
similar between the mass hierarchies, and no bins distinguishing the hierarchies were found.

10.5.2Comparisons With Other Analyses

Figure 10.11 shows some comparisons of the L/E Oscillation Analysis results against other measurements of the same parameters. The left plot compares against the 3-Flavour Zenith Analysis,
the official 3-flavour result from the Super-Kamiokande Collaboration. There was relatively more
asymmetry over sin2 θ23 in the Zenith Analysis, compared to the symmetrical shape from the
L/E Analysis. The Zenith minimum points were also fairly distant from sin2 θ23 = 0.5, changing octant with hierarchy, whereas a minimum close to 0.5 was favoured in this analysis. In
243

600
Number of events

Precise L/E Data

500

Precise Best MC

400

2nd Sample Data300

2nd MC Best

200
100
0
1

2

3

10
10
10
L/E (km/GeV) (SK 1+2+3+4)

10

4

12
Bin by Bin χ2

Figure 10.10: The bin-by-bin χ2 distributions
are shown in the lower left plot, with the corresponding L/E distributions shown above. The
precise sample is shown by the black histogram,the second resolution sample in blue, where thesolid curves are for the Normal Hierarchy andthe dashed for Inverted. Not much difference
was observed, but a few bins contributing to asmall difference in the χ2 can be seen.

Precise L/E

10Second Sample

8
6
4
2
0
1

10

2

10
10
L/E Bins (km/GeV)

3

10

4

the Zenith analysis, one octant was clearly preferred at about the 68% C.L., depending on theneutrino mass hierarchy. In the L/E case, the ∆χ2 distributions were quite flat for sin2 θ23 , forboth hierarchies. It is likely that the Zenith Analysis has better sensitivity for determining a
difference in the octant of sin2 θ23 , as some difference was expected from the electron neutrino
sample at low energies (less than 1 GeV) [265]. However, due to the less precise reconstructionof the L/E variable in such samples, electron samples and low energy samples (below ∼1 GeV by
the L/E resolution requirement) were excluded from the L/E analysis.
Both analyses saw little difference between the Normal and Inverted Mass Hierarchy assumptions. However, the Zenith Analysis does see a switch in the sin2 θ23 octant of the best fit fromeach hierarchy when sin2 θ13 was fixed [266], which was not seen with the L/E distributions. The
3-Flavour L/E analysis suggests a more constrained region for ∆m231 . It should be noted that
the Zenith Analysis scans over δcp , however this parameter was not included in the 3-Flavour
L/E calculation. Inclusion of δcp could, perhaps, lead to a slightly more relaxed allowed region.
The right plot in Figure 10.11 shows the allowed regions from multiple analyses from 2012,
with ∆m231 plotted against sin2 2θ23 . The 2-Flavour L/E Analysis gave a slightly more constrained region in ∆m2 than the 2-Flavour Zenith equivalent. A similar trend seems to be visible
in the 3-flavour variations of the analyses, with the Zenith Analysis allowing a larger range of∆m2 values, particularly at smaller sin2 2θ23 . The slightly narrower allowed region of ∆m2 fromthe L/E analyses was thought to come from considering the neutrino energy on an event-by-event
basis, as opposed to the energy range binning used in the Zenith analyses. The 2012 MINOS
results were more sensitive to ∆m2 than SK, but their allowed range of sin2 2θ23 was quite large.
A comparison of the best fit points and allowed regions for the 2-Flavour and 3-Flavour L/E
and Zenith analyses from 2012 can be seen in Table 10.1. The 2F L/E Analysis favoured larger∆m2 compared to the 2F Zenith, and the 3F L/E favoured a slightly smaller value than the 3F
Zenith. The 3F Zenith Analysis saw a small difference in the sin2 θ23 octant between the mass
hierarchies, and the 3F L/E Analysis saw a little difference in the ∆m231 between the hierarchies.
2440.0040.005∆ m231 (eV2)∆ m231 (eV2)0.005
L/E Normal 90% C.L.
L/E Inverted 90% C.L.
NH Min.
IH Min.
Zen. Normal 90% C.L.
Zen. Inverted 90% C.L.

0.004

0.003

0.003

0.002

0.002

0.001

0.001

0
0.2

0.4

0.6

0
0.8

0.8
Sin2θ23

90% C.L. Allowed Regions
L/E 3 Flavour NH
L/E 3 Flavour IH
L/E 2 Flavour
Zenith 3 Flavour NH
Zenith 3 Flavour IH
Zenith 2 Flavour
MINOS All Data

0.85

0.9

0.95

1
Sin2 2θ23

Figure 10.11: The left plot shows a comparison of the 2D 90% C.L. allowed region measured with
the L/E Analysis in blue, compared with the Zenith Analysis in red. The minimum point for
the Normal Hierarchy is shown with a star, Inverted with a circle, where blue is for L/E and red
for Zenith. The L/E contours were quite symmetric horizontally, while a slight skew was seenfor Zenith, as well as a flip of the minimum point with mass hierarchy. The right plot compares
the 90% C.L. of the L/E, Zenith and Minos Analyses in 2012.

10.5.3

Improved Sensitivity

The main purpose of the analysis in this chapter was to attempt to do an L/E analysis with
3-flavours and matter effects, and to check whether the analysis was stable under the influenceof these additional considerations. The analysis appeared to be reliable, and so more significant
measurements of the atmospheric oscillation parameters could be made, as there were fewer
assumptions. However, this analysis did not have much sensitivity to the neutrino mass hierarchy,or to the octant of θ23 .
In terms of an L/E analysis, as mentioned earlier, the main sensitivity to the mass hierarchy
was in the L/E range 1,200∼4,000 km/GeV. Greater statistics or finer granularity in this range
could perhaps reveal the sign of ∆m231 . In this analysis, at least, the measurement of sin2 θ23
was mostly symmetrical around sin2 θ23 ∼ 0.5. Some sensitivity to the octant could be achieved
with lower energy events, in which the L/E resolution is low. So an L/E analysis is not a good
choice to measure the octant of θ23 . Hyper-Kamiokande [147], the proposed successor to SuperKamiokande, will have much better sensitivity to smaller observations by collecting much greater
statistics.
However, for further improved sensitivity to the mass hierarchy, event-by-event separationof neutrino and anti-neutrino would be preferable. This can be achieved either by accelerator
experiments with a high purity of either neutrinos or anti-neutrinos, or with a precisely understood flux; or with a detector with strong charge discrimination. For accelerators, T2K [37]and NOνA [154] are improving sensitivity to these remaining parameters with dedicated νµ orν̄µ beam time. For a precisely understood neutrino beam flux, the Neutrino Factory [142] hasbeen proposed. For massive-scale charge discrimination, the India-based Neutrino Observatory
245

(INO) [263] are planning to build a 50 kt atmospheric neutrino detector that is magnetised. This
would have much improved sensitivity to the mass hierarchy, as well as other measurements thatdiffer for neutrino and anti-neutrino, such as δcp .

10.5.4

The 3-Flavour L/E Result

The 3-Flavour L/E analysis faithfully calculated the oscillation probabilities, with fewer assumptions than the 2-flavour analyses. It reconstructed one of the key defining features of oscillations,which is its dependence on the L/E variable. This allowed the reconstruction of the first oscillation maximum, and clear regions of average oscillations and no oscillations, leading to a very
accurate measurement of the atmospheric ∆m231 . This is the best measurement of the parameterthat can currently be achieved from atmospheric neutrinos. Accelerator experiments can makemore confined measurements with a well defined human-made flux and baseline, however atmospheric neutrinos can probe a much wider range of energies and baselines, and have much greater
sensitivity to matter effects. This is also the first attempt to make an oscillation measurementby its L/E characteristic in 3-flavours and including matter effects.
So the measurement made in this thesis is very significant, as it considers a wide range of theoscillation parameter space; 3-flavour effects; and matter effects. It is important to have thesewide-ranging measurements, in conjunction with precise observations with tuned-sensitivity toa smaller range of the parameter space. One of the main strengths of an experiment like SuperKamiokande is that it can potentially observe unexpected effects outside the tuned sensitivityof other experiments. One such possibility is the observation of Non-Standard Neutrino Interactions, which is the topic of the following chapter.
2-Flavour Analysis
2F L/E Standard Analysis
2F Zenith Analysis
3-Flavour Analysis
3F L/E Normal
3F L/E Inverted
3F Zenith Normal
3F Zenith Inverted

sin2 2θ23
1.00
0.99
sin2 θ23
0.548
0.536
0.425
0.575

(90% C.L.)> 0.931> 0.96(90% C.L.)
0.396 – 0.644
0.390 – 0.631
0.391 – 0.619
0.393 – 0.630∆m2 (eV 2 , 1σ)(2.5 ± 0.27) × 10−3−3
(2.3+0.16−0.22 ) × 10
∆m2 (eV 2 , 1σ)−3(2.60+0.22−0.21 ) × 10
+0.25
(2.45−0.16 ) × 10−3−3
(2.66+0.15−0.40 ) × 10
+0.17
(2.66−0.23 ) × 10−3

Table 10.1: Oscillation Parameter Results Summary246

χ2
171.7 / 169
555.0 / 478
χ2
69.56 / 73
69.44 / 73556.7 / 477
555.5 / 477

ID

Description

Normal Hierarchyσ% % (σ)Pull
Systematic Errors in Neutrino Interactions
9 QE Scattering Total Cross-Section 10.0
4.50 (0.450)
0.202910 Single Meson Production (x-sec)
20.0 -13.33 (-0.667) 0.4442
11 DIS (Total Cross-Section)
5.0-0.41 (-0.082) 0.0066
12 Coherent π Production
100.0 -18.70 (-0.187) 0.0350
13 NC/CC Ratio
20.0
0.39 (0.020)
0.000421 Nuclear Effect30.0
0.02 (0.001)
0.000024 Axial-Vector Mass (MA )20.0
4.45 (0.223)
0.049425 DIS (Bodek Correction)
20.0
14.80 (0.740) 0.5472
27 Single-Meson Production (π 0 /π ± )40.09.89 (0.247)
0.0611
Table 10.2: Neutrino Interaction Uncertainties

247

Inverted Hierarchy % (σ)Pull
4.80 (0.480)-14.42 (-0.721)-0.41 (-0.082)-20.56 (-0.206)-0.22 (-0.011)
0.29 (0.010)
3.79 (0.190)
15.48 (0.774)
10.53 (0.263)

0.23050.51950.00660.04230.00010.00010.03600.59900.0693

ID

Description

Normal Hierarchy % (σ)Pull

σ%
Systematic Errors in Neutrino Flux
1 Normalisation–
3 ν̄µ /νµ Ratio
6.0
4 Up/Down Ratio
Single-Ring < 400 MeV
0.3
Single-Ring > 400 MeV
0.5
Single-Ring Multi-GeV
0.2
Multi-Ring Sub-GeV
0.2
Multi-Ring Multi-GeV
0.2
PC
0.2
5 Horizontal/Vertical
Single-Ring < 400 MeV
0.1
Single-Ring > 400 MeV
1.9
Single-Ring Multi-GeV
2.3
Multi-Ring Sub-GeV
1.3
Multi-Ring Multi-GeV
1.5
PC
1.7
6 Neutrino Flight Path Length
10.0
7 Energy Spectrum
5.0
8 Sample Normalisation Multi-GeV
SK1
5.0
SK2
5.0
SK3
5.0
SK4
5.0
2 FC Multi-GeV µ / PC
SK1
0.6
SK2
0.5
SK3
0.9
SK4
0.02
26 Solar Activity
SK1
20.0
SK2
50.0
SK3
20.0
SK4
10.0

Inverted Hierarchy % (σ)Pull

5.92
0.36 (0.060)-0.01 (-0.010)

0.00000.00370.00025.91
0.35 (0.058)-0.04 (-0.040)

0.00000.00350.0013-0.12 (-0.120)

0.0138-0.09 (-0.090)

0.00900.65 (0.065)-0.80 (-0.160)

0.00430.02590.23 (0.023)-0.76 (-0.152)

0.00050.0234-3.19 (-0.638)-2.30 (-0.460)-1.11 (-0.222)
1.26 (0.252)

0.40780.21190.04940.0636-3.37 (-0.674)-2.48 (-0.496)-1.26 (-0.252)
1.12 (0.224)

0.45400.24650.06300.04980.01 (0.017)
0.00 (0.000)
0.05 (0.056)
0.00 (0.000)

0.00060.00010.00250.00000.02 (0.033)
0.00 (0.000)
0.05 (0.056)
0.00 (0.000)

0.00070.00000.00280.00002.05 (0.103)-3.16 (-0.063)-0.25 (-0.013)-0.31 (-0.031)

0.01050.00400.00020.00091.99 (0.100)-3.04 (-0.061)-0.24 (-0.012)-0.33 (-0.033)

0.00990.00370.00010.0011

Table 10.3: Neutrino Flux Uncertainties

248

ID

DescriptionSK1σ% % (σ)Pull
σ%
Systematic Errors from Reconstruction - Normal Hierarchy
16 Ring Counting-2.47 (-0.247) 0.0610
Single-Ring < 400 MeV 0.7
2.3
Single-Ring > 400 MeV 0.7
0.7
Single-Ring Multi-GeV
1.7
1.7
Multi-Ring Sub-GeV
-4.5
-8.2
Multi-Ring Multi-GeV
-4.1
-0.8
17 Particle ID Single-Ring-0.02 (-0.020) 0.0004
Sub-GeV-0.1-0.4
Multi-GeV
0.2-0.1
18 Particle ID Multi-Ring
0.20 (0.020) 0.0004
Sub-GeV-3.9
2.2
Multi-GeV-2.9-3.4
19 Energy Calibration
1.1 -0.38 (-0.345) 0.1165
1.7
20 Up-Down Asym of E Calib. 0.6
0.03 (0.050) 0.0022
0.6
23 PC Stop/Through Top
46.1 1.81 (0.039) 0.0015 19.37
28 PC Stop/Through Bottom
22.7 -3.26 (-0.144) 0.0206 12.9
29 PC Stop/Through Barrel
7.01 0.17 (0.024) 0.0006 9.44
ID Description
SK3
16 Ring Counting
1.49 (0.149) 0.0221
Single-Ring < 400 MeV 3.0-3.0
Single-Ring > 400 MeV 0.6
0.6
Single-Ring Multi-GeV
1.0-1.2
Multi-Ring Sub-GeV
-2.6
-2.3
Multi-Ring Multi-GeV-2.1
2.4
17 Particle ID Single-Ring
0.00 (0.000) 0.0000
Sub-GeV-0.4-0.4
Multi-GeV-0.5-0.4
18 Particle ID Multi-Ring
0.48 (0.048) 0.0023
Sub-GeV
3.1
2.2
Multi-GeV
4.5
6.8
19 Energy Calibration
2.7
0.94 (0.348) 0.1217
2.3
20 Up-Down Asym of E Calib. 1.3
0.07 (0.054) 0.0026
0.3
23 PC Stop/Through Top
86.6 -4.45 (-0.051) 0.0026 43.3
28 PC Stop/Through Bottom
12.1 0.78 (0.064) 0.0041 11.6
29 PC Stop/Through Barrel
28.7 7.72 (0.269) 0.0724
7.4SK2 % (σ)Pull-0.41 (-0.041)

0.0017-0.09 (-0.090)

0.00760.37 (0.037)

0.0014-0.31 (-0.182)
0.07 (0.117)-0.32 (-0.017)-0.33 (-0.026)-0.97 (-0.103)
SK4
3.65 (0.365)

0.03370.01470.00030.00070.0106-0.08 (-0.080)

0.0069-0.62 (-0.062)

0.00380.07 (0.030)-0.05 (-0.167)
1.88 (0.043)-0.17 (-0.015)
0.08 (0.011)

0.00080.02950.00190.00020.0001

Table 10.4: Neutrino Reconstruction Uncertainties for Normal Hierarchy

249

0.1332

ID

DescriptionSK1σ% % (σ)Pull
σ%
Systematic Errors from Reconstruction - Inverted Hierarchy
16 Ring Counting-2.49 (-0.249) 0.0619
Single-Ring < 400 MeV 0.7
2.3
Single-Ring > 400 MeV 0.7
0.7
Single-Ring Multi-GeV
1.7
1.7
Multi-Ring Sub-GeV
-4.5
-8.2
Multi-Ring Multi-GeV
-4.1
-0.8
17 Particle ID Single-Ring-0.02 (-0.020) 0.0004
Sub-GeV-0.1-0.4
Multi-GeV
0.2-0.1
18 Particle ID Multi-Ring
0.19 (0.019) 0.0004
Sub-GeV-3.9
2.2
Multi-GeV-2.9-3.4
19 Energy Calibration
1.1 -0.35 (-0.318) 0.0990
1.7
20 Up-Down Asym of E Calib. 0.6
0.04 (0.067) 0.0049
0.6
23 PC Stop/Through Top
46.1 1.67 (0.036) 0.0013 19.37
28 PC Stop/Through Bottom
22.7 -3.25 (-0.143) 0.0205 12.9
29 PC Stop/Through Barrel
7.01 0.24 (0.034) 0.0012 9.44
ID Description
SK3
16 Ring Counting
1.45 (0.145) 0.0211
Single-Ring < 400 MeV 3.0-3.0
Single-Ring > 400 MeV 0.6
0.6
Single-Ring Multi-GeV
1.0-1.2
Multi-Ring Sub-GeV
-2.6
-2.3
Multi-Ring Multi-GeV-2.1
2.4
17 Particle ID Single-Ring
0.00 (0.000) 0.0000
Sub-GeV-0.4-0.4
Multi-GeV-0.5-0.4
18 Particle ID Multi-Ring
0.38 (0.038) 0.0015
Sub-GeV
3.1
2.2
Multi-GeV
4.5
6.8
19 Energy Calibration
2.7
0.96 (0.356) 0.1270
2.3
20 Up-Down Asym of E Calib. 1.3
0.09 (0.069) 0.0044
0.3
23 PC Stop/Through Top
86.6 -4.49 (-0.052) 0.0027 43.3
28 PC Stop/Through Bottom
12.1 0.78 (0.064) 0.0042 11.6
29 PC Stop/Through Barrel
28.7 8.22 (0.286) 0.0820
7.4

SK2
 % (σ)Pull-0.51 (-0.051)

0.0026-0.09 (-0.090)

0.00740.44 (0.044)

0.0019-0.30 (-0.176)
0.08 (0.133)-0.30 (-0.015)-0.33 (-0.026)-0.89 (-0.094)
SK4
3.88 (0.388)

0.03090.01880.00020.00070.0090-0.09 (-0.090)

0.0076-0.78 (-0.078)

0.00610.10 (0.043)-0.05 (-0.167)
1.89 (0.044)-0.18 (-0.016)
0.11 (0.015)

0.00190.02770.00190.00030.0002

Table 10.5: Neutrino Reconstruction Uncertainties for Inverted Hierarchy

250

0.1502

ID

DescriptionSK1σ% % (σ)Pull
Systematic Errors from Reduction - Normal Hierarchy
14 FC Reduction
0.2 -0.01 (-0.050) 0.0029
15 PC Reduction
2.4 -0.20 (-0.083) 0.0067
22 Non-ν Background (µ-like)
0.00 (0.000) 0.0000
Cosmic-Ray FC Sub-GeV
0.1
Cosmic-Ray FC Multi-GeV 0.1
Cosmic-Ray PC
0.2
ID Description
SK3
14 FC Reduction
0.8
0.00 (0.000) 0.0000
15 PC Reduction
0.5
0.01 (0.020) 0.0006
22 Non-ν Background (µ-like)
0.26 (0.260) 0.0696
Cosmic-Ray FC Sub-GeV
0.1
Cosmic-Ray FC Multi-GeV 0.2
Cosmic-Ray PC
1.8

σ%
0.2
4.8

SK2
 % (σ)Pull

0.00 (0.000)-2.69 (-0.560)
0.09 (0.090)

0.00050.31500.0088SK40.02 (0.067)
0.06 (0.060)-0.27 (-0.270)

0.00350.00370.0716

0.1
0.1
0.7
0.3
1.0
0.1
0.8
4.9

Table 10.6: Neutrino Reduction Uncertainties for Normal Hierarchy

ID

DescriptionSK1σ% % (σ)Pull
Systematic Errors from Reduction - Inverted Hierarchy
14 FC Reduction
0.2 -0.01 (-0.050) 0.0029
15 PC Reduction
2.4 -0.17 (-0.071) 0.0048
22 Non-ν Background (µ-like)
0.00 (0.000) 0.0000
Cosmic-Ray FC Sub-GeV
0.1
Cosmic-Ray FC Multi-GeV 0.1
Cosmic-Ray PC
0.2
ID Description
SK3
14 FC Reduction
0.8 -0.01 (-0.013) 0.0001
15 PC Reduction
0.5
0.01 (0.020) 0.0007
22 Non-ν Background (µ-like)
0.28 (0.280) 0.0797
Cosmic-Ray FC Sub-GeV
0.1
Cosmic-Ray FC Multi-GeV 0.2
Cosmic-Ray PC
1.8

σ%
0.2
4.8

SK2
 % (σ)Pull

0.00 (0.000)-2.65 (-0.552)
0.11 (0.110)

0.00040.30570.0111SK4
0.02 (0.067)
0.06 (0.060)-0.23 (-0.230)

0.00370.00370.0546

0.1
0.1
0.7
0.3
1.0
0.1
0.8
4.9

Table 10.7: Neutrino Reduction Uncertainties for Inverted Hierarchy

251

Chapter 11

Analysis of NSI with Oscillationin the νµ–ντ Sector
Non-Standard Neutrino Interactions (NSI) are a class of interactions of neutrinos that rely onsome mechanism outside of the Standard Model. The type of NSI that this analysis was concernedwith was Neutral-Current (NC) Matter NSI, which allows for the possibility of non-standard
neutral current interactions of neutrinos with matter during transit. This is the type of NSI that
atmospheric neutrinos at SK would be most sensitive to, and could be observed by variationsin the neutrino flavours observed, while retaining the expected total neutrino flux. This is asimilar type of observation as neutrino oscillations, but with different dependencies on neutrino
properties. NSI could potentially occur at the creation or detection of the neutrino, which would
be CC Matter NSI. However, any neutrinos that interacted by such mechanisms before reaching
SK, would disappear or change form and not be observed in the detector. This results in a
reduced sensitivity to such effects, and so CC Matter NSI have not been considered here.
As NSI theories are numerous, and it is difficult to favour one model over another, the analysis
in this thesis takes a model-independent approach. In this way, the strength of NC Matter NSI
can be characterised by terms such as εµτ , which allows for flavour transitions between νµ andντ during transit. This analysis simply observes the potential for such interactions allowed by
the data, without being restricted to the mechanisms of any one model. Measurements of suchparameters can be taken as input or constraints by theorists who wish to propose an NSI model.
Pure NSI has already been ruled out to explain the νµ deficit in atmospheric neutrinos [57,59, 60], so NSI were expected to be sub-dominant at best. This analysis makes use of a hybrid
model of 3-Flavour Neutrino Oscillations that allows for NSI. Neutrino Oscillations are widely
accepted as the dominant mechanism for flavour transitions in neutrinos. So this analysis could
potentially observe some NSI in conjunction with Oscillations, or characterise the amount of NSI
that is potentially allowed by the data in the presence of Oscillations.
This is the first analysis of atmospheric neutrino data, that allows for 3-flavour oscillations
and NC matter NSI in the µτ –sector. The model in this analysis is sensitive to the expected∆m231 dependence on the εµτ NSI term, which was not observed in the 2-flavour hybrid models
used in the past [45, 57]. This includes the SK Zenith NSI analysis [73], which also used a
2-flavour model in the µτ –sector. So the results in this analysis should be the most meaningfulmeasurements on these NSI parameters, at least from atmospheric neutrinos.
There are many advantages to an analysis of neutrinos created by cosmic rays in the at252

mosphere. Since the vast majority of these neutrinos pass through the entire Earth, the wholeatmosphere of the Earth acts as a source of neutrinos for Super-Kamiokande. This provides
a wide range of neutrino energies, and baselines that extend from ∼10 km to over 10,000 km.
More importantly, these baselines are filled with the Earth’s crust, mantle, and core, providing
an excellent setting to potentially observe matter effects generated by NSI. In addition, the L/E
Analysis gives the best constraint on ∆m231 from atmospheric neutrinos, making it an ideal choicefor this ∆m231 –dependent measurement.

11.1

NSI Background and Treatment

Non-Standard Neutrino Interactions (NSI) refer to a collection of non-standard interactions postulated to be possible. For this analysis, the interest is in NC Matter NSI that could have an
influence on the flavour ratios observed at SK. Examples of categories of NSI are Flavour Changing Neutral Current (FCNC) [61–64], and Lepton Non-Universality (NU) [65]. These can beillustrated as:να + f −→ νβ + f(FCNC),(11.1)

να + f and νβ + f amplitudes differ(NU).

(11.2)In FCNC, there are non-standard NC interactions with fermions in matter that can resultin a flavour change. As represented in Equation (11.1), through a NC interaction of a neutrinoof flavour “α” with a fermion, f , a transition to flavour “β” is allowed. In NU, the NC forward
scattering amplitudes are allowed to be different between flavours, represented in Equation (11.2).
It has already been shown that NSI alone cannot describe the full energy range of atmospheric
neutrino data [57, 59, 60]. So NSI can be investigated as a sub-dominant effect, with a hybrid
model including both Neutrino Oscillations and NSI. In such a model, the Hamiltonian forνα → νβ can be written as:εee ε∗eµ ε∗eτ
0
0
0
√
1
†
0  UkβHαβ =± VM SW ± 2GF Nf (~r) εeµ εµµ ε∗µτ  , (11.3)
Uαj 0 ∆m221
2E
0
0
∆m231
εeτ εµτ ετ τwhere the first term is the vacuum oscillation term; the second is from MSW matter effects
on electrons; and the third allows for NSI. While U is the Unitary Lepton Mixing Matrix,
GF = 1.166 × 10−5 GeV−2 is the Fermi constant, and Nf the fermion number density, taken
to be down-quarks, calculated along the neutrino path from production to the detector. Detailsof the oscillation term were written in Section 10.1.1. The additional interaction potentials are
positive for neutrinos and negative for anti-neutrinos. So the dominant mechanism for the flavour
transitions come from the vacuum oscillation term, while allowing for possible non-standardtransitions between flavours by NC interactions with matter in a model-independent way. Thesetransitions are characterised by the εαβ parameters, which are like coupling constants for thestrength of the NSI.
In this thesis, NSI in the µτ –sector were of interest, as these would have the largest effect on
atmospheric neutrinos. In this case, εµτ would represent possible FCNC interactions, while thequantity ετ τ − εµµ would represent NU. Here, εµµ = 0 was set, allowing NU to be consideredwith the ετ τ parameter. These are general quantities that could be used to constrain nonstandard theories of NC neutrino interactions with matter. In general, these matter NSIs depend
253

(a) P (νµ → νµ ) Neutrino εµτ = 0.02(b) P (ν̄µ → ν̄µ ) Anti-Neutrino εµτ = 0.02(c) P (νµ → νµ ) Neutrino, Oscillation only(d) P (ν̄µ → ν̄µ ) Anti-Neutrino, Oscillation only(e) P (νµ → νµ ) Neutrino ετ τ − εµµ = 0.08(f) P (ν̄µ → ν̄µ ) Anti-Neutrino ετ τ − εµµ = 0.08

Figure 11.1: P (νµ → νµ ) survival probabilities for neutrinos and anti-neutrinos and various NSI parameters. In the above, 2-flavour oscillations were assumed with the parameters(sin2 θ23 ,∆m231 ) = (0.5,2.1 × 10−3 eV2 )254

on the density of matter on the neutrino flight path, while not showing the E −1 dependence
seen in neutrino oscillations [45]. Details of the derivation of Equation (11.3) were written in
Section 1.5.1. An idea of the effect of these NSI parameters can be found in Figure 11.1.

11.2

Assumptions in This Analysis

For the NSI analysis in this thesis, the NSI terms εµτ and ετ τ were varied while the remaining
NSI terms were set to zero. Atmospheric neutrinos are sensitive to the quantity (ετ τ − εµµ ),sometimes referred to as ε0 , which was probed with the ετ τ parameter while setting εµµ = 0. Or
in other words, in this analysis ετ τ ≡ ε0 .
The NSI interactions were assumed to be with down-quarks in the transition probability
calculation. This means that the matter density calculated along the neutrino flight path was
the expected density of down-quarks. This is a normalisation, one used in some of the first
NSI analyses with atmospheric neutrinos, and the convention followed here. So the strength of
the NSI parameters measured here should be interpreted with this down-quark normalisation inmind. The values reported here could be re-weighted to any other normalisation. A common
normalisation used in more recent years is the electron-number density, in which case the strengthof NSI can be compared to the strength of the MSW matter effect. Here, the density of downquarks was simply assumed to be three times the expected number density of electrons.
The µτ –oscillation terms, ∆m231 and sin2 θ23 , were allowed some freedom, while the remaining oscillation parameters were set at the best fit from a global analysis [44] published after
the Neutrino 2012 conference. Specifically, the values were fixed at ∆m221 = 7.62 × 10−5 eV 2 ,sin2 θ12 = 0.320, and sin2 θ13 = 0.0246. Normal neutrino mass hierarchy was assumed.

11.3

3-Flavour Oscillation and NSI Results

A four-dimensional parameter space was scanned including 23 parameters each for εµτ and ετ τ ,and 13 parameters each for ∆m231 and sin2 θ23 . The NSI ranges ran from -0.32 to -0.001 and0.001 to 0.32, with logarithmic steps. With an additional parameter corresponding to zero NSI
also included. The oscillation ranges were (1.5 → 4.5) × 10−3 eV2 for ∆m231 , and 0.2 → 0.8 forsin2 θ23 . This formed a 4D parameter space of around ∼90,000 points, that took over 24 hours
to calculate the χ2 running on ∼200 CPUs. This analysis was computationally intensive, so amuch finer parameter space could not be scanned without the processing time quickly growing tomany days and weeks (e.g. doubling the number of each parameter would then take more than
2 weeks to run).

11.3.1Best Fit and Allowed Regions

After scanning the above parameter space, the bestfit point was found at:
χ2 = 69.94/71 d.o.f,−3εµτ = 1.0+07.9−12.2 × 10 ,
−3ετ τ = 5.6+24.8−33.3 × 10 ,(11.4)sin θ23 = 0.55+0.07−0.13 ,

+0.4
2∆m31 = 2.5−0.2 × 10−3 eV2 ,(11.7)

2255(11.5)(11.6)(11.8)

600

Precise L/E Data
Precise Best MC
Precise Null MC

500

2nd Sample Data
2nd Sample MC
2nd Null MC

Number of events

Figure 11.2: The bestfit L/E distributions forthe 3-Flavour Oscillation and NSI analysis.
Since the bestfit point is close to pure oscillations, these distributions are very similar to the
3-flavour pure oscillation case. The precise resolution samples are drawn in red, and the second resolution sample in magenta. The dashedhistograms show the respective null-oscillation
Monte Carlo. The points are for data, black forthe precise sample and blue for the second resolution sample.

400

300

200

100

0
1

2

3

10
10
10
L/E (km/GeV) (SK 1+2+3+4)

10

4

where the errors correspond to 1σ (68% C.L. for 1 d.o.f.). The bestfit L/E distributions canbe seen in Figure 11.2, and these have been normalised by the null oscillation expectation in
Figure 11.3. As the minimum point was very close to zero NSI, these distributions do notdiffer much from pure 3-Flavour neutrino oscillations. For illustration, the bestfit L/E overnull oscillation plots were compared with a few NSI parameters with a fit just outside the 99%
C.L. in Figure 11.4. These distributions do not have any pulls from systematics, and show thedifferences of the probability calculation. All of the plots have oscillations at the minimum pointsin2 θ23 = 0.55 and ∆m232 = 2.5 × 10−3 eV2 , with only the stated NSI parameters varied. The
green curve corresponds to the best fit parameters.
The 2D allowed region of the NSI parameter space, with the oscillation parameters marginalised
out, can be seen on the left of Figure 11.5; and the 2D oscillation parameter allowed region with
the NSI parameters marginalised out on the right. As noted previously, any mention of marginalisation in this thesis means that the marginalised parameters were minimised over. Here the ∆χ2
was drawn from the global minimum and the 68%, 90% and 99% C.L. correspond to a ∆χ2 of
2.30, 4.61, and 9.21, respectively. The 1D ∆χ2 distributions for εµτ and ετ τ can be seen in
Figure 11.6, where the undisplayed parameters were marginalised out. The 1D distributions forthe oscillation parameters with the undisplayed parameters marginalised out can be seen in Figure 11.7. The 68%, 90% and 99% C.L. correspond to a ∆χ2 of 1.00, 2.71, and 6.63, respectively.
These distributions were used to calculate the 68% and 90% C.L. allowed regions. The scanned
points were well-fitted with a smooth spline interpolated curve, and the position where the ∆χ2crossed the C.L. boundaries was found, as shown by the intersecting lines in Figure 11.6 and 11.7.
The 90% C.L. allowed regions were:−1.97 × 10−2 <εµτ< 1.45 × 10−2 ,(11.9)

<

ετ τ

−2

(11.10)0.39 <

2sin θ23< 0.65,(11.11)

2.21 × 10−3 eV2 <∆m231< 3.15 × 10−3 eV2 .

(11.12)−2−4.54 × 10

256

< 4.47 × 10

,

1.8

1.8
Precise L/E Data

1.6

2nd Sample L/E Data

1.6

Sample Bestfit MC

Data/Prediction (null oscillation)

Data/Prediction (null oscillation)

Bestfit MC

1.4
1.2
1
0.8
0.6
0.4
0.2

1.4
1.2
1
0.8
0.6
0.4
0.2

0
1

2

3

10
10
10
L/E (km/GeV) (SK 1+2+3+4)

10

0

4

1

2

3

10
10
10
L/E (km/GeV) (SK 1+2+3+4)

10

4

Figure 11.3: The NSI bestfit L/E distributions normalised by the null oscillation expectation, forthe precise resolution sample on the left and the second resolution sample on the right. These
are similar to the 3-flavour pure oscillation case.

1.8

1.8

Precise L/E Dataεµτ = 0.001

1.6

Precise L/E Dataεττ = 0.0056

1.6εττ = +0.1

Data/Prediction (null oscillation)

Data/Prediction (null oscillation)εµτ = +0.056
εµτ = -0.056

1.4
1.2
1
0.8
0.6
0.4
0.2

εττ = -0.1

1.4
1.2
1
0.8
0.6
0.4
0.2

0
1

2

3

10
10
10
L/E (km/GeV) (SK 1+2+3+4)

10

0

4

1

2

3

10
10
10
L/E (km/GeV) (SK 1+2+3+4)

10

4

Figure 11.4: Bestfit L/E distribution in green compared with various εµτ parameters on the left
and various ετ τ parameters on the right. The blue histograms show an increase in the parameterand the red histograms show a decrease. These parameters were chosen just outside the 99% C.L.
boundary to demonstrate considerable variation in the distributions from the NSI parameters.
The systematic pulls have not been applied to these distributions, only the survival and transition
probabilities.

257

+10-1

90% C.L.

∆ m231 (eV2)∈τ τ

99% C.L.

68% C.L.

0.005

99% C.L.
90% C.L.
68% C.L.
0.004

+10-2

0.003
± 10

-3

0.002
-10-2

0.001
-10-1

-10-1

-10-2

± 10

-3

+10-2

0
0.2

+10-1 ∈µ τ

0.3

0.4

0.5

0.6

0.7

0.8

sin2 θ23

Figure 11.5: The 2D allowed regions for the NSI parameters on the left, and oscillation parameterson the right. The undisplayed parameters from the 4D parameter space have been marginalisedout. In the left plot, the εµτ was drawn on the x-axis and ετ τ ≡ ε0 on the y-axis. The axes weredrawn in log scale, with the right panels for positive εµτ , and the left panels for negative εµτ ;postive ετ τ on the top, and negative ετ τ on the bottom. The bestfit point is indicated by a staron the top right panel. In the right plot, the oscillation parameter space with the single-angle
sin2 θ23 is shown. The ∆m231 allowed region was slightly elongated to higher values, due to the
influence of the εµτ NSI parameter.

258

min

χ2-χ2

min

χ2-χ2

20

99% C.L.
18

20

99% C.L.
18

90% C.L.
16

90% C.L.
16

68% C.L.

14

14

12

12

10

10

8

8

6

6

4

4

2

2

0

68% C.L.

0
-1

-10

-2

-10

± 10

-3

+10-2

+10-1

∈µ τ

-10-1

-10-2

± 10

-3

+10-2

+10-1

∈τ τ

Figure 11.6: The 1D allowed regions for the NSI parameters, εµτ in the left plot and ετ τ inthe right, with the undisplayed parameters marginalised out. The NSI parameters are drawn inlog scale, with positive values on the right panels and negative values in the left panels. The
points show the scanned points, which have been fitted with a spline function to find the C.L.
boundaries. The calculated C.L. intervals have been shown by the intersecting lines. These
distributions were flat around the minimum.

11.3.2The εµτ and ∆m231 Dependence

Considering the 1D plots in Figure 11.6, the ∆χ2 distributions were very flat around the minimum, suggesting a considerable range of NSI parameters fitted the data equally well. Whenthe magnitude of the parameters were slightly larger than 10−2 , NSI was rapidly disfavoured.
These distributions show that the allowed region for εµτ was slightly more relaxed in the negative
direction, while ετ τ was fairly symmetrical. Assuming normal hierarchy, for negative εµτ larger
values of ∆m231 are allowed for neutrinos, and smaller values for anti-neutrinos. The data appear
to have slightly more allowance for this scenario, rather than the opposite tendency. In otherwords, there was more allowance for εµτ with the opposite sign of ∆m231 , and this means moreleeway for a larger effective ∆m231 for neutrinos and smaller for anti-neutrinos.
In Figure 11.7, it can be seen that the allowed region for sin2 θ23 was very similar to that ofpure 3-Flavour oscillations. However, in the presence of NSI, the allowed values for ∆m231 wereconsiderably relaxed. Especially, a greater range of larger ∆m231 was allowed, as seen in the rightof Figures 11.5 and 11.7. This corresponds to the less restricted region for negative εµτ , andto the statistics being dominated by neutrino, rather than anti-neutrino. These distributions
demonstrate that the presence of NC matter NSI in nature could allow for a discrepancy in theobservation of ∆m231 between neutrino and anti-neutrino by an analysis assuming pure vacuum
oscillations alone. The effect of the εµτ parameter on the ∆m231 region can be seen in Figure 11.8.
For normal hierarchy, a negative value for εµτ favours larger ∆m231 ; while positive values favour
smaller ∆m231 . The data consisted of combined neutrino and anti-neutrino contributions, but
was statistically dominated by neutrinos, which is the reason for this observed trend.
259

18
16

χ2- χ2minχ2- χ2min

20

99% C.L.
90% C.L.
68% C.L.

20

16

14

14

12

12

10

10

8

8

6

6

4

4

2

2

0
0.2

99% C.L.
90% C.L.
68% C.L.

18

0
0.3

0.4

0.5

0.6

0.7

0.00150.8

0.002

0.00250.003

0.00350.004

sin2 θ23

0.0045∆ m231 (eV2)∆ m231 (eV2)

Figure 11.7: The oscillation 1D allowed regions from the 4D NSI analysis. sin2 θ23 is shown on
the left, and ∆m231 on the right. The sin2 θ23 regions was similar to the pure oscillation case.
However, the ∆m231 allowed region was enlarged, particularly in the positive direction due to theeffect of the εµτ NSI parameter.

0.005

99% C.L.
90% C.L.
68% C.L.
0.004

0.003

Figure 11.8: The 2D ∆m231 vs εµτ allowed region. The εµτ parameter is drawn in log scale,with positive values on the right panel and negative values in the left panel. The bestfit point
is indicated by a star on the right panel. Larger
values of ∆m231 are allowed as εµτ becomes morenegative.

260

0.002

0.001

0

-10-1

-10-2

± 10

-3

+10-2

+10-1

∈µ τ

11.3.3

3F Oscillation and NSI Systematic Uncertainties

The 1 σ estimates for the systematic uncertainties were the same for all of the L/E analyses,
and has been shown in Figure 9.14. The contribution from these systematics, minimised in thefitting, have been drawn in Figure 11.9. These values have been displayed as the fraction of the1 σ estimates that were actually used in the best fit. As in the other L/E analyses, the pulls
for the systematic errors were all below 1 σ, and most were below 0.5 σ. The shape of these
distributions, of which systematics contribute, were very stable across all the analyses, with only
small differences in2 the magnitude of each error seen. The contribution of the uncertainties tothe χ2 was1 Σ σ = 3.32. This was essentially the same contribution as seen in the 3-flavour
normal hierarchy analysis, and was a minimal dependence on the systematic errors.
The observation of very similar uncertainties was likely due to the L/E data being common
between all the analyses, and because the best fits were very close to pure normal neutrino oscillations. Despite allowing for considerable freedom for independent neutrino and anti-neutrino
parameters, or additional NSI flavour transitions, the best fits were for essentially pure oscillations. A very nice fit was found from this parameter space, with the minimum point having
non-zero NSI. However, as it is clear from the allowed regions, this result was consistent withzero NSI. Neutrino Oscillations appears to be stable and the best description of atmospheric
neutrino observations, even in the presence of flavour transitions allowed by NSI.

11.4

Discussion

11.4.1Comparisson With Atmospheric NSI Measurements

There has been a few analyses of NSI in the µτ –sector of atmospheric neutrinos. Most ofthem have taken a phenomenological approach, depending on officially published data fromexperiments like SK or MACRO. Since pure NSI had already been ruled out to explain the νµdeficit in atmospheric neutrinos [57, 59, 60], analyses had to involve hybrid models of oscillations
and NSI. All of these analyses relied on a 2-flavour model [45, 57], including a first measurementof atmospheric data with the SK Zenith distributions [73].
These 2-flavour hybrid models provided stringent constraints on the allowed NSI parameters.
Some of the 90% C.L. allowed regions for εµτ have been written below for comparison [45, 73]:
−1.97 × 10−2 <−2−1.3 × 10

<

εµτ
εµτ
|εµτ |< 1.45 × 10−2
< 1 × 10−2
−2

< 1.1 × 10

(L/E, this thesis),(11.13)(2004 Gonzalez-Garcia),(11.14)(2011 Mitsuka, SK Zenith).

(11.15)For εµτ , the constraints from these past analyses appear to be more confined than the results ofthis analysis. However, it should be noted that these past analyses made use of 2-flavour hybrid
models to probe various new physics, and so are not directly comparable with this analysis.
There has been some controversy over these 2-flavour atmospheric analyses, which exclude thecomplications of traversing the mantle and core [13]. This may lead to unrealistically constrained
allowed regions. The treatment in this thesis implemented a 3-flavour model, removing some ofthe symmetries seen in simpler models.
1

The sum here is over all the systematic uncertainties.

261

Normalisation

Common
SK1
SK2
SK3
SK4

FC Multi-GeV µ / PCν µ/ ν µ Ratio
Up / Down Ratio
Horizontal / Verticalν Flight Path Length
Energy Spectrum
Normalisation Multi-GeV
QE Scattering Total x-sec
Single Meson Production (x-sec)
DIS
Coherent π Production
NC / CC Ratio
FC Reduction
PC Reduction
Ring Counting
PID Single-Ring
PID Multi-Ring
Energy Calibration
Up / Down Asym of E. Calib.
Nuclear EffectNon-ν Background (µ-like)
PC Stop / Through (Top)
Axial-Vector Mass (M )A

DIS (Bodek Correction)
Solar Activity
Single Meson Production (π0 / π± )PC Stop / Through (Bottom)
PC Stop / Through (Barrel)

-1.0

-0.8

-0.6

-0.4 -0.2
0.0 0.2
0.4
Systematics at Best Fit (σ)

0.6

0.8

1.0

Figure 11.9: Systematics at the best fit for the 3F Oscillation and NSI L/E Analysis

262

ε’

90% C.L. Allowed
SK L/E+10-1

SK Zenith 2011
Fornengo 2001

+10-2

± 10-3

-10-2

-10-1
-10-1

-3

-10-2

± 10

+10-2

+10-1

εµ τ

Figure 11.10: Comparison of NSI measurements on the εµτ ×ε0 plane, where ε0 ≡ (ετ τ −εµµ ). The
results from this analysis are shown in solid green; the 2-flavour SK Zenith NSI measurement [73]in dashed blue; and an NSI phenomenological study of atmospheric neutrinos by Fornengo et.al.
in dash-dotted red [57]. Only the L/E analysis properly considers 3-flavour effects, so thesemeasurements are not directly comparable.
The ∆m231 dependence on εµτ , as seen this analysis in Figure 11.8, was not observed in thesepast models. In the 2-flavour SK Zenith NSI analysis [73], the oscillation parameter space wasoverlapping with and without NSI, which was shown in the left of Figure 1.13. So a 3-flavour
approach should give a more accurate representation of the effect of these NSI parameters. The
L/E NSI allowed regions have been compared with the SK Zenith NSI analysis [73] and thephenomenological analysis by Fornengo et.al. [57] in Figure 11.10. The 2D allowed regions forthe 2004 Gonzalez-Garcia analysis [45] were not available, so the comparison was made with the2001 Fornengo analysis, which made use of a similar analysis approach.
The 2-flavour models seem to observe more shape on the εµτ × ε0 plane, where ε0 ≡ (ετ τ −εµµ ). Particularly, the εµτ parameter was considerably constrained for small values of ε0 . Thisbehaviour was not seen in the L/E analysis, leading to larger allowed regions for εµτ . However,with fewer assumptions in this analysis, the L/E ranges should be more reliable.
Some of the 90% C.L. allowed regions for ε0 have been written below for comparison [45, 73]:
−4.54 × 10−2 <−2−4.9 × 10<

ε0< 4.47 × 10−2

(L/E, this thesis),(11.16)|ε0 |< 2.9 × 10−2

(2004 Gonzalez-Garcia),(11.17)0−2(2011 Mitsuka, SK Zenith).

(11.18)ε< 4.9 × 10

Again, the Gonzalez-Garcia analysis was very restrictive on this parameter, while the Zenith
range was quite similar to that measured by the L/E distributions. The slightly more confined
region from the L/E analysis comes from scanning negative values of εµτ , which were omitted
263

from the Zenith analysis. In Figure 11.10, a subtle tendency of the ε0 allowed range to become
narrower with increasingly negative εµτ can be seen in both the L/E and Fornengo analyses. The
best fit points of each analysis were in different quadrants of the parameter space. Generally, the∆χ2 were fairly flat, with some of the 2-flavour analyses having shallow dips away from zero NSI.
Without a consistent and statistically significant non-zero measurement of these parameters, allof these results are consistent with no observation of NSI.
Although the 2-flavour analyses report more confined allowed regions and had more structure
over the parameter space than was visible in this analysis, the L/E 3-flavour oscillation and
NSI result should be the most realistic and reliable. It has considered 3-flavour effects andaccurately simulates the neutrino flight path through matter, including the mantle and core.
The expected characteristic dependence of εµτ on ∆m231 was seen in the L/E analysis, but notin the previous analyses, making this result the current best measurement of εµτ and ε0 fromatmospheric neutrinos.

11.4.2Comparisson With MINOS Measurements

In Figure 11.11, the 90% C.L. allowed regions on the εµτ × ∆m231 parameter space has beenshown. Displayed on the plot are the results from several analyses: 3F Oscillation and NSI L/E;
a phenomenological analysis of the MINOS distributions by Mann et.al. in 2010 [13]; and a 2013
measurement by MINOS Collaboration [69]. Since these MINOS related measurements madeuse of an electron number density normalisation for their calculation, the magnitudes of the εµτhave been re-scaled to match the L/E NSI analysis normalisation. The electron number density
is 3 times less than the down-quark number density used in this analysis, which allows muchlarger values of εµτ to describe the same strength of NSI. Specifically, the Mann and MINOS εµτvalues were scaled by 1/3 in Figure 11.11.
Each of these analyses have assumed normal mass hierarchy, and so see the same trendof allowing larger ∆m231 with increasingly negative εµτ . It is clear that this effect is much
more pronounced in the L/E NSI analysis. This is likely due to the much greater sensitivity tomatter effects from atmospheric neutrinos, which would be relatively weaker at the fixed ∼730 km
baseline through the crust in the MINOS experiment. MINOS is essentially insensitive to theετ τ and 3-flavour effects [68, 69], while an analysis with atmospheric neutrinos should take thesefactors into account. However, MINOS does gain sensitivity due to the event-by-event neutrino
and anti-neutrino separation.
After considering 3-flavour effects, the L/E analysis still provides strong constraints on εµτ ,favouring a steeper angle through the parameter space. In particular, the best fit points from
the Mann and MINOS analyses were excluded at the 90% C.L., and the best fit from the L/E
analysis had close to zero NSI. Quantitatively, the 90% C.L. allowed range for εµτ measured byMINOS was:
−0.20 <
−6.7 × 10

−2

<

εµτ
εµτ

< 0.07,(11.19)−2< 2.3 × 10

(divided by 3),(11.20)where the second line has scaled the magnitudes to the d-quark normalisation. The MINOS
result allowed just over 3 times larger εµτ in the negative direction; while the positive range wasonly slightly larger than the L/E result.
264

0.00400.00380.0036

90% C.L. Allowed
SK L/EMINOS 2013
Mann et.al. 2010

∆ m231 (eV2)0.00340.00320.00300.00280.00260.00240.00220.0020-0.14 -0.12-0.1 -0.08 -0.06 -0.04 -0.02
εµ τ

00.02 0.04

0.06

Figure 11.11: Comparison of NSI measurements on the εµτ ×∆m231 parameter space. The resultsfrom this analysis are shown in solid green; a 2010 NSI phenomenological study of MINOS
distributions by Mann et.al. in dash-dotted grey [13]; and a 2013 MINOS NSI measurement ofdata in dashed red [69]. The MINOS related regions have been re-scaled to match the L/E NSI
normalisation. The L/E region is much more confined in εµτ , and takes a steeper angle throughthe parameter space.

26511.4.3

Improving the Sensitivity to NSI

The analysis in this thesis has included 3-flavour effects, which can largely alleviate the controversy over the highly constraining 2-flavour measurements from atmospheric neutrinos. However,this analysis has not yet considered the effect of εeτ , which has been shown [66] to relax the limitson ετ τ even up to O(1), which can be seen in Figure 1.10. This analysis was already computationally intensive, but given time, the effect of the εeτ parameter could be studied with the samemodel used in this analysis (the other NSI parameters were simply set to 0).
Atmospheric neutrinos have great sensitivity to matter effects such as NSI, and accelerator
experiments often have detectors with good charge discrimination to separate neutrino and antineutrino on an event-by-event basis. These are the best advantages of these experiments. An
experiment like Hyper-Kamiokande [147], the proposed successor to SK, will be able to collect
much greater statistics, improving sensitivity to many measurements. However, there is a planned
atmospheric neutrino experiment which will have a 50 kT magnetised detector: INO [263]. This
experiment could have considerably more sensitivity by applying both of these advantages. The
other difficulty would be decoupling standard and non-standard physics, which could perhaps bebetter resolved with a precision beamline experiment like the Neutrino Factory [142], which also
has magnetised detectors as a part of some of its designs, and potentially baselines in the orderof many thousands of kilometres.

11.4.4

The 3-Flavour Oscillation and NSI L/E Result

Atmospheric neutrinos have unparalleled sensitivity to matter effects such as those seen in NC
NSI, due to the huge neutrino flight paths through the Earth – including the mantle and core.
This property has given this analysis potentially the best sensitivity to µτ –sector NSI. This is
the first analysis of this variety of NSI to include 3-flavour effects and an accurate treatment ofmatter effects using atmospheric neutrino data.
These results go a long way to resolve some of the controversy on the highly constrained
results from 2-flavour NSI atmospheric analyses. While the constraints from this analysis are lessstrong as those from previous measurements, the NSI parameters are still strongly confined when
including 3-flavour effects. Since this analysis involved less assumptions, and has been extensivein the scanned parameter space, these results are perhaps the most significant measurements onµτ –sector NSI – at least from atmospheric neutrinos.
The best fit was close to zero-NSI, and these results were consistent with no observation of
NSI. Strong constraints have been put on the εµτ and ε0 ≡ (ετ τ − εµµ ) NSI parameters. Since
this analysis was model independent, these constraints can be used to define, restrict or exclude
any number of models including non-standard NC interactions of neutrinos with matter. So
these measurements are highly valuable to theorists. Another implication of these results is that
Neutrino Oscillations have been shown to be robust, even in the presence of 3-flavour µτ –sector
NSI.

266

ID

Description

Systematic Errors in Neutrino Interactions
9 QE Scattering Total Cross-Section
10 Single Meson Production (Total Cross-Section)
11 DIS (Total Cross-Section)
12 Coherent π Production
13 NC/CC Ratio
21 Nuclear Effect
24 Axial-Vector Mass (MA ) [QE and Single Meson]
25 DIS (Bodek Correction)
27 Single-Meson Production (π 0 /π ± )σ%Systematics
 % (σ)

Pull

10.0
20.0
5.0
100.0
20.0
30.0
20.0
20.0
40.0

4.53 (0.453)-13.13 (-0.657)-0.41 (-0.083)-19.81 (-0.198)
0.11 (0.006)
0.16 (0.005)
3.51 (0.175)
14.79 (0.739)
10.15 (0.254)

0.2056
0.43130.00690.03920.00000.00000.03070.54650.0644

Table 11.1: Neutrino Interaction Uncertainties

ID

Descriptionσ%Systematic Errors from Reduction
14 FC Reduction
0.2
15 PC Reduction
2.4
22 Non-ν Background (µ-like)Cosmic-Ray FC Sub-GeV
0.1
Cosmic-Ray FC Multi-GeV 0.1
Cosmic-Ray PC
0.2
ID Description
14 FC Reduction
0.8
15 PC Reduction
0.5
22 Non-ν Background (µ-like)Cosmic-Ray FC Sub-GeV
0.1
Cosmic-Ray FC Multi-GeV 0.2
Cosmic-Ray PC
1.8SK1 % (σ)Pull

σ%-0.01 (-0.056)-0.18 (-0.076)
0.00 (0.000)

0.00310.00570.0000

0.2
4.8

SK2
 % (σ)Pull

0.00 (0.021)-2.68 (-0.557)
0.10 (0.098)

0.00050.31070.0097SK40.02 (0.063)
0.06 (0.059)-0.26 (-0.256)

0.00390.00350.0656

0.1
0.1
0.7
SK3
0.00 (-0.006)
0.01 (0.026)
0.27 (0.272)

0.00000.00070.0740

0.3
1.0
0.1
0.8
4.9

Table 11.2: Neutrino Reduction Uncertainties

267

ID

Descriptionσ%Systematic Errors in Neutrino Flux
1 Normalisation–
3 ν̄µ /νµ Ratio
6.0
4 Up/Down Ratio
Single-Ring < 400 MeV
0.3
Single-Ring > 400 MeV
0.5
Single-Ring Multi-GeV
0.2
Multi-Ring Sub-GeV
0.2
Multi-Ring Multi-GeV
0.2
PC
0.2
5 Horizontal/Vertical
Single-Ring < 400 MeV
0.1
Single-Ring > 400 MeV
1.9
Single-Ring Multi-GeV
2.3
Multi-Ring Sub-GeV
1.3
Multi-Ring Multi-GeV
1.5
PC
1.7
6 Neutrino Flight Path Length
10.0
7 Energy Spectrum
5.0
8 Sample Normalisation Multi-GeV
SK1
5.0
SK2
5.0
SK3
5.0
SK4
5.0
2 FC Multi-GeV µ / PC
SK1
0.6
SK2
0.5
SK3
0.9
SK4
0.02
26 Solar Activity
SK1
20.0
SK2
50.0
SK3
20.0
SK4
10.0

Systematics
 % (σ)Pull

5.46
0.37 (0.062)-0.04 (-0.036)

0.00000.00380.0013-0.15 (-0.146)

0.02120.15 (0.015)-0.79 (-0.158)

0.00020.0248-3.24 (-0.648)-2.33 (-0.467)-1.11 (-0.222)
1.27 (0.253)

0.41950.21780.04930.06420.02 (0.025)
0.00 (-0.007)
0.05 (0.051)
0.00 (0.000)

0.00060.00000.00260.00002.13 (0.107)-3.05 (-0.061)-0.24 (-0.012)-0.33 (-0.033)

0.01140.00370.00010.0011

Table 11.3: Neutrino Flux Uncertainties

268

ID

DescriptionSK1σ% % (σ)Systematic Errors from Reconstruction
16 Ring Counting-2.54 (-0.254)
Single-Ring < 400 MeV 0.7
Single-Ring > 400 MeV 0.7
Single-Ring Multi-GeV
1.7
Multi-Ring Sub-GeV-4.5
Multi-Ring Multi-GeV-4.1
17 Particle ID Single-Ring-0.02 (-0.018)
Sub-GeV-0.1
Multi-GeV
0.2
18 Particle ID Multi-Ring
0.20 (0.020)
Sub-GeV-3.9
Multi-GeV-2.9
19 Energy Calibration
1.1 -0.35 (-0.316)
20 Up-Down Asym of E Calib. 0.6
0.04 (0.061)
23 PC Stop/Through Top
46.1 1.69 (0.037)
28 PC Stop/Through Bottom
22.7 -3.20 (-0.141)
29 PC Stop/Through Barrel
7.01 0.19 (0.026)
ID Description
SK3
16 Ring Counting
1.45 (0.145)
Single-Ring < 400 MeV 3.0
Single-Ring > 400 MeV 0.6
Single-Ring Multi-GeV
1.0
Multi-Ring Sub-GeV-2.6
Multi-Ring Multi-GeV-2.1
17 Particle ID Single-Ring
0.00 (-0.001)
Sub-GeV-0.4
Multi-GeV-0.5
18 Particle ID Multi-Ring
0.49 (0.049)
Sub-GeV
3.1
Multi-GeV
4.5
19 Energy Calibration
2.7
0.98 (0.364)
20 Up-Down Asym of E Calib. 1.3
0.07 (0.051)
23 PC Stop/Through Top
86.6 -4.71 (-0.054)
28 PC Stop/Through Bottom
12.1 0.79 (0.065)
29 PC Stop/Through Barrel
28.7 7.71 (0.268)

Pull

σ%

0.0643Pull-0.45 (-0.045)

0.0021-0.09 (-0.086)

0.00730.35 (0.035)

0.0012-0.30 (-0.176)
0.08 (0.128)-0.31 (-0.016)-0.32 (-0.025)-0.97 (-0.103)
SK4
3.88 (0.388)

0.03100.01650.00030.00060.0105-0.09 (-0.089)

0.0080-0.62 (-0.062)

0.00380.14 (0.060)-0.05 (-0.175)
1.64 (0.038)-0.16 (-0.014)
0.06 (0.009)

0.00360.03070.00140.00020.0001

2.3
0.7
1.7
-8.2
-0.8
0.0003-0.4-0.1
0.00040.09990.00370.00140.01990.0007

2.2
-3.4
1.7
0.6
19.37
12.9
9.44

0.02100.1503

-3.0
0.6
-1.2
-2.3
2.4
0.0000-0.4-0.4
0.00240.13260.00260.00300.00420.0721

2.2
6.8
2.3
0.3
43.3
11.6
7.4

Table 11.4: Neutrino Reconstruction Uncertainties

269

SK2
 % (σ)Chapter 12Conclusion
The Standard L/E analysis was updated to include over 10 years of atmospheric neutrino data, up
until summer 2012. This analysis provided the strongest constraint on ∆m231 from atmospheric
neutrinos, by accurately reconstructing L/E, a characteristic quantity from Neutrino Oscillations,and by selecting a sample of data with precise resolution in L/E. This made it an ideal approachfor making ∆m231 –dependent measurements (Chapter 8).
Sensitivity to sub-dominant effects was improved by the addition of a second resolution sample, and by merging the SK1–SK4 analysis bins at the χ2 calculation stage, while properly
estimating the systematics for each detector configuration separately. By allowing for independent neutrino and anti-neutrino oscillations in the simulation, there was no evidence for different∆m232 and sin2 2θ23 between neutrinos and anti-neutrinos, as expected by CPT symmetry (Chapter 9).
The first 3-Flavour L/E Analysis was performed, removing the assumptions from a 2-flavour
analysis. The parameters of interest that were varied were sin2 θ23 and ∆m231 , which the L/E
analysis is primarily sensitive to. The other oscillation parameters were fixed at the values:∆m221 = 7.66 × 10−5 eV 2 , sin2 θ12 = 0.304, and sin2 θ13 = 0.025 for both hierarchies. The L/E
Analysis was confirmed to be a proper method for studying Neutrino Oscillations, even with
the inclusion of 3-flavour and matter effects. In addition, the Standard 2-Flavour L/E Analysis
was confirmed to be an excellent approximation for Oscillations. While the constraint on ∆m231
by the 3-flavour zenith angle analysis was substantially weakened, that from 3-Flavour L/E was
not affected much (Chapter 10). The allowed oscillation parameters at the 90% C.L. for eachneutrino mass hierarchy were:0.396 < sin2 θ23 < 0.644, and 2.275 × 10−3 eV2 < ∆m231 < 2.995 × 10−3 eV2(Normal),0.390 < sin2 θ23 < 0.631, and 2.182 × 10−3 eV2 < ∆m231 < 2.842 × 10−3 eV2(Inverted).

Included in this thesis was the first analysis of 3-Flavour Neutrino Oscillations with NeutralCurrent Non-Standard Neutrino Interactions of atmospheric data in the µτ –sector. The extensive
baselines across the Earth, including the mantle and core, and wide range of available energies
make atmospheric neutrinos have unparalleled sensitivity to matter neutrino NSI. The best fit wasclose to zero-NSI, and was consistent with no observation of NSI (Chapter 11). Strong constraints
have been put on the following NSI parameters at the 90% C.L. (down-quark normalisation):−1.97 × 10−2 <εµτ< 1.45 × 10−2 ,−4.54 × 10−2 <ε0 ≡ (ετ τ − εµµ )< 4.47 × 10−2 .

270

These measurements are model independent constraints that can be used to define, restrict or
exclude any number of models including non-standard NC interactions of neutrinos with matter.
Since this result has properly considered 3-flavour and matter effects, it is much stronger andreliable than past 2-flavour measurements with atmospheric neutrinos. With access to muchgreater baselines and more dense matter than accelerator experiments, this result is currently
the most significant and realistic measurement in the world.
Finally, the implication of these results is that Neutrino Oscillations have been shown to berobust, even in the presence of independent neutrino and anti-neutrino oscillations or 3-flavourµτ –sector NSI.

271

Appendix A

L/E Analysis Distributions
The energy fitting for each of the event samples from SK1 to SK4 can be seen in Figure A.1.
The energy resolution for several samples is shown in Figures A.2 and A.3. The angular resolution for several samples is shown in Figures A.4 and A.5. Distributions for the true L/E overreconstructed L/E ratios are shown in Figure A.6. The resolution contours for each of the event
samples in the (cos θ,Eν ) plane can be seen on the right of Figure A.7.

272

(a) SK1(b) SK2(c) SK3(d) SK4

Figure A.1: Eν Reconstruction from fitting Evis2 to the true neutrino energy estimated by theMonte Carlo. The fitting for each of the event samples is shown, and the PC samples were
further divided into single-ring and multi-ring. The fitted result for each SK period is shown.

273

700003000

SK 1 FC 1-Ring Sub-GeV

60000

σ = 24%

2500

50000

SK 1 FC Multi-Ring Sub-GeV

2000

SK 1 PC Stopping

σ = 30%1750

σ = 20%

1500

2000

1250

40000
1500

1000

1000

750

30000
Number of Events

20000

500

10000

500

0

0

-1

22500

-0.5

0

0.5

1

σ = 14%

15000
12500
10000
7500
5000
2500
-1

-0.5

0

0.5

1

SK 2 FC 1-Ring Sub-GeV

60000

-0.5

0

0.5

1

0

σ = 24%

4000

3000

3000

2000

2000

1000

1000
-1

-0.5
0
0.5
(Erec-Etrue)/Etrue1

σ = 24%

1500

40000

2000

1250

30000

1500

20000

1000

Number of Events

2500

1

-1

-0.5

0

0.5

1

0.5

1

SK 2 PC Stopping
σ = 20%

1000
750
500

500

10000
-1

22500

-0.5

0

0.5

1

0

250
-1

6000

SK 2 FC 1-Ring Multi-GeV

20000

σ = 14%

-0.5

0

0.5

1

SK 2 FC Multi-Ring Multi-GeV

0

-1

-0.5

0

SK 2 PC Through-Going5000

σ = 24%

5000

17500

σ = 47%

4000

15000

4000

12500

3000

3000

10000
7500

2000

2000

5000

1000

1000

2500
0

0.5

σ = 44%

1750

σ = 30%

50000

0

0

2000

SK 2 FC Multi-Ring Sub-GeV

3000

0

-0.5

SK 1 PC Through-Going

5000

4000

0

-1

6000

SK 1 FC Multi-Ring Multi-GeV

5000

17500

0

-1

6000

SK 1 FC 1-Ring Multi-GeV

20000

250

-1

-0.5

0

0.5

1

0

-1

-0.5
0
0.5
(Erec-Etrue)/Etrue

1

0

-1

-0.5

0

0.5

1

Figure A.2: L/E energy resolution for SK1 and SK2. The difference of the reconstructed energy
and the true energy with respect to the true energy is shown. Events to the right of zero havebeen reconstructed with an energy larger than the true energy and vice versa for the left. The
black solid histogram is for events satisfying the 70 % resolution cut, and the green shaded area
corresponds to 68% of the events.
274

70000SK 3 FC 1-Ring Sub-GeV

60000

σ = 24%3000

σ = 20%

1750
1500

2000

40000

1250
1500

30000

1000
750

1000

20000
Number of Events

SK 3 PC Stopping

2000

σ = 30%

2500

50000

2250

SK 3 FC Multi-Ring Sub-GeV

10000

500

0

0

-1

22500

-0.5

0

0.5

1

SK 3 FC 1-Ring Multi-GeV

20000

500
250
-1

0

0.5

1

SK 3 FC Multi-Ring Multi-GeV

6000

σ = 14%

17500

-0.5

5000

-0.5

0

0.5

1

SK 3 PC Through-Goingσ = 44%

5000

4000

4000

10000

3000

3000

7500

2000

2000

1000

1000

12500

-1

6000

σ = 24%

15000

0

5000
2500
0

-1

-0.5

0

0.5

70000

SK 4 FC 1-Ring Sub-GeV

60000

σ = 24%

1

0

-1

-0.5
0
0.5
(Erec-Etrue)/Etrue

1

-1

2500

SK 4 FC Multi-Ring Sub-GeV

3000

0

σ = 30%

0

0.5

1

0.5

1

SK 4 PC Stopping
σ = 19%

2000

2500

-0.5

50000
2000

40000

1500

30000
Number of Events

1000

1000

20000

500

500

10000
0

1500

-1

-0.5

0

0.5

22500

SK 4 FC 1-Ring Multi-GeV

20000

σ = 14%

1

0

-1

-0.5

0

0.5

1

5000

5000

4000

4000

10000

3000

3000

7500

2000

2000

1000

1000

12500

5000
2500
0

-1

-0.5

0

0.5

1

0

-1

-0.5
0
0.5
(Erec-Etrue)/Etrue

-0.5

0

SK 4 PC Through-Going6000

σ = 24%

15000-1

SK 4 FC Multi-Ring Multi-GeV

6000

17500

0

1

0

σ = 44%

-1

-0.5

0

0.5

1

Figure A.3: L/E energy resolution for SK3 and SK4. The difference of the reconstructed energy
and the true energy with respect to the true energy is shown. Events to the right of zero havebeen reconstructed with an energy larger than the true energy and vice versa for the left. The
black solid histogram is for events satisfying the 70 % resolution cut, and the green shaded area
corresponds to 68% of the events.
27545000SK 1 FC 1-Ring Sub-GeV

4000

SK 1 FC Multi-Ring Sub-GeV

35000

3500

σ = 45.0°

30000

3000

25000

2500

800

20000

2000

600

15000

1500

10000

1000

5000

500

Number of Events

40000

0

σ = 61.0°

0

16000

30

60

90 120 150 180

400
200
0

30

60

90

120 150 180

SK 1 FC Multi-Ring Multi-GeV

σ = 18.5°σ = 29.5°

4000

12000
10000

σ = 20.0°

1000

5000

SK 1 FC 1-Ring Multi-GeV

14000

0

SK 1 PC Stopping

1200

0

0

14000

10

20

30

40

50

60

SK 1 PC Through-Going

12000

σ = 11.3°

10000

3000

8000

8000
6000

2000

6000

4000

4000

1000

2000

2000
0

0

45000

15

30

45

60

75

SK 2 FC 1-Ring Sub-GeV

40000

0

90

σ = 61.0°

35000

4000

SK 2 FC Multi-Ring Sub-GeV

3500

σ = 45.0°

25000
20000

10

20

30

40

50

60

SK 2 PC Stopping
σ = 20.0°

1000
800

2000

600

1500

10000

1000

5000

500

Number of Events

1200

2500

15000

0

16000

30

60

90 120 150 180

0

5000

SK 2 FC 1-Ring Multi-GeV

14000

σ = 18.5°

400
200
0

30

60

10000

90

120 150 180

SK 2 FC Multi-Ring Multi-GeV
σ = 29.5°

4000

12000

0

0

12000

10

20

30

40

50

60

SK 2 PC Through-Goingσ = 11.3°

10000
8000

3000

8000

6000
2000

6000

4000

4000

1000

2000

2000
0

0

3000

30000

0

0

0
15 30 45 60 75 90
Angular Resolution (degrees)

0

15

30

45

60

75

90

0

0
15 30 45 60 75 90
Angular Resolution (degrees)

0

0

10

20

30

40

50

60

Figure A.4: Angular resolution in SK1 and SK2 for the L/E samples. The distribution of angles
w.r.t. the true neutrino direction is shown. The black solid histogram is for events satisfying the
70 % resolution cut, and the green shaded region corresponds to 68% of events.

276

45000SK 3 FC 1-Ring Sub-GeV

40000

σ = 61.0°

35000

4000

SK 3 FC Multi-Ring Sub-GeV

3500

σ = 45.0°

1000

2500

25000

800

2000

20000

600

1500

10000

1000

400

5000

500

200

Number of Events

15000

0

0

16000

30

60

90 120 150 180

SK 3 FC 1-Ring Multi-GeV

14000

0

0

12000

30

60

90

120 150 180

SK 3 FC Multi-Ring Multi-GeV

5000

σ = 18.5°σ = 29.5°

0

16000

30

40

50

60

SK 3 PC Through-Goingσ = 11.0°

8000
6000

2000

4000

4000

1000

2000
0

45000

15

30

45

60

75

90

SK 4 FC 1-Ring Sub-GeV

40000

0

2000

SK 4 FC Multi-Ring Sub-GeV

4000

σ = 61.0°00
15 30 45 60 75 90
Angular Resolution (degrees)

3500

30000

3000

25000

2500

20000

2000

15000

1500

10000

1000

400

5000

500

200

0

16000

30

60

SK 4 FC 1-Ring Multi-GeV

30

60

90

50

60

σ = 19.7°

0

0

16000

10

20

30

40

50

60

SK 4 PC Through-Going14000

σ = 10.7°

12000

4000

6000

40

SK 4 PC Stopping

1200

120 150 180

σ = 29.0°

12000

8000

30

600

5000

10000

20

800

SK 4 FC Multi-Ring Multi-GeV

σ = 18.5°

10

1000

0
90 120 150 180
0
6000

14000

0

1400

σ = 45.0°

35000

Number of Events

20

10000

6000

10000

3000

8000

2000

6000

4000

4000
1000

2000
0

10

14000

3000

8000

0

0

12000

4000

10000

0

σ = 19.7°

1200

3000

30000

SK 3 PC Stopping

1400

0

15

30

45

60

75

90

0

2000
0
15 30 45 60 75 90
Angular Resolution (degrees)

0

0

10

20

30

40

50

60

Figure A.5: Angular resolution in SK3 and SK4 for the L/E samples. The distribution of angles
w.r.t. the true neutrino direction is shown. The black solid histogram is for events satisfying the
70 % resolution cut, and the green shaded region corresponds to 68% of events.

277

Figure A.6: log10 of the ratio of True L/E over Reconstructed L/E, in sets of plots for SK1 to SK4
as labelled in the x-axis. Within each set, the ratio is shown for each of the four event samples.
The solid black histogram has had the <70% resolution cut applied, while the dashed blue has
not. The high resolution samples have a tighter distribution on this L/E true/reconstructeddouble ratio.

278

Figure A.7: L/E Resolution Cut by Sample

279

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