学位論文

Development of Neutron-Tagging Techniques andApplication to Atmospheric Neutrino Oscillation
Analysis in Super-Kamiokande(スーパーカミオカンデにおける中性子検出手法の開発と大気ニュートリノ振動解析への適用)

平成26年7月博士(物理学)申請
東京大学大学院理学系研究科
物理学専攻
アーヴィン トリスタン ジェームズ
by Tristan James Irvine
2014Abstract
Neutrino oscillation theory is now well established. The neutrino mixing angles andmass-squared splittings have been precisely measured, by a variety of experiments. However our understanding of this field is still not complete - the neutrino mass hierarchy
and CP violating phase continue to elude us.
Sensitivity to neutrino mass hierarchy in the water Cherenkov detector Super-Kamiokande
is dependent on its ability to distinguish anti-neutrino and neutrino interactions. This
is inherently a difficult task in water Cherenkov detectors, but one possible method isthrough the detection of neutrons. In a typical charged-current anti-neutrino interaction, ν̄e + p → n + e+ , a neutron is ejected, however this is not true of the corresponding
neutrino interaction. In water, these neutrons then thermalize and are captured by hydrogen. A 2.2 MeV γ-ray is emitted, which can be used to tag the neutrons, and thus
infer that the preceding interaction was that of an anti-neutrino.
This thesis describes the development of neutron tagging methods and application to
the Super-Kamiokande atmospheric neutrino sample. This information is then used tohelp distinguish between neutrino and anti-neutrino interactions, thus increasing oursensitivity to the neutrino mass hierarchy. A neutron-tagging efficiency of 20.5% isachieved, with a background of 1.8% per atmospheric neutrino event. Enriched neutrino
and anti-neutrino samples are constructed, improving the sensitivity of the detector toneutrino mass hierarchy by ∆χ2 = 0.06. Three-flavour neutrino oscillation analysis
is performed using all data from SK-I to IV (4581.5 days). The normal hierarchy isfavoured, with a significance of ∆χ2 (NH-IH) = −0.9.

Acknowledgements
Writing this thesis would not have been possible without the support of many wonderfulpeople.
Firstly I’d like to thank my family. There is no way I can list here everything you havedone for me. Most of all, thank you for supporting me in all the crazy things I decide
to do. I am sure I would not manage so well with a son like this!
Thank you to everyone who persevered in educating me up until this point. My firstteacher, Pauline Wilson, for accepting me as a student under such absurd circumstances.
My university tutors, Prof. Weidberg, Laura, Georg, and many others, for pushing me
to the level I needed to succeed. Special thanks to my classmates and friends, Dan and
Nick, without whom I am sure I would not have maintained my sanity until graduation.
I want to thank all my friends in RCCN. Thomas, Andrew, Maggie, Honda-san, Nishimurasan, Euan and Akutsu-kun, thank you for making my life so enjoyable while I have been
here. Also Okumura-san, who in addition to being a good friend, has given me so muchsupport and guidance in my time here, both in physics and life. Thank you also to allof my friends in Kashiwa campus, Yoshida-san, and all of my Japanese teachers, whohelped me to remember that life is not just about physics!
I must also thank all the members of the SK collaboration, specifically the ATMPD
group, for both your advice, and of course your unrelenting work to keep this experiment
operational. Thank you to Shiozawa-san, Ed, Mark, Hayato-san and many more, foryour critiques and advice that eventually managed to mould me into some sort of ascientist (I hope)! Thank you especially to Roger, for your tireless efforts in dealing
with my constant barrage of problems, both in physics, and in real life. Without you I
would never have been able to graduate on time.
I want to thank everyone especially for putting up with my health issues over the last
few years. Especially to my doctors, Takagi-sensei and Dr. Wheldon, who were able toset me on the path to treating an illness which I thought I would be stuck with for life.
I am happy to say today my health is almost totally recovered, a position I would never
have thought possible 2 years ago.
My final thanks, must of course go to my advisor, Kajita-sensei, who gave me theopportunity to come here and study this amazing experiment. Thank you for giving mesuch an interesting and impactive analysis to work on, for letting me have the freedom
to explore it, and helping me whenever I got lost. I could not have hoped for such aknowledgeable and kind person to advise me.

ii

Contents
Abstracti

Acknowledgements

ii

Contents

iii

List of Figures

viii

List of Tablesxx

1 Introduction
1.1 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Author Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
2

2 Neutrinos
2.1 A History of Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Neutrino Flavours . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 The Solar Neutrino Problem . . . . . . . . . . . . . . . . . . . . .
2.1.3 The Atmospheric Neutrino Anomaly . . . . . . . . . . . . . . . . .
2.1.4 The Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Neutrino Oscillation Theory . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Oscillations in a Vacuum . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Oscillations in Matter . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 CP Violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Solar Parameters θ12 , ∆m212 . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Atmospheric Parameters θ23 , ∆m223 . . . . . . . . . . . . . . . . . .
2.3.3 Reactor Experiments, θ13 . . . . . . . . . . . . . . . . . . . . . . .
2.4 Summary, Remaining Issues . . . . . . . . . . . . . . . . . . . . . . . . . .

3
3
3
5
7
8
10
10
13
15
16
17
18
20
22

3 Super-Kamiokande23
3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.1.1 SK Periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Detector Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Cherenkov Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
iii

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iv

3.4
3.5
3.6
3.7

27
28
29
30
30
33

Photo-multiplier tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Water System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Air Purification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Front-end Electronics and Data Acquisition . . . . . . . . . . . . . . . . .
3.7.1 Data Acquisition System for SK-I to SK-III . . . . . . . . . . . . .
3.7.2 Data Acquisition System for SK-IV . . . . . . . . . . . . . . . . .

4 Detector Calibration
35
4.1 PMT Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.1.1 Determining High Voltage for each PMT . . . . . . . . . . . . . . 36
4.1.2 Relative Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.1.3 Absolute gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.1.4 Quantum Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.1.5 Timing Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2 Water Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2.1 Passage of Photons through Water . . . . . . . . . . . . . . . . . . 42
4.2.2 PMT Surface Reflections, Black Sheet . . . . . . . . . . . . . . . . 45
4.3 Energy Scale Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3.1 High Energy Cosmic-Rays . . . . . . . . . . . . . . . . . . . . . . . 47
4.3.2 Low Energy Cosmic-Rays . . . . . . . . . . . . . . . . . . . . . . . 47
4.3.3 π 0 Invariant Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.3.4 Decay Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3.5 Energy Scale Summary . . . . . . . . . . . . . . . . . . . . . . . . 50
5 Monte Carlo Simulation of Atmospheric Neutrinos
52
5.1 Atmospheric Neutrino Flux . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.2 Neutrino Interaction Generator . . . . . . . . . . . . . . . . . . . . . . . . 55
5.2.1 Elastic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.2.2 Single Meson Production . . . . . . . . . . . . . . . . . . . . . . . 57
5.2.3 Deep Inelastic Scattering . . . . . . . . . . . . . . . . . . . . . . . 58
5.2.4 Nuclear Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.3 Detector Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6 Event Reconstruction
64
6.1 Vertex Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.1.1 Ring Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6.1.2 Precise Single Ring Fitting . . . . . . . . . . . . . . . . . . . . . . 67
6.2 Particle Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.2.1 Electron-like and Muon-like rings . . . . . . . . . . . . . . . . . . . 70
6.3 Momentum Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.4 Reconstruction of Single Ring π 0 events . . . . . . . . . . . . . . . . . . . 74
6.5 Neutron Reconstructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.5.1 Bonsai Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.5.1.1 Bonsai Vertex Reconstruction . . . . . . . . . . . . . . . 76
6.5.1.2 Bonsai Energy Reconstruction . . . . . . . . . . . . . . . 76
6.5.2 Neut-Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.6 Upward-going Muon Reconstruction . . . . . . . . . . . . . . . . . . . . . 79

Contentsv

7 Data Reduction
81
7.1 Fully Contained Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7.1.1 1st Reduction (FC1) . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7.1.2 2nd Reduction (FC2) . . . . . . . . . . . . . . . . . . . . . . . . . 83
7.1.3 3rd Reduction (FC3) . . . . . . . . . . . . . . . . . . . . . . . . . . 84
7.1.3.1 Through-going muon cut . . . . . . . . . . . . . . . . . . 84
7.1.3.2 Stopping-muon cut . . . . . . . . . . . . . . . . . . . . . 86
7.1.3.3 Cable Hole Muons . . . . . . . . . . . . . . . . . . . . . . 86
7.1.3.4 Flashers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7.1.3.5 Coincidence Muon . . . . . . . . . . . . . . . . . . . . . . 87
7.1.3.6 Low Energy Events . . . . . . . . . . . . . . . . . . . . . 88
7.1.4 4th Reduction (FC4) . . . . . . . . . . . . . . . . . . . . . . . . . . 89
7.1.5 5th Reduction (FC5) . . . . . . . . . . . . . . . . . . . . . . . . . . 89
7.1.5.1 Invisible Muons . . . . . . . . . . . . . . . . . . . . . . . 89
7.1.5.2 Coincidence Muons . . . . . . . . . . . . . . . . . . . . . 90
7.1.5.3 Long Tail Flashers . . . . . . . . . . . . . . . . . . . . . . 90
7.1.6 Fiducial Volume Selection . . . . . . . . . . . . . . . . . . . . . . . 90
7.1.7 FC Reduction Summary . . . . . . . . . . . . . . . . . . . . . . . . 92
7.2 Partially Contained Reduction . . . . . . . . . . . . . . . . . . . . . . . . 92
7.2.1 1st Reduction (PC1) . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7.2.2 2nd Reduction (PC2) . . . . . . . . . . . . . . . . . . . . . . . . . 94
7.2.2.1 PC2: SK-I . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.2.2.2 PC2: SK-II . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.2.2.3 PC2: SK-III and SK-IV . . . . . . . . . . . . . . . . . . . 96
7.2.3 3rd Reduction (PC3) . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.2.4 4th Reduction (PC4) . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.2.4.1 PC4: SK-I and II . . . . . . . . . . . . . . . . . . . . . . 97
7.2.4.2 PC4: SK-III and IV . . . . . . . . . . . . . . . . . . . . . 98
7.2.5 5th Reduction (PC5) . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7.2.5.1 PC5: SK-I and SK-II . . . . . . . . . . . . . . . . . . . . 100
7.2.5.2 PC5: SK-III and SK-IV . . . . . . . . . . . . . . . . . . . 102
7.2.6 Fiducial Volume Selection . . . . . . . . . . . . . . . . . . . . . . . 105
7.2.7 PC Reduction Summary . . . . . . . . . . . . . . . . . . . . . . . . 106
7.3 Upward-going Muon Reduction . . . . . . . . . . . . . . . . . . . . . . . . 106
7.3.1 1st Reduction (UPMU1) . . . . . . . . . . . . . . . . . . . . . . . . 107
7.3.2 2nd Reduction (UPMU2) . . . . . . . . . . . . . . . . . . . . . . . 107
7.3.3 Interlude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
7.3.3.1 Showering Algorithm . . . . . . . . . . . . . . . . . . . . 109
7.3.4 3rd Reduction (UPMU3) . . . . . . . . . . . . . . . . . . . . . . . 110
7.3.5 4th Reduction (UPMU4) . . . . . . . . . . . . . . . . . . . . . . . 112
7.3.6 Background Subtraction . . . . . . . . . . . . . . . . . . . . . . . . 112
7.3.7 UPMU Reduction Summary . . . . . . . . . . . . . . . . . . . . . . 114
8 Neutron Tagging
116
8.1 Neutron Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
8.2 Neutron Tagging Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 119
8.2.1 Initial Candidate Selection . . . . . . . . . . . . . . . . . . . . . . 120

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vi

8.2.2

8.3
8.4

Final Candidate Selection . . . . . . . . . . . . . . . . . . . . . . . 121
8.2.2.1 Number of Hits in 10 ns: N10 . . . . . . . . . . . . . . . . 122
8.2.2.2 Number of Clustered Hits: Nc . . . . . . . . . . . . . . . 123
8.2.2.3 Number of Hits on Low Probability PMTs: Nlow . . . . . 124
8.2.2.4 Number of Hits in 300 ns: N300 . . . . . . . . . . . . . . 125
8.2.2.5 Hit Vector RMS: φrms . . . . . . . . . . . . . . . . . . . . 126
8.2.2.6 Mean Opening angle: θmean . . . . . . . . . . . . . . . . . 126
8.2.2.7 Root-mean-square of hit timing: trms . . . . . . . . . . . 126
8.2.2.8 Minimum Root-mean-square of hit timing: min − trms . 128
8.2.2.9 Bonsai Fit Variables: BSenergy and BSwall . . . . . . . . 128
8.2.2.10 Neut-Fit Variables: N Fwall , ∆N10 , ∆trms . . . . . . . . . 130
8.2.2.11 Fit Agreement variables: (N F − BS)dis , (N F − AP )dis . 131
8.2.3 Neural Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
8.2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Atmospheric Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Americium-Beryllium Systematic Error Study . . . . . . . . . . . . . . . . 136
8.4.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 138
8.4.2 Data Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
8.4.3 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

9 Oscillation Analysis
142
2
9.1 Calculation of χ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
9.2 Analysis Binning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
9.2.1 Neutrino and Anti-neutrino separation . . . . . . . . . . . . . . . . 146
9.2.2 Multi-Variate Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 151
9.2.2.1 Multi-GeV, Single-Ring, e-like sample . . . . . . . . . . . 152
9.2.2.2 Multi-GeV, Multi-Ring, e-like samples . . . . . . . . . . . 153
9.2.2.3 Multi-GeV, Single-Ring, µ-like samples . . . . . . . . . . 154
9.2.2.4 Multi-GeV, Multi-Ring, µ-like samples . . . . . . . . . . 155
9.3 Systematic Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
9.3.1 Systematic Errors related to Neutrino Flux . . . . . . . . . . . . . 158
9.3.1.1 Flux Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . 158
9.3.1.2 K/π ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
9.3.1.3 Solar Activity . . . . . . . . . . . . . . . . . . . . . . . . 159
9.3.1.4 Neutrino Production Height . . . . . . . . . . . . . . . . 159
9.3.1.5 Matter Effect . . . . . . . . . . . . . . . . . . . . . . . . . 160
9.3.1.6 Absolute normalization . . . . . . . . . . . . . . . . . . . 160
9.3.2 Systematic Errors related to Neutrino Interactions . . . . . . . . . 161
9.3.2.1 CCQE Cross Section . . . . . . . . . . . . . . . . . . . . 161
9.3.2.2 Axial Mass . . . . . . . . . . . . . . . . . . . . . . . . . . 161
9.3.2.3 Single Meson Production . . . . . . . . . . . . . . . . . . 161
9.3.2.4 Coherent Pion Production . . . . . . . . . . . . . . . . . 163
9.3.2.5 Deep Inelastic Scattering . . . . . . . . . . . . . . . . . . 163
9.3.2.6 Other Cross Section Uncertainties . . . . . . . . . . . . . 163
9.3.3 Systematic Errors related to Reduction and Reconstruction Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
9.3.3.1 FC Reduction . . . . . . . . . . . . . . . . . . . . . . . . 164

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9.4
9.5
9.6
9.7

vii

9.3.3.2 PC Reduction . . . . . . . . . . . . . . . . . . . . . . . . 164
9.3.3.3 UPMU Reduction . . . . . . . . . . . . . . . . . . . . . . 165
9.3.3.4 Reconstruction errors . . . . . . . . . . . . . . . . . . . . 165
9.3.4 Event Selection errors . . . . . . . . . . . . . . . . . . . . . . . . . 165
9.3.4.1 Multi-GeV Neutrino and Anti-Neutrino sample errors . . 165
Mass Hierarchy Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
Data Fit Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
Discussion and Comparison to other Experiments . . . . . . . . . . . . . . 174
The Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

10 Conclusion

180

A Appendix: Systematic Error Tables

182

Bibliography

185

List of Figures
2.1
2.2
2.3

2.4

2.5

2.6

2.7

Energy spectrum for neutrinos produced in the pp chain, predicted by
Bachall’s Standard Solar Model . . . . . . . . . . . . . . . . . . . . . . . .
Hadronic shower and atmospheric neutrinos produced by cosmic rays.
Figure from Los Alamos Science . . . . . . . . . . . . . . . . . . . . . . .
Ratio of the data to the MC events as a function of L/E, from SuperKamiokande. The solid black histogram represents the best fit for oscillation, whilst the disfavoured blue dashed, and dotted red lines are the best
fit expectations for neutrino decay, and neutrino decoherence respectively.
This figure shows the oscillation probability of νe → νe , for a 3MeV
neutrino. It may be thought of as the probability that an initial electron
neutrino will be observed as an electron neutrino after travelling distance
L. For this plot, ∆m223 = 2.4 × 10−3 eV 2 , and ∆m212 = 7.6 × 10−5 eV 2 .
The high frequency oscillations are due to ∆m223 , and if a detector isplaced at 1km then you may approximate the mixing matrix to its 2dimensional form considering only this mixing. Likewise, for the low
frequency oscillations, the driving factor is ∆m212 , so to optimally measure
this, a detector may be placed at 50km, and again the mixing matrix may
be reduced to an oscillation between these two states. . . . . . . . . . . .
This figure represents the difference between probability of oscillation
νe ↔ νµ , between normal and inverted mass hierarchies, for neutrinos incident on Super Kamiokande. The y-axis is zenith angle, such that the upper half of the plot represents downward going neutrinos, and the bottom
half upward going neutrinos. The dominant area of resonant enhancement can be seen around 2 − 10GeV . If we instead look at anti-neutrino
oscillation, the signs in this plot are switched, so the enhancement occurs
in inverted, rather than normal hierarchy. . . . . . . . . . . . . . . . . . .
The two possibilities for neutrino mass states, known as the normal andinverted hierarchies. The green forward-slash fill represents νe , red backslash is νµ , and vertical blue lines is ντ . The difference between the massstates ∆m2 is well known, thanks to neutrino oscillation experiments, but
at the time of writing, which mass state is the heaviest is still unknown.
This plot is not to scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
This figure shows the combined result of 8 B solar neutrino flux measurements, from the salt phase of SNO [1], and ES result from SK in 2002 [2].
The dotted line represents the standard solar model prediction, and thecontours represent a joint fit between the two experiments. Figure source[1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

viii

6
7

9

12

15

16

18

List of Figures

ix

2.8

On the top is the energy spectrum for events reconstructed as CC, at the
MINOS far detector. The bottom figure shows the ratio of data to a null
oscillation prediction, with the disfavoured decay and decoherence models
also shown. [3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.9 This plot shows 90% confidence level contours for T2K, MINOS, and SK
superimposed. Taken from [4] . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.10 This plot shows the result from Daya-Bay in 2012. The observed energy
spectrum, and ratio with the null-oscillation hypothesis are displayed on
top and bottom figures respectively. Taken from [5] . . . . . . . . . . . . . 21
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9

A schematic of the layout of Super-Kamiokande . . . . . . . . . . . . . . . 24
The frame structure of the PMT support system in the SK tank . . . . . 26
The structure of an ID PMT in SK. . . . . . . . . . . . . . . . . . . . . . 27
The wavelength dependent quantum efficiency of the bialkali photo-cathode. 28
A schematic of the water purification system used in SK. . . . . . . . . . 29
The varying water temperature across the SK tank. Water input is at thebottom of the tank (negative z). . . . . . . . . . . . . . . . . . . . . . . . 30
A schematic showing the layout of the DAQ used in SK-I to SK-III . . . . 31
Overview of the trigger system in SK-I to III . . . . . . . . . . . . . . . . 32
Block diagram for the new DAQ in SK-IV. Taken from [6] . . . . . . . . . 33

4.1

Xe light emitting from a scintillator ball to calibrate the PMT high voltage in the tank. The red dots illustrate positions of the pre-calibrated
reference PMTs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 The relative gain difference of each PMT from calibration using a nitrogenlaser-driven dye laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 The charge distribution from Ni-Cf single p-e calibration source, after
relative gain corrections had been applied. . . . . . . . . . . . . . . . . . .
4.4 This plot shows the relative PMT hit probability dependent on z (cm)
position in the tank. The red points are from Ni-Cf source, and the blue
points are the MC, before QE correction. . . . . . . . . . . . . . . . . . .
4.5 The setup of the dye laser used to calibrate the timing and relative gainof the ID PMTs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Charge vs Time plot for readout channel 00010. The black line is a fitted
polynomial used for calibration, referred to as the ”TQ map”. . . . . . . .
4.7 The laser system used to calibrate light scattering and absorption throughwater. B1-5 represent different regions of the tank in which analysis was
performed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8 The results of the fit calibration for water scattering parameters, for April
2009. The dots represent values measured in data, and the lines are thecorresponded fitted functions. . . . . . . . . . . . . . . . . . . . . . . . . .
4.9 The split up of data taken from the laser beam (405 nm) used for waterand PMT reflection calibration. Shown is the timing (ns) of each PMT
hit, after ToF correction, for 6 different regions of the tank (the top plot
is of the tank top wall, and the remaining 5 are barrel regions shownin figure 4.7). The hits between the two left-most blue lines, are used
for water transparency calibration, and those between the two right-mostlines are used for reflection calibration. . . . . . . . . . . . . . . . . . . . .
4.10 The experimental set-up used to tune the black sheet reflectivity. . . . . .

36
38
39

40
41
42

43

44

45
46

List of Figures
4.11 High energy cosmic ray stopping muon sample, ratio between momentum
Ppe , calculated from the total number of photo-electrons collected by each
PMT, and the range of the particle in cm. The discrepancy between data
and MC is used as energy calibration in this momentum range. Datashown is from the SK-II period. . . . . . . . . . . . . . . . . . . . . . . . .
4.12 Low energy cosmic ray muon sample, ratio between momentum Ppe , calculated from the total number of photo-electrons collected by each PMT,
to Pθ , calculated from the Cherenkov opening angle. The discrepancy
between data and MC is used as energy calibration in this momentum
range. Data shown is from the SK-I period. . . . . . . . . . . . . . . . . .
4.13 Reconstructed π 0 invariant mass, for data and MC in black and red respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.14 The summary of all absolute energy scale data-MC discrepancies, for SK-I
to IV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1
5.2

5.3
5.4
5.5

5.6

5.7

5.8

x

48

48
49
51

East-west asymmetry in neutrino flux at SK, due to deflection of primarycosmic rays by the Earth’s geomagnetic field. Plots taken from [7]. . . . . 53
Zenith angle distribution for neutrino flux incident on SK. You may notethe peak at the horizon (around cos θ = 0), and the slight asymmetry between upward and downward going fluxes. These effects are more prominent in low energy particles, which are more sensitive to magnetic field
deflection effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Additional cosmic ray vectors now contribute to the horizon neutrino fluxin the 3-d calculation of Honda flux, compared to the previous 1-d version. 54
Flux comparison with cosmic ray muon data. From left to right, Tsukuba(ground), Norikura (mountain) and Ft. Summer (balloon) . . . . . . . . . 55
Atmospheric neutrino flux incident on SK, taken from the Honda 2011
paper. Differences to other models are shown, and taken as a systematic
error in analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
The breakdown of expected neutron multiplicity for each neutrino interaction mode. The plot shows SK-IV atmospheric neutrino MC Multi-GeV
1-ring electron like sample, normalized to the livetime of the equivalent
data-set (shown in black dots). This is the most relevant sample for theneutrino mass hierarchy. In this sample the dominant production modes
are CCQE and single-pion production. . . . . . . . . . . . . . . . . . . . . 57
The agreement of NEUT predictions (black line) of free nucleon charged
current quasi-elastic scattering cross section, with experimental data. The
experiments considered in figure (a) are ANL (triangle) [8], GGM (square)[9] and Serpukov (circle) [10]. In figure (b), these are GGM (square) [11]and Serpukov (circle) [10]. Figures taken from [12]. . . . . . . . . . . . . . 58
The agreement of NEUT predictions (black line) of single pion production
cross section, with experimental data. The experimental data consideredis from ANL (triangle) [13], and GGM (circle) [14]. The figures are takenfrom [12]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

List of Figures
The agreement of the total neutrino and anti-neutrino charged current
cross sections, between the NEUT model and experimental data. In figure(a), the data used is from experiments GGM73 [15], GGM79 [16], GGM81[17], SKAT79 [18], BEBC79 [19], IHEP79 [20], ANL79 [21], BNL80 [22],BNL81 [23] and IHEP96 [24]. For the anti-neutrino figure (b), the experiments used are GGM79 [25], GGM81 [17], IHEP79 [20] and IHEP96 [24].
The figures are taken from [12]. . . . . . . . . . . . . . . . . . . . . . . . .
5.10 Comparison between NEUT simulation, and experimental data. Cross
section contributions from charged current elastic scattering, single pion
production, and deep inelastic scattering are shown. Experimental data
is overlaid from [26]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.11 Tuning of nuclear effect cross sections to π − 12 C scattering data. The
dots are taken from experimental data in [27]. The dashed and solid lines
represent before and after tuning. . . . . . . . . . . . . . . . . . . . . . . .
5.12 Absolute neutron yields after impact of 256 MeV protons on a sphere of
238 U. Lines are simulated by CALOR, and experimental data is from [28].
This plot is taken from [29]. . . . . . . . . . . . . . . . . . . . . . . . . . .
5.13 Neutron multiplicity simulated by HETC (open dots), compared to experimental data (solid triangles) for a 2 GeV proton interacting in Au.
The solid dots are before detector efficiency is taken into account. This
study and plot are from [30]. . . . . . . . . . . . . . . . . . . . . . . . . .

xi

5.9

6.1
6.2

6.3

6.4
6.5
6.6

6.7

6.8

A depiction of the co-ordinate system used in SK. θ (zenith angle) is
measured downwards from positive z. . . . . . . . . . . . . . . . . . . . . .
On the left, is a 2-d depiction of ring candidate searching. Each PMT
has a charge-weighted cone drawn out from it at 42◦ , and the points withmaximal charge are taken as ring candidates. The plot on the right is an
example of the result of this, in a spherical co-ordinate system centredon the TDC-fit vertex. The angles with peaks of charge represent mostprobable rings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Output of L(2 rings) − L(1 ring) for SK-IV, < 1330 MeV neutrino events.
The fraction of different event modes in each sample is shown by thecoloured lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Vertex resolution for SK-IV sub-GeV single-ring events, separated intoµ-like and e-like. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Angular resolution for SK-IV fully contained sub-GeV single-ring events,
separated into µ-like and e-like. . . . . . . . . . . . . . . . . . . . . . . . .
Example Cherenkov ring patterns for an electron (left), and a muon(right). The top and bottom circles represent the top and bottom of
the SK tank, and the long rectangle represents the barrel. Auxiliary
event information is provided around the edges (but it is not importantto understand). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A depiction of how the muon track length is treated when calculating the
expected charge. When a muon travels distance dx, Cherenkov photons
are emitted to an area of 2πr sin θ(dx sin θ + rdθ). . . . . . . . . . . . . . .
Output particle identification likelihoods (definition here 6.12) for SK-IV,
sub-GeV fully contained single-ring events. The left and right plots show
true CC-quasi elastic electron and muon events respectively. . . . . . . . .

59

60

60

62

62

65

66

67
68
69

69

72

73

List of Figures
The timing residual probability density function measured by LINAC
calibration data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.10 The energy distributions of upward-going muon showering, non-showering,and stopping events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.11 The vertex resolution of upward-going muon fitter precise-fit, based off a
40 year MC study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.12 The calculation of the end-point of a stopping muon track. The dottedline shows the 40% of the average value in the first 1.5 m of the track. . .

xii

6.9

7.1
7.2
7.3

77
79
80
80

The different reduction types for selecting atmospheric neutrino events . . 82
The energy spectrum for each sample of events, after reduction. . . . . . . 82
FC2 distributions of NHITA800 , for data, MC, and then the final selection
with data and MC superimposed. The MC is a pure atmospheric neutrino
sample, so the second plot shows the efficiency of this cut. In the final
sample the events which remain in the cut region have either PEtotal >100, 000, or the OD trigger did not fire. . . . . . . . . . . . . . . . . . . . 84
7.4 FC2 distributions of PEmax /PE300 . Plots of data, MC and then the final
selection with data and MC superimposed are shown. The MC consistsof pure atmospheric neutrino events. . . . . . . . . . . . . . . . . . . . . . 85
7.5 A schematic of the cable hole veto counters in SK . . . . . . . . . . . . . . 87
7.6 ID timing distributions of a typical flasher (i), and a typical neutrino (ii)event. The two lines indicate the timing region scanned in calculating
NMIN100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
7.7 FC5 Coincidence muon cut variable PElate . From top to bottom, the plotsshow the data, atmospheric neutrino MC, and then the final selection of
events with data and MC superimposed. It is also required that PE500 <300 p-e, so there are a few events remaining in the final sample that have
PElate > 20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7.8 The event rate for each of FC1-5, and FCFV reductions, for all of SK-IV.
The increasing event rate is due to the PMT degradation causing gainincrease of 2% per year, and is accounted for in analysis. The event rate
spike around March 2009 is due to faults in the HV system. . . . . . . . . 92
7.9 Explanation of OD clustering algorithm. The charge in each patch is
proportional to the size of the circle. The charge is added to a cluster inthe direction of the largest charge gradient. . . . . . . . . . . . . . . . . . 95
7.10 The number of hits in the 2nd largest OD charge cluster. If there are twolarge clusters, they likely represent entrance and exit points in an event.
As PC events must originate in the ID, these events may be cut. The
MC shows a simulated pure atmospheric neutrino sample, and the final
sample is the agreement between data and MC for this variable after the
final PC selection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7.11 These are the selection criteria in PC4 for SK-III and IV, used to rejectmuon events that originated outside the inner detector. The dotted line
shows the data, after the output of PC3. The solid line shows MC, which
may be thought of as the expectation for a pure PC neutrino sample. . . 98
7.12 These plots show the distance from the reconstructed muon vertex to thenearest corner of the tank. MC is a sample of pure atmospheric neutrino.
This cut is used to remove corner-clipping muons. The bottom plot shows
data-MC agreement after the final selection of PC events. . . . . . . . . . 104

List of Figuresxiii

7.13 NCLSTA5(x) gives the number of OD PMTs inside the x-th highest
charge cluster. Large values for two clusters suggests a through-going
muon event, which is then cut. . . . . . . . . . . . . . . . . . . . . . . . . 104
7.14 Total ID p-e as a function of track length. Data taken with down-goingcosmic ray muon calibration sample. . . . . . . . . . . . . . . . . . . . . . 108
7.15 Primary methods of muon energy loss, when travelling through water.
The line (a) shows energy loss via ionization, (b) shows energy loss via
Bremsstrahlung, and (c) is the total. . . . . . . . . . . . . . . . . . . . . . 109
7.16 The thru-mu showering vs non-showering selection criteria, shown herewith SK-IV dataset. Events above the red line are classified as showering,and below are classified as non-showering. . . . . . . . . . . . . . . . . . . 111
7.17 The zenith and azimuthal dependence of Stopping and through-going
muons for SK-III. Some above horizon events have been included fordemonstration of this effect. . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.18 The 1-dimensional projection of the azimuthal dependence of UPMU
background. Events coming from above the horizon are shown with nofill, and upward-going muons are shown with hashed fill. . . . . . . . . . . 113
7.19 Extra background contamination in stop-mu and thru-mu samples, fitted
to a constant + exponential function of zenith angle. The triangles represent the approximately constant background in azimuthal region 1, andthe circles are the exponentially increasing background in region 2. . . . . 114
8.1
8.2

8.3
8.4

8.5
8.6
8.7

8.8

8.9

Construction of neutron MC. After 18µs dummy trigger data is convoluted
with simulated neutrons produced by an atmospheric neutrino interaction. 117
Number of 2.2 MeV γ candidates (before neural network selection) per
day, as a function of run number in SK-IV. The dummy trigger datataking period is shaded in red. The average candidates per event for thedummy trigger period is 5.34/event, and the average for all of SK-IV is
5.54/event. The gradient is consistent with a 2% increase in PMT gain
per year. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
Neutron travel distance from the initial neutrino interaction vertex, according to the atmospheric neutrino MC simulation. . . . . . . . . . . . . 118
Showing the time resolution of final selection of neutrons in 500 years of
MC. The blue line shows the cut for calculating efficiency and backgroundof the neutron tagging algorithm. . . . . . . . . . . . . . . . . . . . . . . . 119
The increase in number of γ candidates due to the PMT after-pulse.
Scanning is preformed after 18µs. . . . . . . . . . . . . . . . . . . . . . . . 121
Values of N10 used to select initial 2.2 MeV candidates. Backgroundincreases exponentially as the required N10 is reduced. . . . . . . . . . . . 121
N10 variable as a neural network input. The plot shows a comparison ofneutron capture signal (green), background (blue), data (black dots) and
MC total (red line). The normalization is explained in section 8.2.2 . . . . 122
Clusters of hits variable, N10 − Nc . The plot shows a comparison ofneutron capture signal (green), background (blue), data (black dots) and
MC total (red line). The normalization is explained in section 8.2.2 . . . . 123
The varying acceptance requirements for the Nlow cut, shown as a functionof tank coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

List of Figuresxiv

8.10 Hits on low probability PMTs, N10 − Nlow . The plot shows a comparisonof neutron capture signal (green), background (blue), data (black dots)
and MC total (red line). The normalization is explained in section 8.2.2 . 125
8.11 The utility of the N300 cut. Figure (a) shows a typical neutron capture
event, and figure (b) shows a hypothetical background signal to be rejected. The blue lines correspond to PMT dark noise, and the red is somesignal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
8.12 Number of hits in 300 ns, N300 − N10 . The plot shows a comparison ofneutron capture signal (green), background (blue), data (black dots) and
MC total (red line). The normalization is explained in section 8.2.2 . . . . 126
8.13 Root-mean-square of the azimuthal angle of 2.2 MeV candidate hit vectors φrms . The plot shows a comparison of neutron capture signal (green),background (blue), data (black dots) and MC total (red line). The normalization is explained in section 8.2.2 . . . . . . . . . . . . . . . . . . . . 127
8.14 Mean value of hit opening angles, θmean . The plot shows a comparison ofneutron capture signal (green), background (blue), data (black dots) and
MC total (red line). The normalization is explained in section 8.2.2 . . . . 127
8.15 The root-mean-square of the timing of candidate hits, trms . The plotshows a comparison of neutron capture signal (green), background (blue),data (black dots) and MC total (red line). The normalization is explained
8.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
8.16 The selection of min − trms hit clusters. Optimal selections for clustersof 3-6 hits are shown, for an example candidate where N10 = 9. Possible
background hits on the fringes of the candidate time are ignored. . . . . . 128
8.17 min − trms of clusters of 6 hits (left) and 3 hits (right). The plot shows
a comparison of neutron capture signal (green), background (blue), data
(black dots) and MC total (red line) comparison. The normalization is
explained 8.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
8.18 Distributions of BSwall and BSenergy for 2.2 MeV γ candidate events.
The normalization of BSenergy is different for signal and background, as
many background events fail to reconstruct at this stage. The plots show
a comparison of neutron capture signal (green), background (blue), data
(black dots) and MC total (red line). The normalization is explained in
section 8.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
8.19 The distance to the wall from Neut-Fit vertex, N Fwall . The plot shows
a comparison of neutron capture signal (green), background (blue), data
(black dots) and MC total (red line). The normalization is explained in
section 8.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
8.20 The variables ∆trms and ∆N10 are shown in the left and right figures,
respectively. The plots show a comparison of neutron capture signal(green), background (blue), data (black dots) and MC total (red line).
The normalization is explained in section 8.2.2 . . . . . . . . . . . . . . . 131
8.21 The variables (N F − BS)dis (distance between Bonsai reconstructed candidate vertex, and Neut-Fit reconstructed candidate vertex) and (N F −AP )dis (distance between Neut-Fit reconstructed candidate vertex, andAPFit vertex), are shown in the left and right figures, respectively. The
plots show a comparison of neutron capture signal (green), background(blue), data (black dots) and MC total (red line). The normalization is
explained in section 8.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

List of Figuresxv

8.22 The neural network output for selecting final neutron candidates. Everything to the right of the dotted cut line is selected as a neutron capture
event. The small peak in the background histogram at NN output ∼1 isdue to events where the secondary particle bank is overfilled. For these
events some true neutron capture information is not stored, and somecandidates are mis-classified as background. The data-MC discrepancy
in the far left bin looks drastic on a log-scale plot, but this represents< 0.1% of total background events. The plot shows a comparison of neutron capture signal (green), background (blue), data (black dots) and MC
total (red line). The normalization is explained in section 8.2.2 . . . . . . 133
8.23 The dependence of 2.2 MeV detection efficiency on the distance travelledby the neutron (left), and the visible energy from the neutrino interaction(right). Efficiency is defined as the number of selected candidates corresponding to true neutrons, per number of true neutron capture events ineach bin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
8.24 The efficiency of detecting neutrons as a function of neutrino (APFit) vertex position throughout the ID. Only neutrino events passing the fiducial
volume cut (dW all > 200 cm) are considered. . . . . . . . . . . . . . . . . 135
8.25 Final selected 2.2 MeV gamma candidates timing distribution. The primary trigger is at t = 0. Blue curve is a best exponential fit using chisquared method. The neutron capture lifetime is calculated to be 205.2± 3.7 µs. The background rate is set to 0.0184 / neutrino event. Data istaken from all of SK-IV (1775.6 days) . . . . . . . . . . . . . . . . . . . . 136
8.26 These plots show data-MC comparisons for neutrons tagged in the SK-IV
1775.6 days dataset. The top left plot shows the average multiplicity ofneutrons, the top right plot shows the number of neutrino events with
≥ 1 neutrons, the bottom left plot shows the neutron multiplicity in SubGeV events (Evis < 1330 MeV), and the bottom right plot shows neutron
multiplicity in Multi-GeV events (Evis ≥ 1330 MeV). The multiplicity
plots are normalized to the number of neutrino events. The number of
events plot (top right) is normalized to data livetime and oscillated undera 2 flavour approximation: ∆m2 = 2.5 × 10−3 and sin2 2θ = 1.0. . . . . . . 137
8.27 Am-Be crystal embedded in a 5 cm cube of BGO scintillator. This is held
in an acrylic case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
8.28 The total p-e from 4.4 MeV γ scintillation events, with the Am-Be source
positioned at the centre of the tank. Events with 750 < p-e < 1050 are
selected. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
8.29 The timing distribution of tagged 2.2 MeV γ candidates, after triggering
on a prompt 4.4 MeV γ ray from the Am-Be neutron source. The dots
are data-points, and the blue line is a best fit to a constant background+ exponential decay, where τ is the neutron capture lifetime in µs, andsignal fraction is the fraction of all tagged events expected to be neutrons. 141

List of Figures
9.1

9.2

9.3

9.4

9.5

9.6

9.7

9.8

xvi

The multiplicity of neutrons following neutrino interactions in the MultiGeV 1-ring electron-like sample (left) and MultiGeV Multi-ring electronlike sample (right). The multi-ring sample consists of more deep inelastic
scattering events, so neutrons are significantly less useful in this sample.
However the excess in anti-neutrino 1-neutron events from CCQE interactions is still apparent. Only CC events corresponding to sample typeare shown (i.e. for e-like samples, only CC νe and CC ν̄e ), and all histograms are normalized to 1. For use in analysis, any events producing>5 neutrons are added into the 5 neutron bin. . . . . . . . . . . . . . . . 148
The number of decay electrons, following neutrino interactions in theMultiGeV 1-ring electron-like sample (left) and MultiGeV 1-ring muonlike sample (right). In the muon sample, many more decay-electrons
are produced in general, however the excess in the neutrino sample isstill apparent. Only CC events corresponding to sample type are shown(i.e. for e-like samples, only CC νe and CC ν̄e ), and all histograms arenormalized to 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
The number of Cherenkov rings, following neutrino interactions in theMultiGeV Multi-ring electron-like sample. There is some excess in thetrue CC ν sample, due to the additional charged hadrons produced inthis interaction. Only CC events corresponding to sample type are shown(i.e. for e-like samples, only CC νe and CC ν̄e ), and all histograms arenormalized to 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
This plot shows the farthest decay-electron travel distance |neutrino vertex - decay electron vertex|, in the Multi-Gev 1-ring µ-like sample. Only
CC events corresponding to sample type are shown (i.e. for e-like samples,
only CC νe and CC ν̄e ). All histograms are normalized to 1. . . . . . . . . 150
This plot depicts the fraction of reconstructed energy in the first ring
divided by the total energy in the event for the MultiGeV Multi-Ring
e-like sample. Only CC events corresponding to sample type are shown(i.e. for e-like samples, only CC νe and CC ν̄e ) and all histograms arenormalized to 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
The particle identification likelihood output, for the Multi-Gev 1-ring elike sample. PID < 0 is required to classify the ring as e-like. Only CC
events corresponding to sample type are shown (i.e. for e-like samples,
only CC νe and CC ν̄e ), and all histograms are normalized to 1. . . . . . . 151
The decay time of the decay-electron reconstructed farthest from the neutrino interaction, for the MultiGeV Multi-Ring µ-like sample. Only CC
events corresponding to sample type are shown (i.e. for e-like samples,
only CC νe and CC ν̄e ), and all histograms are normalized to 1. . . . . . . 151
These plots show the data-MC agreement in SK-IV for the number ofneutrons and number of decay-electrons in the MultiGeV 1 ring e-likesample. The colours represent the different event types each sample consists of: green and green-hashed are νe and ν̄e respectively. Purple istrue νµ events, and yellow is neutral current events. The MC events arenormalized to the livetime of the dataset, and oscillated in a 2 flavour
framework with sin2 2θ = 1.0, and ∆m2 = 2.5 × 10−3 eV 2 . Systematic
errors are not included here. . . . . . . . . . . . . . . . . . . . . . . . . . . 153

List of Figures
9.9

xvii

These plots show the data-MC agreement, and the νe -ν̄e separation in
the multi-variate analysis output. The event sample is the MultiGeV
Multi-ring e-like sample for SK-IV. The cut position is shown here as adotted blue line at 0.72, chosen automatically (see 9.2.2). Events >0.72
are classified as νe -like, and <0.72 are ν̄e -like. For the right-hand plot,
both histograms are normalized to 1. In the left-hand plot, the MC events
are normalized to the livetime of the dataset, and oscillated in a 2 flavour
framework with sin2 2θ = 1.0, and ∆m2 = 2.5 × 10−3 eV 2 . Systematic
errors are not included here. . . . . . . . . . . . . . . . . . . . . . . . . . . 155
9.10 These plots show the data-MC agreement, and the νµ -ν̄µ separation in
the multi-variate analysis output. The event sample is Multi-GeV singlering µ-like, for SK-III. The cut value is chosen automatically (describedat 9.2.2), shown here as a dotted blue line at 0.62. Events >0.62 areclassified as νµ -like, and <0.62 are ν̄µ -like. For the right-hand plot, both
histograms are normalized to 1. In the left-hand plot, the MC events arenormalized to the livetime of the dataset, and oscillated in a 2 flavour
framework with sin2 2θ = 1.0, and ∆m2 = 2.5 × 10−3 eV 2 . Systematic
errors are not included here. . . . . . . . . . . . . . . . . . . . . . . . . . . 156
9.11 These plots show the data-MC agreement, and the νµ -ν̄µ separation in
the multi-variate analysis output. The event sample is Multi-GeV multiring µ-like, for SK-II. The cut value is chosen automatically (describedat 9.2.2), shown here as a dotted blue line at 0.75. Events >0.75 areclassified as νµ -like, and <0.75 are ν̄µ -like. For the right-hand plot, both
histograms are normalized to 1. In the left-hand plot the MC events arenormalized to the livetime of the dataset, and oscillated in a 2 flavour
framework with sin2 2θ = 1.0, and ∆m2 = 2.5 × 10−3 eV 2 . Systematic
errors are not included here. . . . . . . . . . . . . . . . . . . . . . . . . . . 157
9.12 The flavour ratio between the Honda flux and competing flux models
used for systematic error calculation. The solid black line shows the ratio
FLUKA/Honda flux, and the dashed red line depicts the Bartol/Honda
flux. The spike around 8 GeV in the FLUKA flux is due to a flux calculation technicality, and has been corrected for in the error evaluation. . . 158
9.13 The anti-neutrino / neutrino double ratio between the Honda flux andcompeting flux models used for systematic error calculation. The solid
black line shows the ratio FLUKA/Honda flux, and the dashed red line
depicts the Bartol/Honda flux. The left plot shows the double ratio ofν̄e /νe and the right plot shows ν̄µ /νµ . . . . . . . . . . . . . . . . . . . . . 159
9.14 The neutrino path length, as a function of zenith angle. The solid line
represents the path lengths provided in the default Honda flux model,
and the dotted line shows the path-lengths after a 10% change in theatmospheric density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
9.15 The top plot shows the relative uncertainty in each aspect of the flux
model, where δπ , δK , δσ , and δair are the uncertainties on pion production,
kaon production, hadronic interaction cross sections and atmospheric density profile respectively. The bottom plot is the calculated uncertainty on
the absolute normalization of the neutrino flux. . . . . . . . . . . . . . . . 162

List of Figuresxviii

9.16 These plots show the ratio of the CCQE cross section calculated by the
Nieves model to that calculated by the fermi gas model used in SK. The
top left plot displays νe + ν̄e and νµ + ν̄µ in the solid and dashed lines
respectively. The top right plot shows the ratio (νµ + ν̄µ )/(νe + ν̄e ), andthe bottom plot is ν̄/ν. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
9.17 Projected χ2 plots for the mass hierarchy sensitivity. The true hierarchy istaken to be normal, and the black and green lines represent fits assuming
the normal and inverted hierarchy respectively. The minimum ∆χ2 valueof the inverted hierarchy at ∆χ2 = 1.87 represents our sensitivity to themass hierarchy in this case. . . . . . . . . . . . . . . . . . . . . . . . . . . 168
9.18 Assuming the normal hierarchy, these plots show fitted χ2 values forpoints across the oscillation parameter space. The minimum value of χ2
is subtracted from every point. From top right going clockwise the plots
show: sin2 θ23 − ∆m223 contour plot, minimum ∆m223 , minimum δCP, andminimum sin2 θ23 . The coloured lines on the plots represent the confidence level contours, 99% (orange), 90% (blue) and 68% (red). In the 1dplots, these correspond to ∆χ2 = 6.6, 2.7 and 1.0 respectively. In the 2dcase, these contours correspond to ∆χ2 = 9.2, 4.6 and 2.3. The black linein the 1d plots represents the projected minimum χ2 values. The dataset
is all of SK-I to IV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
9.19 Assuming the inverted hierarchy, these plots show fitted χ2 values forpoints across the oscillation parameter space. The minimum value of χ2
is subtracted from every point. From top right going clockwise, the plots
show: sin2 θ23 − ∆m223 contour plot, minimum ∆m223 , minimum δCP, andminimum sin2 θ23 . The coloured lines on the plots represent the confidence
level contours, 99% (orange), 90% (blue) and 68% (red). The black linein the 1d plots represents the projected minimum χ2 values. The data-set
is all of SK-I to IV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
9.20 The data-MC agreement for each sub-sample, after fitting, with combined
data from SK-I to IV. The oscillation parameters used can be seen in table
9.9. Multi-GeV refers to the Multi-GeV 1-ring sample, and Multi-Ringrefers to the Multi-GeV Multi-Ring sample. The inverted hierarchy best
fit is in red, and normal hierarchy in green (these are mostly overlaid andcannot be distinguished). The 4 samples on the left are not divided into
zenith bins during analysis, so they are shown in momentum bins. The
remaining 4 columns of samples, are divided into momentum and zenithbins in analysis, but here only the projection onto zenith bins is shown. . 172
9.21 Data vs MC for the new ν-like, ν̄-like, and Multi-Ring Other samples.
The dataset shown includes all of SK-I to IV. The MC is oscillated using the best fit parameters shown in table 9.9. Multi-GeV refers to theMulti-GeV 1-ring samples, and Multi-Ring to the Multi-GeV Multi-Ringsamples. The x-axis shows the zenith angle cos θ, where cos θ = -1 represents upward-going events. The blue line represents the normal hierarchy
expectation, and the red line the inverted hierarchy. The momentum
shown at the top of each plot is the lower end of the momentum bin. . . . 173
9.22 Comparison of the 90% confidence level contour between the current SK
official analysis (green) and analysis with the updated samples presented
in this thesis (blue). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

List of Figuresxix

9.23 The contribution to ∆χ2 (NH-IH) for each sample in the analysis. The
samples favouring normal hierarchy are primarily the Multi-GeV 1-ringe-like samples, and the Multi-GeV Multi-Ring Other sample. . . . . . . . 175
9.24 Comparison of the 90% confidence level contours between the analysis
presented in this thesis (blue) and published results from the MINOS(green) [31] and T2K (red) [32] experiments. . . . . . . . . . . . . . . . . 175
9.25 Expected future significance of neutrino mass hierarchy in Super-Kamiokande.
The green and red dotted lines indicate 1σ and 2σ significance respectively. The oscillation parameters are as described at 9.7. . . . . . . . . . 176
9.26 Expected significance of neutrino mass hierarchy in the NOνA experiment, after 3 years each of νµ and ν̄µ beam. The blue (dashed red) curves
show the significance assuming the normal (inverted) hierarchies. Plottaken from [33]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
9.27 Expected sensitivity to the neutrino mass hierarchy of the proposed INO
experiment. The blue shaded region represents different values for truesin2 θ13 . Plot taken from [34]. . . . . . . . . . . . . . . . . . . . . . . . . . 178
9.28 Expected sensitivity to the neutrino mass hierarchy of the Hyper-Kamiokande
experiment. The different colours and shaded regions represent the expected hierarchy sensitivity given different true values of other neutrino
oscillation parameters. Plot taken from [35]. . . . . . . . . . . . . . . . . . 179
9.29 Expected mass hierarchy sensitivity of various future experiments. The
widths represent uncertainty due to oscillation parameters. Plot takenfrom [36]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

List of Tables
3.1
3.2
3.3

A summary of the differences between each of the SK periods . . . . . . . 25
Trigger thresholds for each trigger type in SK-I to III . . . . . . . . . . . . 32
Trigger information for SK-IV . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.1

Absolute gain conversion factors between single photo-electron events,and output PMT charge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

6.1

Vertex and angular resolutions for MS-Fit reconstruction of fully contained neutrino events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

7.1

Summary of the efficiencies, remaining background, and number of events
remaining after the FC reduction process. . . . . . . . . . . . . . . . . . . 93
PC Summary Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Approximate events remaining after each UPMU reduction step. . . . . . 114
Summary of the remaining events, and background subtraction for thedata sample remaining after UPMU reduction . . . . . . . . . . . . . . . . 115

7.2
7.3
7.4

8.1

8.2
8.3

8.4

9.1
9.2
9.3

The differences in neutron tagging efficiency, and training time, for eachneural network layout that was tested. The efficiencies shown are relative
to the result given by the 16:14:7:1 architecture. . . . . . . . . . . . . . . 133
Final efficiency and background rate of the neutron tag algorithm, after
each stage of selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
A comparison of the expected and measured neutron capture events in theatmospheric neutrino SK-IV data-set (1775.6 days). Most of the data-MC
discrepancy is from very high energy (>30 GeV events). The MC is normalized by data livetime, and oscillated under a 2 flavour approximation:∆m2 = 2.5 × 10−3 and sin2 2θ = 1.0. . . . . . . . . . . . . . . . . . . . . . 136
Estimated efficiency of neutron tagging, using Am-Be data, compared tothe expectation from a 2.2 MeV γ-ray MC. The efficiencies here are higher
than those quoted previously, as the neutron capture vertex is now known
to < 20 cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Summary for selection of Fully Contained Sub-GeV event samples . . . . 145
Summary for selection of Fully Contained Multi-GeV event samples . . . 146
Multi-GeV 1-ring e-like sample, split into νe -like and ν̄e -like. The top
half of the table shows the estimated event breakdown of each sample,
constructed with MC information. The bottom half shows a total dataMC comparison. The data used is all of SK-I to IV. . . . . . . . . . . . . 153

xx

List of Tables
9.4

xxi

Multi-GeV Multi-ring e-like sample, split into νe -like and ν̄e -like. The top
half of the table shows the estimated event breakdown of each sample,
constructed with MC information. The bottom half shows a total dataMC comparison. The data used is all of SK-I to IV. . . . . . . . . . . . . 154
9.5 Multi-GeV 1-ring µ-like sample, split into νµ -like and ν̄µ -like. The top
half of the table shows the estimated event breakdown of each sample,
constructed with MC information. The bottom half shows a total dataMC comparison. The data used is all of SK-I to IV. . . . . . . . . . . . . 155
9.6 Multi-GeV Multi-ring µ-like sample, split into νµ -like and ν̄µ -like. The
top half of the table shows the estimated event breakdown of each sample,
constructed with MC information. The bottom half shows a total dataMC comparison. The data used is all of SK-I to IV. . . . . . . . . . . . . 157
9.7 Chosen “true” oscillation parameters for the mass hierarchy sensitivity
study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
9.8 Improvement in sensitivity to neutrino mass hierarchy due to the new ν-ν̄
samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
9.9 Oscillation analysis best fit point for normal and inverted hierarchy assumptions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
9.10 Oscillation analysis 90% confidence levels for normal and inverted hierarchy assumptions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

To my brotherxxii

Chapter 1

Introduction
Of the particles predicted by the Standard Model, the neutrino is undoubtedly among
the most elusive. As leptons with no charge or mass, interacting with their surroundings
only via the weak force, it is a considerable challenge to observe these particles at all.
After neutrino mixing was observed for the first time in the 1990s, this experimental
challenge was extended to a theoretical one - the otherwise well validated Standard Model
could not justify neutrino mixing, unless the neutrinos had non-zero mass. Even today,
while many important neutrino mixing parameters have now been precisely observed,
still mysteries remain - what is the absolute mass and mass hierarchy of neutrinos?
Can we see evidence of CP violation in the lepton sector, and possibly account for thematter-antimatter asymmetry of our universe? It is an exciting time to be a part of theneutrino physics community.

1.1

Thesis Overview

In chapter 2, basic neutrino physics will be explained, including a brief history of neutrino
experiments, the current state of the art, and neutrino oscillation theory. The maincontent of this work relates to atmospheric neutrinos, which are explained here, as well
as some overview of other prominent neutrino sources. The reader may thus gain the
background necessary to understand the topics presented in later analysis.
Chapter 3 describes Super-Kamiokande, the light water Cherenkov detector used tocollect data for this analysis, and chapter 4 goes into the work done to calibrate thisdetector. The various systems that come together to make the detector functional are
explained - basic Cherenkov detector theory, photo-multiplier tubes, the water system,
detector electronics, and others.
1Chapter 1. Introduction

2

Chapter 5 covers neutrino event simulation, including the initial atmospheric neutrino
flux prediction, neutrino interaction generation, and the simulation of the detector.
In chapter 6 are the reconstruction procedures, the methods by which the timing andcharge information recorded by the photo-multiplier tubes are reconstructed into energy,
direction, neutrino flavour.
The processes used to reduce the large amount of data observed, into a more manageablesample of pure neutrino events, is detailed in chapter 7.
In chapter 8, one of the focuses of this thesis is introduced - neutron tagging. Applicationand development of this technique to be applied to atmospheric neutrinos is discussed.
Finally, chapter 9 contains descriptions of the new anti-neutrino data samples which canbe constructed by use of neutron information. Systematic errors and analysis methods
are described, and then full neutrino oscillation analysis and sensitivity study are performed. The best fit neutrino mixing parameters are calculated here. The thesis is thenconcluded in chapter 10.

1.2

Author Contributions

Super-Kamiokande is a well established experiment, having been in operation for almost
20 years now. Thus, many of the analysis techniques are already mature, and it isimportant to identify my (the author’s) personal contribution to the experiment. I
have been responsible for updating the Fully-Contained neutrino dataset and reduction
process, performing maintenance and updates that were required during my time here.
I also worked to upgrade the atmospheric neutrino flux calculation.
The bulk of my contribution to the experiment, is in developing the neutron tagging
software that may effectively identify neutrons, as well as Monte-Carlo modifications
to correctly simulate and accurately predict neutron capture events. The basis for thisstudy was pioneered by Haibing Zhang for study of relic supernova neutrinos, however
this is the first time to apply and develop this technique for >100 MeV analyses. My
software is currently being used by other members of Super-Kamiokande, for proton
decay studies, and other high energy analysis, as well as preliminary work for usage in
T2K.
For neutrino oscillation analysis, my contribution mainly consists of improvements to
the Multi-GeV ν − ν̄ separation, by use of neutron information and other techniques.
Improved separation of ν and ν̄ events increases the detector’s sensitivity to neutrinomass hierarchy, a prevalent topic in neutrino physics at this time.

Chapter 2

Neutrinos
2.1

A History of Neutrinos

2.1.1

Neutrino Flavours

The first prediction that a neutrino may exist, was made by Wolfgang Pauli, in December 1930. The problem which prompted this prediction, was observation of β-decay
interactions, whereby a neutron decays to a proton inside a nucleus, emitting an electron,
for example:
14
14−
6 C→7 N+e .

(2.1)

Of course, the electron produced was expected to be mono-energetic, similar to gamma
or alpha-producing decays, however a continuous energy spectrum was observed [37].
This led Pauli to postulate that an additional, charge-less particle was being producedin this decay, sharing some of the energy of the electron, without being observed. Thisparticle was initially named ”neutron”, however after what we now know as a neutron
was observed by Chadwick in 1932 [38], Enrico Fermi distinguished this mysterious
particle by renaming it a neutrino [39], the appellation which has stuck to this day.
The first direct observation of neutrinos however, did not come until 1956, when Fred
Reines and Clyde Cowan measured electron anti-neutrinos via the inverse β-decay interaction
ν e + p → e+ + n(2.2)

in a water tank. The e+ annihilated, producing two γ-rays in the water, which were amplified by surrounding organic scintillator, and could be observed. In addition, cadmium

3Chapter 2. Neutrinos

4

was added to the water, increasing the efficiency of neutron capture events. These neutron capture events would produce a further, delayed γ-ray, and requiring observationof this in addition to the primary signal allowed background to be greatly reduced.
At this point there were only two known charged leptons, the electron and the muon,
along with their corresponding anti-particles. However, it was still unknown whether
both these leptons could produce different neutrinos, or if the anti-neutrino was distinct
from the neutrino.
To solve the latter of these problems, a similar inverse beta decay experiment was conducted by Ray Davis [40], attempting to observe the interaction
ν + 37 Cl → 37 Ar + e−

(2.3)

from neutrinos produced by a nuclear reactor. This interaction is perfectly acceptablefrom a charge-conservation perspective, however in hindsight we now know that reactors
produce ν e , when νe is required to conserve lepton number in this interaction. So
unfortunately, Davis was unable to observe the desired interaction. However, due tothis work and subsequent experiments by Davis, we are able to establish that neutrino
and anti-neutrino are separate entities, and that lepton number must be conserved in
weak interactions.
The remaining uncertainty then, is whether the neutrino associated with an electron,is the same as the neutrino associated with a muon. This was studied using a π-decay
based accelerator experiment, in Brookhaven, 1962, led by Lederman, Schwartz andSteinberger [41]. Charged pions dominantly decay via the mode:π → µ + νµ .

(2.4)

Decay to electron is possible, but heavily suppressed due to helicity. Therefore using
15GeV protons fired at a Beryllium target, charged pions were produced, which decayed
into νµ , creating a beam of these neutrinos. From previous reactor experiments, we
already know that a neutrino can cause an inverse β-decay reaction, and produce anelectron. Therefore if νµ = νe , we expect electrons to be observed when this νµ beaminteracts in a detector.
However, as we now know, this was not the case, and the experiment detected muons
being produced as a result of the beam interaction, thus we may conclude that νµ , andνe are two distinct particles, and that lepton family number must be conserved in theirinteractions.

Chapter 2. Neutrinos

5

The last piece of the puzzle, ντ , was not discovered until much later. The τ lepton itself
was revealed by Martin Perl et al., using an e+ − e− collider at SLAC in 1974-1977[42]. ντ was finally detected in 2000, by the DONUT collaboration, from study of theTevatron (Fermilab) fixed target proton beam experiment[43].
So, up until now, it appears that we have a consistent model of neutrino physics, andthe nature of these chargeless, massless, lepton number conserving particles is well understood. However, this is not all there is to this story.

2.1.2

The Solar Neutrino Problem

The Standard Solar Model[44] (SSM), as calculated by John Bachall, posits that the
Sun generates its energy through processes of nuclear fusion, primarily the fusion ofhydrogen atoms, to create helium. This fusion interaction can occur in two ways, thecarbon-nitrogen-oxygen (CNO) cycle, or the far more dominant proton-proton (pp) chain(accounting for ∼ 98.4% of the Sun’s total luminosity). In the pp-chain, five distinct
reactions produce neutrinos of various energies:

p + p → 2 H + e+ + νe (< 0.42M eV )p + e− + p → 2 H + νe (1.442M eV )7

Be + e− → 7 Li + νe (0.861, 0.383M eV )8

3

B → 8 Be* + e+ + νe (< 14.06M eV )He + p → α + e+ + νe (< 18.77M eV ).

Neutrinos are also produced in the CNO cycle, however the resultant flux is insignificant
compared to the pp-chain. The SSM is constrained by the luminosity and radius of theSun, as well as the observed ratio of heavy elements on the Sun’s surface [45], and it isgenerally well accepted.
The Homestake experiment [46] was conducted in the 1960s, led by Ray Davis, to measure this flux of νe coming from Sun, and validate Bachall’s SSM. The experiment was
conducted similarly to his previous reactor neutrino experiment, again utilising the reaction:
νe + 37 Cl → 37 Ar + e− .

(2.5)

This time, it was a success, as the Sun does produce neutrinos, rather than antineutrinos. The experiment consisted of 615 tons of C2 Cl4 , in a tank located in theChapter 2. Neutrinos

6

Figure 2.1: Energy spectrum for neutrinos produced in the pp chain, predicted by
Bachall’s Standard Solar Model

Homestake mine, South Dakota. The 37 Ar produced is radioactive, with a half-life of
35 days, and could be siphoned out of the tank in bubbles of Helium gas. The quantity
could then be ascertained by counting radioactive decays of the extracted argon, thus
inferring the number of initial electron neutrino interactions.
However a problem arose. The number of νe detected by the Homestake experiment,
was only around one third of the number predicted by the SSM [47]. Extensive investigation was performed, questioning both the efficiency of the detector, and the validityof the SSM, however nothing was found that could explain this discrepancy. Homestake
continued running until 1994, consistently measuring the same deficit of neutrinos compared to prediction. A similar deficit was later observed by Kamiokande in 1989, by the
GALLEX collaboration in 1992 [48], and at the SAGE experiment in 1994 [49].
Therefore we are led to the conclusion, that something must be wrong with the underlying particle physics model. The neutrinos are undergoing some unknown process which
reduces the number of νe . Perhaps, as suggested by Pontecorvo in 1969, the neutrinos
are oscillating between states [50]. While we now think this to be true, other explanations to the solar neutrino problem are possible, such as neutrinos decaying mid-flight[51]. More information is needed to confirm the cause of this anomaly.

Chapter 2. Neutrinos

7

Figure 2.2: Hadronic shower and atmospheric neutrinos produced by cosmic rays.
Figure from Los Alamos Science

2.1.3

The Atmospheric Neutrino Anomaly

The solar neutrino problem was later compounded by confusing measurements fromanother dataset - atmospheric neutrinos. So called as they are neutrinos produced bycosmic rays interacting in the atmosphere. A cosmic ray (typically a proton) enters
our atmosphere, interacting with nuclei and creating a hadronic shower. Amongst thishadronic shower are many charged pions, which decay to muons, and then to electrons,
producing multiple neutrinos in the process:

π + → µ+ + νµ

(2.6)µ+ → e+ + νe + ν µ(2.7)

Chapter 2. Neutrinos

8along with the corresponding interactions for π − . Therefore our prediction for the ratio
R = (ν µ + νµ )/(ν e + νe ) is ∼ 2 for neutrinos < 1GeV . For high energy cosmic rays,
some of the muons may reach the surface of the earth before decaying, so this ratio will
be increasing with energy.
The Kamiokande experiment, a 3000 ton underground water Cherenkov detector, wasoriginally designed to search for proton decays - a similarly rare (still unobserved) phenomenon. This required a large volume of matter, and very low background, to have achance at observation. Coincidentally, these are also the qualities required for an effective neutrino detector, therefore Kamiokande was propelled into the limelight in 1987
for detecting neutrinos from the 1987a Large Magellanic Cloud supernova [52].
Kamiokande was also able to detect atmospheric neutrinos, reporting the above ratio
R = (ν µ + νµ )/(ν e + νe ), in the form Rdata /RM C = 0.60+0.07−0.06 (stat) ± 0.05(syst)[53].
Interestingly, this deficit of muon neutrinos had a strong energy and zenith angle dependence [54], suggestive of some travel distance dependent oscillation or decay effect.
A similar discrepancy was detected by IMB (another proton-decay water-Cherenkov detector) [55], and Soudan-2 (an iron tracking-calorimeter experiment), leading this to be
accepted as the ”atmospheric neutrino anomaly”.

2.1.4

The Solution

The cause of these neutrino anomalies could not be identified, until the successor to the
Kamiokande experiment, ”Super-Kamiokande” (SK), was constructed in 1996. As the
topic of this thesis, this experiment will be described in detail in the following chapters,
but for now it suffices to say that it is a large water Cherenkov detector, allowing muchgreater sensitivity to physics parameters than its predecessor.
Using a distribution of the ratio of reconstructed path length (L) to reconstructed energy(E), using only events where this reconstruction is possible at high resolution (> 70%),
SK was able to observe a dip corresponding to the first maximum of neutrino oscillation
probability [56]. This was the first direct evidence for atmospheric neutrino oscillation(in this case νµ ↔ ντ ) as the favoured explanation for the atmospheric neutrino anomaly.

Concurrently, the Sudbury Neutrino Observatory (SNO), announced direct observationof solar neutrino mixing [57][58] (SK measurements were also included to obtain thisconclusion). SNO is a heavy water (D2 O) Cherenkov detector, designed to solve thesolar neutrino problem by measurements of νe flux independently from the total neutrino
flux.

Chapter 2. Neutrinos

9

Figure 2.3: Ratio of the data to the MC events as a function of L/E, from SuperKamiokande. The solid black histogram represents the best fit for oscillation, whilst the
disfavoured blue dashed, and dotted red lines are the best fit expectations for neutrino
decay, and neutrino decoherence respectively.

This is accomplished due to the differing nature of charged current (CC), and neutral
current (NC) interactions in heavy water. In this context a CC interaction (which
exchanges a charged W boson)νe + d → p + p + e−

(2.8)

only occurs for electron neutrinos, as solar neutrinos (∼ 10M eV ), regardless of flavour,are too low energy to produce a muon (∼ 106M eV /c). The electron from CC interactions
then produces Cherenkov light, and can be observed by the PMTs surrounding thedetector.
By contrast a NC interaction (exchanging a neutral Z boson), can occur for any neutrino
flavour:

νx + d → p + n + νx .

(2.9)

Following this the produced neutron will be captured and emit a gamma ray, providing
a distinct signature compared to the CC interaction. Thus the total number of neutrinos
can be measured independently from the number of electron-neutrinos. You may haveguessed by now, the results of SNO were consistent with νe ↔ νµ /ντ oscillation, and thesolar neutrino problem could also be solved, using a single consistent theory.

Chapter 2. Neutrinos

10

2.2

Neutrino Oscillation Theory

2.2.1

Oscillations in a Vacuum

Neutrino oscillations are based on the idea that the neutrino mass eigenstates do notexactly match the weak interaction (flavour) states, but instead each flavour state isconstructed by a combination of different mass eigenstates. That is to say, flavour states|να i (α = e, µ, τ ), may be expressed as a superposition of three mass eigenstates |νi i,with mass mi (i = 1, 2, 3) [59].

|να i =

3X∗
Uαi
|νi i ,

(2.10)

i=1

where U represents the rotation matrix between these states. If U is not diagonal,
then each flavour state will be a mixture of multiple mass states. This rotation matrix
is known as the Pontecorvo-Maki-Nakagawa-Sakata (MNS) mixing matrix [60]. The
total number of neutrino flavours is strongly limited to 3, by measurements of the Zboson total width [61]. Thus the matrix U may be characterized by only 4 independent
parameters:


1
0
0
C13





U = 0 C23 S23  
00 −S23 C23−S13 eiδcp

0 S13 e−iδcp



1

0

0

C13


 −S12 C12

0
0

C12

S12


0

0
,
1

(2.11)

where Cij = cos θij and Sij = sin θij represent the mixing between states i and j, and δcp
is the charge-parity (CP) symmetry violating phase (δcp 6= 0 implies that CP violation
is present).
You may consider a neutrino produced by a lepton capture interaction:α− + p → n + να .

(2.12)As the neutrino was produced in a weak interaction, its state at the instant of creation
is known be one of the three flavour eigenstates, να .
The time propagation of a particle is governed by the Schrödinger equation, and if we
assume propagation through a vacuum, may be simply solved as:|νi (t)i = e−iEi t |νi (0)i .

(2.13)

Chapter 2. Neutrinos

11

However recall that Ei =q
p2 + m2i , so if a neutrino is made up of a superposition ofmass states, they will each have an independent time evolution. Therefore to understandthe state of our neutrino after some time t, we must first re-write it in its mass states:|να (t)i =

3
X

∗ −i
Uαi
e

√2

p +m2i t|νi i .

(2.14)

i=1

This gives us the time evolution of the neutrino. However, as a neutrino must interact
via the weak force, we must recast this equation in the flavour basis to understand ourobservations. The amplitude of finding the neutrino να in flavour state νβ after time t
is thenAνα →νβ (t) = hνβ |να (t)i =3X∗ −iEi tUαiehνβ |νi i

i=1

=

3 X
3
X

∗
Uαi
Uβj e−iEi t hνj |νi i

i=1 j=1

=

3
X

∗
Uαi
Uβi e−iEi t .

i=1

The transition probability can be expressed asPνα →νβ (t) = |Aνα →νβ (t)|2 =

3 X
3
X

∗
∗ −i(Ei −Ej )te.
Uαi
Uβi Uαj Uβj(2.15)

i=1 j=1

So you can see, if the energy of any two neutrino mass states is different we will be able
to see oscillatory effects in the transition probability. However it is still quite difficult tounderstand in this form. q
If we assume that the neutrinos are relativistic, we may makethe approximation Ei =p2 + m2i ' p + m2i /2p ' p + m2i /2E.

Pνα →νβ (t) =

3 X
3
X

∗
∗ −i
Uαi
Uβi Uαj Uβj
e

∆m2
ij
2E

t

(2.16)

i=1 j=1

where ∆m2ij = m2i − m2j . Typically it is more useful to express this in terms of the
distance travelled L by the neutrinos, rather than time t

Pνα →νβ (L) =3 X
3X
i=1 j=1∆m2ij L∗∗Uαi
Uβi Uαj Uβj
exp −i2E!
.

(2.17)

Chapter 2. Neutrinos

12

So, just from this equation, you may form an understanding of the physics underlying
neutrino oscillation. For oscillation to exist, it is necessary for flavour states to becomprised of a superposition of different mass states, where ∆m2ij 6= 0. Also, you may
Lnote that the oscillation probability depends on E
, which may be helpful to explain someissues from the previous chapter -the atmospheric neutrino deficit that only appears in
upwards travelling neutrinos, which have a larger L than their downward travellingcounterparts, thus some oscillation effects have been revealed.

Figure 2.4: This figure shows the oscillation probability of νe → νe , for a 3MeV
neutrino. It may be thought of as the probability that an initial electron neutrino
will be observed as an electron neutrino after travelling distance L. For this plot,∆m223 = 2.4 × 10−3 eV 2 , and ∆m212 = 7.6 × 10−5 eV 2 . The high frequency oscillations
are due to ∆m223 , and if a detector is placed at 1km then you may approximate themixing matrix to its 2-dimensional form considering only this mixing. Likewise, for the
low frequency oscillations, the driving factor is ∆m212 , so to optimally measure this, adetector may be placed at 50km, and again the mixing matrix may be reduced to an
oscillation between these two states.

In some cases, it is possible to simplify mixing matrix U into an oscillation between
just two mass states. This is possible as the frequency of the oscillations is essentiallydetermined by ∆m2ij , so for example, if ∆m223 is significantly larger than ∆m212 , theoscillations caused by ∆m212 will require a larger L/E to be observed. It so happens,that ∆m212 is of an appropriate magnitude that large values of L/E, corresponding
to solar neutrinos, are optimal for viewing the oscillation effect. Similarly, ∆m223 isChapter 2. Neutrinos

13

the dominant factor in oscillations for L/E values similar to that which we find inatmospheric neutrinos. Hence it is often suitable in these cases to simplify the problem
to that of two flavour oscillation.

U=

cos θ

!

sin θ

− sin θ cos θ(2.18)

Here U becomes the familiar 2d rotation matrix, and in a similar manner to above, it
may be shown that the transition probability becomes

P (να → νβ ) = sin2 2θ sin

∆m2 L
.
4E

(2.19)

This simple case demonstrates that both the ∆m2 term and mixing angle θ must be
non-zero for oscillations to occur. To avoid confusion, it is worth remembering that θ isa constant, and sin θ is not the term causing the oscillation effect. The goal of neutrino
oscillation experiments then, is to measure the different parameters which control thiseffect: ∆m2 , θ, and ∆cp .

2.2.2

Oscillations in Matter

Until now, we have been operating under the assumption that neutrinos move only
through vacuum. However, as our detectors are generally built very close to a large
amount of matter, we cannot always rely on this assumption. Neutrino transport throughmatter was first explored by Wolfenstein in 1978[62], and elaborated on by Smirnov andMikheyev [63].
Neutrino scattering in matter can occur via a neutral current Z boson, in which caseall neutrino flavours interact in the same way, so this just results in an additional common phase factor in the propagation Hamiltonian. Interactions may also take place via
charged current bosons (W ± ), but due to the high density of electrons in matter, aflavour asymmetry is introduced, enhancing electron neutrino scattering in a forward
direction. The effective potentials of the different neutrino flavours are
Ve = VCC + VN C

(2.20)

Vµ,τ = VN C ,(2.21)

where the charged and neutral current potentials are√
VCC = ± 2GF ne(2.22)

Chapter 2. Neutrinos

14√
VN C = ∓2
GF n n .
2(2.23)

GF is the weak coupling constant, and ne,n are the number densities of electrons andneutrons in the surrounding material. The sign flips for anti-neutrinos.
If we take a simple 2-flavour mixing case, recall that the time evolution of the massstates is given by the Schrödinger equation

ν1

d
i
dt

!
=

ν2

E1

0

0

E2

!

ν1
ν2

!
.

(2.24)

Converting this into the flavour basis

d
i
dt

νe

!

νµ

=U

E1

0

0

E2

!
U

νe

†

νµ

2

!
=

− ∆m
4E cos 2θ
∆m2
4E sin 2θ

∆m2
4E sin 2θ
∆m2
4E cos 2θ

!

νe
νµ

!
,

(2.25)

where ∆m2 , E, and θ are as defined in the previous section. The common phase is notincluded here, as it has no effect on neutrino oscillations. To add the effect of neutrino
oscillations in matter, we must include the potentials defined above. VN C is a common
term applied to the diagonal components for both νe and νµ , thus may be taken asa phase shift, and safely ignored. The extra VCC term must be added for νe → νescattering.

d
i
dt

νe
νµ

2

!
=

− ∆m
4E cos 2θ ±√2GF ne∆m2
4E sin 2θ

∆m2
4E sin 2θ
∆m2
4E cos 2θ

!

νe
νµ

!
(2.26)

We may convert this back to the standard mixing matrix form of (2.25) by making thesubstitutions∆m2M = ∆m2
andsin2 2θM =
where

q

sin2 2θ + (cos 2θ − ACC /∆m2 )2(2.27)

sin2 2θ
,sin2 2θ + (cos 2θ − ACC /∆m2 )2(2.28)√
ACC = ±2 2GF ne E.

(2.29)

Chapter 2. Neutrinos

15

You may note that as ne → 0, these formulae return to that of the vacuum. A resonanceoccurs whenACC
= cos 2θ∆m2

(2.30)

is satisfied, representing the maximal mixing for neutrinos in matter. This is known as
the Mikheyev-Smirnov-Wolfenstein (MSW) condition (the effect of matter on neutrino
oscillation is also known as the MSW effect, after the same three scientists). The matter
effect also allows us sensitivity to the sign of ∆m2 . However, as the sign of ACC changes
dependent on whether the neutrino is an antiparticle, this resonant enhancement will
occur in either anti-neutrino or neutrino, depending on the mass hierarchy. A full threeflavour matter effect has been calculated [64], however the math is considerably more
involved.

Figure 2.5: This figure represents the difference between probability of oscillation
νe ↔ νµ , between normal and inverted mass hierarchies, for neutrinos incident on
Super Kamiokande. The y-axis is zenith angle, such that the upper half of the plot
represents downward going neutrinos, and the bottom half upward going neutrinos. The
dominant area of resonant enhancement can be seen around 2 − 10GeV . If we instead
look at anti-neutrino oscillation, the signs in this plot are switched, so the enhancement
occurs in inverted, rather than normal hierarchy.

2.2.3

CP Violation

C and P refer to the transformations of charge conjugation (particle ↔ anti-particle),and parity (inversion of particle position). Symmetry of C and P may be either conserved
or violated, dependent on whether these quantities are conserved in particle interactions.

Chapter 2. Neutrinos

16

Figure 2.6: The two possibilities for neutrino mass states, known as the normal andinverted hierarchies. The green forward-slash fill represents νe , red back-slash is νµ , andvertical blue lines is ντ . The difference between the mass states ∆m2 is well known,
thanks to neutrino oscillation experiments, but at the time of writing, which mass stateis the heaviest is still unknown. This plot is not to scale.

CP violation therefore, refers to the symmetry-breaking of the quantity C × P . Thisquantity is particularly interesting to measure, as it may provide an explanation to theoutstanding problem of matter-antimatter asymmetry in our universe.
In terms of measurable quantities, CP invariance will be observed, only if the neutrino
mixing matrix U is real, that is, U ∗ = U [65]. To rephrase this, an off-diagonal non-zero
imaginary phase δ would cause CP to be violated. The magnitude of the CP violationin neutrino mixing may be expressed as∗∗
JCP = Im(Uµ3 Ue3
Ue2 Uµ2)=

1
cos θ13 sin 2θ12 sin 2θ23 sin 2θ13 sin δCP .
8(2.31)

CP violation in the quark sector is well established, having been measured in neutralkaon decay from 1964 [66], and later B (bottom quark) decay [67]. This stems from theimaginary phase in the analogous CKM (Cabibbo-Kobayashi-Maskawa) quark mixing
matrix [68]. Recent discovery of non-zero θ13 allows for potential CP violation measurement in the neutrino sector, however at present no experimental information is available.

2.3

State of the Art

Assuming the above oscillation model is correct, the next task is to measure all theparameters of neutrino mixing, as accurately as possible. This includes all θij mixing
angles, ∆m2ij mass splitting, and δCP CP violation phase. As stated previously, most,Chapter 2. Neutrinos

17

but not all measurable predictions made by the oscillation model are insensitive tothe sign of ∆m2 , therefore it is potentially possible to measure the mass hierarchy ofneutrinos. Here the best measurements of these quantities to date will be discussed.

2.3.1

Solar Parameters θ12 , ∆m212

These are the parameters that primarily control the oscillation of long baseline, lowenergy neutrinos, which were originally observed in solar neutrino data.
Solar neutrino flux has now been measured in many experiments - radiochemical detectors, such as Homestake, SAGE and GALLEX, the water Cherenkov detectors Kamiokande
and Super-Kamiokande, and heavy water detector SNO. The liquid scintillator Borexino experiment has recently pioneered measurements of low energy solar neutrinos,
sensitive down to 250keV, allowing observation of the 0.862 MeV 7 Be neutrinos φ =3.10 ± 0.15 × 109 cm−2 s−1 [69].
For the 8 B neutrino flux, the SNO experiment, through CC, NC and ES observations,provides results for both the νe flux φνe = 2.35 ± 0.22 ± 0.15(106 cm−2 s−1 ) [1], and the6−2 −1
total solar neutrino flux φall ν = 5.25 ± 0.16+0.11−0.13 (10 cm s ) [70]. Super Kamiokande

is able to observe ES interactions, measuring direction sensitive electron neutrino flux
2.32 ± 0.04 ± 0.05(106 cm−2 s−1 ) [71]. Borexino also provides a νe 8 B measurement of
φ = 2.4 ± 0.4 ± 0.1(106 cm−2 s−1 ).
The KamLAND experiment also provides important data when considering these mixing parameters. KamLAND is a 1-kTon liquid scintillator detector, located at the samesite as Super-Kamiokande in Japan. The observation target here is not solar neutrinos,
but the ν e produced by nuclear reactors. Reactor neutrinos have an energy spectrum< 8M eV , and are detected via the interaction ν e + p → e+ + n. The neutron is thencaptured on a proton, emitting a characteristic 2.2MeV γ-ray. This delayed coincidence
allows us to distinguish from the solar νe , which have a similar energy spectrum. KamLAND reported a deficit of ν e [72] events compared to no-oscillation expectation ofNobs − NBG= 0.611 ± 0.085 ± 0.041
NN oOscillation(2.32)

which confirmed the neutrino oscillation hypothesis. KamLAND collaboration produced
a combined fit for oscillation mixing parameters, including solar neutrino information,
along with up to date θ13 results from short-baseline reactor experiments [73]. The22−5
results give tan2 θ12 = 0.436+0.029−0.025 , |∆m21 | = (7.53 ± 0.18) × 10 eV .

Chapter 2. Neutrinos

18

Figure 2.7: This figure shows the combined result of 8 B solar neutrino flux measurements, from the salt phase of SNO [1], and ES result from SK in 2002 [2]. The dottedline represents the standard solar model prediction, and the contours represent a joint
fit between the two experiments. Figure source [1]

.

2.3.2

Atmospheric Parameters θ23 , ∆m223

High energy, long baseline neutrinos are sensitive to these parameters, which includes
atmospheric neutrinos and accelerator experiments.
As the topic of this thesis, Super Kamiokande and atmospheric neutrinos will be discussed in detail later. SK provides strong evidence for zenith angle dependent atmospheric νµ disappearance [74], and using an L/E binning scheme, favours the oscillation model over decay and decoherence hypotheses. Recently, ντ appearance has been
demonstrated at a 3.8σ level, using neural network analysis [75].
Accelerator neutrinos are also valuable to probe these mixing parameters. Utilizing awell known neutrino spectrum, and high statistics, accelerators can potentially provide
tight constraints. The first accelerator-based experiment was K2K (KEK to Kamioka),
constructed in Japan in 1999, running for a period of 5 years. K2K collided a 12GeV
proton beam with a target, producing pions, which would then decay to muons and νµ ,with an energy Eν ∼ 1.3GeV. These then travelled over a distance of 250km, to reach
the Super-Kamiokande detector. To better understand the neutrino energy spectrum,a near detector was used 300m from the beam target. K2K measured a deficit of νµ ,detecting 112 events, compared to the expectation of 158.1+9.2−8.6 with no oscillation disfavouring statistical fluctuation by 4.3σ [76].

Chapter 2. Neutrinos

19

Figure 2.8: On the top is the energy spectrum for events reconstructed as CC, at the
MINOS far detector. The bottom figure shows the ratio of data to a null oscillation
prediction, with the disfavoured decay and decoherence models also shown. [3].
Following this in 2005, MINOS began data taking. Using neutrinos produced by a120GeV proton beam at Fermilab, MINOS aimed to measure νµ disappearance in a
5.4kTon iron-scintillator in the Soudan mine, 735km away. Neutrino energy may betuned by varying horn current and target position of the beam, but most data wastaken with Eν ∼ 1 − 5GeV. MINOS results disfavour the decoherence and decay modelsof νµ disappearance, by 7σ and 9σ respectively [3]. MINOS was also the first experimentto observe ν µ disappearance [77]. By tuning the magnetic field of the beam line after
the target, it is possible to focus either positive or negative mesons (and thus νµ or ν µ )towards the far detector. Using all of this data, MINOS reported |∆m223 | = (2.41+0.09−0.10 ) ×
+0.03510−3 eV2 , and sin2 2θ23 = 0.950−0.036. MINOS is also able to oscillate νµ and ν µ events
2−3
independently, resulting in a best fit of |∆m223 | − |∆m223 | = (0.12+0.24−0.26 ) × 10 eV [78].

K2K’s successor, the T2K (Tokai to Kamioka) experiment, started data taking in 2010.
Utilizing an off-axis narrow-band 0.6GeV νµ beam, it is tuned to the first oscillation
maximum at SK, 295km away. Coherent with a νµ disappearance signal, 58 νµ -like eventswere observed at SK, compared to the expectation of 205 ± 17 for the no oscillation case

Chapter 2. Neutrinos

20[79]. Assuming θ13 and solar mixing parameters from other experiments, T2K best-fit
2−3
values give sin2 θ23 = 0.514 ± 0.082, and ∆m223 = (2.44+0.17−0.15 ) × 10 eV .

Figure 2.9: This plot shows 90% confidence level contours for T2K, MINOS, and SK
superimposed. Taken from [4].
A further experiment that should be noted in this category is OPERA. OPERA is thefirst accelerator experiment designed to measure ντ appearance, and runs from a muon
neutrino beam at CERN, over 730km to a detector at Gran Sasso. A high energy beam(Eν ∼ 17GeV) is required for the ντ to produce τ particles upon interaction. Upon
production, τ leptons quickly decay to µ or hadrons, giving a distinctive kink in the
track after a few millimetres. OPERA has thus far observed 3 ντ candidates, giving a
3.2σ significance over null observation [80].

2.3.3Reactor Experiments, θ13

The primary method of measurement for θ13 mixing parameters is by use of short baseline(∼ 1km), low energy (∼ 3MeV) reactor neutrino experiments. Nuclear reactors typically
produce many ν e , from β-decay of heavy particles in the fission decay chain. Over the
past two years, three similar reactor neutrino experiments reported observation of ν edisappearance, consistent with a non-zero θ13 . |∆m213 | is generally not reported, as ofcourse there are only two independent mass splittings between three different masses.
As |∆m212 |  |∆m223 |, we may infer that |∆m213 | ' |∆m223 |.
The Daya-Bay experiment in China, measured ν e from six nuclear reactors, with six identical Gd-doped liquid scintillator detectors arranged in 0-2km from the reactors. The

Chapter 2. Neutrinos

21

Gadolinium maximizes neutron capture efficiency, which is useful to reduce backgroundrates. These are surrounded by an outer (non-Gd loaded) scintillator detector. The
fiducial volume is thus precisely defined by requiring that events are followed by a corresponding neutron capture γ-ray signal. Multiple detectors are used to reduce the errors
due to nuclear reactor flux predictions. Daya-Bay reported a deficit in observed ν e in thefar detector, compared to the null oscillation prediction, of R = 0.944 ± 0.007 ± 0.003.
This fits to a mixing angle sin 2θ13 2 = 0.089 ± 0.010 ± 0.005, excluding null-oscillations
by 7.7σ [81].

Figure 2.10: This plot shows the result from Daya-Bay in 2012. The observed energy
spectrum, and ratio with the null-oscillation hypothesis are displayed on top and bottom
figures respectively. Taken from [5].
The RENO experiment, in Yonggwang Korea, is a similar setup, again with 6 nuclear
reactors producing neutrinos, which can be then measured by a near and far liquid
scintillator detector. In this case, the reactors are arranged symmetrically in a line, thus
a sufficiently small systematic error is obtainable with just two detectors, contrasting
Daya Bay’s six. In 2013, RENO reported fitted values of sin2 2θ13 = 0.100±0.010±0.015[82].

Chapter 2. Neutrinos

22

A similar experiment named Double Chooz is active in France, detecting ν e from twonuclear reactors. The construction of a near detector will be completed in 2014, but theresults from just a single far detector at 1050m give sin2 2θ13 = 0.109 ± 0.030 ± 0.025
after 227.9 days livetime [83].
While our most precise values for θ13 come from reactor experiments, the accelerators
mentioned previously can also search for this value, using the νe appearance channel.
However, as this appearance channel is also influenced by the yet unknown δCP , theuncertainty is higher on these measurements.
First came T2K, who presented initial results of νµ → νe analysis in 2011, reportingsix νe candidate events, above an expectation of 1.5 ± 0.3 for θ13 = 0. The most recent
result includes the observation of 28 νe appearance candidates, over a background of
4.64, excluding θ13 = 0 at 7.5σ [84]. MINOS also observed a νe appearance signal,
detecting 62 candidate events, over a θ13 = 0 expectation of 49.6 ± 7.0 ± 2.7. This may
be interpreted as a restriction on sin2 2θ13 < 0.12 at the 90% CL [85].
Solar and atmospheric neutrino experiments also have some sensitivity to θ13 , and SNO[86], SK [71], and KamLAND [73] have all published fit results consistent with thereactor and accelerator experiments.

2.4

Summary, Remaining Issues

Various global analyses have been performed, attempting to combine all of these results
in a set of best fit parameters. Here the results are presented from a 2012 analysis by
Fogll et. al. [87].
Parameter∆m221 (10−5 eV2 )|∆m232 |(10−3 eV2 )sin2 θ12
sin2 θ23
sin2 θ13

Best-fit
7.54+0.26−0.22
2.43+0.06−0.10
0.307+0.018−0.016
0.386+0.024−0.0210.0241 ± 0.0025

Much is as of yet undetermined in the neutrino sector. The nature of neutrinos as either
Majorana or Dirac particles (respectively whether the neutrino is its own antiparticle,
or not), is unknown. The absolute scale of neutrino mass is not well constrained. Of
particular interest to this thesis, are the CP violating phase δCP , and the sign of thelarge mass splitting parameter, ∆m32 . After the recent high precision measurements ofa relatively large θ13 , the possibility opens up for accelerator and atmospheric neutrino
experiments, such as SK, T2K and NOνA to measure these parameters. Measuring
these, is the objective of this thesis.

Chapter 3

Super-Kamiokande
3.1

Overview

Super-Kamiokande (SK) is a 50kTon water Cherenkov detector, located in Kamioka
mine beneath Mt. Ikenoyama, in Gifu Prefecture, Japan [88]. It is the successor of
the Kamiokande experiment, which was originally named after its location and purpose:”Kamioka Nucleon Decay Experiment”. After the experiment became famous for neutrino observation from supernova SN1987, the detector was often referenced colloquiallyas “Kamioka Neutrino Detection Experiment”, conveniently having the same acronym.
As the mine is horizontal, the detector is in fact 370m above sea level, however it is stillshielded by 1000m of rock in the mountain above. This causes a reduction of cosmic ray
muon background by ∼5 orders of magnitude, down to 2.2Hz. The detector is comprisedof a large cylindrical tank of light water (H2 O), split into an inner and outer detector(ID and OD) for additional reduction of extraneous background. Neutrinos interact inthe water, producing charge particles, which in turn produce Cherenkov light that canbe detected by the surrounding photomultiplier tubes (PMTs).
Water Cherenkov detectors benefit from a low energy threshold, allowing a large variety
of different analyses to be performed at SK. Atmospheric neutrinos, solar neutrinos,
supernova neutrinos, and artificial neutrinos from accelerators and nuclear reactors may
all potentially be detected. On top of this, the detector is very competitive as a probe
for various dark matter candidates, and search for different modes of proton decay.
Super Kamiokande and Kamioka Observatory are owned and operated by the Universityof Tokyo’s Institute for Cosmic Ray Research (ICRR), receiving the bulk of it’s funding
from Japan’s Ministry of Education, Culture, Sports, Science and Technology (MEXT),the U.S. Department of Energy and National Science Foundation. There are over 100
23

Chapter 3. Super Kamiokande

24

Figure 3.1: A schematic of the layout of Super-Kamiokande

collaborators to SK throughout the world, including scientists from Japan, Korea, USA,
Canada, the UK, China and Spain.

3.1.1

SK Periods

The operation of SK is typically split into four different periods, referred to as SK-I
to SK-IV. The first period of data-taking, SK-I, began in April 1996. The initial build
consisted of 11,146 PMTs, and 1,885 PMTs in the ID and OD respectively. SK-I ran
stably for 5 years, until April 2001, when the detector was stopped to replace faulty
PMTs and other maintenance. Regrettably, while re-filling the tank with water one of
the ID PMTs shattered, resulting in a shockwave that destroyed more than half of the
PMTs. To avoid a repeat of this incident, a fibre reinforced plastic and acrylic casewas used to protect the surviving PMTs. New PMTs were installed in the OD, andthe remaining ID PMTs were redistributed to allow even coverage of the tank, before
the detector was restarted in October 2002, as SK-II. During the SK-II running period,
replacement PMTs were manufactured, and in October 2005 the detector was stopped,
to mount these in the tank. The installation process carried on until July 2006, whendata taking resumed, beginning SK-III. The final stop of the detector in August 2008
was to upgrade the data aquisition system (DAQ) and electronics. The SK-IV datataking period began on September 2008, and continues to run to this day.

Chapter 3. Super Kamiokande
SK Period
Begin
End
Livetime (days)
No. ID PMTs
No. OD PMTs
Photo Coverage
Front-end Electronics
Trigger25
SK-I
Apr. 1996
Apr. 2001
1489.2
11,146
1,885
40%
ATM
Hardware

SK-II
Oct. 2002
Oct. 2005
798.6
5,182
1,885
19%
ATM
Hardware

SK-III
Jul. 2005
Aug. 2008
518.1
11,129
1,885
40%
ATM
Hardware

SK-IV
Sep. 2008
Ongoing1775.6
11,129
1,885
40%
QBEE
SoftwareTable 3.1: A summary of the differences between each of the SK periods

3.2

Detector Design

The Super-Kamiokande detector is primarily made up of a cylindrical tank, 41.4m tallwith a diameter of 39.3m. This tank is filled with 50kTon of ultra-pure water, and split
into an outer, and inner detector (OD and ID), which are concentric cylinders. The OD
extends ∼2m out from the ID, and acts as a veto for incoming cosmic ray muons. The
walls of the cavity are covered in 40-50cm of reinforced concrete, to reduce radioactive
backgrounds from the surrounding rock. A large dome above the tank is used to store
DAQ electronics, calibration equipment and other related items. The surrounding area
is covered in “Mineguard”, a polyurethane material used to reduce radon gas emanationfrom the rock wall.
The ID consists of 32kTon of water, surrounded by 11,146 inwardly facing 20 inch
PMTs. The PMTs are evenly distributed 70cm from each other, allowing for ∼40%
photo-coverage. A supporting frame separates the ID and OD, housing all the PMTs
and their cabling. This is covered with a black polyethylene terephthalate sheet, which
prevents light leaks between the two detector regions. The OD side is covered in 1,885
outward facing 8 inch PMTs. As these are just used for vetoing, a lower photo-coverageis acceptable.
Before the start of SK-III, the OD was optically segmented from the ID, by covering the
OD walls (barrel, top and bottom of tank) with reflective Tyvek sheeting (reflectivity ∼90% at 400nm). This improved the collection efficiency of the OD PMTs, and allowed
some previously difficult backgrounds such as corner-clipping muons to be removed moreeasily.

Chapter 3. Super Kamiokande

26

Figure 3.2: The frame structure of the PMT support system in the SK tank

3.3

Cherenkov Radiation

Upon interacting in the water tank, energy from the neutrino will be imparted into
leptons, and any hadrons which are ejected from the interaction nucleus. These willthen produce photons, through a process known as Cherenkov radiation, which canthen be detected by the surrounding PMTs. Cherenkov radiation is generated when acharged particle passes through a material at a speed faster than that of light in thatmaterial, that is, v > c/n, where n is the refractive index of the medium. The Cherenkov
light is emitted in a cone pattern, in the direction of travel of the particle, where theChapter 3. Super Kamiokande

27

critical cone half angle is given bycos θC =

1
,
βn

(3.1)

where β = v/c. Assuming passage through water, n ∼ 1.33 at 589nm, leading to a
critical angle of ∼ 42◦ , and a threshold energy of 0.76MeV, and 158.7MeV for electrons
and muons respectively.
The number of Cherenkov photons produced per unit wavelength dλ per unit distancedx is given by
2πα
d2 N
= 2
dxdλ
λ


1−

1
(n(λ)β)2=2πα
sin2 θC ,
λ2

(3.2)

where α is the fine structure constant. This relation allows us to reconstruct initial
particle energy, by counting the number of detected Cherenkov photons.

3.4

Photo-multiplier tubes

The 11,146 inward facing ID PMTs (model R3600 developed by Hamamatsu Photonics[89]), each have a 20-inch diameter, resulting in total photo-coverage of ∼ 40% (excluding
SK-II). These PMTs are sensitive to wavelengths in a range 300-600nm, with a quantum
efficiency peaking around 400nm of ∼ 21%.

Figure 3.3: The structure of an ID PMT in SK.

Photons are collected by the bialkali photo-cathode, which then emits single or multiple photo-electrons. This photo-electron signal is then amplified by a venetian-blind
multiplier, comprising of an 11-stage chain of dynodes, with a resultant gain of ∼ 107 .
A large external magnetic field is liable to divert electrons out of the dynode chain,Chapter 3. Super Kamiokande

28

Figure 3.4: The wavelength dependent quantum efficiency of the bialkali photocathode.

lowering collection efficiency. To prevent the PMTs from being affected by the Earth’s
geomagnetic field, 26 Helmholtz coils are set up around the walls of the tank. These
reduce the magnetic field effect from 450mG to 50mG.
In the OD, 1,885 8-inch Hamamatsu R1408 PMTs are used. These are covered by awavelength shifting acrylic plate, which increases the wavelength of the incoming photons
to better match the peak quantum efficiency of the PMT, increasing overall collection
efficiency by 50%. Time resolution becomes worse, but since the OD is only used as aveto counter, this is of no serious consequence.

3.5

The Water System

It is very important to maintain the SK tank water at a constant and high level of purity.
Firstly, a pure water supply for the tank ensures that the water has the highest possible
transparency, and Cherenkov photons are less likely to be absorbed or scattered before
they are detected at the PMTs. If the water transparency cannot be maintained at an
approximately constant level, calibrating the energy reconstruction of particles would
be difficult. In addition to this, it is important to remove any radioactive contaminants
such as radon, which can constitute a significant source of background, especially forlow energy neutrinos.
The water in SK is continuously circulated through a purification system, with a flowof ∼ 60 tons/hour. Water entrance and removal points have been chosen to reduce

Chapter 3. Super Kamiokande

29

convection as much as possible, which can allow radioactive contaminants to seep into
the surrounding tank structure. To further reduce convection, heat exchangers (HE) areused throughout the water purification system to maintain the supply at a very constanttemperature (∼ 13◦ C, with a variation of 0.01◦ C). This temperature is low, to suppress
bacterial growth, and in addition a UV sterlizer is used to kill any present bacteria.

Figure 3.5: A schematic of the water purification system used in SK.

Various filters and systems are in operation for the removal of impurities. An Ion
exchanger is used to remove Na+ , Cl− and others. Radon gas is removed from the waterby a vacuum degasifier and membrane degasifier, reducing the water Rn concentration to< 0.4 ± 0.2mBq/m3 . Size specific filters are used for further removal of various particles- the ultra filter (UF) removes very small particles down to 10nm, the 1µm filters are
suitable for removing dust, and the reverse osmosis (RO) filter is used to remove heavy
particles, of over 1000 molecular weight.

3.6

Air Purification

The natural mine air is heavily contaminated with radon gas, which is continuously
produced in the surrounding rocks from natural uranium decay. This represents botha potential background for low energy neutrino analyses, and a health risk for mine
workers. The air flow patterns through the mine change with season, blowing air outof the mine during summer, and into the mine during winter. This results in a large
seasonal variation in the radon contamination, with ∼ 30 Bq/m3 in winter, which rises
to ∼ 1500 Bq/m3 in summer [90].
To reduce this peak summer level, an air purification system was constructed. Air iscollected from outside the mine, passed through a system of filters, and then pumped
into the mine, at a rate of up to 50 m3 /minute [91], maintaining radon levels in theexperimental areas at < 40 Bq/m3 , all year round.

Chapter 3. Super Kamiokande

30

Figure 3.6: The varying water temperature across the SK tank. Water input is at
the bottom of the tank (negative z).

3.7

Front-end Electronics and Data Acquisition

The data acquisition system (DAQ) used from SK-I to SK-III was upgraded in September
2008 for SK-IV, so these two systems will be described separately.

3.7.1

Data Acquisition System for SK-I to SK-III

An outline of the DAQ system used in SK-I to III can be seen here 3.7. Signal from the
PMTs was first processed by front-end electronics ”Analog-Timing-Modules” (ATMs),
based on the TRISTAN KEK Online (TKO) standard [92]. Each ATM took inputs fromup to 12 PMTs, requiring a total of ∼ 1000 ATMs for the whole detector. The signal
input from each PMT is initially amplified 100x, and then split into 4 separate signals.
The first of these was sent to a discriminator, checking the pulse height recorded fromthe PMT. If it exceeded a threshold of 0.23 photo-electrons, a trigger was sent allowing
the other three signals to proceed.
In the second signal, a 15mV 200ns square wave is generated for each PMT, and thensummed for every PMT connected to the ATM. This signal is known as HITSUM. The
HITSUM result from each ATM module are then sent to a global trigger module, andChapter 3. Super Kamiokande

31

Figure 3.7: A schematic showing the layout of the DAQ used in SK-I to SK-III

added up, effectively counting up the number of PMT hits within this period. If found
to exceed specific thresholds, a global trigger was executed. This signal was chosen to
be 200ns wide, as this corresponds to the maximum time it would take light to travel
across the tank.

Chapter 3. Super Kamiokande

32

Figure 3.8: Overview of the trigger system in SK-I to III

The final two signals are sent to the charge-to-analog converter (QAC), and time-toanalog converter (TAC) to register the integrated charge over 400 ns for each PMT, and
the PMT hit timing respectively.
There were three separate types of trigger derived from the HITSUM signal - Super
Low Energy (SLE), Low Energy (LE) and High Energy (HE) triggers. The thresholdfor these varies dependent on SK period according to the varying amount of PMTs, as
shown in table 3.2. An outer detector (OD) trigger also exists, for triggering on heavy
activity in the OD.
SK Period
SLE
LE
HE

SK-I
186 mV
320 mV
340 mVSK-II
110 mV152 mV
180 mV

SK-III
186 mV
302 mV
320 mV

Table 3.2: Trigger thresholds for each trigger type in SK-I to III

The minimum threshold 186mV, approximately corresponds to a 4.6MeV electron.

Chapter 3. Super Kamiokande

3.7.2

33

Data Acquisition System for SK-IV

This electronics system was upgraded for SK-IV in 2008 [6]. This upgrade granted many
improvements - every hit from PMTs is now quickly digitized, allowing a software-based
trigger system to replace the hardware trigger of SK-I to III, and essentially remove
electronics down time. Another aim of the upgrade was to allow different event widths
to be stored for different trigger types. This is useful, because disk space is prohibitivein storing SLE trigger data, which occurs at a rate of ∼ 3kHz, however for the higher
energy triggers, we want to store a large amount of data, capturing any possible preactivity, and allowing a more comprehensive view of decay electrons and neutron capture
gammas following the initial interaction.
With this in mind, the ATM system was replaced by a new QBEE system (QTCBased Electronics with Ethernet). QTC refers to an ASIC developed as a high-speed
charge (Q)-Time converter. One QBEE module may take input from 24 PMTs, quickly
digitalizing the PMT signal using QTC and multi-hit time-to-digital converters (TDC).
Similarly to the previous system, PMT charge is integrated over a 400ns gate, however
here to expand the potential range of measurable charge, three different input channels
are used, each with different gain. These channels have gains of 1, 1/7, and 1/49,
allowing a range of charge from 0.2 to 2500pC to be observed.
The output from the QTC modules is then sent to the TDCs, which measure the width of
the QTC output pulse. This data is then input to an FPGA, which digitalizes the charge
and timing information, before sending it to the front end PCs (this is the ethernet partof QBEE). The front end PCs pass the hit information onto the merging PCs, which
apply the trigger thresholds, before storing any selected events onto the disk for further
reduction.

Figure 3.9: Block diagram for the new DAQ in SK-IV. Taken from [6]Chapter 3. Super Kamiokande

34

In SK-IV, the trigger is applied emulating the HITSUM signal of SK-I to III, by calculating the number of PMT hits in a 200ns sliding window. If this value exceeds a trigger
threshold, the event will be recorded. Under the ATM system, all triggers recorded fora period of 1.3µs surrounding the event. However using the QBEEs, the SLE trigger
can be set to save only 1.5µs of data, while the much less frequently triggering LE andHE triggers can save data from −5 → +35µs surrounding the trigger time.
A special high energy (SHE) trigger was also introduced in SK-IV. This has a similar hit
threshold as the existing HE trigger, but if a SHE triggered event is not accompanied
by an OD trigger, an additional ”After Trigger” (AFT) is issued, saving an additional
500µs of data, which can be used for analysing neutron captures. SHE was initiallyintroduced with a threshold of 70 hits, but this was lowered to 58 hits in September
2011.
SK-IV Triggers
OD
SLE
LE
HE
SHE
AFT

Hits/200ns Threshold
22
34
47
50
70 - 58
SHE, no OD

Event Width (µs)-0.5→ 1.0
-5→ 35-5→ 35-5→ 3535 → 535

Table 3.3: Trigger information for SK-IV

Chapter 4

Detector Calibration
It is important to properly calibrate our detector, so that we may be confident in our
knowledge of its performance, and thus apply smaller systematic errors during the reconstruction and analysis of data. The types of calibration performed may be split into
three general areas: PMT calibration, water transparency calibration, and energy scalecalibration. For further details, please see [93].

4.1

PMT Calibration

Calibrating PMTs is necessary to understand the charge and timing output of each
PMT, and how it relates to the energy and timing of photons that were collected on thesurface of the photo-cathode. PMT charge calibration consists largely of two factors:
the PMT QE (quantum efficiency), and its gain.
Quantum efficiency (QE) refers to the efficiency with which photons are collected, andgenerate photo-electrons. For purposes of the calibration, this may be combined withthe collection efficiency, which gives the efficiency for photo-electrons that have beenreleased from the photo-cathode, to arrive at the first dynode. The product of the two,
will be referred to as quantum efficiency. The second factor - gain - will be defined as theconversion factor between initial photo-electrons, to the charge output from the PMT
in pC.
For Low-energy events, most PMT hits consist of a single photo-electron, so the charge
output from each PMT is largely uniform, and the most important calibration is the
QE. For higher energy events, where PMTs are likely to release more than a single
photo-electron per hit, gain calibration is also necessary. Both of these must be known
individually for each PMT. The calibration of PMT gain may be thought of as split into
35

Chapter 4. Detector Calibration

36

an “absolute gain”, the average amplification factor applied to all PMTs, and a “relativegain”, which details the fluctuations on a PMT-by-PMT basis.

4.1.1

Determining High Voltage for each PMT

Before any calibration of output can be done, we must determine an appropriate voltage
to apply to each PMT. To decide this, it was required that each PMT must produce thesame output charge, for the same incident light intensity. As SK is not spherically symmetric, using the same light source we would expect to get different charges dependent
on the PMT distance from the light source. So to compensate for this, 420 reference
PMTs were individually calibrated before installation, so they could be mounted in thetank and serve as references for the un-calibrated PMTs.

Figure 4.1: Xe light emitting from a scintillator ball to calibrate the PMT high voltage
in the tank. The red dots illustrate positions of the pre-calibrated reference PMTs.

The light source used is a Xe lamp, the output from which is then passed through an
optical fibre, and into a scintillator ball placed at the centre of the tank. This scintillatorball the scatters the light isotropically throughout the tank. The HV input of each
PMT was then adjusted such that the output charge would match the pre-calibrated
reference PMTs in similar geometrical locations. The Xe lamp and scintillator ball area permanent fixture in the middle of the tank, for continuous monitoring of PMT gain.

Chapter 4. Detector Calibration

4.1.2

37

Relative Gain

The relative gain calibration is performed to compensate for any differences in response
between PMTs, and apply adjustments to each so the result will look uniform. To
perform this calibration, a light source (nitrogen-laser-driven dye laser → optical fibre→ diffuser ball in tank) is placed in the tank, and two measurements are performed.
Firstly, using high-intensity flashes (Is ), to record the average charge observed Qobs (i)for each PMT i, and secondly, low intensity flashes (Iw ) which are designed to produce
single photo-electron ejection on every PMT. These single p-e events are counted, togive Nobs (i) for each PMT. To change the intensity of the light source, a filter wheelwith neutral density filters was set up between the output of the laser and the opticalfibre. A diagram of this may be seen at 4.5.

Qobs (i) ∝ Is × a(i) × qe (i) × G(i)(4.1)

Nobs (i) ∝ Iw × a(i) × qe (i),(4.2)

where qe (i) is the QE, a(i) is the acceptance and G(i) is the gain of each PMT. As thelight source is in the same position for both of these tests, the acceptance is the same
in both equations, and we may cancel out the unknown QE factor, to find the gain ofeach PMT

G(i) ∝

Qobs (i).
Nobs (i)(4.3)

This is then normalized to the average gain, to find the relative gain difference for each
PMT. The RMS was found to be 5.9%, and is used as a PMT-by-PMT correction factor
on the conversion between observed charge and photo-electrons. As the HV system
has already been calibrated to create uniform output intensity Qobs on every PMT, we
can conclude that these relative gain fluctuations are due to differences in QE between
PMTs.

4.1.3

Absolute gain

Now that the relative gain corrections for each PMT are known, we can apply these
corrections to align the single p-e distributions of all PMTs. Adding these distributions
together for each PMT results in the average single p-e response of the detector. From
this we can determine the absolute gain of all PMTs.

Chapter 4. Detector Calibration

38

Figure 4.2: The relative gain difference of each PMT from calibration using a nitrogenlaser-driven dye laser.

A stable source of single p-e producing light is required for this calibration, for whichnickel-californium (Ni-Cf) was chosen. The Cf decays via spontaneous fission, 3.8% of
the time producing an average multiplicity of 3.8 neutrons (lifetime 2.65 years). These
neutrons are then captured by the surrounding Nickel, and 6.1-9.0 MeV gamma rays are
released isotropically. This was positioned at the centre of the tank, and the resultant
PMT signals were 99% single p-e hits. To compensate for dark hits, the same distribution
was taken without the Ni-Cf source, and then subtracted from the on-timing data. The
relative gain corrections were applied, resulting in the plot 4.3.
The average value from plot 4.3 was taken to be the conversion factor between pC andsingle p-e signals.

SK-I
SK-II
SK-III
SK-IV

Absolute Gain (pC/1p-e)
2.055
2.297
2.243
2.645

Table 4.1: Absolute gain conversion factors between single photo-electron events, andoutput PMT charge.

Chapter 4. Detector Calibration

39

Figure 4.3: The charge distribution from Ni-Cf single p-e calibration source, after
relative gain corrections had been applied.

This calculation was performed again before the start each SK period, and values found
are recorded in table 4.1. The PMT gain is found to increase by 2-3% per year. The
break in this pattern between SK-II and III is due to the installation of new PMTs inthis interval. Although the reason for this increasing gain is currently unknown, it istaken account of in physics analyses.

4.1.4

Quantum Efficiency

To properly simulate PMT action in our MC simulation, it is required to know the QEof each PMT. To do this, the same Ni-Cf source was used to provide a sample of single
p-e signals. We can then calculate the difference in MC predicted number of hits, to theobserved number in data, and adjust the simulated QE to account for any discrepancy.
The geometry of the tank is adjusted by applying the following correction to the MC

Nobs (i) × R(i)2 /a(θ(i)),(4.4)

Chapter 4. Detector Calibration

40

where R(i) is the distance from the source to PMT, and a(θ(i)) is the acceptance of the
PMT for a given incident angle θ. Further corrections for light propagation were applied,
including modelling of the black sheet, reflections from PMT surface, and scattering or
absorption in water. The remaining MC-data differences are attributed to varying QE
between PMTs.

Figure 4.4: This plot shows the relative PMT hit probability dependent on z (cm)
position in the tank. The red points are from Ni-Cf source, and the blue points are theMC, before QE correction.

It is of interest to note, that PMTs added between SK-II and SK-III (replacing thoselost in the accident), were found to have a higher QE than the remaining original PMTs.
Similarly, even out of the originally used PMTs, one batch produced between 1992 and1995 were found to have 4.5% lower QE than the second batch, made in 1996-7. These
improvements were attributed to improvements made by Hamamatsu in the PMT glasstransparency.

4.1.5

Timing Calibration

To ensure accurate reconstruction of event vertices and track directions, it is important tounderstand the response time of each PMT and electronics channel. There are a numberof effects which must be considered - length of PMT cabling, electronics processing time,
PMT time-walk. The latter refers to an effect in which the response time of a PMT is
reduced for large pulses, as the pulse rise time (and thus time to exceed trigger threshold)is faster.
To perform timing calculation, the dye laser is used again, the setup of which you may
see in figure 4.5. The input nitrogen laser produces pulses 0.4ns wide at 337 nm, which
are then used as input for the dye laser. This light is also observed by a monitoring
PMT, to define the time of laser injection. The dye laser further shortens the pulse
to 0.2 ns, and outputs a wavelength of 398 nm, better matching the optimal QE ofChapter 4. Detector Calibration

41

Figure 4.5: The setup of the dye laser used to calibrate the timing and relative gainof the ID PMTs.

our PMTs. This pulse is transmitted to the tank via a 400 nm optical fibre, where an
isotropic distribution is produced using a diffuser ball.
Time-Charge (TQ) plots are then constructed for each readout channel, showing the
response time of each channel as a function of its charge. The charge dependence isdue to the aforementioned time-walk effect. To use this information in calibration, a”TQ-map” is constructed for each channel, by fitting a polynomial to each TQ plot.

Chapter 4. Detector Calibration

42

Figure 4.6: Charge vs Time plot for readout channel 00010. The black line is a fitted
polynomial used for calibration, referred to as the ”TQ map”.

The selection for events used in this plot, is performed by subtracting time-of-flight(ToF) from the diffuser ball for each PMT in the tank, and then selecting all hits within±50 ns of the monitor PMT hit timing.

4.2

Water Calibration

To best model the passage of photons through water in our MC, we must calibrate to
the properties of the SK tank water. Firstly, work on measuring the absorption andscattering of light is presented, followed by measurements of light reflection at the PMT
surface, and properties of the black sheet.

4.2.1

Passage of Photons through Water

To calibrate the absorption and scattering of light in water, we fire a laser through an
optical fibre and into the tank, at various different wavelengths. The output is recorded,
and compared with MC estimates to fit the parameters for absorption and scattering.
A depiction of the experimental setup may be seen in figure 4.7, and for a more detailed
description, you may see [94].
The intensity of light of wavelength λ as it passes through a medium may be given by

I(λ) = I0 (λ)ex
− L(λ),(4.5)

Chapter 4. Detector Calibration

43

Figure 4.7: The laser system used to calibrate light scattering and absorption throughwater. B1-5 represent different regions of the tank in which analysis was performed.

where L(λ) is the attenuation length (commonly referred to as water transparency withinSK), and x is the distance travelled. Typically the attenuation length is described interms of absorption and scattering, however in SK it is defined as

L(λ) =

1αabs (λ) + αsym (λ) + αasy (λ)

,

(4.6)

where αabs (λ), αsym (λ) and αasy (λ) are empirically fitted parameters, corresponding toabsorption, symmetric and asymmetric scattering.
Symmetric scattering includes Rayleigh scattering, and the symmetric component of
Mie scattering, described by 1 + cos2 θ. The asymmetric scattering term is used for theasymmetric component of Mie scattering, where scattering probability increases linearly
for forward scattering angles. These terms are each represented by different functions,Chapter 4. Detector Calibration

44

for a total of 8 different fitting parameters, which are adjusted to minimize the χ2
between the laser data, and MC fit:

P1
+ C(λ)λ4P5P4
αsym (λ) = 4 × 1.0 + 2
λ
λ


P7
2
αasy (λ) = P6 × 1.0 + 4 × (λ − P8 ) .
λαabs (λ) = P0 ×

(4.7)
(4.8)
(4.9)

The term C is altered depending on wavelength: for λ ≥ 464 nm, the function is based
off an experimental measurement by Pope and Fry [95], and for λ < 464 nm, it is givenbyC(λ) = P0 × P2 ×

λ
500

P3
.

(4.10)An example of the results of this fit from April 2009 data can be seen in figure 4.8.

Figure 4.8: The results of the fit calibration for water scattering parameters, for April
2009. The dots represent values measured in data, and the lines are the corresponded
fitted functions.

These fits are updated in real time, so we may constantly measure the water quality. Symmetric scattering is mostly stable in time, however asymmetric scattering andabsorption have a time dependence of ∼ 20 − 40% and ∼ 20 − 60% respectively.

Chapter 4. Detector Calibration

4.2.2

45

PMT Surface Reflections, Black Sheet

Light reflection at the surface of the PMT is tuned, by varying the refractive index of
the bialkali photo-cathode until best fit is achieved. The same laser setup as in thewater transparency calibration is used here (figure 4.7). However the samples used for
determining reflections or scattering varies between the studies. All of the hits are ToFcorrected to the vertex of laser beam emission. After this, PMT hits that occur relatively
early, are taken to be direct laser hits, and used in light transparency calibration. PMT
hits coming later, are assumed to be due to reflections, and are used in this study. The
exact timing criteria for this split is different depending on tank region, as depicted infigure 4.9.

Figure 4.9: The split up of data taken from the laser beam (405 nm) used for waterand PMT reflection calibration. Shown is the timing (ns) of each PMT hit, after ToF
correction, for 6 different regions of the tank (the top plot is of the tank top wall,and the remaining 5 are barrel regions shown in figure 4.7). The hits between the twoleft-most blue lines, are used for water transparency calibration, and those between thetwo right-most lines are used for reflection calibration.

The bialkali refractive index was taken to be nimg + nreal , and fitted to the laser data asnimg = 1.667, and nreal = 2.31, 2.69, 3.06 and 3.24 for laser wavelengths λ = 337, 365,
400, and 420 nm respectively.
The black sheet used to stop light leaks between the ID and OD must also have its
reflectivity and absorptivity calibrated. For this purpose a sample of black sheet wasplaced inside the SK tank, and a laser was directed at it, reflecting into the tank. A
diagram of this set-up can be seen in figure 4.10. The reflected charge (Qscattered ) wasChapter 4. Detector Calibration

46

then measured by the PMTs. As a comparison, the charge was also measured with noblack sheet present (Qdirect ). The ratio of these numbers, R = Qscattered /Qdirect wasused to calibrate the black sheet reflectivity. This experiment was repeated for threedifferent incident angles (30◦ , 45◦ and 60◦ ), and three different laser wavelengths (337
nm, 400 nm and 420 nm).

Figure 4.10: The experimental set-up used to tune the black sheet reflectivity.

4.3

Energy Scale Calibration

The conversion from observed photo-electrons to the output charge is now well calibrated, but for a reliable reconstruction we must also calibrate how many photo-electrons
are produced by particles for a given energy. Knowing particle energy precisely is very
important for any neutrino oscillation analysis, as we know the probability of flavourchange is dependent on the neutrino energy (see chapter 2).
For energy scale calibration in high energy analyses, four separate calibration sources
are considered, across a wide energy range.

• Track range of high energy cosmic rays (1∼10 GeV)
• Cherenkov angle of low energy cosmic rays (200∼500 MeV)
• Invariant mass of π 0 produced in neutrino interactions (∼130 MeV)
• Decay electron momentum (∼50 MeV)

Chapter 4. Detector Calibration

47

A data vs MC comparison is performed for each of these event samples, and the simulation is calibrated to attain uniformity.

4.3.1

High Energy Cosmic-Rays

In this context, “cosmic-rays” refer to muons produced in the atmosphere by π-decay,
which have successfully passed through the rock above SK and come to a stop within
the detector. Electrons are generally thermalized in the atmosphere or rock, so there
is no equivalent electron sample. Several selections must be passed for an event to be
identified as a cosmic ray muon for this calibration:• Enters through the top wall of the detector.
• Reconstructed direction must be downward (cos θ > 0.94, where θ is the zenith
angle).
• A single decay electron is detected.
• The track length of the muon must be (7 < L < 30)m.
Energy calibration may be done using the range of the muons. The track length ofthe muon is approximately proportional to its initial momentum, thus the track length
selection listed above may be thought of as a high energy selection, with the added
criteria that the muon must stop inside the detector. The muon track length is calculated
by taking the distance between the reconstructed entry point, and the reconstructeddecay electron vertex.
The data-MC discrepancies before calibration are 0.7%, 1.1%, 2.0% and 2.2% for SK-I
to IV respectively. The factor used for calibration is dependent on the track length ofthe particle, as shown in figure 4.11.

4.3.2

Low Energy Cosmic-Rays

This sample is similar to the previous, selecting down going cosmic muon events. However in the low energy case, energy reconstruction is performed based on the opening
Cherenkov angle. Recall

11=
cos θC =
nβ
n

s
1+

m2
,
p2

(4.11)

Chapter 4. Detector Calibration

48

Figure 4.11: High energy cosmic ray stopping muon sample, ratio between momentum
Ppe , calculated from the total number of photo-electrons collected by each PMT, andthe range of the particle in cm. The discrepancy between data and MC is used asenergy calibration in this momentum range. Data shown is from the SK-II period.

where θC is the critical Cherenkov opening angle, and n is refractive index. This sample
of cosmic rays is selected the same as previously, except the track length requirement is
now replaced by:• Total ID photoelectrons < 1500 (750 for SK-II).

Figure 4.12: Low energy cosmic ray muon sample, ratio between momentum Ppe ,calculated from the total number of photo-electrons collected by each PMT, to Pθ ,calculated from the Cherenkov opening angle. The discrepancy between data and MC
is used as energy calibration in this momentum range. Data shown is from the SK-I
period.

The expected momentum for each event is then compared, as calculated from PMT
photo-electrons, to the momentum calculated from Cherenkov opening angle. The ratioof these momenta Ppe /Pθ is compared for data and MC, and the discrepancy between

Chapter 4. Detector Calibration49

the two is used to calibrate the energy scale. You may see this in figure 4.12. The
disagreement between data and MC pre-calibration is 0.7%, 1.3%, 2.1% and 2.1% for
SK-I to IV respectively.

4.3.3

π 0 Invariant Mass

In some neutrino interactions, single π 0 are produced. These interactions are not used
for oscillation analysis, so they may be used as a calibration source. Around 99% of π 0
decay to γγ, and this signal may be identified by selecting events with:• Event vertex reconstructed within tank fiducial volume (> 2 m from ID wall, nolarge clusters of hits in OD). This is a standard criteria for selecting any fullycontained neutrino events.
• Two electron-like Cherenkov Rings. All reconstructed Cherenkov rings in SK areclassified as electron-like or muon-like. This will be discussed in greater detail inchapter 6, but it suffices to say that γ-rays produce Cherenkov rings very similar
to those produced by electrons, so this criteria is just a search for the expected γγ
signature from π 0 decay.
• No decay electron (as this would imply a muon was present in the interaction).

Figure 4.13: Reconstructed π 0 invariant mass, for data and MC in black and red
respectively.

The π0 invariant mass is calculated byChapter 4. Detector Calibration

Mπ 0 =

50q

2Pγ1 Pγ2 (1 − cos θ)(4.12)

and fitted with a gaussian function. The difference between the fitted peak positions of
the invariant mass for data and MC is used as an energy calibration. Before calibration,
the discrepancies between data and MC are 0.7%, 1.3%, 0.3% and 1.7% for SK-I to IV.

4.3.4

Decay Electrons

As a low energy calibration, the reconstruction of decay electron energy is used. Only
decay electrons produced by cosmic muon events are considered, to avoid interferencewith any neutrino analysis. The selection criteria for these is as follows:• Must occur between 2 µs and 8 µs following a stopping cosmic ray event.
• More than 60 (30 for SK-II) PMT hits in a 50 ns window. This is required toreject signal from µ− capture on oxygen.
• Vertex fit goodness > 0.5• Vertex reconstructed inside the fiducial volume.
Similarly, the reconstructed decay electron momentum spectra are compared between
data and MC, and the energy scale is calibrated by the difference. Pre-calibration
discrepancy for SK-I to IV is 0.6%, 1.6%, 0.8% and 1.6% respectively.

4.3.5

Energy Scale Summary

These various calibrations are all effective for different momentum ranges, and may be
combined to find an overall uncertainty of the absolute energy scale. This is estimatedto be 0.88%, 0.55%, 1.79% and 2.19% for SK-I to IV respectively. A summary of theabsolute energy scale calibrations can be found in figure 4.14.

Chapter 4. Detector Calibration

Figure 4.14: The summary of all absolute energy scale data-MC discrepancies, for
SK-I to IV.

51

Chapter 5Monte Carlo Simulation of
Atmospheric Neutrinos
Generating a prediction for the number of atmospheric neutrino interactions observedin Super Kamiokande is no simple task. Results depend heavily on detector tuning,
flux predictions, particle reconstruction and so forth. To create a reliable expectation
on a bin-by-bin basis, it is necessary to simulate the entire process. Using a Monte
Carlo (MC) method, inputs are taken from random sampling of atmospheric neutrino
flux predictions, processed through the simulation, and finally expected output event
rate distributions for each sample are generated. As the simulation is improved, we canbe more confident (our degree of “confidence” is represented by the systematic error
applied in the analysis) that differences between data and MC are due to real physics
effects.
Our MC simulation is constructed in two stages - firstly the initial neutrino interaction
in water is modelled, and then the particles output from this are input to a detector simulation, where we model their kinematics across the detector, and the expected numberof hits in each PMT. However before any of this is possible, we must first discuss theatmospheric neutrino flux model, which is used as an input to the simulation.

5.1

Atmospheric Neutrino Flux

The flux model used to generate atmospheric neutrino events in our simulation is basedon M. Honda’s 2011 model [7]. The Honda flux does not support neutrino energies
above 10 TeV, so it is replaced by the Volkova flux in this region. Two additional flux

52

Chapter 5. Simulating Neutrino Interactions

53

models, those by G. Battistoni (Fluka flux) [96] and G. Barr (Bartol flux) [97] are used
to evaluate the systematic uncertainties on the Honda flux.
The flux model calculates the expected energy spectrum of incident neutrinos on SK.
To reach this stage, broadly two processes must be understood - the primary flux ofcosmic rays, and then simulation of the hadronic showers they cause, to allow estimation of resultant neutrino multiplicity. The primary flux of cosmic rays is directly
measured by high altitude experiments, such as AMS (Alpha Magnetic Spectrometer,in the International Space Station) [98], and balloon-borne experiment BESS [99].

Figure 5.1: East-west asymmetry in neutrino flux at SK, due to deflection of primarycosmic rays by the Earth’s geomagnetic field. Plots taken from [7].

Almost all incoming cosmic rays are protons, and as the Earth’s magnetic field points tothe south, these cosmic rays will be deflected in an eastward direction. This creates aneast-west asymmetry in the cosmic ray flux (see fig. 5.1), and a rigidity cut off for low
energy cosmic rays, below which they are simply deflected back into space. Similarly,
around Kamioka the Earth’s magnetic field is slightly stronger than average, so weobserve a slight asymmetry in upward and downward going neutrino flux. The neutrino
flux is peaked near the horizon (fig. 5.2), as muons approaching from this direction
have more time to decay into neutrinos. The flux is also significantly affected by solar
activity. At solar maximum the strong magnetic field due to solar winds reduces theflux of low energy cosmic rays to ∼half of its value at solar minimum. For higher energy
cosmic rays, > 10 GeV, these deflection effects become negligible.
In 2004 Honda updated his flux model to include 3-dimensional interaction modelling,in which the outgoing particles may be deflected at a different angle than the incident
cosmic ray [100]. This led to a larger enhancement of flux at the horizon comparedto the 1-d flux, due to the larger effective area of cosmic rays which could contributeto these events (depiction in figure 5.3). In the Honda 2011 version, the 3-d scheme isChapter 5. Simulating Neutrino Interactions

54

Figure 5.2: Zenith angle distribution for neutrino flux incident on SK. You may notethe peak at the horizon (around cos θ = 0), and the slight asymmetry between upwardand downward going fluxes. These effects are more prominent in low energy particles,
which are more sensitive to magnetic field deflection effects.

used to calculate flux up to 32 GeV (after which required computational power becomesprohibitive).

Figure 5.3: Additional cosmic ray vectors now contribute to the horizon neutrino fluxin the 3-d calculation of Honda flux, compared to the previous 1-d version.

The second stage of modelling neutrino flux, is the simulation of hadronic showers produced when a cosmic ray interacts in the atmosphere. In the Honda flux 2011, twomodels are used: DPMJET-III [101] for hadronic interactions above 32 GeV, and JAM[102] for interactions between 0.2 and 32 GeV. The JAM model is an addition in the2011 update, as it was found to agree more favourably with low energy cosmic ray muon
data taken by the HARP experiment in 2008 [103]. For further tuning, the model wasChapter 5. Simulating Neutrino Interactions

55

compared to cosmic ray muon data was taken at three altitudes by the BESS group:
Tsukuba (30 m) [104], Mt. Norikura (2770 m) [105] and Fort Summer (balloon) [106],
as shown in figure 5.4. Modifications were made to the JAM model to optimize agreement with data at balloon altitudes, as this is thought to be where most neutrinos areproduced.

Figure 5.4: Flux comparison with cosmic ray muon data. From left to right, Tsukuba(ground), Norikura (mountain) and Ft. Summer (balloon)

The total atmospheric neutrino flux incident on SK can be seen in figure 5.5.

Figure 5.5: Atmospheric neutrino flux incident on SK, taken from the Honda 2011
paper. Differences to other models are shown, and taken as a systematic error inanalysis.

5.2

Neutrino Interaction Generator

Using the previously described flux model as input, a neutrino interaction generator isused to calculate how often, and in what way these neutrinos will interact in our detector
and the surrounding rock. For this purpose, we are using a software library called NEUT

Chapter 5. Simulating Neutrino Interactions

56(for details see [12]). NEUT was originally developed for the Kamiokande experiment,
just for modelling atmospheric neutrino interactions in water, and has continuously
expanded its functionality since then. Only neutrino-nucleon interactions are considered,as the cross-section for neutrino-electron interactions is a factor 103 smaller, so may besafely neglected.
The function of NEUT is divided into two distinct steps. Firstly, the initial interaction
between incoming neutrino and target nucleon is simulated. Following this, for composite
nuclei such as 16 O, output hadrons’ passage through the nucleus is simulated, as there
is a high chance for additional interactions before it escapes.
There are four basic types of interaction modes simulated inside NEUT:

CC/NC elastic scattering: νl + N → l + N 0
CC/NC single meson production: νl + N → l + N 0 + meson
CC/NC deep inelastic scattering (DIS): νl + N → l + N 0 + hadrons
CC/NC coherent pion production: νl + 16 O → l + 16 O + π

where l is a lepton, and N , N 0 are nucleons.
As the main topic of this thesis relates to using information about the number of neutrons in each event, it is important that this is simulated in each mode as best as possible.
Figure 5.6 shows the expected breakdown of neutrons produced by different interaction
modes in the sample selection most sensitive to the neutrino mass hierarchy. The total
number of neutrons comes from a combination of those produced at the initial interaction (simulated by NEUT, described in this section), and those produced in secondary
hadronic interactions in the tank (simulated by SKDetsim and GEANT, as described in
the following section).
For each interaction mode, the models used are described, and where available comparisons with experimental data are shown.

5.2.1

Elastic Scattering

Elastic scattering in NEUT is modelled separately for free, and bound nucleons. For freenucleons, the calculation used is described by Llewellyn-Smith [107]. For bound nucleons,(neutrino interactions in 16 O), we must consider the whole nucleus in the calculation,

Chapter 5. Simulating Neutrino Interactions

57

Figure 5.6: The breakdown of expected neutron multiplicity for each neutrino interaction mode. The plot shows SK-IV atmospheric neutrino MC Multi-GeV 1-ringelectron like sample, normalized to the livetime of the equivalent data-set (shown inblack dots). This is the most relevant sample for the neutrino mass hierarchy. In thissample the dominant production modes are CCQE and single-pion production.

which is done using the model of Smith and Moniz [108]. This model approximates thesurrounding nucleons as a relativistic Fermi gas, taking the momentum distribution of
the nucleons to be flat, up to a Fermi momentum pf = 225 MeV/c. To incorporate thePauli-blocking effect, it is required that the momentum of the recoiling nucleus should
be greater than pf . NC interaction cross sections are calculated using the more wellknown charged current cross section [109][110]. The axial vector mass MA is set basedon experimental results from K2K and MiniBooNE [111][112].
The agreement between the modelled result and experimental data for neutrino andanti-neutrino interactions can be seen in figure 5.7.

5.2.2

Single Meson Production

The Rein and Sehgal model [113][114] is used to simulate single meson production ofπ, K and η. An intermediate baryon resonance with mass < 2 GeV/c2 , is assumed,which then decays to a meson and nucleon. Most of the mesons produced are pions,Chapter 5. Simulating Neutrino Interactions

58

Figure 5.7: The agreement of NEUT predictions (black line) of free nucleon charged
current quasi-elastic scattering cross section, with experimental data. The experiments
considered in figure (a) are ANL (triangle) [8], GGM (square) [9] and Serpukov (circle)[10]. In figure (b), these are GGM (square) [11] and Serpukov (circle) [10]. Figures
taken from [12].

but a small amount of K and η mesons represent a significant source of proton decay
background. Most baryon resonances produce pions isotropically, with the exception of∆(1232).
Another possibility is that the neutrino may react with the entire 16 O nucleus, rather
than an individual nucleon. This is known as coherent pion production. As in thiscase the target is much heavier, the resulting decay products have a larger fraction of
the neutrino momentum, and thus tend to be scattered in the forward direction. This
interaction is described by Rein and Sehgal in [115].
Good agreement is seen between the single meson production cross sections and experimental data, as demonstrated in figure 5.8.

Figure 5.8: The agreement of NEUT predictions (black line) of single pion production
cross section, with experimental data. The experimental data considered is from ANL(triangle) [13], and GGM (circle) [14]. The figures are taken from [12].

5.2.3

Deep Inelastic Scattering

Deep inelastic scattering refers to the interaction when a neutrino (or other particle)
interacts with a constituent quark inside a nucleon, causing the nucleon itself to breakup, and potentially generate multiple output hadrons. This process is considered for invariant mass W > 1.3 GeV, and becomes the dominant mode in multi-GeV interactions.

Chapter 5. Simulating Neutrino Interactions

59

The parton distribution function (PDF) for this mode is taken from GRV98 [116], andthen a correction by Bodek and Yang is applied to use the PDF at low Q2 [117].
For invariant mass 1.3 < W < 2.0 GeV/c2 , single pion production interactions are
generated as per the previous section, so nπ > 1 is required for DIS. In this energy
range, the only outgoing mesons considered are pions, and their multiplicity is tuned
using the results of a Fermilab hydrogen bubble chamber experiment [118]hnπ i = 0.09 + 1.83 ln W 2

(5.1)

.
For energies W > 2.0 GeV/c2 , the hadronic final states are computed by PYTHIA JETSET [119].
No experimental comparison of neutron production following deep inelastic scattering
events was made, however the agreement of the total charged current cross section (which
includes quasi-elastic scattering, single meson production and deep inelastic scattering)can be seen in figure 5.9. Another comparison of total cross section on a higher energy scale is in figure 5.10, showing good agreement in the DIS-dominated high energy
interactions.

Figure 5.9: The agreement of the total neutrino and anti-neutrino charged current
cross sections, between the NEUT model and experimental data. In figure (a), thedata used is from experiments GGM73 [15], GGM79 [16], GGM81 [17], SKAT79 [18],BEBC79 [19], IHEP79 [20], ANL79 [21], BNL80 [22], BNL81 [23] and IHEP96 [24].
For the anti-neutrino figure (b), the experiments used are GGM79 [25], GGM81 [17],IHEP79 [20] and IHEP96 [24]. The figures are taken from [12].

5.2.4

Nuclear Effects

For interactions taking place in 16 O, hadrons that are produced may undergo secondaryinteractions before leaving the nucleus. To simulate these interactions in NEUT, particles
are tracked as they propagate through the nucleus until exiting or absorption, and any
secondary interactions which take place are calculated at each tracking step.

Chapter 5. Simulating Neutrino Interactions

60

Figure 5.10: Comparison between NEUT simulation, and experimental data. Cross
section contributions from charged current elastic scattering, single pion production,and deep inelastic scattering are shown. Experimental data is overlaid from [26].

Figure 5.11: Tuning of nuclear effect cross sections to π − 12 C scattering data. The
dots are taken from experimental data in [27]. The dashed and solid lines represent
before and after tuning.

For interaction probabilities of low momentum pions pπ ≤ 500 MeV/c, a model by
Salcedo is used [120], which was tuned using experimental data from π + −12 C [121]. Later
this was further corrected to a larger value of charge exchange cross-section measured
by Ashery [122]. For momentums pπ > 500MeV/c, nucleons may be treated as freeparticles, so π-free proton scattering data is used, and again tuned by π +12 C results.
For K interactions, the cross sections are tuned to K ± − N scattering experiments[123][124] and [125]. For η mesons, experimental data from the interaction (ηN →N ∗ → πN ) is used [126].

Chapter 5. Simulating Neutrino Interactions

5.3

61

Detector Simulation

After the neutrino interaction has been simulated, the next step is to propagate theoutgoing particles throughout a simulation of the SK tank. This involves tracking any
further interactions and secondary particles that are produced, simulating Cherenkov
photon production and propagation, light collection by PMTs, and the resulting responseof the detector electronics. The simulation is carefully tuned using calibration data
discussed in the previous chapter.
The detector simulation is constructed around a software package called SKDetsim.
SKDetsim interfaces with CERN’s GEANT package (v3.21)[127]. GEANT controls the
production and propagation of Cherenkov light, as well as processing any particle decaysthat are required. As Cherenkov photons propagate through the tank, the primary
scattering effects are through Rayleigh and Mie elastic scattering, which are tuned using
the calibration data previously discussed. Photo-electric absorption is also recently
added to our simulation.
For the simulation of hadronic interaction processes, an interface to package CALOR[128] is used. This is a hybrid package, containing various different interaction models,each of which is applicable to different particle types and energies. The models chosen
for our simulation are as follows:
Criteria
Hadrons > 10 GeV
Hadrons < 10 GeV
Neutrons < 20 MeV
Pions < 500 MeV

Hadronic Interaction Model
FLUKA [129]HETC [130]MICAP [131]Custom routine based off the NEUT cascade model

For nucleon interactions < 3.5 GeV, and charged pion interactions < 2.5 GeV, the HETC(High Energy Transport Code) uses the Nucleon-Meson-Transport-Code (NMTC), which
is based off the Bertini intra-nuclear hadronic cascade model [132]. For energies above
this, the particle energies and multiplicities are scaled to the centre of mass energy of
the interaction.
For low energy neutron propagation the MICAP (Monte Carlo Ionization Chamber Analysis Package) code is employed. This uses all neutron scattering cross section data fromthe Evaluated Nuclear Data File (ENDF/B). For very low neutron energies, thermal
scattering is calculated based on a free gas model, and the final neutron energy is determined by sampling from an isotropic (in neutron centre-of-mass frame) Maxwellianenergy distribution. Neutrons are allowed to track until the lowest allowable energy(10−5 eV) to ensure that neutron capture interactions are simulated correctly.

Chapter 5. Simulating Neutrino Interactions

Figure 5.12: Absolute neutron yields after impact of 256 MeV protons on a sphereof 238 U. Lines are simulated by CALOR, and experimental data is from [28]. This plot
is taken from [29].

Figure 5.13: Neutron multiplicity simulated by HETC (open dots), compared toexperimental data (solid triangles) for a 2 GeV proton interacting in Au. The solid
dots are before detector efficiency is taken into account. This study and plot are from
[30].

62

Chapter 5. Simulating Neutrino Interactions

63

In the context of this thesis, a large amount of the neutrons produced following neutrino
interactions are as a result of secondary hadronic interactions with the water nuclei. It
is difficult to study each of these interactions individually, so the best course of action is
to compare final neutron multiplicity, after passing through a body of matter. Studiesof the agreement between simulated and measured neutron multiplicities can be seen in
figures 5.12 and 5.13.
The electronic response of the detector is simulated down to the signal coming out of the
PMTs. The quantum efficiency, gain and single-photon distributions of the PMTs are
input from calibration data (described in chapter 4). The HITSUM signal is calculated,
and then the same software trigger is applied, as for real data (described in chapter
3). Electronic PMT dark noise is simulated at 5.8kHz. When PMTs degrade and are
disabled in the detector (due to becoming flashers, or suspiciously high or low hit rate,etc.), they are masked in the simulation.

Chapter 6

Event Reconstruction
Event reconstruction refers to the process of starting with the charge and timing information received from the PMTs, and “reconstructing” this into useful physics properties
which can be used for analysis. You may recall from chapter 2, neutrino oscillation probability is dependent on the energy E, and the distance travelled L of the neutrino. For
any sort of oscillation analysis, it is crucial to know what flavour of neutrino we areobserving - if this cannot be discerned, we cannot measure anything but total neutrino
flux. To identify whether a particle track is entering the tank from outside, or has been
generated in the centre of the tank by a neutrino interaction, it is necessary to haveprecise vertex fitting. Reconstruction of all these properties, and more, is discussed inthis chapter. To understand the co-ordinate system used in SK, please see figure 6.1.
The software library encompassing neutrino event reconstruction is named “APFit”.

6.1

Vertex Fitting

General vertex fitting of events is calculated in a three-step process. Firstly, the mostbasic fitting procedure, termed ”Point-fit” is used. Point-fit assumes that all the PMT
hits in an event come from a single point. Considering a test point, time of flight (ToF)
is subtracted from the hit timing on each PMT, forming a distribution of the residual
timings. This distribution is fit to a gaussian for each test point, and that which returns
the maximum goodness-of-fit, is taken as the event vertex. Particle direction is thencalculated by taking the average of the charge-weighted direction vectors, from thisvertex to each hit PMT.
The second step, is to identify the primary Cerenkov ring, using this initial vertex. Using
the direction calculated by Point-Fit as a starting point, various Cherenkov ring opening
64Chapter 6. Event Reconstruction

65

Figure 6.1: A depiction of the co-ordinate system used in SK. θ (zenith angle) is
measured downwards from positive z.

angles θ are tested. The amount of charge collected in a ring is q(θ). The goodness-of-fit
for a given opening angle θt is defined as
R θt
G=

0

(θt −θC )2
q(θ)dθ
× e− 2σ2 ,
sinθt

(6.1)

where θC is the critical Cherenkov opening angle, and σ is the resolution of PMT hitsaround θt . Different directions and angles are tested, and the set with maximum G ischosen to be the first ring fit.
Following this, we attempt to refit the vertex, using only the PMT hits designated inside
this chosen Cherenkov cone. The algorithm that performs this fit is called TDC-fit. Ina similar fashion to Point-Fit, the timing residuals are computed, however this timethey are allowed to come from any point along the particle track. Fixing the fitted
Cherenkov ring, the vertex and direction of the particle are allowed to vary, then the fit
with maximal goodness is selected.
For events with only a single Cherenkov ring, a more precise fitter is used, incorporating
particle type information. This will be discussed in a later section.

Chapter 6. Event Reconstruction

6.1.1

66

Ring Counting

Once the primary ring has been determined, further rings are searched for using a Houghtransformation[133] and likelihood technique. Ring searching is performed by assuming
Cherenkov opening angle of 42◦ , and then for every hit PMT, a charge-weighted virtualcone is drawn facing towards the vertex calculated by TDC-fit. Using a spherical coordinate system centred around this vertex, the weighted charge from each PMT is
added up for every Θ and Φ. Directions identified with peaks of expected charge, areselected as ring candidates.

Figure 6.2: On the left, is a 2-d depiction of ring candidate searching. Each PMT hasa charge-weighted cone drawn out from it at 42◦ , and the points with maximal charge
are taken as ring candidates. The plot on the right is an example of the result of this,in a spherical co-ordinate system centred on the TDC-fit vertex. The angles with peaksof charge represent most probable rings.

Each ring is tested iteratively by a likelihood method, by comparing the N + 1 ring
hypothesis to the N ring hypothesis. If N + 1 rings are found to be a better fit, N + 2
ring hypothesis is tested, and so forth, up to a maximum of 5 rings. The likelihood
is calculated based on the probability of seeing observed charge qiobs in each PMT,
compared to the expected charge qiexp given a hypothesis of N rings:
LN =X

log(P (qiobs ,

N
X

exp
αn qi,n
)),

(6.2)

n=1

i

where αn is a weight for ring n. The weights αn are allowed to vary to find maximum
likelihood. The probability function P is defined as
obs −q exp )2
i
2σ 2

(qi
1
P (qiobs , qiexp ) = √ e−
σ 2π

for qiexp > 20p-e,(6.3)

Chapter 6. Event Reconstruction

67

where α is the resolution for q exp . For PMTs with expected charge less than 20 photoelectrons, the probability is calculated from a convolution of the single p-e function witha Poisson distribution.

Figure 6.3: Output of L(2 rings) − L(1 ring) for SK-IV, < 1330 MeV neutrino events.
The fraction of different event modes in each sample is shown by the coloured lines.

6.1.2

Precise Single Ring Fitting

Fitting is further optimized for single ring events, by optimizing the vertex and direction
based off the expected light pattern, given that the ring is either e-like or µ-like (determining this will be explained in the next section, please be patient). This algorithm iscalled MS-Fit.
To evaluate the performance of these fits, the Atmospheric Monte-Carlo sample describedin chapter 5 is used. Distributions are made for the distance between true vertex andreconstructed vertex (and similarly for direction), then the fit resolution is defined as
the width where 68% of events are included.

Chapter 6. Event Reconstruction

68

Single Ring Vertex Resolution (cm)
SK-I SK-II SK-III SK-IV< 1 GeV events
e-like 31.2
35.6
31.1
31.3
µ-like 23.8
20.2
23.9
23.2
≥ 1 GeV events
e-like 33.5
34.3
33.4
33.5
µ-like 24.8
26.9
25.1
23.7

Single Ring Angular Resolution (◦ )SK-I SK-II SK-III SK-IV< 1 GeV events
e-like
3.1
3.2
3.0
3.0
µ-like 1.9
2.1
1.9
1.9
≥ 1 GeV events
e-like
1.2
1.3
1.2
1.2
µ-like 0.8
1.0
0.9
0.9
Table 6.1: Vertex and angular resolutions for MS-Fit reconstruction of fully contained
neutrino events.

Figure 6.4: Vertex resolution for SK-IV sub-GeV single-ring events, separated intoµ-like and e-like.

6.2

Particle Identification

In CC neutrino interactions, the flavour of the neutrino corresponds to the flavour of
lepton produced in the interaction. Therefore to perform any sort of oscillation analysis,
we must have some sensitivity as to the particle type of, at least the most energetic ring(which usually corresponds to the initial lepton). Muon and electron Cherenkov ringpatterns look very different, due to their respective interaction methods in water.
Electrons (or positrons) passing through water will generate an electro-magnetic shower,
through the Bremsstrahlung (e− → e− γ) and then photon pair production (γ → e+ e− ).
As electrons and positrons produced in the shower go on to generate Cherenkov lightChapter 6. Event Reconstruction

69

Figure 6.5: Angular resolution for SK-IV fully contained sub-GeV single-ring events,
separated into µ-like and e-like.

of their own, the resulting ring pattern is very diffuse. High energy gamma rays also
create electro-magnetic showers and are indistinguishable from electrons.
The cross section for Bremsstrahlung is ∝ m12 , so for muons the primary energy loss isdue to particle ionization. This means that scattering and showering effects are muchless pronounced, and the muon tends to have a much sharper ring pattern.

Figure 6.6: Example Cherenkov ring patterns for an electron (left), and a muon(right). The top and bottom circles represent the top and bottom of the SK tank,and the long rectangle represents the barrel. Auxiliary event information is provided
around the edges (but it is not important to understand).

An additional distinguishing factor is the opening Cherenkov angle. Comparatively lightelectrons tend to have β = v/c ∼ 1, resulting in an opening angle of 42◦ . In comparison
the heavier muon has lower values of β at the same energy, resulting in a smaller opening
angle.
Tau lepton interactions can occur in SK, but it is difficult to distinguish these fromother ring types, as τ has a very short lifetime of 2.9 × 10−13 s, before which it decays
to primarily pions and other leptons. Recently SK published work on using a neuralChapter 6. Event Reconstruction

70

network analysis to make statistical measurements of τ appearance [75], however on an
event by event basis we are still unable to distinguish these.
It is possible to identify π 0 . The main decay mode is π 0 → γγ. As γ rings shower similarly to e± searching for event with two e-like rings, and further cuts on invariant mass
can lead to pure samples of π 0 . These are useful to reduce neutral current background,much of which consists of ν + N → ν + N 0 + π 0 type interactions. Charged pions π ±look very similar to muon rings, and thus cannot be separated. Protons from scatterednuclei can produce Cherenkov light, but they are heavy, and usually do not take a large
fraction of the momentum in a neutrino interaction, so they produce little light and are
hard to discern.
The primary type of particle identification computed in SK is the splitting of event
samples into e-like and µ-like rings, so first this algorithm will be discussed.

6.2.1

Electron-like and Muon-like rings

Each ring in an event is categorized as either an electron-like (e-like) or muon-like (µlike) ring. This is done by calculating expected charge on each PMT, for the assumptionthat the ring is e-like or µ-like, and then comparing to observed charge. Whichever ofthese fits to a maximal likelihood is taken as the Particle Identification (PID) of thatring.
The expected charge distribution for PMT i is calculated as
qiexp (e) = αe × Qexp (pe , θi ) × 1.51
R× r /L × f (Θi ) + qiscatt ,
ri
ei

(6.4)

where the variable definitions are as follows:• αe : a normalization factor• ri : the distance from fit vertex to i-th PMT• θi : the opening angle between the i-th PMT and the ring direction• L: the light attenuation length in water• f (Θi ): the i-th PMT acceptance as a function of incident photon angle• R: the radius of the virtual sphere (16.9 m)• Qexp (pe , θi ): The expected charge distribution for an electron as a function of
Cherenkov ring opening angle and momentumChapter 6. Event Reconstruction

71• qiscatt : Expected charge on the i-th PMT due to scattered photons
The expected charge Qexp (pe , θi ) and qiscatt are calculated using a MC simulation. The
1.5 is used to correct for the distance dependence of light intensity.
factor ( RL)

The equivalent predicted charge for muons is obtained as follows:
qiexp (µ) =sin2 θxi+ qiknock
αµ ×dθri (sin θxi + ri dx |x=xi )!
×1× f (Θi ) + qiscatt ,eri /L(6.5)

where the additional variable definitions are:• αµ : a normalization factor• x: muon track length• xi : the estimated track length of the muon at which Cherenkov photons received
by PMT i were emitted.
• θ: the Cherenkov opening angle of the muon at track position x• θi : the Cherenkov opening angle of the muon at track position xi• qiknock : Expected charge on the i-th PMT due to knock-on electrons
Knock-on electrons refer to electrons ionized by the passing muon. qiknock is calculated
via MC simulation.
Electrons generate an electromagnetic shower, and expelling all of their energy in a very
short distance, so the origin of Cherenkov photons is treated as point-like. For muons
which can potentially travel all across the tank, the track length must be considered in
calculation of expected charge. Hence the additional complication in the muon case.
Next, a likelihood distribution for the n-th ring is constructed as follows:Ln (e or µ) =Yexp
P qiobs , qi,n(e or µ) +θi <(1.5×θC )


X

exp 
qi,n
.
0

(6.6)

n0 6=nexp
Function P is defined in equation 6.3. qi,n0 gives expected charge in PMT i due to other

Cherenkov ring n0 , without assuming particle type.
This is then combined with another estimator based on the Cherenkov opening angle.
To use this combination we convert our likelihood value to a χ2 parameter, and then aprobability value.
χ2n (e or µ) = −2 log Ln (e or µ) + constant(6.7)

Chapter 6. Event Reconstruction

72

Figure 6.7: A depiction of how the muon track length is treated when calculating the
expected charge. When a muon travels distance dx, Cherenkov photons are emitted toan area of 2πr sin θ(dx sin θ + rdθ).

Then the probability of having an e-like or µ-like ring based on charge pattern isχ2 (e or µ) − min[χ2n (e), χ2n (µ)]2
Pnpattern (e or µ) = exp − n2σχ2 2

!
,

(6.8)

n

where σχ2n is the resolution of the χ2 function, defined by√

2N , where N is the numberof PMTs included in the ring.
The second estimator using Cherenkov opening angle is calculated as:
Pnangle (e or µ) = exp

2 !
θnobs − θnexp (e or µ)−.
2(δθn )2

(6.9)

Variables θnobs and θnexp (e or µ) are the observed and expected Cherenkov opening angles
for ring n, with the latter being calculated from reconstructed momentum assuming
either e-like or µ-like. δθn is the fitting error on the opening angle.

Chapter 6. Event Reconstruction

73

These estimators are then combined, to form the final probability functions, separately
for single-ring and multi-ring events,
angle
P1 ring (e, µ) = P1patternring (e, µ) × P1 ring (e, µ)(6.10)

pattern
Pmulti-ring (e, µ) = Pmulti-ring(e, µ).

(6.11)

Only the ring pattern estimator is used for multi-ring events, as it is difficult to estimate
the opening angle precisely. The final PID likelihood is then taken to be
PP ID =pp
− log P (µ) − − log P (e),(6.12)

where P is taken from 6.10 or 6.11 depending on how many rings the event has.

Figure 6.8: Output particle identification likelihoods (definition here 6.12) for SK-IV,
sub-GeV fully contained single-ring events. The left and right plots show true CC-quasi
elastic electron and muon events respectively.

6.3

Momentum Reconstruction

Particle momentum is reconstructed first by counting the number of p-e recorded ineach ring, and then converting this value to a momentum using the absolute energy
scale discussed in chapter 4. Firstly, for PMTs which contribute to multiple rings, theexpected p-e from the n-th ring is calculated as:
exp
qi,n
obs
obs
qi,n = qi × P exp ,n0 qi,n0(6.13)

obs is the fraction of p-e expected to belong to the n-th ring in the i-th PMT. All
where qi,np-e within a 70◦ half angle cone centred on the ring are taken to make up it’s momentum.

Chapter 6. Event Reconstruction

74

The timing of the hits is also restricted to be within −50 < t < +250 ns surrounding thepeak of ToF subtracted timing distribution. This is to avoid potential contaminationfrom delayed hits such as decay electrons. When summing these p-e, we correct forwater attenuation, PMT acceptance, and contribution from scattered light:
GM C
RT OTn =
Gdata

α×

NX
P MT 
icos Θiiobs
qi,n× e( L ) ×
f (Θi
r


−

NX
P MT

!
Si

.

(6.14)i

The variables are defined as:• α: normalization factor• Gdata,M C : the relative PMT gain for data and MC• θi,n : half-angle for PMT i assuming it is a member of ring n.
• ti : ToF subtracted hit timing for PMT i.
• NP M T : Number of PMT hits which pass the cuts θi,j < 70◦ and −50 < ti < +250ns• L: light attenuation length in water• ri : distance from vertex position to i-th PMT• f (Θi ): PMT light acceptance as a function of incident angle Θi .
• Si : expected p.e. for the i-th PMT from scattered photons
The calibration of Gdata,M C and L are discussed in chapter 4. RTOT is converted to a
momentum value using the absolute energy scale calibration. The reconstructed momenp
tum resolution for electron and muon events is estimated to be 1.7 + 0.7/ P (GeV/c)p% and 0.6 + 2.6/ P (GeV/c) % respectively [134].

6.4

Reconstruction of Single Ring π 0 events

You may recall that π 0 almost always decay to γγ, and this potentially gives us a nicemethod to distinguish these events. However, a common problem, especially for highenergy π 0 , is that the Cherenkov rings of the two γ may be very overlapping, or theenergy may be very asymmetric (that is, one ring with very high energy compared tothe other). It is possible in these cases that only a single ring will be found, and thus
the π 0 would be mis-classified as an electron. This is a problem, because a large numberof NC events, which are undesirable in oscillation analysis, have interaction modes suchas ν + N → ν + N + π 0 .

Chapter 6. Event Reconstruction

75

To try and identify these elusive π 0 events, an algorithm called POLfit was developed[135]. POLfit is applied to all single ring e-like events. Using the initial 1st ring, thebest candidate for a second ring is searched for. Using these two rings, the invariant
mass is calculated, and if this is similar to a π 0 (mπ0 = 135 MeV), the event will beclassified as such.
Using the same vertex as is calculated during APfit, POLfit searches for an additionalring by varying three parameters - the direction of the new ring (θ, φ in the co-ordinate
system of the first ring), and its energy fraction γ. For each step of these parameters,
POLfit calculates the expected charge on every hit PMT, and compares to the observed
charge. Then, a likelihood distribution is constructed, comparing the fits with a single
electron assumption, and with the π 0 → γγ assumption. The best fit second ring is
calculated by maximizing the likelihood difference:∆L(θ, φ, γ) = log(Lπ0 (θ, φ, γ)) − log(L1e (θ, φ, γ)).

(6.15)

Note that ∆L is not required to be positive, a second ring is always selected. Whenthe new ring is chosen, its momentum is calculated using the more sophisticated APfit
momentum reconstruction. The invariant mass is then calculated. For atmospheric neutrino analysis, if mπ0 > 100 MeV, and the 2-ring likelihood passes an energy dependent
cut, the particle is classified as a π 0 .

6.5

Neutron Reconstructions

Two additional independent fitters are used in neutron-tagging analysis.

Neutron-tagging refers to the algorithm that searches for 2.2 MeV γ-rays produced by neutron
capture on hydrogen. 2.2 MeV γ-rays on average produce 7-8 PMT hits, and are difficult
to reconstruct accurately. The vertex resolution of both of the neutron fitters is worse
than the average distance the neutrons have travelled from the neutrino interaction before capture. Thus for best vertex resolution, the APfit reconstructed neutrino vertex
should be used. However, as neutron capture events are known to not travel far fromthe neutrino vertex, these additional fits are valuable in reducing sources of background.

6.5.1

Bonsai Fit

The Bonsai (Branch Optimization Navigating Successive Annealing Iterations) fit is theprimary fitting tool used in Low Energy analysis in SK. For a more detailed description,
you may consult [136]. Bonsai is not optimized for fitting energy as low as 2.2MeV γ

Chapter 6. Event Reconstruction

76

rays - in fact the minimum amount of PMT hits supported to be fit by Bonsai is 10,
which many 2.2 MeV gamma candidates do not have. So in the context of neutron
tagging, all the PMT hits in a 1.3 µs window surrounding the 2.2 MeV γ candidate are
given as input to Bonsai.

6.5.1.1

Bonsai Vertex Reconstruction

Low energy particles will only travel a short distance from their source before thermalization, so it is assumed that all the PMT hits come from a point source. The fit must
be done entirely using hit timing information, as at low energy almost every PMT hit isa single photo-electron, and any charge variation is due to noise in the gain. The timing
residuals of each hit are calculated for all potential vertices, and the accepted vertex
is selected by maximizing a likelihood function based off these residuals. The timing
residual of a hit is calculated bytres = t − ttof − t0 ,(6.16)

where t is the raw PMT hit timing, ttof is the time-of-flight from that PMT to theprospective vertex, and t0 is the time of the interaction. The likelihood function isconstructed as

L(~x, t0 ) =

N
hit
X

log(P (tres )),(6.17)iwhere ~x is the prospective vertex, and P (tres ) is the probability density function of
the timing residuals, measured by LINAC calibration data (see figure 6.9). ~x is theniteratively varied, until the maximum value of L is found.

6.5.1.2

Bonsai Energy Reconstruction

As the neutron capture gamma ray is mono-energetic (2.2 MeV), energy reconstruction
is a potentially useful method of distinguishing signal from background.
As mentioned before, at low energy the charge collected by a PMT is not very useful
information, so energy reconstruction is calculated based off variables Nhit and Nef f the number, and effective number of hit PMTs in a 50ns time window. Nef f is definedChapter 6. Event Reconstruction

77

Figure 6.9: The timing residual probability density function measured by LINAC
calibration data.

as:
Nef f =

N
hit 
X
i


ri
1
Nall
λef f
(Xi + tail − dark ) ×××e× Gi (t) ,Nalive S(θi , φi )(6.18)

where the variables are as follows:• Xi - a factor used to correct for PMTs which are likely to observe more than single
photon. The definition differs depending on xi = fraction of hit PMTs in a 3x3
patch around PMT i. If xi < 1, Xi =1
log 1−xxii, and if xi = 1, Xi = 3.

• tail - accounts for hits that have scattered or reflected and thus fall outside the50ns window:
tail =

N100 − N50 − Nalive × Rdark × 50 ns,
N50(6.19)

where Nj is the number of hits in j ns, Nalive is the number of functioning PMTs,
and Rdark is the dark noise rate.

Chapter 6. Event Reconstruction78• dark - correction applied due to dark noise:dark =

Nalive × Rdark × 50 ns.
N50(6.20)• Nall /Nalive - corrects for the number of dead PMTs.
• S(θi , φi ) - corrects for the acceptance of the i-th PMT, given an incident angle of
θi , φ i .
• ri /λef f - ri is the distance travelled from reconstructed vertex to the i-th PMT,and λef f is the water transparency, the measurement of which is described inchapter 4.
• Gi (t) - this factor accounts for the 2-3% per year increase in PMT gain.
Using this calculated value of Nef f , the reconstructed energy can be calculated. The
relationship between these two variables was precisely calibrated by LINAC and MC
data.
2
−9 4
−6 3
Erec = 0.82 + 0.13Nef f − 1.11 × 10−4 Nef
f + 1.25 × 10 Nef f − 3.42 × 10 Nef f (6.21)

Bonsai energy fit returns a failure state for event vertices reconstructed out of the innerdetector.

6.5.2

Neut-Fit

Neutron Fit (Neut-Fit) is a simple vertex fitter designed to produce a vertex fit withvery low numbers of PMT hits. As an input, it takes a neutron candidate (this will be
described in detail in chapter 8), which consists of N10 = 7 − 15 hits in a 10 ns window.
Each candidate is selected by ToF subtracting hits with respect to the APfit vertex, so alarge bias exists towards this vertex. Using the initial N10 hits, a new vertex is searchedfor. For each prospective vertex, ToF subtraction for all the hits is recalculated. The
best vertex is selected as the one with the minimum trms for all N10 hits:strms (~x) =where tmean =

PN10
i

PN10
i

(ti − tmean )2,
N10(6.22)

ti /N10 , and ti is the hit timing after ToF subtraction to vertex ~x.

This process is then repeated with an increasingly fine grid search, until the best fit isdetermined to within 0.5 cm.

Chapter 6. Event Reconstruction

6.6

79

Upward-going Muon Reconstruction

The upward-going muon event sample is broken down into two sub-samples: stoppingmuons, which use up all their energy before leaving the inner detector, and through-going
muons, which pass through the inner detector, and exit through the top of the detector.
Through-going muons are further divided into those which produce electro-magnetic
showers, and those which do not, the former having a higher energy spectrum.

Figure 6.10:

The energy distributions of upward-going muon showering, nonshowering, and stopping events.

The re-construction of these events is done using a hybrid-fitter, named Precise-fit.
Precise-fit is only applied to events with greater than 8000 p.e., below which there is
too little light for a precise fit, and less than 175,000 p.e., above which the ID PMTs
become saturated.
Precise fit uses adaptations of MS-fit and TDC-fit, as well as a third package, OD-fit.
OD-fit is used for high energy showering muons, where it is difficult to reconstruct a
Cherenkov ring due to a very large amount of energy being deposited in the ID. In
these cases, an OD cluster finding algorithm is used, and then the particle vertex anddirection are estimated based on clusters at the presumed entrance and exit positions

Chapter 6. Event Reconstruction

80of the particle. The result from the most appropriate of these three fitters, is output by
precise-fit.

Figure 6.11: The vertex resolution of upward-going muon fitter precise-fit, based off
a 40 year MC study.

The track length of upward going muons is calculated by monitoring dE/dx, along themuon track. The end of the muon track is defined as the point where the energy loss in
a 1 m interval drops to below 40% of the average value from the first 1.5 m of the track.

Figure 6.12: The calculation of the end-point of a stopping muon track. The dottedline shows the 40% of the average value in the first 1.5 m of the track.

Chapter 7

Data Reduction
The Super-Kamiokande inner detector high energy trigger fires ∼ 106 times each day.
The vast majority of these events are background - cosmic ray muons, radioactive gammas from radon contamination, spallation events, PMT flasher events and more. The
goal of the reduction process is to remove these background events, and leave behind apure sample of atmospheric neutrinos.
There are three main reduction processes, depending on the topology of the events.
Fully Contained (FC) reduction relates to events that are fully contained within the
inner volume of the detector. That is, the neutrinos interact somewhere in the ID, andthe produced lepton does not have enough energy to travel outside. Partially Contained(PC) reduction searches for events which have a vertex inside the inner detector, butsome of the produced particles have crossed over into the outer detector. These are
generally higher energy muon events, as the electron deposits all its energy in a very
short distance.
The final sample of atmospheric neutrinos is named UPMU, a shorthand for “UPwardgoing MUons”. This sample is comprised of muons, which enter from the bottom of
the tank. As there is no feasible chance a cosmic ray muon could pass all the way
through the Earth and interact with SK, these muon events must have been producedby neutrino interactions in the rock, shortly before entering the SK tank. This is limited
to upward going muons, because similar events going downwards cannot be distinguishedfrom cosmic ray background. The UPMU sample is further split into through-going andstopping muons, indicating whether the muon stopped in the tank (having deposited allits energy), or passed through. The highest energy events we observe are in the UPMU
through-going sample.

81

Chapter 7. Atmospheric Neutrino Data Reduction

82

A graphical depiction of each of these event types is shown in figure 7.1. The neutrinoenergy spectrums of each of these event categories can be seen in figure 7.2.
In some cases the reduction process is different, depending on the SK generation. The
primary reasons for this are the reduced number of ID PMTs installed in SK-II (after
the accident), and the improved front-end electronics in SK-IV.

Figure 7.1: The different reduction types for selecting atmospheric neutrino events

Figure 7.2: The energy spectrum for each sample of events, after reduction.

Chapter 7. Atmospheric Neutrino Data Reduction

7.1

83

Fully Contained Reduction

The fully contained reduction is split up into 5 steps, each effective at removing a
different category of background events.

7.1.1

1st Reduction (FC1)This reduction step is designed to filter out most of the obvious low energy and cosmic
ray muon backgrounds. The requirements to pass this stage are as follows:• PEtotal > 200 p-e (100 p-e for SK-II). PEtotal represents the total charge collectedby ID PMTs inside a 300 ns time window surrounding the trigger time.
• NHITA800 ≤ 50. NHITA800 is the total number of OD hits within an 800 ns
window surrounding the trigger time. Many OD variables are labelled with “A”,
which stands for “anti-detector”.
A 200 p-e event is roughly equivalent to a 22 MeV/c electron, so this cut removes all
solar neutrinos, reactor neutrinos, and many other low energy backgrounds. The lowest
energy atmospheric neutrinos may produce around 30 MeV/c electrons, due to the flux
rigidity cut off.
The output of FC1 is typically ∼ 1500 events/day.

7.1.2

2nd Reduction (FC2)

The reduction criteria for the second step are:
• NHITA800 ≤ 25, for events where PEtotal < 100, 000 p-e (50,000 p-e for SK-II).
• PEmax /PE300 < 0.5 - PEmax is the maximum p-e observed in a single PMT within
a 300 ns timing window around the event, and PE300 is the total p-e from all PMTs
in this period.
The first of these cuts removes further cosmic ray muon events, which may have been too
low energy to be identified by FC1. The second cut removes events where the majorityof the charge in the event is coming from a single PMT. These type of events are likely
to be due to flashing PMTs, or electronics noise.
The output for FC2 is ∼ 200 events/day.

Chapter 7. Atmospheric Neutrino Data Reduction

84

Figure 7.3: FC2 distributions of NHITA800 , for data, MC, and then the final selection
with data and MC superimposed. The MC is a pure atmospheric neutrino sample, so
the second plot shows the efficiency of this cut. In the final sample the events which
remain in the cut region have either PEtotal > 100, 000, or the OD trigger did not fire.

7.1.3

3rd Reduction (FC3)

The third reduction consists of a series of more specifically targeted cuts, focusing on
removal of cosmic ray muon events that have little OD activity, and further reductionsof electronic noise events. After FC3 has been applied, ∼ 45 events/day remain.

7.1.3.1

Through-going muon cut

Muons events that go through the tank without depositing much energy in the OD may
be rejected here. These are generally high energy events, so the pre-criteria for this cut
to be considered is that at least 1000 ID PMTs are hit, with each detecting more thanChapter 7. Atmospheric Neutrino Data Reduction

85

Figure 7.4: FC2 distributions of PEmax /PE300 . Plots of data, MC and then the
final selection with data and MC superimposed are shown. The MC consists of pureatmospheric neutrino events.

230 p-e. If this condition is met, a through-µ fitter is applied. The fit assumes that theevent is a through-going muon, setting the entrance point as the first hit PMT in theentrance cluster, and the exit point as the PMT at the centre of the exit cluster. For
these entrance and exit points, the expected hit timing Ti for each PMT is calculatedusing a MC simulation. From this a goodness-of-fit parameter is given by:X
1
goodness = P 1 ×
i σi2

i





1
(ti − Ti )2
exp
−
,
σi2
3σi2

(7.1)

where ti and σi are the i-th PMT timing and timing resolution respectively. The requirements to pass this cut are:• goodness > 0.75

Chapter 7. Atmospheric Neutrino Data Reduction

86• NHITAin ≥ 10 or NHITout ≥ 10
where NHITAin and NHITAout are the numbers of hit OD PMTs in 800 ns, within 8 mof the entrance and exit PMTs respectively.

7.1.3.2

Stopping-muon cut

Stopping muons are cut in almost the same way as the through-going muons, except they
do not have an exit point. To perform the fit this time, we must reconstruct the muon
direction, which is found by maximizing the number of p-e observed by PMTs in a 42◦
Cherenkov opening angle from the entrance point. The goodness of fit is calculated again
by equation 7.1. If either of the following cut criteria are met, the event is discarded asa stopping muon:• goodness of stopping muon fit ≥ 0 and NHITAin ≥ 10
• goodness of stopping muon fit > 0.5 and NHITAin ≥ 5

7.1.3.3

Cable Hole Muons

To connect to the PMTs around the tank, SK has twelve holes through which cablingis thread. OD PMTs cannot be installed in these positions, so it is possible for cosmic
ray muons to pass into the inner detector without interacting with enough PMTs in the
OD to be vetoed. To circumvent this, plastic scintillator detectors are installed in fourof the twelve cable holes. The event is cut if:• Cable hole veto counter is hit• Vertex reconstruction is within 4m of the cable hole counter.
To account for the holes where no veto counters are installed, an alternative set of cuts
was introduced in SK-IV:• goodness ≥ 0.4 (as defined by 7.1)
• PEtotal > 4000 p-e• reconstructed as downward going (cos θ < −0.6)• distance from reconstructed vertex to cable hole < 2.5 m

Chapter 7. Atmospheric Neutrino Data Reduction

87

Figure 7.5: A schematic of the cable hole veto counters in SK

7.1.3.4

Flashers

Flasher refers to the phenomena where a PMT may misfire and result in many repetitive
internal discharges, of both itself and nearby PMTs. Flasher events tend to have abroader hit timing distribution than neutrino events, so the cut to remove these is:• SK-I: NMIN100 ≥ 15 or NMIN100 ≥ 10 if Nhit ≤ 800• SK-II to SK-IV: NMIN100 ≥ 20
where NMIN100 is the minimum number of hits in a 100 ns sliding time window from+300 to +800 ns after the trigger. Nhit is the number of ID PMT hits. Later reduction
steps have other, more specific cuts for removing flashers.

7.1.3.5

Coincidence Muon

Sometimes, an event will trigger from a low energy event, but then shortly afterwards
a cosmic ray muon will also pass through the tank. These events are referred to as

Chapter 7. Atmospheric Neutrino Data Reduction

88

Figure 7.6: ID timing distributions of a typical flasher (i), and a typical neutrino (ii)event. The two lines indicate the timing region scanned in calculating NMIN100 .

coincidence muons. They avoid the previous low energy cuts, as the muon contributes alarge amount of charge to the ID, but also the OD veto cut is circumvented, because themuon arrives later, outside of the cut timing window. The cuts to remove these eventsare:• NHITAof f ≥ 20
• PEof f > 5000 p-e (2500 for SK-II)
where NHITAof f is the number of OD hit PMTs in the off-timing window, +400 to+900 ns after the trigger time. PEof f is the total ID hit PMTs in the same off-timing
window.

7.1.3.6

Low Energy Events

This is a more tight cut to remove low energy events than we saw previously. Firstly
hits are ToF corrected to a reconstructed vertex. Using a 50 ns scanning time window,
the largest value of NHIT50 is recorded. Cut the event if:• NHIT50 < 50 (25 for SK-II)Chapter 7. Atmospheric Neutrino Data Reduction

7.1.4

89

4th Reduction (FC4)

Even after FC3, a significant amount of PMT flasher events remain in the data-sample.
Often after a PMT becomes a flasher, it will continue to produce flashing events whichhave a highly correlated charge pattern, over a short space of time. FC4 attempts to
reduce these, by using a pattern matching algorithm. This algorithm is run as follows:

1. The ID tank wall is divided up into 1450 partitions, containing between 6 and 9
PMTs each.
2. Correlation r between two events is calculated for each patch, by
r=

A
B
B
1 X (QAi − hQ i) × (Qi − hQ i),
N
σA × σB(7.2)

iwhere N is the total number of patches. A and B are the two events being compared. hQi is the average charge, and σ is the charge resolution.
3. The distance DISTmax between the PMT with the maximum charge in each event
is calculated. If DISTmax < 0.75 m then an offset is applied to r = r + 0.15.
4. If r > rth , the two events are marked as a match. rth is a variable quantity basedon the total ID charge for each event.

This procedure is carried out attempting to match with 10,000 events surrounding thetarget event. Events are cut depending on the number of total matches Nmatch , andcorrelation r. After FC4, an average of ∼ 18 events/day remain.

7.1.5

5th Reduction (FC5)FC5 removes a final set of very specific background types - invisible muons, coincidence
muons, and long tail flashers. After FC5 has been applied, ∼ 16 events/day remain.

7.1.5.1

Invisible Muons

Invisible muons refer to muons which have entered the detector some time before thetrigger, and decayed to an electron. SK triggers on the signal from the decay electron,
without being aware of the initial muon. These events can be identified as there is likely
to be a signal in the OD from the muon, before the primary trigger. The cuts are as
follows:

Chapter 7. Atmospheric Neutrino Data Reduction

90• PEtot < 1000 p-e (500 for SK-II). PEtot is the total number of p-e in the ID.
• NHITACearly + NHITAC500 ≥ 10 if DISTclust < 500 cm.
NHITACearly > 9 otherwise.
NHITAC is the number of hit OD PMTs in the largest cluster, and NHITACearlyis calculated in a sliding 200 ns time window between -8800 ns and -100 ns relative
to the primary trigger. NHITAC500 is the largest number of hits in an OD clusterfrom -100 to +400 ns. DISTclust is the distance (cm) between these two OD hit
clusters.

7.1.5.2

Coincidence Muons

This is a tighter cut for the same coincidence muons defined in FC3, that is, muons
which pass through the detector it a late timing period after it has triggered off a low
energy event. The cut criteria for the removal of these are• PE500 < 300 p-e (150 for SK-II). PE500 is the total of ID p-e in the time interval-100 to +400 ns.
• PElate ≥ 20 p-e. PElate is the maximum amount of p-e recorded in the OD, using
a 200 ns scanning time window from +400 to +1600 ns.

7.1.5.3

Long Tail Flashers

Similar to the flasher cut in FC3, but with tighter criteria. The simple vertex fitter
Point-Fit’s (as described in chapter 6) goodness output is used - a well fit Cherenkov
ring is unlikely to be a flasher event. The criteria are as follows:• NMIN100 ≥ 6 if the goodness of Point Fit is < 0.4. As before, NMIN100 is theminimum number of ID PMT hits in a 100 ns scanning time window between +300
and +800 ns.
• SK-II to SK-IV only: NMIN100 ≤ 5 if the goodness of Point Fit < 0.3.

7.1.6

Fiducial Volume Selection

There are three final cuts applied to the FC sample, to ensure our neutrino sample is as
pure as possible:

Chapter 7. Atmospheric Neutrino Data Reduction

91

Figure 7.7: FC5 Coincidence muon cut variable PElate . From top to bottom, theplots show the data, atmospheric neutrino MC, and then the final selection of eventswith data and MC superimposed. It is also required that PE500 < 300 p-e, so there area few events remaining in the final sample that have PElate > 20.

• dWall > 200 cm, where dWall is the distance from the reconstructed vertex to theclosest tank wall. This is to further veto events which have entered from outside
the tank, and events due to radioactive impurities in the tank wall.
• NHITAC < 16 (10 for SK-I). NHITAC is the number of hits in the largest chargecluster in the OD. Again, the purpose is to remove any events entering from outside
the tank.
• Evis > 30 MeV, where Evis is the sum of the reconstructed momentum of all rings,
assuming they are produced by electrons.
After this final selection, the remaining event rate is around 8 atmospheric neutrinos /day. This sample is known as the FCFV (fully-contained fiducial volume) sample.

Chapter 7. Atmospheric Neutrino Data Reduction

92

Figure 7.8: The event rate for each of FC1-5, and FCFV reductions, for all of SK-IV.
The increasing event rate is due to the PMT degradation causing gain increase of 2%
per year, and is accounted for in analysis. The event rate spike around March 2009 isdue to faults in the HV system.

7.1.7

FC Reduction Summary

The efficiency of each reduction step is calculated by applying the procedure to a purely
atmospheric neutrino MC, as described in a chapter 5. The systematic error for the
reduction efficiency is estimated by comparing distributions of data vs MC for eachreduction step. These errors are 0.2%, 0.2%, 0.8% and 0.3% for SK I-IV respectively.
The background in the final sample is determined by eye-scanning all of the selected
events. The primary sources of background are flasher events remaining in the e-likesamples, and cosmic ray muon events remaining in the µ-like samples. These samples arefurther split into sub-GeV, where Evis < 1330 MeV, and multi-GeV, where Evis ≥ 1330
MeV.
A summary of the FC reduction process can be seen in table 7.1.

7.2

Partially Contained Reduction

The partially contained sample consists of neutrino events, whose vertex is reconstructedin the ID, but also have some activity in the OD that prevents them from being fully
contained. OD segmentation was installed after SK-II, so there are some differences in
the reduction from SK-III onwards. Cosmic ray muon and upward-going muon background is significantly more difficult to remove in PC, as there are expected to be somehits in the OD. This is reflected in the complexity of the reduction steps.

Chapter 7. Atmospheric Neutrino Data Reduction
FC
Reduction
FC1
FC2
FC3
FC4
FC5
FCFV
Sub-GeV e-like
Sub-GeV µ-like
Multi-GeV e-like
Multi-GeV µ-like
Livetime (days)
MC Prediction
Data
Events / day

93

SK-I

SK-II
SK-III
SK-IV
Selection Efficiency (%)
100.0
99.97
100.0
100.0
100.0
99.92
99.98
99.99
99.93
99.78
99.81
99.82
99.29
99.38
99.30
99.00
99.26
99.30
99.24
98.95
99.25
99.95
99.62
99.19
Final Sample Background (%)
0.1
0.4
0.1
0.1
0.2
0.2
0.3
0.1
0.5
0.2
0.2
0.1
0.3
0.2
0.3
0.1
Total Number of Events
1489.2798.6
518.2
1775.611878.76407.64198.514251.2
12299
6610
4355
14444
8.26±0.07 8.28±0.10 8.41±0.13 8.13±0.07

Table 7.1: Summary of the efficiencies, remaining background, and number of events
remaining after the FC reduction process.

7.2.1

1st Reduction (PC1)

Similarly to FC1, PC1 rejects through going cosmic ray muon events, and low energy
events. For SK-I and II the criteria are as follows:• PEtot ≥ 1000 p-e (500 for SK-II)• TWIDA ≤ 260 ns (170 for SK-II)• NCLSTA ≤ 1 (SK-I only)
PEtot is the total charge in the ID, which for a muon is equivalent to a track length of∼ 2 m. TWIDA is the width of the hit timing distribution in the OD PMTs. This isused to reject events which have entered the tank through the OD, and subsequently
exited, which will have a broader timing distribution. This criteria is tightened in SK-IIdue to the increased quantum efficiencies of OD PMTs at this time. NCLSTA is thenumber of clusters of hits in the OD, an additional cut to reject through-going cosmic
rays.
From SK-III onwards, the OD segmentation made through going particle rejection more
efficient, so the cut criteria were altered:• PEtot ≥ 1000 p-e

Chapter 7. Atmospheric Neutrino Data Reduction

94• NHITAtop < 10 or NHITAbottom < 10
• NHITAendcap < 25 or NHITAside < 70
• ODRmean < 2100 cm, if NHITA < 20 in a 500 ns time window.
NHITAlocation is the number of hits in that area of the OD. “Endcap” refers to the
top + bottom segments. Thus the NHITAx cuts are used to remove muons that travel
through the top and bottom of the tank, and also muons which travel through the top
or bottom and side of the tank (referred to as through-going muons and corner-clippingmuons respectively).
The final cut variable, ODRmean is the average distance between every pair of hits in
the OD.

ODRmean =1
Npair

NHITA−1
X NHITAX
i|~xi − ~xj |

(7.3)

j=i

This is another measure of the spatial spreading of the PMT hits, specifically useful for
events with less deposited energy.

7.2.2

2nd Reduction (PC2)PC2 is used in combination with a clustering algorithm, using the number of hit clusters
to reject through-going, corner clipping and stopping muons. The OD walls are divided
into 11 × 11 patches, and for each of these the total charge is added up. Clusters of hits
are selected based on the charge gradient between neighbouring patches. This is also
defined for the ID, with 21 × 21 patches. A graphical depiction of this algorithm can beseen here 7.9.
These cuts are fairly involved, so the variables relevant for these reduction cuts will be
described first, as follows:• NCLSTA2[x] - the number of clusters including > x PMTs.
• NCLSTA2(2)[x] - the number of hits in the 2nd OD hit cluster requiring > x
PMTs.
• NHITACmin - the minimum number of clustered hit PMTs in the top or bottom,
and side regions of the OD.

Chapter 7. Atmospheric Neutrino Data Reduction

95

Figure 7.9: Explanation of OD clustering algorithm. The charge in each patch is
proportional to the size of the circle. The charge is added to a cluster in the directionof the largest charge gradient.

• PE200 - the number of p-e within 200 cm of the highest charge PMT, in the ID hit
cluster which is closest to the OD hit cluster.
• NHITAlocation - the number of OD PMT hits in that region.
• MAX(NHITAside ) = exp(a − b × NHITAside ). If NHITAside < 75, a = 5.8, b =0.023. If NHITAside ≥ 75, a = 4.675, b = 0.008.
• NHITAC2 - the number of OD hit PMTs in the 2nd cluster.
The exact cut criteria are quite different across the SK generations, so will be described
separately.

7.2.2.1

PC2: SK-I• NCLSTA2[6] ≤ 1• NCLSTA2(2)[6] ≤ 1• NHITACmin < 7• If NCLSTA2[6] = 1, require that PE200 > 1000 p-e

7.2.2.2

PC2: SK-II• NCLSTA2[6] ≤ 1

Chapter 7. Atmospheric Neutrino Data Reduction

96• NCLSTA2(2)[10] ≤ 1• NHITAendcap < min[20, MAX(NHITAside )]• NHITAC2 < 12 + 0.085 × PE200

7.2.2.3

PC2: SK-III and SK-IV• NCLSTA2(2)[10] ≤ 1• NHITAendcap < min[20, MAX(NHITAside )]For SK-III and IV, OD segmentation makes PC1 very efficient, so the PC2 cuts can besomewhat relaxed.

Figure 7.10: The number of hits in the 2nd largest OD charge cluster. If there are
two large clusters, they likely represent entrance and exit points in an event. As PC
events must originate in the ID, these events may be cut. The MC shows a simulated
pure atmospheric neutrino sample, and the final sample is the agreement between data
and MC for this variable after the final PC selection.

Chapter 7. Atmospheric Neutrino Data Reduction

7.2.3

97

3rd Reduction (PC3)

PC3 consists of two specific cuts. Similar to FC3, the first is to remove PMT flashers
from the data sample. Flasher events tend to have a more broad timing distribution
than neutrino events. The second cut, applied only for SK-I and II, is used to reject
stopping muon events. For SK-III and IV, this cut has been moved into PC5. Thus, thecriteria for rejection at this stage is:• NMIN100 > 14 or, if the number of hit ID PMTs < 800, NMIN100 > 9.
NMIN100 is defined as the minimum number of ID PMTs hit in a 100 ns scanningwindow from +300 ns to +800 ns after the primary trigger.
• NHITAin > 10 (SK-I and II only)
NHITAin is defined as the number of hit OD PMTs within 8 m and 500 ns of thefitted ID vertex (as fitted by Point-fit).

7.2.4

4th Reduction (PC4)PC4 has two very different versions for SK-I and II, and SK-III and IV. They will be
described separately.

7.2.4.1

PC4: SK-I and II

Despite many efforts so far to remove them, there are still persistent stop, throughgoing and corner-clipping muons contaminating this sample, typically with relatively
little OD activity. This reduction step uses information from multiple fit programs totry and extract these events from the PC neutrino sample. The through-going muon
fitter is applied, with the expectation that the fit result will be sub-optimal and havelower goodness for true PC events, as their vertices are not necessarily at the tankwall. Point-fit is also used, which gives a reasonable estimate for the vertex of boththrough-going muons and PC events. Events may criteria are as follows:• d~pf it .d~P M T > −0.8,
where d~pf it is the direction as reconstructed by Point-fit, and d~P M T is the direction
from the reconstructed vertex, to the earliest saturated PMT. This cut removes
events where the earliest hit PMT is in the opposite direction to that the primary
Cherenkov cone is pointing in, implying the particle may have entered from outside
the tank.

Chapter 7. Atmospheric Neutrino Data Reduction

98• DCORN < 150 cm.
DCORN is the distance between the Point-fit vertex, and the nearest tank corner.
This aims to reject corner-clipping muons.
• TLMU > 30 m, and goodness of the through-muon fit > 0.85.
TLMU is the estimated track length of the event, when modelling it as a throughgoing muon. Events with long track lengths are rejected.

Figure 7.11: These are the selection criteria in PC4 for SK-III and IV, used to rejectmuon events that originated outside the inner detector. The dotted line shows the data,
after the output of PC3. The solid line shows MC, which may be thought of as the
expectation for a pure PC neutrino sample.

7.2.4.2

PC4: SK-III and IV

Here a new muon fitting program is used, which attempts to classify muon events as oneof: stopping muon, through-going muon, corner-clipping muon, or multiple muons. To
this end, five selection criteria are calculated:
1. angleµ < 90◦ .
This is the angle between the fitted direction, and the direction between Point-fit
vertex and the centre of the largest OD charge cluster.
2. dotprodµ > −0.8.
The dot product of the fitted direction, with the direction between Point-fit vertex
and the earliest saturated ID PMT.

Chapter 7. Atmospheric Neutrino Data Reduction99

3. lengthµ < 1750 cm.
The length of the fitted muon track.
4. goodnessµ < 0.52.
The goodness of fit returned by the muon fitter.
5. cornerµ ≥ 300 cm.
Distance between the fitted entrance point and the tank corner.
An additional selection criteria applies for only stopping muon events:• goodnessµ < 0.5 or ehitµ < 10,where ehitµ is the number of OD hits within 8 m and 500 ns of the fitted muon
ID entrance point.
For classification as a through-going muon, the event must pass 4/5 of these selections.
For stopping muon classification, 4/5 passes are still required, however these must include selection 2), and also the additional stopping-muon selection. Corner-clipping andmultiple muon classifications require only 2/5 of the above criteria to be passed. If an
event passes the selection for any of these types of muon events, it is rejected from thePC sample.
An additional cut imposed in PC4 is the following:• PEtot < 2900 p-e,
where PEtot is the total number of p-e in the ID.
This corresponds to a ∼ 500 MeV/c muon, and serves to reduce more low energy background events. In a true PC event, the muon must have a minimum momentum of ∼ 700
MeV/c to even reach the OD, so there is not much efficiency loss by this cut.

7.2.5

5th Reduction (PC5)In PC5, further cuts are imposed to reduce specific subsets of each background whichhave failed to be removed thus far. Again, these cuts are quite different for SK-I and II,
compared to SK-III and IV, so they will be discussed in separate sections.

Chapter 7. Atmospheric Neutrino Data Reduction
7.2.5.1

100

PC5: SK-I and SK-II

Low energy event cut
Reduction of low energy background that was not performed in PC4.
• PEtot < 3000 p-e,
where PEtot is the total number of p-e in the ID.
Through-going muon cut A
Events satisfying these additional criteria are removed as through-going muons. The
OD cluster finding algorithm from PC2 is used except this time the OD is divided into
6 × 6 patches instead of 11 × 11.

• DISTclust > 20 m,
where DISTclust is the distance between the two OD clusters with the highest
charge.
• PEAC2nd ≥ 10 p-e,
where PEAC2nd is the number of p-e in the 2nd largest OD hit cluster.
• NCLSTA5 ≥ 2,
where NCLSTA5 is the number of clusters in the OD with more than 9 hits.

For events that have passed all cuts up until now, a full reconstruction is performed, so
final rejections can be calculated from variables output from this.
Through-going muon cut B
This cut is to reject cosmic ray muon events which travel from the top of the tank tothe bottom, staying close to the ID wall for the whole journey. The light collection forevents near the tank wall is poor, so they often escape previous reductions. If all thefollowing criteria are satisfied, the event is rejected:• NHITAtop ≥ 7 and NHITAbottom ≥ 7,
where NHITAlocation is the number of OD PMTs hit within an 8 m radius sphereof that location.
• PEAtop ≥ 10 p-e and PEAbottom ≥ 10 p-e.
PEAlocation is defined the same as NHITAlocation , but counts the total p-e insteadof the number of hits.

Chapter 7. Atmospheric Neutrino Data Reduction

101• 0.75 < TDIFFA × c/40 m < 1.5,
where TDIFFA is the difference between the average hit time for the two spheres
defined by NHITAlocation .
Through-going muon cut C
The final cut for through-going muons, is still looking for charge clusters near the entrance and exit points, but now the event reconstruction is based off the MS-fit result.
The entrance and exit points for the muon are found by extrapolation from the MS-fit
vertex and direction. If both of the following conditions are met, the event is cut:
• NHITAin ≥ 5 and NHITAout ≥ 5.
NHITAin,out gives the number of hit OD PMTs within an 8 m radius sphere of theentrance or exit point.
• 0.75 < TDIFFA × c/TRACK < 1.5. TRACK is the distance between the entrance
and exit points, by MS-fit calculation.
Stopping muon cut A
The entrance point is reconstructed by extrapolating from the MS-fit vertex and direction. Any event satisfying the following criteria is rejected as a stopping muon.
• NHITAin ≥ 10.
NHITAin gives the number of hit OD PMTs within an 8 m radius sphere of theentrance point.
Stopping muon cut B
Using both TDC-fit, and MS-fit, the Cherenkov opening angle is calculated. Stopping
muon events are expected to have higher energy, thus a larger opening angle than PC
events. If the event passes this cut, it is rejected:• θT DC > 90◦ or θM S > 90◦ ,where θx is the Cherenkov opening angle as calculated by fit x.
Stopping muon cut C
The stopping mu fitter is used to try and fit the event. As PC event vertices are
significantly displaced from the tank wall, but stopping muons require that the vertex
is on the wall, the fit will be more likely to reconstruct incorrectly for PC events. If thefollowing criteria are all satisfied, the event is rejected as a stopping muon:• stopping muon fit goodness > 0

Chapter 7. Atmospheric Neutrino Data Reduction

102• PEcone /PEtot ≥ 0.6,
where PEcone is the total p-e count within a 42◦ Cherenkov cone from the stopping
muon fit vertex. PEtot is the total ID p-e If the Cherenkov cone contains a large
amount of the total charge, the event is more likely to be a stopping muon.
• NHITAin > 6,
where NHITAin is the number of hit OD PMTs within 8 m of the fit vertex.
Cable hole muon cut
If a cable hole veto counter is hit, and the event reconstruction matches the direction of
the cable hole, the event is rejected:• a veto counter is hit• d~T DC .d~veto−vertex > −0.8,
where d~T DC is the reconstructed ring direction by TDC fit, and d~vertex−veto is the
direction from the hit veto counter to the reconstructed vertex.
Corner clipping muon cut A
Corner clipping muons typically only have a small cluster of charge in the ID, before they
exit through the tank wall. Thus the vertex reconstruction assuming it is a primarily ID
event, will often be very poor. This cut compares the predicted track length using the
MS-fit reconstruction, to the predicted track length by visible energy (recall that muonslose ∼ 2 MeV/cm. Thus the cut criteria to reject these events is:• if (TRACK > 1500 cm) and Evis /2 < TRACK - 1500,
where TRACK is the track length estimated as the distance from MS-fit vertex
to extrapolated exit point, and Evis /2 is an approximation for the track length incm.

7.2.5.2

PC5: SK-III and SK-IV

The final PC reduction for SK-III and IV consists of two types of cuts - “hard” cut and“soft” cuts. All events are required to pass all of the hard cuts. If events fail > 1 softcut, they will be removed from the sample. The hard cuts are as follows:

1. Through-going muon cut AChapter 7. Atmospheric Neutrino Data Reduction

103

2. Through-going muon cut B
3. Stopping muon cut B
4. Cable hole muon cut
5. Corner-clipping muon cut B

The first four of these are identical to those described in the SK-I and II reductions.
The final cut is specific to SK-III and IV. The soft cuts are:

1. Through-going muon cut C
2. Through-going muon cut D
3. Stopping muon cut A
4. Stopping muon cut C
5. Stopping muon cut D
6. Stopping muon cut E
7. Corner clipping muon cut A
8. Decay electron cutof which if two or more are failed, the event is cut. Soft cuts (1), (3), (4) and (7) havealready been covered in the SK-I and II PC5 reduction. The hard and soft cuts whichhave not yet been introduced will be defined below.
Corner clipping muon cut B
This corner clipping cut is used to remove muons which just tag the corner of the tank.
This is a hard cut, and must be passed for the event to be accepted.
• DCORN > 150 cm,
where DCORN is the distance between the vertex reconstructed by point-fit, and thenearest tank corner.
Through-going muon cut DThis cut removes through-going muons with two large clusters of charge in the OD.

Chapter 7. Atmospheric Neutrino Data Reduction

104

Figure 7.12: These plots show the distance from the reconstructed muon vertex tothe nearest corner of the tank. MC is a sample of pure atmospheric neutrino. This cut
is used to remove corner-clipping muons. The bottom plot shows data-MC agreement
after the final selection of PC events.

Figure 7.13: NCLSTA5(x) gives the number of OD PMTs inside the x-th highest
charge cluster. Large values for two clusters suggests a through-going muon event,which is then cut.

• NCLSTA5(1) ≥ 10 and NCLSTA5(2) ≥ 17,
where NCLSTA5(x) is the number of OD PMTs inside the x-th highest chargecluster.

Chapter 7. Atmospheric Neutrino Data Reduction

105

Stopping muon cut D
Stopping muon cut D is similar to that used in PC3 for SK-I and II events, but has
been moved to PC5 for SK-III and IV. This is because MS-fit information is needed,
and the full reconstruction is not applied until PC5. Stopping muons are removed if
there are enough hits surrounding the entrance point of the particle in the OD, but only
if point-fit and MS-fit vertex positions are in agreement.
• NHITAin > 6, applied only if |P~pf it − P~M S | < 1500 cm,
where NHITAin is the number of hit OD PMTs within 8 m from the particle entrance point. P~pf it and P~M S are the vertices from point-fit and MS-fit respectively.
Stopping muon cut E
This is identical to a selection criteria used in PC4 to remove upward-going muon events.
It was not necessary to pass in PC4, and it is included again here as a soft cut. A muon
fitter result is compared to the assumption that the muon originated from an OD chargecluster. If the directions of the two vectors match well, the event is removed.
• angleµ < 90◦ ,where angleµ is the difference between the fitted direction from the µ-fitter described in PC4, and the vector between the point-fit vertex and the highest OD
charge cluster.
Decay electron cut
If a neutrino is very high energy (Evis > 25 GeV), it is very likely to produce a decay
electron, via pions ejected in the initial interaction. If no decay electron is detected, theevent is cut, as it is likely that the interaction producing the observed event occurred
outside of the inner detector.

7.2.6

Fiducial Volume Selection

A final set of cuts are performed on the remaining PC sample:• dWall > 200 cm, where dWall is the distance from the reconstructed vertex to theclosest tank wall. This is to further veto events which have entered from outside
the tank, and events due to radioactive impurities in the tank wall.
• NHITAC ≥ 16 (10 for SK-I). NHITAC is the number of hits in the largest chargecluster in the OD. This cut ensures there is no overlap of events between the PC
sample and the FC sample (which requires NHITAC < 16).

Chapter 7. Atmospheric Neutrino Data Reduction

106• Evis > 350 MeV, where Evis is the sum of the reconstructed momentum of all rings,
assuming they are e-like. This cut is higher than the equivalent FC cut, because
the particles need this minimum amount of energy to reach and pass through the
ID-OD tank wall.

7.2.7

PC Reduction Summary

The efficiency of each reduction step is determined by performing the reduction on
an atmospheric neutrino MC. The systematic error on these reduction efficiencies iscalculated by comparing data-MC distributions for each cut variable, and is determined
to be 2.4%, 4.8%, 0.5% and 1.0% for SK-I to IV respectively.
Background in for final PC reduction comprises predominantly of cosmic ray muon
events. The background rate is estimated by eye-scanning the final sample. A tablecontaining the reduction efficiencies, backgrounds, and final event rates is shown in 7.2.
SK-I

PC
Reduction
PC1
PC2
PC3
PC4
PC5
PCFV
Background in PCFVLivetime (days)
MC Prediction
Data
Events / day

SK-II
SK-III
SK-IV
Selection Efficiency (%)
98.98
98.58
99.09
99.63
96.74
93.43
98.52
98.73
95.69
92.32
98.51
98.68
89.86
84.60
97.87
97.42
88.66
82.63
96.61
96.15
80.98
74.80
88.80
86.30
Final Sample Background (%)
0.2%
0.7%
1.1%
0.5%
Number of Events
1489.2798.6
518.2
1775.6
913.8
448.6
356.0
1206.3
902
427
344
1105
0.66±0.04 0.65±0.04 0.62±0.03 0.62±0.02

Table 7.2: PC Summary Table

7.3

Upward-going Muon Reduction

Upward-going muons (UPMU) are muons which enter the detector tank from below
the horizon. These muons are created by neutrino interactions in the rock below SK.
There are analogous downward going muons generated from neutrino events, but these
are impossible to separate from cosmic-ray muon background, so are not consideredin neutrino analyses. Each UPMU event may be classified as a stopping muon (stopmu), which stops inside the SK tank, or a through-going muon (thru-mu), which has

Chapter 7. Atmospheric Neutrino Data Reduction

107

not stopped before it exits through the top of the detector. Through-going muons arefurther split into showering (produces Bremsstrahlung and electro-magnetic showers),
and non-showering events.
The main aim for UPMU reduction is to remove cosmic ray muons and other backgrounds, then separate the remaining sample into one of the three event categories. The
reduction steps will be covered briefly here, for more details see [137].

7.3.1

1st Reduction (UPMU1)Firstly, a low energy cut, and a very high energy cut are applied:• PEtot > 8000 p-e (3000 for SK-II)• PEtot < 1.75 × 106 p-e (8 × 105 for SK-II)
PEtot is the total number of photo-electrons recorded in the ID. At energies above
1.75 × 106 p-e, the tank PMTs become saturated, and the thus the event is difficult toreconstruct. These high energy events are very rare - only one was observed in all of
SK-I. The low energy cut corresponds to a minimum track length of ∼ 7 m.
An OD cluster finding algorithm is used to broadly categorize events as either stop-mu
or thru-mu, as well as eliminating some other backgrounds such as calibration events.
An UPMU fitter is then used, to estimate the vertex, direction and track length of allremaining events. Further cuts are made based on the output of this fit.
• fit goodness > 0.3• fit direction cos θ > 0,
where θ is the zenith angle.
The latter criteria rejects events that are not going upwards. The output of this reduction
step is ∼ 9, 000 events/day.

7.3.2

2nd Reduction (UPMU2)

The remaining events are fit again, with three separate fitters. The first attempts to fit
the event as a stop-mu, the second a non-showering thru-mu, and the third attempts to
fit as a showering thru-mu or multiple muon event. Based on the goodness of these fits,a decision is made as to whether the event is a stop-mu or thru-mu. The fit results also
categorize the event as one of six types:

Chapter 7. Atmospheric Neutrino Data Reduction

108

Figure 7.14: Total ID p-e as a function of track length. Data taken with down-goingcosmic ray muon calibration sample.

• failed fit• down-going (cos θ < −0.12)• near-horizontal (−0.12 ≤ cos θ < 0)
• down-going, entry from top of tank• near-horizontal, entry from top of tank
• up-mu
After combining the results from all the fits, any events recorded as down-going or failed
are removed. Horizontal and up-mu events continue onto the next reduction, comprisingof ∼ 250 events/day.

7.3.3

Interlude

After the second reduction, the most precise muon fitter is applied, aptly named precisefit and described in chapter 6. As a subroutine of the precise-fit, a showering algorithm isChapter 7. Atmospheric Neutrino Data Reduction

109

run, to determine whether the event should be labelled as showering, or non-showering,which shall be briefly described here.

7.3.3.1

Showering Algorithm

The showering algorithm is applied to through-going muons, and aims to identify events
which primarily lose their energy via an electromagnetic shower, rather than ionization.
An electromagnetic shower occurs as a result of Bremsstrahlung interactions along themuon track, which produce photons that create EM showers. Of course the muonis also producing Cherenkov photons all the time, but these are too low energy to
induce a showering effect. The switch from energy loss by ionization to energy loss byBremsstrahlung occurs around 1 TeV [138].

Figure 7.15: Primary methods of muon energy loss, when travelling through water.
The line (a) shows energy loss via ionization, (b) shows energy loss via Bremsstrahlung,and (c) is the total.

Energy loss by ionization occurs at a relatively constant rate of 2.2 MeV/cm. Showeringmuons can be distinguished from this, as their energy loss is much faster, as can be seenin figure 7.15.

Chapter 7. Atmospheric Neutrino Data Reduction

110

The muon track is broken up into segments of 50 cm, and for each segment i, the observed
charge Qicorr is compared to the average charge observed in all segments hQcorr i (tojudge if the rate of energy loss is changing). In addition, the average observed charge iscompared to the expectation for a non-showering event hQexp i. The corr subscripts forthese variables refer to the charge correction performed for every PMT hit to accountfor attenuation in water, and acceptance for a given incident angle.
Two values are calculated based on these parameters:

2

χ =

N
−2
X
i=3

(Qicorr − hQcorr i)2 (hQcorr i − hQexp i)2
+
2
2
σQ
σhQ
i
exp i

!
(7.4)

corr

∆ = hQcorr i − hQexp i − δQ,(7.5)

where σx is the resolution of quantity x. Track segments near the edges of the detector
are ignored (i = 1, 2, N − 1, N ), as it is difficult to calculate the corrected charge
accurately here. δQ is a tuning parameter calibrated in each SK period, to set the peakvalue of ∆ to 0 for the MC.
Using these two variables, showering muons are selected if the following criteria aresatisfied:• ∆ > 0.5, for χ2 > 50.0
• ∆ > 4.5 − 0.08χ2 , for χ2 ≤ 50.0
The separation between showering and non-showering events can be seen in figure 7.16

7.3.4

3rd Reduction (UPMU3)

From here, separate cuts will be applied to stop-mu and thru-mu events. The cuts
are based on the output of precise-fit, which was performed on every candidate before
beginning this reduction step. For through going events, the requirements are as follows:• UM EHIT8M > 10 (16 for SK-II),
where UM EHIT8M is the number of OD PMTs hit within 8 m of the tank entrance
point.
• UM OHIT8M > 10 (16 for SK-II),
where UM OHIT8M is the number of OD PMTs hit within 8 m of the tank exit
point.

Chapter 7. Atmospheric Neutrino Data Reduction

111

Figure 7.16: The thru-mu showering vs non-showering selection criteria, shown herewith SK-IV dataset. Events above the red line are classified as showering, and below
are classified as non-showering.

• Track length ≥ 700 cm (precise-fit)• Fit direction cos θ ≥ 0 (precise-fit),
where θ is the zenith angle.
For stopping muon events, the criteria are:
• UM EHIT8M > 10 (16 for SK-II),
where UM EHIT8M is the number of OD PMTs hit within 8 m of the tank entrance
point.
• UM OHIT8M < 10 (16 for SK-II),
where UM OHIT8M is the number of OD PMTs hit within 8 m of the tank exit
point.
• Fit momentum ≥ 1.6 GeV/c (precise-fit)• Fit direction cos θ ≥ 0 (precise-fit),
where θ is the zenith angle.

Chapter 7. Atmospheric Neutrino Data Reduction

112

Track length is used in the through-going muon case, as it is simple to calculate between
the entrance an exit points, and is analogous to a momentum requirement. For stoppingmuons, reconstructed momentum of 1.6 GeV/c corresponds to a track length ∼ 700 cm.

7.3.5

4th Reduction (UPMU4)

By this stage almost all of the cosmic ray muon background in these samples has been
removed, however still remaining are events where cosmic ray muons brush the cornerof the inner detector, so called ”corner-clipping” muons. These events are difficult tofit, as the two OD clusters are close together, and the ID track is very short.
From SK-III, after the optical segmentation of the ID and OD was introduced, an improved OD cluster-fitting algorithm was developed. If 2 or more clusters are found using
this algorithm, and Precise-fit is found to perform poorly compared to a simple fit basedon clusters of ID PMT charge, the event is classified as a corner-clipper and removed
from the sample.
All remaining UPMU events are eye-scanned by experts, to remove remaining cornerclipper and cosmic ray background.

7.3.6

Background Subtraction

Mt. Ikenoyama above SK reduces much cosmic-ray background, however for cosmic-rays
entering from a horizontal direction, the amount of rock shielding has azimuthal dependence, and thus the background rate varies depending on the thickness of surroundingrock. This effect can be seen in figure 7.17, and the 1-d projection in figure 7.18.
From the plots we can see that the tank may be roughly split up in to two regions region 1, where little additional background is expected due to significant shielding, andregion 2, where the rock is thin, and additional cosmic ray muons are likely to enter the
sample. For stopping muons
Region 1 :φ < 60◦ and φ > 310◦
Region 2 :60◦ < φ < 310◦

and for through-going muons
Region 1 :φ < 60◦ and φ > 240◦
Region 2 :60◦ < φ < 240◦ ,Chapter 7. Atmospheric Neutrino Data Reduction

113

Figure 7.17: The zenith and azimuthal dependence of Stopping and through-going
muons for SK-III. Some above horizon events have been included for demonstration ofthis effect.

Figure 7.18: The 1-dimensional projection of the azimuthal dependence of UPMU
background. Events coming from above the horizon are shown with no fill, and upwardgoing muons are shown with hashed fill.

where φ is the azimuthal angle. For region 2, the event distributions may be modelledusing an exponential function of zenith angle:

F (cos θ) = P 0 + eP 1+P 2 cos θ .

(7.6)

The results of this fit can be seen in figure 7.19.
For atmospheric neutrino oscillation analysis, for which event-by-event discrimination isless important, a correction for the extra expected background in near-horizontal zenithbins is made:Chapter 7. Atmospheric Neutrino Data Reduction

114

Figure 7.19: Extra background contamination in stop-mu and thru-mu samples, fitted to a constant + exponential function of zenith angle. The triangles represent theapproximately constant background in azimuthal region 1, and the circles are the exponentially increasing background in region 2.

Z cos θ2
NBG =eP 1+P 2 cos θ d(cos θ),(7.7)

cos θ1

where θ1 and θ2 represent the edges of the zenith bin to be corrected. This correction is
performed for through-going muons where −0.1 < cos θ < 0.0, and for stopping muons
where −0.2 < cos θ < 0.0.

7.3.7

UPMU Reduction Summary

A summary of the event rates after each reduction (data shown for SK-III period), canbe seen below (7.3).
Reduction Step
UPMU1
UPMU2
UPMU3
UPMU4
EyescanEvents/day
9,900
244
3.7
1.9
1.5

Table 7.3: Approximate events remaining after each UPMU reduction step.

The final samples, event rates and expected background for SK-I to IV UPMU reduction
are shown in table 7.4.

Chapter 7. Atmospheric Neutrino Data Reduction

UPMU

SK-I

Stop-mu
Thru-mu
Livetime (days)
Total Events/day

462
1866
1645.91.41 ± 0.03

Stop-mu
Thru-mu

36.0
30.4

SK-II
SK-III
Number of Events
214
210
888
735
827.7
635.6
1.33 ± 0.04 1.49 ± 0.05
Background Subtraction
13.0
17.0
37.6
12.1

115

SK-IV
480
2097
1775.61.45 ± 0.03
35.0
14.1

Table 7.4: Summary of the remaining events, and background subtraction for thedata sample remaining after UPMU reduction

Chapter 8

Neutron Tagging
Free neutrons travelling through water are quickly thermalized, and then captured byan oxygen or hydrogen nucleus. The cross sections for these capture processes are 0.19
milli-barns and 0.33 barns respectively, so it is safe to assume ∼ 100% of neutrons arecaptured by hydrogen, in the interactionn + p → d + γ (2.2 MeV).

(8.1)

Neutron tagging refers to the process by which these 2.2 MeV γ-rays are identified, thus
the number of neutrons produced by the neutrino interaction may be inferred.
Detecting neutrons provides a method by which water Cherenkov detectors can distinguish anti-neutrinos from neutrinos, since the charged-current quasi-elastic interaction
ν e + p → e+ + n is prohibited by charge conservation and lepton number for neutrinos.
At energies Eν > 1 GeV, few interactions are pure CCQE, and often additional neutrons
are liberated by secondary hadronic interactions. However it is still possible to produce
anti-neutrino enriched samples.
The neutron capture lifetime is measured to be 204.8µs [139]. To encompass this time
range, an additional trigger “AFT” was introduced at the start of SK-IV, triggering
automatically if a SHE trigger is issued without an OD trigger. The AFT trigger saves
an additional 500µs of data following the 35µs saved by SHE, to allow a total of ∼ 92%
of neutron capture signals to be saved in the data.
The neutron tagging study was first introduced to SK in 2009 by H. Watanabe and H.
Zhang [140] with the intention of reducing the background from a primarily anti-neutrino
based supernova relic neutrino signal. This thesis presents the development of neutron
tagging for use in high energy atmospheric neutrino interactions.

116

Chapter 8. Neutron Tagging

8.1

117

Neutron Simulation

The primary event simulation methods for atmospheric neutrinos are discussed in chapter 5. Some additional modifications have been made to this MC to enable neutron
study.
• To enable production of neutron capture gamma rays, the low energy cut off forsimulating neutral hadrons is reduced to 0.1 × 10−4 eV.
• The simulation time is extended to 535µs to match the SHE+AFT trigger length.
• PMT dark noise is simulated only up until 18µs, after which dummy trigger data
was used instead (figure 8.1). Dummy trigger data simply refers to real data
recorded in SK after a random trigger. Using this circumvents the need to simulate
all possible low energy backgrounds.

Figure 8.1: Construction of neutron MC. After 18µs dummy trigger data is convoluted
with simulated neutrons produced by an atmospheric neutrino interaction.

The last point is necessary as low energy backgrounds, such as radioactive decay products, are not simulated by the MC. Simulated PMT dark noise is used up until 18µs
after primary trigger time, to negate any effect on the existing atmospheric neutrino
analysis. The dummy trigger data used was taken in 2009 near the start of SK-IV. Dueto the increasing PMT gain of 2% per year, the average number of 2.2 MeV γ candidates
in the dummy trigger period - 5.34/event, is 4% less than the average for all of SK-IV 5.54/event (figure 8.2). This difference is accounted for in the background estimation.
Following high energy neutrino interactions, it is common for neutrons to be ejected with
momentum over 100 MeV/c. This means the neutron capture vertex is often displacedby some metres from the neutrino interaction vertex (figure 8.3). This causes a larger
spread in the 2.2 MeV γ timing distribution after Time-of-Flight (ToF) subtraction,
reducing the efficiency of detection at high energies.

Chapter 8. Neutron Tagging

118

Figure 8.2: Number of 2.2 MeV γ candidates (before neural network selection) per
day, as a function of run number in SK-IV. The dummy trigger data taking period
is shaded in red. The average candidates per event for the dummy trigger period is5.34/event, and the average for all of SK-IV is 5.54/event. The gradient is consistent
with a 2% increase in PMT gain per year.

Figure 8.3: Neutron travel distance from the initial neutrino interaction vertex, according to the atmospheric neutrino MC simulation.

The efficiency and background of the neutron tagging algorithm are evaluated using theatmospheric neutrino MC by the following criteria:
δtmin < 100 ns: True 2.2 MeV γ
δtmin > 100 ns: Background,Chapter 8. Neutron Tagging

119

where tmin is defined asδtmin = min[|trec − titrue |](8.2)

the minimum time interval between a true neutron capture at titrue and the reconstructed
neutron capture time trec (figure 8.4).

Figure 8.4: Showing the time resolution of final selection of neutrons in 500 yearsof MC. The blue line shows the cut for calculating efficiency and background of theneutron tagging algorithm.

The secondary particle information storage bank for the MC is limited to 1000 particles.
For events with more particles, all secondary particles and gamma rays are still simulated
and generate signals on PMTs, but information about the timing, particle type and soon is not stored. Therefore MC events with 1000 or more secondary particles are ignored
when calculating efficiency and background of the neutron tagging algorithm.

8.2

Neutron Tagging Algorithm

To distinguish a 2.2 MeV γ signal, we must first consider its characteristics. The 2.2
MeV γ ray is expected to produce all its PMT hits in a narrow timing distribution,and these are expected to be anisotropic. The neutron capture point is expected to be<200 cm from the neutrino vertex, and the energy of the hits should correspond to ∼2.2
MeV. Prominent backgrounds at this energy range include radioactive decays from thesurrounding rock and detector material, radon contamination in the water, PMT darknoise and others. Each background source has different characteristics - some produce
isotropic, seemingly random hit patterns, and others produce very tight clusters of hits.

Chapter 8. Neutron Tagging

120

Many background sources are located along the tank wall, and some have a more spreadout timing signature than 2.2 MeV γ. With this in mind, an algorithm to pick out true
2.2 MeV γ signal from its background was devised.
The identification of 2.2 MeV γ-rays is split up into two stages. The first looks for peaksin the PMT hit timing distribution to select initial 2.2 MeV γ candidates. Afterwards
a set of 16 variables is calculated for each γ candidate, and input into a neural network.
The final selection of neutron capture γ-rays is based on the output of this neuralnetwork.

8.2.1

Initial Candidate Selection

First, all PMT hits are Time-of-Flight (ToF) subtracted to the neutrino vertex, as calculated by APfit, described in chapter 6. A 10 ns sliding time window from 18µs until
535µs is then used, to select 2.2 MeV γ candidates, where time is defined relative tothe primary trigger. Though particle backgrounds from the initial neutrino interaction mostly subside during the first 5µs after the event (with the exception of muon
decay electrons), from 12-18µs the PMTs experience an after pulse, which contributes
significant background to neutron-tagging (figure 8.5). This gives the lower bound of
18µs.
A cluster of PMT hits is selected as a 2.2 MeV γ candidate if
N10 ≥ 7,(8.3)

where N10 is the number of hits in the 10 ns window (figure 8.6). The candidate time,
t0 , is defined as the timing of the first hit of N10 . To avoid multiple counting of a single
neutron capture event, once a candidate is found searching continues until t0 + 20 ns.
The peak value of N10 from this 20 ns window is taken as the 2.2 MeV γ candidate.
Peaks found after t0 + 20 ns are treated as separate candidates.
Additional requirements for the initial selection of neutron capture candidates are
N10 ≤50
N200 ≤200,
where N200 is the number of hits in 200 ns surrounding the candidate at t0 . These cuts are
to remove high energy backgrounds such as cosmic ray muons. After all these selections
are applied 33.2% of neutron capture events are still remaining, with an expected 4.5
mis-tagged background events per neutrino event in 535µs.

Chapter 8. Neutron Tagging

121

Figure 8.5: The increase in number of γ candidates due to the PMT after-pulse.
Scanning is preformed after 18µs.

Figure 8.6: Values of N10 used to select initial 2.2 MeV candidates. Backgroundincreases exponentially as the required N10 is reduced.

8.2.2

Final Candidate Selection

Following this initial selection 16 variables with power to distinguish between 2.2 MeV
γ signals and backgrounds are calculated and input to a neural network. Plots foreach of these variables will be shown - including data comparison with MC, and signalbackground figures. The signal and background histograms, shaded in green and blue
respectively, are both normalized to the integral of the data histogram/4 to make readingChapter 8. Neutron Tagging

122

easier (on log scale plots, this normalization factor is reduced to 1/40) . In the data-MC
comparison, the MC (red line) is normalized by(M Csig + 1.04 × M Cbg )LTdata
,
LTM C

(8.4)

where M Csig is the number of tagged candidates corresponding to a true neutron event,
M Cbg is the number of tagged candidates which do not correspond to a neutron-capture
event, as defined in section 8.1. LTx is the livetime of sample x. The coefficient 1.04 is acorrection for the dummy trigger data period’s PMT gain not matching the average of
SK-IV. This is a somewhat crude correction, as increased gain will lead to a proportional
increase in some background sources, such as PMT dark noise, but for real particle
backgrounds such as radioactive decay products, the frequency will not increase but theshape of the signal may change. This may lead to discrepancies in some variables which
are not easy to compensate for.

8.2.2.1

Number of Hits in 10 ns: N10

This is the same variable that is used to make the initial selection of 2.2 MeV γ candidates. This variable has already been cut, requiring N10 ≥ 7, but the remaining
distribution’s shape still holds discriminatory power, so it is used in the neural network(figure 8.7). The background increases exponentially for smaller values of N10 .

Figure 8.7: N10 variable as a neural network input. The plot shows a comparisonof neutron capture signal (green), background (blue), data (black dots) and MC total(red line). The normalization is explained in section 8.2.2

Chapter 8. Neutron Tagging
8.2.2.2

123

Number of Clustered Hits: Nc

Many background events are found to have a clustered hit pattern, both in time andspace. This may be due to backgrounds which originate close to the PMTs themselves,for instance due to radioactive contaminants in the PMT glass or to PMT flashing. A
requirement of atmospheric neutrino events is that the event vertex lies > 2 m from thetank wall, so it is rare for neutron capture to occur so close to the PMTs as to cause
this type of hit pattern.
To search for clustered hits, the angle between each of the N10 hit vectors is calculated.
These are unit vectors between each hit PMT, and the APfit neutrino vertex. If a pairof hit vectors is less than 14.1◦ apart, they are considered a cluster. If additional hits
are within 14.1◦ of any hit contained in a cluster, they are added to it. This process isiterated until all N10 hits have been considered and the total number of hits in clusters
is counted as Nc .
As N10 for 2.2 MeV γ signal is likely to be larger than for background events, this may
unfairly increase the number of clusters in the signal sample. Hence the variable input
to the neural network is taken as N10 − Nc (figure 8.8).

Figure 8.8: Clusters of hits variable, N10 − Nc . The plot shows a comparison ofneutron capture signal (green), background (blue), data (black dots) and MC total(red line). The normalization is explained in section 8.2.2

Chapter 8. Neutron Tagging
8.2.2.3

124

Number of Hits on Low Probability PMTs: Nlow

For a given neutrino event, most neutron captures occur < 200 cm from the neutrino
interaction point (see figure 8.3). Therefore PMTs which are closer to the neutrino
interaction vertex have a higher chance of being hit by the neutron capture γ-ray. In
contrast, background events may originate from anywhere in the tank. There is very little
spatial bias in the background towards the neutrino vertex, as after cutting δt > 18µs
most related particles have already decayed.

Figure 8.9: The varying acceptance requirements for the Nlow cut, shown as a functionof tank coordinates.

The probability to be hit (termed acceptance) for the i-th PMT is defined as
Ai ∝F (θi ) Ri /L
e
,
Ri2

(8.5)

where F (θi ) is the PMT angular dependence for incident angle θi , Ri is the distancefrom the neutrino interaction vertex to PMT i, and L is the light attenuation length inwater.
To calculate Nlow , an area of high probability PMTs is defined for each neutrino event,which contains PMTs totalling x% of the acceptance for all PMTs. If a PMT hit occurs
outside of this area, it is counted towards Nlow . For neutrino events occuring near thewall this cut is very strict, as a very focused signal on a small number of PMTs. In
contrast, for neutrino vertices near the tank centre, all PMTs are assigned a similar
hit probability. To compensate for this bias, the total hit probability required, x%, isaltered throughout the tank, as shown in figure 8.9.
The variable input to the neural network is in fact N10 - Nlow (figure 8.10).

Chapter 8. Neutron Tagging

125

Figure 8.10: Hits on low probability PMTs, N10 −Nlow . The plot shows a comparisonof neutron capture signal (green), background (blue), data (black dots) and MC total(red line). The normalization is explained in section 8.2.2

8.2.2.4

Number of Hits in 300 ns: N300

A true neutron capture event will have a sharp timing peak, almost all of which will becontained within a 10 ns window after ToF subtraction. Some background events (forexample PMT flashing) may mimic a 2.2 MeV γ signal in 10 ns but in fact be part of alarger, more disperse signal (for an example see figure 8.11). N300 is sensitive to thesetypes of background, and looks for broader scale timing peaks in the 300 ns surrounding
a 2.2 MeV γ candidate. Again, to compensate for γ signals with large values of N10 , thevalue input to the neural network is N300 − N10 (figure 8.12).

Figure 8.11: The utility of the N300 cut. Figure (a) shows a typical neutron capture
event, and figure (b) shows a hypothetical background signal to be rejected. The bluelines correspond to PMT dark noise, and the red is some signal.

Chapter 8. Neutron Tagging

126

Figure 8.12: Number of hits in 300 ns, N300 − N10 . The plot shows a comparisonof neutron capture signal (green), background (blue), data (black dots) and MC total(red line). The normalization is explained in section 8.2.2

8.2.2.5

Hit Vector RMS: φrms

The signal from a neutron capture event is a single 2.2 MeV γ-ray, which is expected
to deposit most of its energy in a single direction. Some sources of background, such as
PMT dark noise, are likely to have a more random hit pattern. φrms calculates the rootmean-square of the azimuthal angle (with respect to the average hit vector direction) of
the hit vectors for each of the N10 candidate hits (figure 8.13). This information helps
to effectively remove these types of random background.

8.2.2.6

Mean Opening angle: θmean

An angle θi is defined as the angle between the fitted direction (average of all hit vectors),
and each individual PMT hit i that makes up the candidate event. Compared to a 2.2
MeV γ-ray, background events are more likely to have clustered hit patterns. θmean is
the mean value of the opening angle θi (figure 8.14).

8.2.2.7

Root-mean-square of hit timing: trms

A distinguishing feature of the 2.2 MeV γ signal is its narrow hit timing distribution.
The PMT timing resolution is ∼ 3 ns [141], so even within a 10 ns window, it is possible

Chapter 8. Neutron Tagging

127

Figure 8.13: Root-mean-square of the azimuthal angle of 2.2 MeV candidate hit vectors φrms . The plot shows a comparison of neutron capture signal (green), background(blue), data (black dots) and MC total (red line). The normalization is explained in
section 8.2.2

Figure 8.14: Mean value of hit opening angles, θmean . The plot shows a comparisonof neutron capture signal (green), background (blue), data (black dots) and MC total(red line). The normalization is explained in section 8.2.2

to pick out particularly sharp concentrations of PMT hits. Variable trms is the rootmean-square of the timing of all N10 hits (figure 8.15).

Chapter 8. Neutron Tagging

128

Figure 8.15: The root-mean-square of the timing of candidate hits, trms . The plotshows a comparison of neutron capture signal (green), background (blue), data (blackdots) and MC total (red line). The normalization is explained 8.2.2

8.2.2.8

Minimum Root-mean-square of hit timing: min − trms

Ideally, for a 2.2 MeV γ event all of the hits in the 10 ns timing window will be due to the
gamma-ray. However many candidates are selected which consist of only a few hits fromthe 2.2 MeV γ, and then the rest are contributed by dark noise or other backgrounds.
To account for these events, min − trms calculates the minimum value of trms when
considering a cluster of x hits out of the total N10 . All possible combinations of x hits
are considered, to find the tightest cluster. A depiction of this method is shown in figure
8.16. Two variables are constructed, for x = 3 and x = 6, and used as neural network
inputs (figure 8.17).

Figure 8.16: The selection of min − trms hit clusters. Optimal selections for clustersof 3-6 hits are shown, for an example candidate where N10 = 9. Possible backgroundhits on the fringes of the candidate time are ignored.

8.2.2.9

Bonsai Fit Variables: BSenergy and BSwall

The Bonsai fit is an unbiased low energy fitter described in chapter 6. For each 2.2 MeV
γ candidate, Bonsai fit is performed on the surrounding 1.3µs of data. BSwall givesChapter 8. Neutron Tagging

129

Figure 8.17: min − trms of clusters of 6 hits (left) and 3 hits (right). The plot shows
a comparison of neutron capture signal (green), background (blue), data (black dots)
and MC total (red line) comparison. The normalization is explained 8.2.2

the distance from the reconstructed Bonsai vertex to the nearest ID tank wall. Manybackground sources originate from the walls of the tank, such as those from radioactive
decay in the detector material, and these events can be removed by the BSwall variable(figure 8.18). BSenergy is the reconstructed energy of the candidate event. Since the
2.2 MeV γ-ray is mono-energetic, so its energy distribution is expected to peak more
sharply at 2.2 MeV (figure 8.18).

Figure 8.18: Distributions of BSwall and BSenergy for 2.2 MeV γ candidate events.
The normalization of BSenergy is different for signal and background, as many background events fail to reconstruct at this stage. The plots show a comparison of neutron
capture signal (green), background (blue), data (black dots) and MC total (red line).
The normalization is explained in section 8.2.2

Chapter 8. Neutron Tagging
8.2.2.10130

Neut-Fit Variables: N Fwall , ∆N10 , ∆trms

These variables relate to Neut-Fit, as described in chapter 6. Neut-Fit acts only on the
N10 hits already selected by ToF-subtraction to the APFit vertex, so it has substantial
bias towards this point. N Fwall is the distance between the Neut-Fit vertex and thenearest tank wall. As Neut-Fit is biased towards the APFit vertex, and the atmospheric
neutrino FC reduction requires that the APFit vertex be > 200cm from the tank wall,
Neut-Fit is unlikely to reconstruct any true neutron events in this region (figure 8.19).

Figure 8.19: The distance to the wall from Neut-Fit vertex, N Fwall . The plot shows
a comparison of neutron capture signal (green), background (blue), data (black dots)
and MC total (red line). The normalization is explained in section 8.2.2

Neut-Fit functions by searching for the point in the SK tank with the minimum valueof trms for all the hits. For neutron capture signal events, an optimal neutron capture
vertex is expected to be found with a small value of trms . Many background events
do not come from a single source (like PMT dark noise), or may come from a source
where the vertex reconstruction is outside the inner detector so the improvement of trms
will be less for these backgrounds than for a true neutron capture signal. Thus ∆trms
examines the difference between trms as calculated with the APFit vertex, and trms
when calculated using the Neut-Fit vertex (figure 8.20).
The final Neut-Fit related variable is ∆N10 . For 2.2 MeV γ candidates, ∼90% of PMT
hits are recorded inside 10 ns, but due to the imperfect reconstruction of the APFit
vertex, some hits may be spread outside this window. Using the Neut-Fit vertex, ToF
subtraction is performed again on 40 ns of hits surrounding the gamma candidate, anda new value of N10 is calculated. If the candidate is a true 2.2 MeV γ, some of theseChapter 8. Neutron Tagging

131

Figure 8.20: The variables ∆trms and ∆N10 are shown in the left and right figures, respectively. The plots show a comparison of neutron capture signal (green), background(blue), data (black dots) and MC total (red line). The normalization is explained in
section 8.2.2

outside hits may be brought into the 10 ns window. The variable ∆N10 is the differencein N10 when calculated using APFit vertex, and Neut-Fit vertex.

8.2.2.11

Fit Agreement variables: (N F − BS)dis , (N F − AP )dis

The final two variables are the distance between the vertex reconstructions of the different neutron fitters. (N F − AP )dis gives the distance between the Neut-Fit, and the
APFit vertex. This is a powerful variable, as we know that neutrons do not travel far(see figure 8.3) from the neutrino interaction point before capture. Thus when Neut-Fit
reconstructs close to the APFit vertex, the candidate is more likely to be a true 2.2 MeV
γ (figre 8.21).
The distance between Bonsai fit and Neut-Fit, (N F −BS)dis is also used. Bonsai decides
the vertex based on all hits within 1.3µs of the candidate event, however Neut-Fit uses
only 10 ns surrounding the candidate. Thus, if the reconstructions are very different,
it suggests that the candidate is a part some larger scale background spread out over
1.3µs that Neut-Fit is not sensitive to (figure 8.21).

8.2.3

Neural Network

Neural networks are a machine-learning based tool, primarily used for pattern recognition in fields such as stock market prediction, image analysis, remote sensing, as well

Chapter 8. Neutron Tagging

132

Figure 8.21: The variables (N F − BS)dis (distance between Bonsai reconstructed
candidate vertex, and Neut-Fit reconstructed candidate vertex) and (N F − AP )dis
(distance between Neut-Fit reconstructed candidate vertex, and APFit vertex), areshown in the left and right figures, respectively. The plots show a comparison of neutron
capture signal (green), background (blue), data (black dots) and MC total (red line).
The normalization is explained in section 8.2.2

as signal-background classifications in particle physics. Here a feed-forward Multi-Layer
Perceptron is used based on the ROOT class TMLP. The network was trained using 500
years of Atmospheric Neutrino MC, which comprised of 1,721,004 true 2.2 MeV candidates, and 15,578,057 background candidates after the initial selection process. This
sample was split into a training sample and a testing sample to avoid overtraining of
the network. The network was trained using the Broyden, Fletcher, Goldfarb, Shanno(BFGS) method [142], for 1000 iterations over the training data.
Deciding the structure and number of nodes in a neural network is a difficult problem,
typically only solved experimentally [143]. The node pattern is chosen as 16:14:7:1,
where 16 represents the number of input nodes, 14 and 7 are the number of nodes in
two hidden layers, and 1 is the output node: 2.2 MeV γ signal or background. A single
hidden-layer network was tested using a proposed optimal number of nodes log T where
T is the number of events in the training sample [144]. For this analysis, the architecture
becomes 16:17:1. This was found to give inferior performance (-11% efficiency), so a 2
hidden-layer network is deemed necessary for this analysis. Three configurations of
networks with 2 hidden-layer networks were tested - 16:8:4:1 and 16:17:16:1, in addition
to the aforementioned 16:14:7:1.
A summary of the results from these is shown in table 8.1. Methods are evaluated bytheir efficiency relative to the 16:14:7:1 architecture. The output of the networks is cut
automatically to retain a signal efficiency (%) to background per neutrino event ratioChapter 8. Neutron Tagging
Network Architecture
16:14:7:1
16:17:16:1
16:8:4:1
16:17:1

133
Relative Efficiency
n/a
+0.3%
-3.7%
-14.8%

Training time
10 days
13 days
8 days
1 day

Table 8.1: The differences in neutron tagging efficiency, and training time, for eachneural network layout that was tested. The efficiencies shown are relative to the result
given by the 16:14:7:1 architecture.

of 14:1. The 16:14:7:1 architecture appears to be close to the optimal solution, and waschosen for this analysis. For this network structure, the neural network cut position isdetermined to be at 0.832, as shown in figure 8.22. This cut position is chosen to achievetagging efficiency/background per event = 14.

Figure 8.22: The neural network output for selecting final neutron candidates. Everything to the right of the dotted cut line is selected as a neutron capture event. The
small peak in the background histogram at NN output ∼1 is due to events where thesecondary particle bank is overfilled. For these events some true neutron capture information is not stored, and some candidates are mis-classified as background. The
data-MC discrepancy in the far left bin looks drastic on a log-scale plot, but this represents < 0.1% of total background events. The plot shows a comparison of neutron
capture signal (green), background (blue), data (black dots) and MC total (red line).
The normalization is explained in section 8.2.2.

Chapter 8. Neutron Tagging

8.2.4

134

Results

The efficiency of 2.2 MeV γ-ray selection is defined to be
Candidates selected which are true neutrons (see 8.1).
All true neutron captures(8.6)

The background is defined by
Candidates selected which are background events (see 8.1).
Total Neutrino Events(8.7)

Events where the secondary particle storage banks are filled are discarded in this calculation, as some true neutron information may be missed. The calculated efficiency andaccidental background values are shown in table 8.2.
Selection stage
Initial Selection
After Neural Net

Efficiency
33.2%
20.5%

Background / Event
4.45
0.018

Table 8.2: Final efficiency and background rate of the neutron tag algorithm, after
each stage of selection

If we consider only neutron captures which fall within the selected timing range, 18 →535µs, the efficiency becomes 24.6%. Compensating for increasing PMT gain, the background changes from 1.77% → 1.84%. The efficiency is heavily dependent on the distancetravelled from the neutrino interaction by the neutron (see figure 8.23). For high energy
neutrino events, the average multiplicity of neutrons increases, which results in many
low energy neutrons that do not travel very far. Therefore, the average efficiency does
not change much (figure 8.23).
The efficiency is also dependent on the position of the neutron in the tank. γ-rays fromneutron capture close to the centre of the tank are attenuated more before reaching the
PMTs on the tank wall so the detection efficiency is reduced in this region. For capture
events very close to the wall the acceptance of the PMTs is reduced, resulting in lower
efficiency. The 2-dimensional efficiency profile can be seen in figure 8.24.

8.3

Atmospheric Neutrinos

Neutron tagging was applied to the SK-IV atmospheric neutrino fully-contained sample,
as selected by the FC reduction process described in chapter 7, and the atmospheric
neutrino MC from chapter 5 (with neutron modifications mentioned in section 8.1). The
MC was normalized to the livetime of the data, 1775.6 days. The MC is oscillated using
a 2-flavour oscillation approximation with ∆m2 = 2.5 × 10−3 eV2 and sin2 2θ = 1.0. At

Chapter 8. Neutron Tagging

135

Figure 8.23: The dependence of 2.2 MeV detection efficiency on the distance travelledby the neutron (left), and the visible energy from the neutrino interaction (right). Efficiency is defined as the number of selected candidates corresponding to true neutrons,
per number of true neutron capture events in each bin.

Figure 8.24: The efficiency of detecting neutrons as a function of neutrino (APFit)
vertex position throughout the ID. Only neutrino events passing the fiducial volume
cut (dW all > 200 cm) are considered.

energies Eν > 30 GeV, significant data-MC discrepancy is present, however this energy
range comprises a relatively small number of events and has little effect on the neutrinoChapter 8. Neutron Tagging

136

oscillation analysis in chapter 9. A summary of the data-MC agreement can be seen intable 8.3.
SampleTotal Neutrons
Events with any neutrons
Events with only one neutron

SK-IV Data (1775.6 days)
8125
4511
2900

MC
8589.94391.82721.0

Table 8.3: A comparison of the expected and measured neutron capture events in theatmospheric neutrino SK-IV data-set (1775.6 days). Most of the data-MC discrepancy
is from very high energy (>30 GeV events). The MC is normalized by data livetime,and oscillated under a 2 flavour approximation: ∆m2 = 2.5 × 10−3 and sin2 2θ = 1.0.

The neutron lifetime is fit to 205.2 ± 3.7 µs, as shown in figure 8.25. This agrees with aprevious measurement of 204.8µs [139]. The total number of events with neutrons, theaverage multiplicity, and the neutron multiplicity split into Sub-GeV and Multi-GeV
samples can be seen in figure 8.26.

Figure 8.25: Final selected 2.2 MeV gamma candidates timing distribution. The primary trigger is at t = 0. Blue curve is a best exponential fit using chi-squared method.
The neutron capture lifetime is calculated to be 205.2 ± 3.7 µs. The background rate
is set to 0.0184 / neutrino event. Data is taken from all of SK-IV (1775.6 days)

8.4

Americium-Beryllium Systematic Error Study

To study the efficiency of neutron tagging, an Americium-Beryllium (Am-Be) source isused. This study was first performed in SK-III [140], and then again at the start of
SK-IV to take advantage of the upgraded electronics.

Chapter 8. Neutron Tagging

137

Figure 8.26: These plots show data-MC comparisons for neutrons tagged in the SKIV 1775.6 days dataset. The top left plot shows the average multiplicity of neutrons,
the top right plot shows the number of neutrino events with ≥ 1 neutrons, the bottom
left plot shows the neutron multiplicity in Sub-GeV events (Evis < 1330 MeV), and thebottom right plot shows neutron multiplicity in Multi-GeV events (Evis ≥ 1330 MeV).
The multiplicity plots are normalized to the number of neutrino events. The numberof events plot (top right) is normalized to data livetime and oscillated under a 2 flavour
approximation: ∆m2 = 2.5 × 10−3 and sin2 2θ = 1.0.

The 241 Am nucleus emits α-particles, which then interact with 9 Be, and produce aneutron in the following fashion:
α + 9 Be →12 C∗ + n
12C∗ →12 C + γ (4.4 MeV)orα + 9 Be →12 C + n.

Chapter 8. Neutron Tagging

138

The 241 Am source intensity for this experiment is 97 µCi, which corresponds to 87 Hz of
4.4 MeV γ-ray emission interaction, and 76 Hz of the direct transition to ground-state.
The 4.4 MeV γ-ray is used to trigger the detector, and then subsequent neutron capture
signals are searched for. The ground-state transition produces an irreducible constantbackground of neutrons, since there is no way for it to produce a primary trigger in thedetector.

8.4.1

Experimental Setup

The Am-Be source is embedded in a 5 cm cube of BGO scintillator (see figure 8.27)
to amplify the light released by the 4.4 MeV γ-ray, such that it will activate the SHE
trigger in the detector. Upon triggering SHE stores 35µs of data, and then an extended
AFT trigger is activated to store an additional 800µs, and grant a more complete viewthe neutron capture time spectrum.

Figure 8.27: Am-Be crystal embedded in a 5 cm cube of BGO scintillator. This isheld in an acrylic case.

This configuration was set up in 3 different locations around the tank: the centre (35.3,-70.7, 0) cm (Centre), near the side of the barrel (35.3, -1201.9, 0) cm (Y12), and near
the top of the tank (35.3, -70.7, 1500.0) cm (Z15). Random data was also taken withthe apparatus in the centre, using a 10Hz trigger, to study the irreducible backgroundfrom the ground-state transition.

8.4.2

Data Selection

The primary selection criteria for 4.4 MeV γ-ray events is based on the number ofphoto-electrons (p-e) recorded in the ID. The exact criteria varies dependent on thesource position, as follows:
Centre: 750 < p-e < 1050
Y12: 850 < p-e < 1150
Z15: 900 < p-e < 1150.

Chapter 8. Neutron Tagging

139

Figure 8.28: The total p-e from 4.4 MeV γ scintillation events, with the Am-Be source
positioned at the centre of the tank. Events with 750 < p-e < 1050 are selected.

After this two further cuts are made. First, to avoid neutron signal contamination froma previous scintillation event• The time to the previous trigger must be > 1.5 ms,
Additionally, events where a second scintillation event has occurred shortly after the
first must be rejected. A 200 ns sliding time window is used to search through the 835µs
of data following a trigger. The maximum number of hits recorded by this window is
max . Ifsaved as N200

max < 50
• N200is not satisfied, the event is rejected.

8.4.3

Data Analysis

Neutron tagging is then performed on the remaining Am-Be dataset, and a 2.2 MeV
γ-ray MC. There are some differences between this study and the atmospheric neutrino study discussed up until now. Neutrons released by Beryllium have an energy
ranging from 2-10 MeV, much less than the average neutron energy following atmospheric neutrino interactions. Thus, we can make the assumption that the location of
the Am-Be apparatus is roughly the same as the neutron capture vertex. This raises
the expected neutron detection efficiency in accordance with figure 8.23. In addition,

Chapter 8. Neutron Tagging

140

the PMT after-pulse observed following high energy atmospheric neutrino events (figure 8.5) is not produced to the same extent after Am-Be events, so neutron searchingis started at 5µs (previously > 18µs) after the primary trigger. The AFT trigger for
Am-Be is also extended, allowing neutron-searching until 835µs after the initial trigger.
The efficiency is calculated by fitting the timing distribution of neutron candidates to a
constant background + exponentially decaying signal representing the neutron capture
lifetime.
To predict the neutron tagging efficiency for the Am-Be study, a MC of pure 2.2 MeV γinteractions was created, with 50,000 events at each of the three Am-Be crystal locations.
Only 2 µs of data surrounding the gamma-ray is analysed, so the effect from mis-tagging
is assumed to be negligible (<3.4 mis-tagged events expected in 50,000 MC events). The
true vertex of γ-ray production was used to perform ToF-subtraction in neutron tagging
algorithm. The efficiency is calculated simply by counting the number of successfully
identified 2.2 MeV γ-rays.
The fitted neutron lifetimes are 206.3 ± 5.3 µs, 201.7 ± 8.2 µs and 207.7 ± 9.5 µs forthe centre, Y12 and Z15 positions respectively (see figure 8.29). This is consistent with
204.8 µs found in previous works [139].
The efficiency of neutron-tagging in Am-Be data compared to the 2.2 MeV γ-ray MC
can be seen in table 8.4.
SampleCentreY12
Z15

Am-Be Efficiency
25.6 ± 0.5%
30.9 ± 0.9%
33.6 ± 1.1%

MC Efficiency
26.9 ± 0.2 %33.5 ± 0.3 %36.8 ± 0.3 %Table 8.4: Estimated efficiency of neutron tagging, using Am-Be data, compared tothe expectation from a 2.2 MeV γ-ray MC. The efficiencies here are higher than thosequoted previously, as the neutron capture vertex is now known to < 20 cm.

The data-MC discrepancy here is larger than previous studies which werre conductedwith a combined cuts and likelihood tagging algorithm [145]. The reason for this may
be that the presence of BGO scintillator is affecting the 2.2 MeV γ signal, and the more
finely tuned multi-variate analysis techniques presented here are more sensitive to thisdifference.
The largest discrepancy, from the Z15 data is found to be 9.5%. Therefore 10% is taken
as the systematic error on neutron-tagging efficiency in the following neutrino oscillation
analysis.

Chapter 8. Neutron Tagging

Figure 8.29: The timing distribution of tagged 2.2 MeV γ candidates, after triggering
on a prompt 4.4 MeV γ ray from the Am-Be neutron source. The dots are data-points,
and the blue line is a best fit to a constant background + exponential decay, where τis the neutron capture lifetime in µs, and signal fraction is the fraction of all tagged
events expected to be neutrons.

141

Chapter 9

Oscillation Analysis
Once the reduction, reconstruction, and neutron-tagging processes have been performedon the data-sets and corresponding MC for each of SK-I to SK-IV, neutrino oscillation
analysis may begin. This analysis specifies a set of oscillation parameters for the MC,
which is oscillated accordingly to calculate the expected number of events in each of 580
analysis bins. A goodness of fit is defined using a χ2 method, to judge the agreement
between the oscillated MC and data. The systematic errors are included in this calculation via a pull method. This χ2 value is calculated over a parameter space of 37,638
different sets of oscillation parameters. The smallest value of χ2 in this parameter spaceis defined as the best fit point. The whole parameter space is calculated twice - once
assuming the normal hierarchy, and then assuming the inverted hierarchy (referring to
the mass hierarchy of the neutrinos, as explained in chapter 2). The mass hierarchy
preference is calculated by the difference in χ2 at the best fit points for each of thehierarchies.
In this thesis only variation in atmospheric neutrino oscillation parameters is considered,across a linear parameter space of∆m223 : 1.0 × 10−3 → 5.0 × 10−3 eV2(41 points)

sin2 θ23 : 0.2 → 0.7(51 points)

δCP : 0 → 360◦

(18 points)

The remaining oscillation parameters are fixed by values set by other experiments:
sin2 θ13 = 0.025 [65], ∆m212 = 7.5 × 10−5 eV , and sin2 θ12 = 0.31 [146]. Their uncertainties are included as systematic errors.

142

Chapter 9. Oscillation Analysis

143

This section will explain the calculation of χ2 , the systematic errors used in the analysis,
the updated analysis binning using neutron information, the sensitivity of the analysis
to the neutrino mass hierarchy, and finally the analysis results.

9.1

Calculation of χ2

The χ2 value for each point in the oscillation parameter space is calculated using a
Poisson likelihood function, given by:

L(N

exp

,N

obs

)=

nBins
Y
i

exp

expNiobs

e−Ni Ni
Niobs !

,

(9.1)

where Niobs and Niexp are the observed and expected events in the i-th bin, and nBins
is the total number of analysis bins. The χ2 value is defined as the log likelihood ratio:nBins X
L(N exp , N obs )=2
χ = −2 ln
L(N obs , N obs )
2

i

N obs
Niexp − Niobs + Niobs ln iexp
Ni


.

(9.2)

At this stage we must also take into account the systematic errors, which are includedin each bin utilizing the “pull” method [147]. This involves allowing the expected valueof each bin to vary depending on each independent error relevant to that bin:Niexp → Niexp (1 +nErr
Xfji j ),

(9.3)

j

where nErr is the total number of systematic errors, and fij is the fractional change ofthe number of events in the i-th bin, due to a 1σ variation of the j-th systematic error,
defined as:

fji =(Niexp )j(+σ) − (Niexp )j(−σ).
2(Niexp )j(0)(9.4)

The values j are fit parameters that are allowed to vary. At every point in the oscillation
parameter space, a fit is performed to find the values of j that give the minimum χ2 atthat point.
If the  values are allowed to move freely, a near perfect fit could be achieved every time,so the range of epsilons must be constrained. This is done by adding a penalty term toChapter 9. Oscillation Analysis

144

the χ2 calculation, assuming gaussian systematic error:
nErr
X
j

j
σj

2
,

(9.5)

where σj is the estimated 1σ error size of the j-th systematic error. Thus, the final valueof χ2 to be minimized is given by:χ2 = 2nBins
XN exp (1 +
i

i

nErr
X
j


nErr
obs
X  j 2
N
i
obs
obs
i
+
.
fj j ) − Ni + Ni ln exp
PσjNi (1 + nErr
fji j )jj(9.6)

The minimum value of χ2 is then obtained by requiring δχ2 /δj = 0 for each of thenErr errors.

9.2

Analysis Binning

Before the oscillation analysis is performed the data sample and MC are divided into
bins to improve sensitivity to the oscillation parameters. First the data is separated into
Upward-going muons (UPMU), Fully Contained (FC) and Partially Contained (PC)events, as selected by their respective reduction processes (these are all samples of neutrino events, as described in chapter 7).
L, where E is the energy and L is
Recall that oscillations are dependent on the fraction E

the distance travelled of the neutrino. The reconstructed energy and zenith angle cos θ(different value of cos θ correspond to different neutrino path lengths through the Earth)
are therefore used to bin each event.
The FC event sample undergoes additional separation. As νe and νµ are constructed fromdifferent fractions of neutrino mass states, the oscillation profiles of these flavours will bedifferent. To best observe this, the FC sample is split up into µ-like and e-like, referringto the reconstructed identity of the most energetic ring. The events of interest to us in
neutrino oscillation analysis are primarily CC (charge current) events, as NC interactions
are insensitive to neutrino flavour. To select more pure CC samples, some of the samples
are further split up into single-ring, and multi-ring samples; the former are more likely tobe pure CC quasi-elastic interactions. The Multi-GeV Multi-Ring electron-like sample
has a large contamination of NC events, so an additional likelihood to improve thesample’s CC νe purity is applied. Events which fail this likelihood selection are placedChapter 9. Oscillation Analysis

145

into the Multi-Ring Other category. For more details on this likelihood selection see[148].
In Sub-GeV interactions, additional cuts are made to remove single π 0 events, which
comprise a significant neutral current background. The π 0 almost always decays to γγ,each of which creates a Cherenkov ring pattern similar to an electron. Thus eventswith 2 e-like rings and no decay electron are labelled as π 0 -like. These events are often
reconstructed as single-ring events, so an additional likelihood is constructed to selectout these events. More details of this selection can be found in chapter 6 (POLFIT),
and in [148]. A summary of the FC sub-GeV event sample selections is shown below, in
table 9.1.
The final FC sample splitting, and focus of this thesis, is the separation of anti-neutrino
and neutrino interactions in multi-GeV events. While event-by-event separation is notpossible, statistically enriched samples can be produced. This is the region most sensitive to neutrino mass hierarchy, which causes an enhancement in upward going electronneutrino, or electron-antineutrino event rate depending on whether it is normal or inverted. In this analysis improved separation is achieved using information from theneutron-tagging procedure, and a neural network method. Separation of µ-like eventsis also introduced, which may provide some sensitivity through a deficit in events corresponding to the νe excess. A summary of the FC Multi-GeV event samples (prior toν-ν̄ selection), can be seen in table 9.2.
Sub-GeVSample Namee-like 0 dcy
e-like 1 dcy

Cherenkov Rings
1
1

Particle ID
e-likee-likeDecay-electrons
0≥1

1-Ring π 0 -like

1e-like02-Ring π 0 -likeµ-like 0 dcyµ-like 1 dcyµ-like 2 dcy

2
1
1
1

e-likeµ-likeµ-likeµ-like00
1≥2Other Criteria
pe > 100 MeV/c
pe > 100 MeV/cpasses π 0 likelihood selectionpe > 100 MeV/c
85 < π 0 mass < 215 MeV
pµ > 200 MeV/c
pµ > 200 MeV/c
pµ > 200 MeV/c

Table 9.1: Summary for selection of Fully Contained Sub-GeV event samples

In tables 9.1 and 9.2, px refers to the momentum of the most energetic ring (MER),
reconstructed assuming it was created by lepton x. Sub-GeV refers events with energy
Evis < 1.33 GeV, and events exceeding this energy are labelled as Multi-GeV.
Excluding the Multi-Ring Other sample, all of the Multi-GeV samples are further split
into ν-like and ν-like, using methods discussed in the following section.

Chapter 9. Oscillation Analysis

146

Multi-GeV
Sample Namee-like 1-Ring (M1E)
e-like Multi-Ring (MME)
Multi-Ring Otherµ-like 1-Ring (M1M)

Cherenkov Rings
1, e-like≥ 2, MER e-like≥ 2, MER e-like
1, µ-likeµ-like Multi-Ring (MMM)≥ 2, MER µ-like

Other Criteria
Pass MultiRing likelihood
Fail MultiRing likelihood
If pµ > 600 MeV/c
Evis < 1.33 GeV is allowed

Table 9.2: Summary for selection of Fully Contained Multi-GeV event samples

The PC event sample is split up into two samples: PC stopping, which come to a halt
within SK’s outer detector, and PC-through-going, which pass through the OD andleave the tank. The PC-through events generally have a higher energy, but also more
reconstruction uncertainty. Similarly the UPMU samples are categorized into stoppingmuons, which stop within the tank, and through-going non-showering, and showering
events (referring to the production of an electro-magnetic shower). Again the energyof these events is progressively higher and more uncertain. The selection of the UPMU
and PC samples is described in detail in chapter 7.
Thus, the entire data-set is split up into 21 sub-samples, for each of SK-I to SK-IV.
Each of these sub-samples is split further into zenith and energy bins for the oscillation
analysis. The bins in each SK period are then merged, after evaluating the systematic
errors:Niobs →

4X

obs
Ni,skxskx

Niexp (1 +nErr
Xfji j ) →

j

4
X

exp
(1 +
Ni,skx

skx

nErr
X

fji j )jThis results in a total of 580 analysis bins, of which 60 come from PC events, 50 fromUPMU events, and 470 from FC events.

9.2.1

Neutrino and Anti-neutrino separation

Distinguishing between neutrino and anti-neutrino is a difficult task for pure water
Cherenkov detectors. Applying a magnetic field and observing the lepton tracks curving
in different directions depending on their charge is not viable here, as for consistent PMT
operation very low magnetic fields are required. Even if we could apply a magnetic field,
this would only help to distinguish µ+ and µ− tracks, as both e+ and e− quickly produce

Chapter 9. Oscillation Analysis

147

an electromagnetic shower, which causes the same Cherenkov light pattern regardless of
the initial particle charge.
Thus, we must look for other means of distinguishing ν and ν̄. This thesis focuses onusing neutron tagging to achieve this goal. Charged current quasi-elastic interactions
are represented asνl + n → l− + p
ν l + p → l+ + n,

where l is a lepton flavour. In high energy interactions, the protons can produce a small
amount of Cherenkov light, but it is difficult to distinguish this from other low energy
charged particles. As discussed in chapter 8, the delayed neutron-capture γ-ray allows
us to observe 20.5% of all neutrons. Unfortunately, for mass hierarchy sensitivity we areinterested mostly in Evis = 2-10 GeV events, where pure CCQE interactions are more
rare. In addition, many neutrino interactions produce more than just a single neutron,
either through secondary hadronic processes in the water, or multi-nucleon production
within the interaction nucleus. Thus it cannot be said simply that neutron observation
directly implies presence of an anti-neutrino. However neutrons still have significant
discriminating power in the CCQE-enhanced single Cherenkov ring samples.
• More neutrons are produced following anti-neutrino interactions (figure 9.1).
Aside from neutrons, there are other clues we can use to determine the neutrino type.
In high energy neutrino interactions, it is typical for many pions to be produced. We
may write these interactions as:

νl + n → l− + N 0 + pions(total hadronic charge = +1)

νl + p → l− + N 0 + pions(total hadronic charge = +2)

ν l + p → l+ + N 0 + pions(total hadronic charge = 0)

ν l + n → l+ + N 0 + pions(total hadronic charge = −1).

Thus we can expect an excess (deficit) in the number of charged hadrons, and in particular positively charged pions following a neutrino (anti-neutrino) interactions. These
positively charged pions will decay to muons and produce decay electrons:Chapter 9. Oscillation Analysis

148

Figure 9.1: The multiplicity of neutrons following neutrino interactions in the MultiGeV 1-ring electron-like sample (left) and MultiGeV Multi-ring electron-like sample(right). The multi-ring sample consists of more deep inelastic scattering events, so neutrons are significantly less useful in this sample. However the excess in anti-neutrino
1-neutron events from CCQE interactions is still apparent. Only CC events corresponding to sample type are shown (i.e. for e-like samples, only CC νe and CC ν̄e ), and all
histograms are normalized to 1. For use in analysis, any events producing >5 neutrons
are added into the 5 neutron bin.

π + →µ+ + νµµ+ →e+ + νe + ν̄µ .

In water, most negatively charged pions will be absorbed, without producing a decay
electron. Therefore a neutrino interaction will tend to produce more decay electrons
than an anti-neutrino interaction. Any events containing > 5 decay-electrons are added
into the 5 decay-electron bin.
• More (fewer) decay electrons are produced following CC ν (ν̄) interactions (figure
9.2).
• More (fewer) Cherenkov rings are observed following CC ν (ν̄) interactions (figure
9.3).
The additional hadrons produced in a neutrino interaction, also have the effect of spreading the interaction energy over a larger number of particles, so we may expect the primary
lepton to be less energetic.

Chapter 9. Oscillation Analysis

149

Figure 9.2: The number of decay electrons, following neutrino interactions in theMultiGeV 1-ring electron-like sample (left) and MultiGeV 1-ring muon-like sample(right). In the muon sample, many more decay-electrons are produced in general, however the excess in the neutrino sample is still apparent. Only CC events corresponding
to sample type are shown (i.e. for e-like samples, only CC νe and CC ν̄e ), and all
histograms are normalized to 1.

Figure 9.3: The number of Cherenkov rings, following neutrino interactions in theMultiGeV Multi-ring electron-like sample. There is some excess in the true CC ν
sample, due to the additional charged hadrons produced in this interaction. Only CC
events corresponding to sample type are shown (i.e. for e-like samples, only CC νe andCC ν̄e ), and all histograms are normalized to 1.

• In CC ν (ν̄) interactions the energy fraction of the most energetic Cherenkov ring
will be smaller (larger) (figure 9.5).

Chapter 9. Oscillation Analysis

150• In CC ν (ν̄) interactions the farthest decay electron will be reconstructed closer
to (farther from) the interaction vertex (“farthest” decay-electron means the reconstructed candidate vertex farthest from the reconstructed neutrino interaction
point. This is chosen as it is most likely to correspond to the decay electron fromprimary lepton) (figure 9.4).

Figure 9.4: This plot shows thefarthest decay-electron travel distance|neutrino vertex - decay electron
vertex|, in the Multi-Gev 1-ring µ-likesample. Only CC events corresponding
to sample type are shown (i.e. for e-likesamples, only CC νe and CC ν̄e ). All
histograms are normalized to 1.

Figure 9.5: This plot depicts the fraction of reconstructed energy in the firstring divided by the total energy in theevent for the MultiGeV Multi-Ring elike sample. Only CC events corresponding to sample type are shown (i.e.
for e-like samples, only CC νe and CCν̄e ) and all histograms are normalized
to 1.

Charged pions and other hadrons are not always perfectly re-constructed into their own
Cherenkov rings. Many times these particles will not produce enough Cherenkov light
to be re-constructed, or their rings may overlap with the light pattern from the primary
lepton, and be mis-reconstructed as a single ring. In these instances, the mis-fitted
Cherenkov ring is more likely to deviate from the expected charge pattern of an electron
or muon. Thus, for CC ν (ν̄) events, the particle identification algorithm may identify
these events as “less (more) e-like” or “less (more) µ-like”.
• In CC ν (ν̄) interactions, the particle identification of the most energetic ring is“less (more) e-like” or “less (more) µ-like” (figure 9.6).
There is one final discriminating variable, which can only be applied to muon-neutrino
events. We know that µ− may be captured by an oxygen nucleus whereas µ+ may not.

Chapter 9. Oscillation Analysis

Figure 9.6: The particle identification likelihood output, for the MultiGev 1-ring e-like sample. PID < 0 is
required to classify the ring as e-like.
Only CC events corresponding to sample type are shown (i.e. for e-like samples, only CC νe and CC ν̄e ), and all
histograms are normalized to 1.

151

Figure 9.7: The decay time of the
decay-electron reconstructed farthestfrom the neutrino interaction, for theMultiGeV Multi-Ring µ-like sample.
Only CC events corresponding to sample type are shown (i.e. for e-like samples, only CC νe and CC ν̄e ), and all
histograms are normalized to 1.

This is due to its negative charge allowing it to orbit very close to the nucleus, similar
to an electron. Hence the effective decay time for the µ− events is reduced.
• After CC νµ interactions, the time between the neutrino interaction, and detectionof the farthest decay electron will be shorter relative to a CC ν̄µ -produced decay
electron (figure 9.7).

9.2.2

Multi-Variate Analysis

Multi-variate analysis is then performed using these variables to separate each sample
into ν-like and ν̄-like. Different sets of variables are applicable to different samples, sothey are described individually here. Neutron tagging is only available is SK-IV, so is not
used in the analysis of SK-I to III. The improved trigger logic of SK-IV also increases
our decay electron tagging efficiency, so generally the ν-ν̄ separation is better in thisperiod.
Most of the neutrino type sensitivity in the MultiGeV 1-ring electron-like sample comesfrom decay-electron and neutron counting, which are both discrete variables, so a simple
cut-based analysis was used in this case. The MultiRing e-like sample, and the µlike samples are separated via a neural network. This is implemented using ROOT’sChapter 9. Oscillation Analysis

152

TMVA (Multi-Variate Analysis) framework, trained using the BFGS method (ROOT
option MLPBNN)[149]. One hidden layer of size N + 5 for each network was used,
where N is the number of input variables. Separate networks were trained for each
SK generation. Alternate multi-variate analysis tools were also considered: a likelihood
method, which was found to have worse performance, and a boosted decision tree, which
had slightly increased performance, but significant over-training (that is, significantly
better performance was found using the training sample than the testing sample).
The cut position on the neural network output is determined automatically in each caseto maximize the value
efficiencyν−like × purityν−like × efficiencyν̄−like × purityν̄−like .

(9.7)

Efficiency is the number of neutrinos (anti-neutrinos) in the ν (ν̄)-like sample, divided
by the total number of neutrinos (anti-neutrinos) in both samples. Purity is the fractionof the the ν (ν̄)-like sample which comprises of ν (ν̄) events.
Determining the optimal cut position in regards to mass hierarchy sensitivity for theseseparations is an interesting problem. It is unclear whether two equally sized samples,or a small but pure sample and a large impure sample is ideal. If using a small and puresample is optimal, then we must consider how small it can be made before the reduction
in hierarchy sensitivity from lower statistics outweighs the improvement from increased
purity. Of course, the optimal way to determine cut position would be to re-calculatemass hierarchy sensitivity for every possible cut position. However this method was not
used here as the required computing power was prohibitive.

9.2.2.1

Multi-GeV, Single-Ring, e-like sample

The Multi-GeV Single-Ring e-like sample is split using a cut on two variables: decayelectron number, and neutron multiplicity. Neutron multiplicity is only used in SK-IV.
An event is selected as ν-like, if it satisfies the following:• ≥1 decay electron• No neutrons (SK-IV only)
Otherwise it is classified as an anti-neutrino. This results in a very small and pure νe like sample, and a large, impure ν̄e -like sample. This cut position was chosen in SK-I to
III as any other cut position results in an un-usably small neutrino sample. In SK-IV
neutron cut was added to increase the purity of the νe -like sample, whilst maintaining aChapter 9. Oscillation Analysis

153

consistent sample size with SK-I to III. The sample breakdown and data-MC agreement
is shown in table 9.3, and figure 9.8.
Interaction type
True CC νe
True CC ν̄e
True CC νµ
True CC ν̄µTrue NC
Total MC
Data (All SK, 4581.5 days)

Total
1409.5
752.5
67.8
14.3
224.3
2468.42463

νe -like
246.4
38.2
35.7
7.5
59.2
387.0
455

ν̄e -like
1163.2
714.3
32.1
6.8
165.1
2081.4
2008

Table 9.3: Multi-GeV 1-ring e-like sample, split into νe -like and ν̄e -like. The top halfof the table shows the estimated event breakdown of each sample, constructed with MC
information. The bottom half shows a total data-MC comparison. The data used is allof SK-I to IV.

Figure 9.8: These plots show the data-MC agreement in SK-IV for the number ofneutrons and number of decay-electrons in the MultiGeV 1 ring e-like sample. The
colours represent the different event types each sample consists of: green and greenhashed are νe and ν̄e respectively. Purple is true νµ events, and yellow is neutral current
events. The MC events are normalized to the livetime of the dataset, and oscillated ina 2 flavour framework with sin2 2θ = 1.0, and ∆m2 = 2.5 × 10−3 eV 2 . Systematic errors
are not included here.

9.2.2.2

Multi-GeV, Multi-Ring, e-like samples

The Multi-GeV Multi-Ring e-like sample separation is done using a neural network
analysis, taking the following variables as input:• Number of decay electronsChapter 9. Oscillation Analysis

154• Number of rings• The most energetic ring’s visible energy fraction• Particle identification likelihood• Number of neutrons (SK-IV only)This sample consists only of events selected after a likelihood function was applied
to remove neutral current events, but there is still a relatively large neutral current
contamination. The NC events are more often classified as ν-like rather than ν̄-like, as
these events are often multiple-pion ejection, a feature shared by neutrino events (asdescribed in the previous section). Figure 9.9 shows the multivariate analysis output forthis sample. Table 9.4 shows the event breakdown.
Interaction type
True CC νe
True CC ν̄e
True CC νµ
True CC ν̄µTrue NC
Total MC
Data (All SK, 4581.5 days)

Total
777.1
267.8
102.1
9.5
235.7
1392.21435

νe -like
453.4
102.6
77.8
6.2
168.9
809.0
862

ν̄e -like
323.7
165.2
24.3
3.3
66.7
583.2
573

Table 9.4: Multi-GeV Multi-ring e-like sample, split into νe -like and ν̄e -like. The top
half of the table shows the estimated event breakdown of each sample, constructed withMC information. The bottom half shows a total data-MC comparison. The data usedis all of SK-I to IV.

9.2.2.3

Multi-GeV, Single-Ring, µ-like samples

The Multi-GeV single-ring µ-like sample is sensitive to all the same variables as the
e-like samples, with the addition of the decay electron distance, and decay electron time
information. Compared to the single-ring e-like case, significant performance increase
was seen using a neural network method rather than a cuts method in this sample. The
variables employed are as follows:• Number of decay electrons• Particle identification likelihood• Decay electron distance• Decay electron time
• Number of neutrons (SK-IV only)Chapter 9. Oscillation Analysis

155

Figure 9.9: These plots show the data-MC agreement, and the νe -ν̄e separation in
the multi-variate analysis output. The event sample is the MultiGeV Multi-ring e-likesample for SK-IV. The cut position is shown here as a dotted blue line at 0.72, chosen
automatically (see 9.2.2). Events >0.72 are classified as νe -like, and <0.72 are ν̄e -like.
For the right-hand plot, both histograms are normalized to 1. In the left-hand plot, theMC events are normalized to the livetime of the dataset, and oscillated in a 2 flavour
framework with sin2 2θ = 1.0, and ∆m2 = 2.5 × 10−3 eV 2 . Systematic errors are notincluded here.

The neural network output here is quite an unusual shape, as the variables with the mostdiscriminating power - neutrons and decay electrons, are both discrete. This feature iscommon in both the data and MC. The data-MC agreement and ν-ν̄ separation can beseen in figure 9.10, and table 9.5.
Interaction type
True CC νe
True CC ν̄e
True CC νµ
True CC ν̄µTrue NC
Total MC
Data (All SK, 4581.5 days)

Total
6.0
2.3
1479.2888.2
4.7
2380.42210

νµ -like
5.4
2.2
930.4
372.8
3.6
1314.31210

ν̄µ -like
0.6
0.1
548.8
515.4
1.1
1066.1
1000

Table 9.5: Multi-GeV 1-ring µ-like sample, split into νµ -like and ν̄µ -like. The top halfof the table shows the estimated event breakdown of each sample, constructed with MC
information. The bottom half shows a total data-MC comparison. The data used is allof SK-I to IV.

9.2.2.4

Multi-GeV, Multi-Ring, µ-like samples

The Multi-GeV Multi-Ring µ-like sample uses the largest amount of input variables,
combining those relevant to muons, and those relevant to multi-ring events:

Chapter 9. Oscillation Analysis

156

Figure 9.10: These plots show the data-MC agreement, and the νµ -ν̄µ separation in
the multi-variate analysis output. The event sample is Multi-GeV single-ring µ-like,for SK-III. The cut value is chosen automatically (described at 9.2.2), shown here as adotted blue line at 0.62. Events >0.62 are classified as νµ -like, and <0.62 are ν̄µ -like.
For the right-hand plot, both histograms are normalized to 1. In the left-hand plot, theMC events are normalized to the livetime of the dataset, and oscillated in a 2 flavour
framework with sin2 2θ = 1.0, and ∆m2 = 2.5 × 10−3 eV 2 . Systematic errors are notincluded here.

• Number of decay electrons• Particle identification likelihood• Number of rings• Most energetic ring’s visible energy fraction• Decay electron distance• Decay electron time
• Number of neutrons (SK-IV only)
Similar to the multi-ring e-like case, this sample has some NC contamination, which has
been mostly labelled as ν-like. The data-MC agreement and ν-ν̄ separation can be seenin figure 9.11, and table 9.6.

9.3

Systematic Errors

This analysis utilizes 185 independent systematic error parameters. Of these, 37 arecommon in SK-I to IV, and an additional 37 x 4 are calculated individually for eachChapter 9. Oscillation Analysis
Interaction type
True CC νe
True CC ν̄e
True CC νµ
True CC ν̄µTrue NC
Total MC
Data (All SK, 4581.5 days)

157
Total
60.6
9.1
1417.1
419.0
113.8
2019.71937

νµ -like
54.9
8.7
942.2
165.1
104.0
1274.81146

ν̄µ -like
5.8
0.4
475.0
254.0
9.8
744.9
791

Table 9.6: Multi-GeV Multi-ring µ-like sample, split into νµ -like and ν̄µ -like. The
top half of the table shows the estimated event breakdown of each sample, constructedwith MC information. The bottom half shows a total data-MC comparison. The data
used is all of SK-I to IV.

Figure 9.11: These plots show the data-MC agreement, and the νµ -ν̄µ separation in
the multi-variate analysis output. The event sample is Multi-GeV multi-ring µ-like,for SK-II. The cut value is chosen automatically (described at 9.2.2), shown here as adotted blue line at 0.75. Events >0.75 are classified as νµ -like, and <0.75 are ν̄µ -like.
For the right-hand plot, both histograms are normalized to 1. In the left-hand plot theMC events are normalized to the livetime of the dataset, and oscillated in a 2 flavour
framework with sin2 2θ = 1.0, and ∆m2 = 2.5 × 10−3 eV 2 . Systematic errors are notincluded here.

SK generation. Each of these error sources contribute an additional fit parameter, j ,which is allowed to vary during the fit to the data. Most of the errors may be classified
as either uncertainty in the neutrino flux, uncertainty in the neutrino interaction, or
uncertainty in the event reconstruction and sample selection. The best fit values of j ,and estimated σj of each error are listed in the appendix.

Chapter 9. Oscillation Analysis

9.3.1

158

Systematic Errors related to Neutrino Flux

This category of errors refers to any uncertainty on the neutrino flux. This includes
uncertainty on the composition of the neutrino flux, and its angular dependence.

9.3.1.1

Flux Ratios

Various ratios of the incoming flux are examined. Firstly, the uncertainty in the neutrino
flavour ratio (νµ + ν̄µ )/(νe + ν¯e ). This is estimated by comparing the result from the
Honda flux used in SK [7], with other neutrino flux models, such as the FLUKA flux [96]and Bartol flux [97]. The flux models diverge at high energy so an energy dependent
error is assigned here. The uncertainty is estimated as 2% for Eν < 1 GeV, 3% for
1 < Eν < 10 GeV, and 5% for 10 < Eν < 30 GeV. From 30 GeV < Eν < 1 TeV theerror increases as a function of log Eν . The deviation between flux models can be seenin figure 9.12.

Figure 9.12: The flavour ratio between the Honda flux and competing flux models used for systematic error calculation. The solid black line shows the ratio
FLUKA/Honda flux, and the dashed red line depicts the Bartol/Honda flux. The
spike around 8 GeV in the FLUKA flux is due to a flux calculation technicality, andhas been corrected for in the error evaluation.

In a similar fashion to the flavour ratio error, there is an error term accounting for
uncertainty in the anti-neutrino/neutrino ratios, ν̄e /νe and ν̄µ /νµ . Again the estimationis conducted by comparing the Honda flux, to the FLUKA and Bartol models.
The ratio of the flux models used to decide this error is shown in figure 9.13.

Chapter 9. Oscillation Analysis

159

Figure 9.13: The anti-neutrino / neutrino double ratio between the Honda flux andcompeting flux models used for systematic error calculation. The solid black line shows
the ratio FLUKA/Honda flux, and the dashed red line depicts the Bartol/Honda flux.
The left plot shows the double ratio of ν̄e /νe and the right plot shows ν̄µ /νµA further two flux ratio errors are the up/down error, and the horizontal/vertical error.
As the names imply, these account for uncertainty in neutrino flux depending on the
direction by which they enter the tank, which is important for zenith angle analysis. The
main cause of uncertainty in these errors is the shape and rigidity cut-off of the Earth’s
geomagnetic field. Similarly, these errors are estimated using the difference between the
Honda flux and the FLUKA and Bartol models.

9.3.1.2

K/π ratio

At energies below 10 GeV, neutrino production is primarily due to pion decay, howeveras the cosmic ray energy increases, the production becomes more dominated by kaons.
This error describes the uncertainty in the ratio of K to π based neutrino production.
This error was estimated using data from the SPY experiment [150] to be 5% for Eν <100 GeV, and then linearly increasing with energy until 20% at 1 TeV.

9.3.1.3

Solar Activity

The solar activity cycles over an 11 year period and the resultant magnetic field changes
have some effect on the cosmic ray flux. The uncertainty of the solar cycle is taken to be±1 year, which corresponds to 20%, 50%, 20% and 10% errors for SK I-IV respectively.

9.3.1.4

Neutrino Production Height

An uncertainty in the neutrino production height will propagate through to uncertainty
in the distance travelled by neutrinos, and thus the oscillation probability of the neutrinos. This error is negligible for upward-going neutrinos but represents a significant

Chapter 9. Oscillation Analysis

160

uncertainty in downward-going and horizontal-going neutrinos. The root of this error isin the uncertainty of the atmospheric density structure, which is estimated to be 10%
by comparison between the US-standard 76 and the MSISE90 experiment [151]. The atmospheric density is then changed by 10% in the neutrino flux model, and the change inneutrino path length is calculated. This change in path length is taken as the systematic
uncertainty.

Figure 9.14: The neutrino path length, as a function of zenith angle. The solid line
represents the path lengths provided in the default Honda flux model, and the dottedline shows the path-lengths after a 10% change in the atmospheric density.

9.3.1.5

Matter Effect

The neutrino oscillation probability in the Earth is dependent on the electron density inthe Earth, and while the total matter density is well known [152], the electron density
depends on the exact chemical composition, and has some uncertainty. The core isassumed to consist of iron and other heavy elements, which have ∼6.8% reduced electron
density compared to if it was constructed primarily of light elements. This difference
is propagated to the oscillation probability uncertainty, and used as a systematic error
source.

9.3.1.6

Absolute normalization

Finally we take a systematic error on the absolute normalization of the neutrino flux.
This was estimated by Honda in [153], taking into account uncertainty in pion production, kaon production, hadronic interaction cross sections and the atmospheric densityChapter 9. Oscillation Analysis

161

profile, as shown in figure 9.15. The final uncertainty is calculated using the two dominant sources, pion production and the hadronic interaction errors. The kaon production
and atmospheric density profile uncertainties are already accounted for in previously
mentioned errors.
However above 10 GeV, this error is not sufficient to account for the differences between
the flux models, Bartol, FLUKA and Honda. Thus, an additional normalization error
of 5% is applied in the FC Multi-GeV, PC, and UPMU samples, whose energy spectra
peak above 10 GeV.

9.3.2

Systematic Errors related to Neutrino Interactions

Systematic errors are applied to the simulation of neutrino interactions including thosethat affect the initial interaction, associated nuclear effects, and the secondary interactions in the water.

9.3.2.1

CCQE Cross Section

Similar to the neutrino flux systematics, the errors applied to the neutrino cross sections
are in the form of an absolute error on νe and νµ , and the ratios (νµ + ν̄µ )/(νe + ν̄e )and ν̄/ν. The CCQE cross section model used in SK is based on the Smith and Moniz
Fermi-gas model [108], and to estimate uncertainty this was compared to the model
calculated by Nieves et al. [154]. The difference between these models for each error
can be seen in figure 9.16.

9.3.2.2

Axial Mass

The axial mass used in the SK MC is set to MA = 1.21 GeV, however there is someuncertainty on this value in recent experiments [155][156]. To account for this, the axial
mass error is taken to be 10%.

9.3.2.3

Single Meson Production

Experimental measurements of the interaction νµ p → µ− pπ + are used to assess the single
meson production cross section error, which is estimated to be 20%. There are fewer
experimental results for π 0 production, so a comparison with the Hernandez model [157]was performed to calculate this error. Thus the π 0 production cross section is assigned
an additional 40% error relative to the π ± uncertainty.

Chapter 9. Oscillation Analysis

Figure 9.15: The top plot shows the relative uncertainty in each aspect of the flux
model, where δπ , δK , δσ , and δair are the uncertainties on pion production, kaon production, hadronic interaction cross sections and atmospheric density profile respectively.
The bottom plot is the calculated uncertainty on the absolute normalization of theneutrino flux.

162Chapter 9. Oscillation Analysis

163

Figure 9.16: These plots show the ratio of the CCQE cross section calculated by the
Nieves model to that calculated by the fermi gas model used in SK. The top left plot
displays νe + ν̄e and νµ + ν̄µ in the solid and dashed lines respectively. The top rightplot shows the ratio (νµ + ν̄µ )/(νe + ν̄e ), and the bottom plot is ν̄/ν.

9.3.2.4

Coherent Pion Production

For coherent pion cross section in NC and CCνe interactions, the uncertainty is estimatedto be 50% following comparison between measurements by SciBooNE [158], and the Rein
and Sehgal model. SciBooNE reported non-existing coherent scattering events, so theuncertainty is taken to be 100%.

9.3.2.5

Deep Inelastic Scattering

The deep inelastic scattering cross section expectation agrees with experiments to within
5%. This uncertainty increases at lower energies, so an additional error is estimated bycomparison to the CKMT model [159]. The uncertainty on the deep inelastic scatteringcross section’s Q2 dependence is also considered, for both W < 1.3 GeV/c2 and W > 1.3
GeV/c2 , by comparison to the GRV98 model.

9.3.2.6

Other Cross Section Uncertainties

The error on the ντ CC cross-section is taken to be 25%, by making a comparison with
the Hagiwara model [160]. The uncertainty of the NC/CC ratio is estimated to be 20%.

Chapter 9. Oscillation Analysis

9.3.3

164

Systematic Errors related to Reduction and Reconstruction Processes

The reduction process refers to the set of cuts applied to reduce the data into background
free neutrino samples. The errors are separate for each of the FC, PC and UPMU event
categories. For more information on the selection of these samples, please refer to chapter
7. Reconstruction refers to the calculation of expected energy, vertex, particle-type, andso forth. This is described in detail in chapter 6.

9.3.3.1

FC Reduction

The uncertainty of the FC reduction efficiency is calculated by comparing data and MC
distributions, for each of the cuts applied in this reduction. This is estimated to be
0.6%, 0.5%, 0.9%, and 0.4% for SK-I to IV, respectively. An error is also calculated
for the background rate in the final sample. The background is estimated by eyescanning the final data-set and contributes a 0.1 − 0.5% uncertainty (differs depending
on reconstructed energy and particle ID. See chapter 7 for details).
FC and PC events are separated by counting the number of PMT hits in the OD hit
cluster with highest charge. By comparing the data and MC expectation for this variable,
the uncertainties in FC/PC separation are estimated as 0.6%, 0.5%, 0.9% and 0.4% for
SK-I to IV.

9.3.3.2

PC Reduction

The PC reduction efficiency uncertainty is estimated by comparing the data and MC for
different cut variables at each stage of the reduction process. This error is taken to be
2.5%, 4.8%, 0.5% and 1.0% for SK-I to IV, respectively. There is also an uncertainty on
the background rate in the final sample, which is estimated by eye-scanning the selected
PC events. This is set to 0.2%, 0.7%, 1.1% and 0.5% for SK-I to IV.
Another PC related uncertainty is the separation between PC stopping events, and PC
through-going events (referring to whether the lepton track stops in the OD or continues out of the detector). This separation is decided by assessing the photo-electrons
deposited in the OD compared to the expectation, (p-eobs /p-eexp ). The uncertainty is
estimated using the difference in this distribution between data and MC, and is assigned
separately for each region of the OD.

Chapter 9. Oscillation Analysis
9.3.3.3

165

UPMU Reduction

Similar to the previous two reductions, the UPMU reduction efficiency uncertainty isassessed by comparing cut variable distributions between data and MC. For stoppingmuons, these uncertainties are assigned to 0.7%, 0.7%, 0.7% and 0.5%, for SK-I to IV.
For through-going muons, these become 0.5%, 0.5%, 0.5% and 0.3% for SK-I to IV.
The separation between stopping and through-going muons is calculated using the number of OD PMT hits within 8m of the expected tank exit point. The data and MC
results for this distribution are compared, and used to calculate the uncertainty, whichis estimated as 0.4%, 0.6%, 0.4% and 0.6% for SK-I to IV.
Uncertainty is also assigned to the separation of UPMU showering and non-showering
events. The average expected charge for non-showering events is calculated, and thedifference between data and MC distributions is taken as the error. This is set as 3.4%,
4.4%, 2.4% and 3.0% for SK-I to IV.

9.3.3.4

Reconstruction errors

The ring separation is based on a likelihood distribution used to determine how many
Cherenkov rings an event is fit best with. The uncertainty in this is calculated bycomparing these likelihood distributions between data and MC.
The particle identification likelihood is similarly used to distinguish between e-like andµ-like events. The uncertainty in this is also calculated by comparing data to MC.
The error decay electron tagging efficiency is calculated by using cosmic muon data to
be 1.5% for SK-I to III. This improves to 0.8% in SK-IV.

9.3.4

Event Selection errors

Event selection errors are related to the division of the dataset into various sub-samples,as described earlier in this chapter. The errors relating to the new samples in thisanalysis will be described below.

9.3.4.1

Multi-GeV Neutrino and Anti-Neutrino sample errors

Multi-GeV samples are split by neutrino/anti-neutrino, single-ring/multi-ring, and elike/µ-like, into a total of 8 samples (plus Multi-Ring Other, which is not discussedhere). Each of these samples is assigned an error on the efficiency, of either CC ν or CC

Chapter 9. Oscillation Analysis

166

ν̄. For example, in the 1-ring e-like sample the errors allow movement of true CC νe andCC ν̄e events between the νe -like and ν̄e -like samples. No additional error is applied to
NC or the opposite neutrino type (CC νµ in this example).
The total error for each sample is calculated as the sum in quadrature of multiple error
sources. The errors utilized are:• Americium-Beryllium 10% error on neutron tagging efficiency: this is a large sourceof uncertainty in the efficiency of the CCQE-enriched single-ring samples, however
it represents a small uncertainty in the multi-ring samples, which are largely deep
inelastic scattering events. In these latter interactions neutron number is a lesspowerful discriminant, and thus contributes less to the total error.
• Decay-electron tagging efficiency ∼ 1%: this error is only significant in SK-I to
III, as in SK-IV the neutron tagging error becomes dominant.
• Pion final state interaction cross sections: 24 sets of weights are created, eachusing an altered set of pion interaction cross-sections to represent an overall 1σchange, as estimated by fits to pion-nucleus scattering data. The parameter set
causing the largest deviation in sample efficiency was chosen as the error in eachcase (∼ 2 − 10%, depending on sample).
• Pion multiplicity error: from comparison between NEUT simulation and CHORUS
νµ beam experimental data. This error results in a ∼ 5% uncertainty on theefficiency of the µ-like samples.
A neutron production systematic error was also considered by comparing neutron multiplicity following a variety of hadronic particle guns, between the FLUKA event generator
with the Geant3 model used in SK. This was however found to be negligible comparedto the Americium-Beryllium neutron tagging efficiency error, so was not included in the
final estimation.

9.4

Mass Hierarchy Sensitivity

Before fitting to data, it is interesting to evaluate our expected sensitivity to the mass
hierarchy with these new samples. This sensitivity can be expressed as ∆χ2 = χ2 (N H)−
χ2 (IH). A lower value χ2 represents a better fit, so a negative value of ∆χ2 would showa preference to the normal hierarchy.
In the sensitivity study, we perform two fits, firstly using a MC generated assuming
normal hierarchy in the place of data, and fitting a MC oscillated according to theChapter 9. Oscillation Analysis

167

inverted hierarchy, and then vice versa. The χ2 value at the best fit point represents the
expected difference between normal and inverted hierarchy.
The “true” parameters for the sensitivity study are chosen as shown in table 9.7.
Oscillation Parameter∆m223
sin2 θ23∆m212
sin2 θ12
sin2 θ13
δCP

Selected True Value
2.5 × 10−3 eV 2
0.53
7.5 × 10−5 eV 2
0.31
0.025
260◦

Table 9.7: Chosen “true” oscillation parameters for the mass hierarchy sensitivity
study.

An identical study was preformed with the binning scheme described in [148], to judge
the effect of the new sample splitting on the mass hierarchy sensitivity.. This binning
does not use neutron tagging, and the Multi-GeV µ-like samples are not split into ν andν̄-like. The results are shown below in table 9.8. Plots showing the projected sensitivity
contours can be seen in figure 9.17.

True Normal Hierarchy
True Inverted Hierarchy

Previous Sample Selection |∆χ2 |
1.81
0.83

New Sample Selection |∆χ2 |
1.87
0.89

Table 9.8: Improvement in sensitivity to neutrino mass hierarchy due to the new ν-ν̄
samples.

Chapter 9. Oscillation Analysis

168

Figure 9.17: Projected χ2 plots for the mass hierarchy sensitivity. The true hierarchy
is taken to be normal, and the black and green lines represent fits assuming the normal
and inverted hierarchy respectively. The minimum ∆χ2 value of the inverted hierarchyat ∆χ2 = 1.87 represents our sensitivity to the mass hierarchy in this case.

9.5

Data Fit Results

Our MC simulation is now fit to the dataset from SK-I to IV, for 37,638 different
sets of oscillation parameters defined at the start of this chapter. At every point in theparameter space a fit was performed over the systematic errors j to find the minimum χ2value for that parameter set. Following this, the minimum χ2 over the whole oscillation
parameter space is selected as best fit point. This procedure was performed twice,assuming a normal or an inverted hierarchy. The following parameters are taken as
constant: sin2 θ13 = 0.025, ∆m212 = 7.5 × 10−5 eV2 , and sin2 θ12 = 0.31. The best fit
points for each hierarchy are summarized in table 9.9.

Chapter 9. Oscillation Analysis∆m223
sin2 θ23
δCP
χ2

Assumed Normal Hierarchy
2.4 × 10−3 eV 2
0.46
260◦
636.2/577 DoF

169
Assumed Inverted Hierarchy
2.3 × 10−3 eV 2
0.56
220◦
637.1/577 DoF

Table 9.9: Oscillation analysis best fit point for normal and inverted hierarchy assumptions.

Thus the normal hierarchy is favoured by ∆χ2 (N H −IH) = −0.9. The error parameters
for the best fit point can be found in the appendix. Contour plots for the normal andinverted hierarchy results are shown in figures 9.18 and 9.19. The 90% confidence levels
for sin2 θ23 , ∆m223 and δCP are defined at χ2 = χ2min + 2.7, and shown in table 9.10.
∆m223
sin2 θ23
δCP

Assumed Normal Hierarchy(2.14 < ∆m223 < 2.91) × 10−3 eV 2
0.397 < sin2 θ23 < 0.623
103.0 < δCP < 352.8◦

Assumed Inverted Hierarchy(2.07 < ∆m223 < 2.83) × 10−3 eV 2
0.406 < sin2 θ23 < 0.617
110.5 < δCP < 333.9◦

Table 9.10: Oscillation analysis 90% confidence levels for normal and inverted hierarchy assumptions.

An overview of the data and MC agreement after fitting can be seen in figure 9.20.
Bin-by-bin plots of the new ν-like, ν̄-like, and Multi-Ring Other samples can be foundin figure 9.21.

Chapter 9. Oscillation Analysis

Figure 9.18: Assuming the normal hierarchy, these plots show fitted χ2 values forpoints across the oscillation parameter space. The minimum value of χ2 is subtracted
from every point. From top right going clockwise the plots show: sin2 θ23 − ∆m223
contour plot, minimum ∆m223 , minimum δCP, and minimum sin2 θ23 . The colouredlines on the plots represent the confidence level contours, 99% (orange), 90% (blue) and
68% (red). In the 1d plots, these correspond to ∆χ2 = 6.6, 2.7 and 1.0 respectively. In
the 2d case, these contours correspond to ∆χ2 = 9.2, 4.6 and 2.3. The black line in the
1d plots represents the projected minimum χ2 values. The dataset is all of SK-I to IV.

170

Chapter 9. Oscillation Analysis

Figure 9.19: Assuming the inverted hierarchy, these plots show fitted χ2 values forpoints across the oscillation parameter space. The minimum value of χ2 is subtracted
from every point. From top right going clockwise, the plots show: sin2 θ23 − ∆m223
contour plot, minimum ∆m223 , minimum δCP, and minimum sin2 θ23 . The colouredlines on the plots represent the confidence level contours, 99% (orange), 90% (blue)and 68% (red). The black line in the 1d plots represents the projected minimum χ2values. The data-set is all of SK-I to IV.

171

Chapter 9. Oscillation Analysis

Figure 9.20: The data-MC agreement for each sub-sample, after fitting, with combined data from SK-I to IV. The oscillation parameters used can be seen in table 9.9.
Multi-GeV refers to the Multi-GeV 1-ring sample, and Multi-Ring refers to the MultiGeV Multi-Ring sample. The inverted hierarchy best fit is in red, and normal hierarchy
in green (these are mostly overlaid and cannot be distinguished). The 4 samples on theleft are not divided into zenith bins during analysis, so they are shown in momentum
bins. The remaining 4 columns of samples, are divided into momentum and zenith bins
in analysis, but here only the projection onto zenith bins is shown.

172

Chapter 9. Oscillation Analysis

Figure 9.21: Data vs MC for the new ν-like, ν̄-like, and Multi-Ring Other samples.
The dataset shown includes all of SK-I to IV. The MC is oscillated using the best fit
parameters shown in table 9.9. Multi-GeV refers to the Multi-GeV 1-ring samples, andMulti-Ring to the Multi-GeV Multi-Ring samples. The x-axis shows the zenith anglecos θ, where cos θ = -1 represents upward-going events. The blue line represents thenormal hierarchy expectation, and the red line the inverted hierarchy. The momentum
shown at the top of each plot is the lower end of the momentum bin.

173

Chapter 9. Oscillation Analysis

9.6

174

Discussion and Comparison to other Experiments

Utilizing the new ν-ν̄ separation the sensitivity to neutrino mass hierarchy has improved
from ∆χ2 = 1.81 to ∆χ2 = 1.87 (assuming true normal hierarchy). After fitting to data
the difference at the best fit points for each hierarchy is ∆χ2 (NH-IH) = -0.9 in boththe current official analysis and this analysis. This represents a slight preference forthe normal hierarchy, but is unfortunately not sufficient to reach a conclusion. Mostof the preference for normal hierarchy comes from the Multi-GeV 1-R e-like ν and ν̄like samples, which have a small excess and deficit respectively in the upward going(cos θ < 0) bins (figure 9.21). The contributions to the value of ∆χ2 on a sample-bysample basis can be seen in figure 9.23. A comparison of the 90% confidence levels of
the previous, and updated SK analyses can be seen in figure 9.22. These contours arelargely unchanged.
The 90% confidence levels of δCP in the normal hierarchy case are found to be 103.0 <δCP < 352.8◦ . The exclusion of δCP = 60◦ is primarily driven by an excess in SubGeV
e-like events. CP-conservation requires sin (δCP ) = 0, which is allowed within the 90%
confidence level in both the normal hierarchy and inverted hierarchy case.
The MINOS result from 2013 reports a best fit point of ∆m223 = 2.41 × 10−3 eV2 , andsin2 θ23 = 0.39 [31]. The T2K result from muon disappearance studies gives a best fit of∆m223 = 2.51 × 10−3 eV2 and sin2 θ23 = 0.514[32]. The agreement of these experimentswith the presented analysis at the 90% confidence level is shown in figure 9.24.

Figure 9.22: Comparison of the 90% confidence level contour between the current SK
official analysis (green) and analysis with the updated samples presented in this thesis(blue).

Chapter 9. Oscillation Analysis

Figure 9.23: The contribution to ∆χ2 (NH-IH) for each sample in the analysis. The
samples favouring normal hierarchy are primarily the Multi-GeV 1-ring e-like samples,
and the Multi-GeV Multi-Ring Other sample.

Figure 9.24: Comparison of the 90% confidence level contours between the analysis
presented in this thesis (blue) and published results from the MINOS (green) [31] andT2K (red) [32] experiments.

175Chapter 9. Oscillation Analysis

9.7

176

The Future

So the questions remains - if we are not able to deduce the neutrino mass hierarchy using
the current generation detector and analysis techniques, how do we proceed from here?
Super-Kamiokande clearly has some sensitivity to the neutrino mass hierarchy, but remains limited by its systematic and statistical errors. If at some point in the future weare able to understand our detector perfectly and reduce systematic error to 0, the mass
hierarchy sensitivity (assuming true NH) improves from ∆χ2 = 1.87 → 2.68. This is agood improvement, however the detector is already mature and well understood, so it isdifficult to significantly reduce systematic errors past the current level. Statistical error
may be reduced simply by running the detector for a longer time period. An additional
10 years of runtime for Super-K would result in a mass hierarchy sensitivity improvement of ∆χ2 = 1.87 → 3.23 (see figure 9.25). Unfortunately even after this time a 2σ
level of sensitivity cannot be achieved, so without major upgrades to the detector it isunlikely Super-K will make this discovery in the near future.

Figure 9.25: Expected future significance of neutrino mass hierarchy in SuperKamiokande. The green and red dotted lines indicate 1σ and 2σ significance respectively. The oscillation parameters are as described at 9.7.

However there are other upcoming experiments that are potentially more sensitive to theneutrino mass hierarchy than Super-K. The NOνA experiment, which started operation
earlier this year (2014), produces a νµ beam with a long baseline of 810 km runningfrom Fermilab to Minnesota [33]. This corresponds to a value of L/E well matching thematter effect resonant enhancement, and thus a good sensitivity to the neutrino mass
hierarchy is expected. After 3(+3) years of running in neutrino (anti-neutrino) mode,Chapter 9. Oscillation Analysis

177∼3σ significance of hierarchy resolution may be achieved depending on the true value ofδCP (figure 9.26).

Figure 9.26: Expected significance of neutrino mass hierarchy in the NOνA experiment, after 3 years each of νµ and ν̄µ beam. The blue (dashed red) curves show thesignificance assuming the normal (inverted) hierarchies. Plot taken from [33].

The Indian Neutrino Observatory (INO) is a proposed experiment to further the studyof atmospheric neutrinos [161]. It will use a 50 kton magnetized iron calorimeter, whichallows for the separation of neutrino from anti-neutrino events based on the lepton
track curvature. NOνA is sensitive to uncertainty in the true value δCP , however measurements of the atmospheric neutrino matter effect enhancement in INO are virtuallyindependent of this [34]. Depending on the true value of the neutrino oscillation parameters, a ∼ 3σ significance in the mass hierarchy determination after 10-20 years of
running time (figure 9.27).
The direct successor to Super-Kamiokande is the proposed Hyper-Kamiokande experiment. Hyper-K is a 0.99 Mton water Cherenkov detector, with a fiducial volume ∼25times larger than Super-K [35]. Compared to Super-K, Hyper-K will have a lower
photo-coverage of just 20%, which will make the neutron-tagging study of this thesis
more difficult to perform, but mass hierarchy sensitivity can still be achieved from therelative cross sections and fluxes of neutrino and anti-neutrino. Thus a ∼3σ sensitivity
is expected within 5 years of operation, depending on the true oscillation parametervalues (figure 9.28).
The final experiment that should be discussed, is the PINGU upgrade to the IceCube
experiment [162]. PINGU will add a compact set of 40 strings (each used to hold PMTs)
to the centre of the experiment, reducing the detector’s energy threshold down a few
GeV. No direct distinction between neutrino and anti-neutrino is possible; differences influxes and cross sections are relied upon for measurements of neutrino mass hierarchy.

Chapter 9. Oscillation Analysis

178

Figure 9.27: Expected sensitivity to the neutrino mass hierarchy of the proposed INO
experiment. The blue shaded region represents different values for true sin2 θ13 . Plottaken from [34].

Due to a large predicted atmospheric neutrino flux and relative ease of construction,
PINGU may be the first experiment to exceed 3σ mass hierarchy sensitivity (figure9.29).

Chapter 9. Oscillation Analysis

Figure 9.28: Expected sensitivity to the neutrino mass hierarchy of the HyperKamiokande experiment. The different colours and shaded regions represent the expected hierarchy sensitivity given different true values of other neutrino oscillation
parameters. Plot taken from [35].

Figure 9.29: Expected mass hierarchy sensitivity of various future experiments. The
widths represent uncertainty due to oscillation parameters. Plot taken from [36].

179Chapter 10

Conclusion
Following atmospheric neutrino events in SK-IV, neutrons have been successfully identified via the characteristic 2.2 MeV γ ray produced after their capture on hydrogen.
Using a neural network analysis the achieved neutron detection efficiency is 20.5%, andthe expected mis-tagging rate is 1.8% per neutrino event. Neutron capture gamma-rays
were effectively simulated and integrated into an atmospheric neutrino Monte-Carlo simulation, and the predicted number of neutrons was found to agree well with SK-IV data.
Using this information and other improvements, the splitting of the Multi-GeV e-likesamples into ν-like and ν̄-like was updated. Similar separation was also implemented in
Multi-GeV µ-like samples.
With this updated event binning, the neutrino mass hierarchy sensitivity was found to
improve by ∆χ2 = 0.06 (∼5%), in both the true NH and true IH cases. This result
represents the current world best neutrino mass hierarchy sensitivity. A three-flavour
oscillation analysis was performed using all data from SK-I (1489.2 days FC data),
SK-II (798.6 days FC data), SK-III (518.1 days FC data) and SK-IV (1775.6 days FCdata). The fit values of ∆m223 and sin2 θ23 are consistent with results from T2K [32]and MINOS [31]. The results favour the normal hierarchy by ∆χ2 (N H − IH) = −0.9,
which is insufficient to definitely determine the mass hierarchy. The fit results disfavour
δCP = 60◦ , however CP conservation is allowed within the 90% confidence level forboth hierarchies.
Despite almost 20 years of operation, statistics still remain the major limiting factor in
the measurement of the PMNS oscillation parameters at Super-Kamiokande. Study is
ongoing into the possibility of adding a gadolinium compound to SK [163], which would
produce an 8 MeV γ cascade following neutron capture. With an absorption efficiencyof 90%, and potentially very high detection efficiency, this could further improve theneutrino anti-neutrino separation, and thus the detector’s sensitivity to neutrino mass
180

Chapter 10. Conclusion

181

hierarchy. A new reconstruction algorithm, fiTQun [164], is also in development for use
in SK, which could grant additional improvements to event sample purities. However,it may be that we must wait for the next generation of experiments, until the neutrinomass hierarchy can be conclusively determined.

Appendix A

Appendix: Systematic Error
Tables
The estimated σ values, and  corresponding to the best fit point for every (185) systematic error are shown here. Of the errors, 37 are common to all SK generations, and37 (x4) are defined separately for SK-I to IV.
Flux Errors
Absolute NormalizationEν <1 GeVEν >1 GeVνµ /νeEν <1 GeV
1 < Eν <10 GeV
Eν >10 GeVν̄e /νeEν <1 GeV
1 < Eν <10 GeV
Eν >10 GeVν̄µ /νµEν <1 GeV
1 < Eν <10 GeV
Eν >10 GeV
Up/Down ratio
Horizontal/vertical ratio
K/π ratio
Neutrino path length
Relative Normalization
FC MultiGeV
PC, UPMU Stop
Matter EffectSolar Activity
SK-I
SK-II
SK-III
SK-IV

182

Estimated σ (%)
25.0
15.0
2.0
3.0
5.0
5.0
5.0
8.0
2.0
6.0
15.0
1.0
1.0
10.0
10.0
5.0
5.0
6.8
20.0
50.0
20.0
10.0

Best Fit  (%)
32.0
13.5
-1.9
2.8
6.9
6.0
4.4
-1.8
-0.3
-0.8
2.6
-1.8
0.5
-7.5
5.9
-6.9
-3.7
- 1.5
-0.5
12.5
1.6
0.7

Appendix A. Systematic Error Tables

183

Neutrino Interaction Errors
Axial MassCC ντ Cross Section
CCQE ν̄/ν ratio
CCQE νµ /νe ratio
CCQE cross section model
Coherent π production
DIS cross section model (Eν < 10 GeV)DIS cross section
DIS Q2 spectrum (W < 1.3 GeV/c2 )DIS Q2 spectrum (W > 1.3 GeV/c2 )Hadronic Simulation
NC/CC ratio
Single π production ν̄/ν ratio
Single π production π 0 /π ± ratio
Single meson production cross section
Decay Electron tagging (π interaction)

Oscillation Parameter Errors
θ13
θ12

Sample Migration Errors
Multi-GeV 1-Ring e-like ν-like
Multi-GeV 1-Ring e-like ν̄-like
Multi-GeV 1-Ring µ-like ν-like
Multi-GeV 1-Ring µ-like ν̄-like
Multi-GeV Multi-Ring e-like ν-like
Multi-GeV Multi-Ring e-like ν̄-like
Multi-GeV Multi-Ring µ-like ν-like
Multi-GeV Multi-Ring µ-like ν̄-like
Multi-GeV Multi-Ring Other

Reconstruction Errors
Single-Ring PID
Multi-Ring PID
Ring Separation
Single-Ring π 0 selection
Two-Ring π 0 selection
Decay-e tagging efficiency 10.0
Energy Calibration
Energy Calibration Up/down ratio

Estimated σ (%)
10.0
25.0
10.0
10.0
10.0
100.0
10.0
5.0
10.0
10.0
10.0
20.0
10.0
40.0
20.0
10.0

Estimated σ (%)
1.0
2.1Best Fit  (%)-17.7- 33.8
15.3
5.74
3.65
19.5
-0.2
-2.1
4.2
0.0
0.8
3.7
-6.6
-20.2
11.5
-9.1

Best Fit  (%)
-0.1
-0.1

SK-I
σ

10.0
6.0
0.2
-0.1
1.7
0.4
1.8
-0.2
1.1
0.1
1.0
0.0
1.9
-1.7
2.8
0.8
10.0 -10.4

SK-II
σ

7.9
0.8
0.5
0.1
1.5
0.1
1.8 -0.1
0.8
0.1
1.2 -0.1
1.7 -0.9
3.1
0.7
10.0 -4.9

SK-III
σ

9.3
4.7
0.2
-0.1
1.6
0.2
1.8
-0.1
1.0
-0.1
1.1
0.0
2.6
0.1
2.5
0.0
10.0 -12.3

SK-IV
σ

12.5 4.9
0.3 -0.1
3.1 -4.1
2.6
1.7
1.0
0.1
1.3 -0.1
3.6 -5.2
4.0
1.6
10.0 -1.5

SK-I

SK-II
σ

1.0
0.2
10.0 8.5
10.0 -8.0
10.0 2.1
4.4 -2.7-2.9 10.0
1.7 -0.5
0.6
0.0

SK-IIIσ
1.0 -0.1
10.0 -1.2
10.0 2.1
10.0 2.9
5.9 -1.6
1.5 10.0
2.7
0.6
1.3
0.3

SK-IV
σ

1.0
0.0
10.0 3.5
10.0 -2.2
10.0 4.6
5.6
1.4
0.1
2.3 -0.2
0.3 -0.2

σ
1.0
10.0
10.0
10.0
5.6
-5.5
1.1
0.6


-0.1
6.5
4.8
2.8
-0.5
10.0
-0.1
0.3

Appendix A. Systematic Error Tables

Reduction Errors
FC Reduction
Fiducial Volume
Non-ν BG (µ-like)
Non-ν BG (SubGeV e-like)
Non-ν BG (MultiGeV 1R e-like)
Non-ν BG (MultiGeV Multi-Ring e-like)
FC/PC Separation
PC Reduction
PC Stop-Thru Separation (Barrel)
PC Stop-Thru Separation (Bottom)
PC Stop-Thru Separation (Top)UPMU Stop-thru separation
UPMU Reduction
UPMU Stop BG
UPMU Stop Energy cut
UPMU Thru Path cut
Non-Showering BG
Shower Non-shower separation
Showering BG

184

SK-I
σ

0.2
0.0
2.0
-0.6
1.0
0.3
1.0
0.1
13.2
0.9
12.1 11.6
0.6
0.0
2.4
-1.6
7.0
1.1
22.7 -27.0
46.1
3.2
0.4
0.0
1.0
-0.1
16.0 16.3
0.9
0.1
1.5
0.2
18.0 -5.9
3.4
5.6
18.0 -7.3

SK-II
σ

0.2
0.0
2.0
-0.4
1.0
0.2
1.0
0.0
38.1 -9.9
11.1
6.4
0.5
0.1
4.8
-4.3
9.4
-6.7
12.9 -0.9
19.4 -4.3
0.6
0.0
1.0
-0.2
21.0 -1.5
1.3
0.2
2.3
0.8
14.0 -0.2
4.4-1.3
14.0 -13.0

SK-III
σ

0.8
0.1
2.0
-0.6
1.0
-0.1
1.0
0.0
26.7
2.4
11.4
5.5
0.9
0.0
0.5
-0.1
28.7 -12.5
12.1 -0.9
86.6 19.2
0.4
0.0
1.0
0.2
20.0
3.3
2.0
0.8
2.8
-1.2
24.0
0.3
2.4
2.9
24.0
2.8

SK-IV
σ

0.3
0.0
2.0
-0.6
1.0
0.0
1.0
0.0
17.6
0.9
11.6
4.8
0.0
0.0
1.0
-0.6
8.5
-2.4
6.8
-2.3
40.3 -17.5
0.6
-0.1
1.0
0.2
17.0 -0.5
1.7
-0.2
1.5
-1.8
17.0
2.1
3.0
4.1
24.0 -1.5

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