Study of the neutrino mass hierarchy with theatmospheric neutrino data observed in Super-Kamiokande

By
Lee Ka Pik
October 2012

Abstract

This thesis presents the full three flavor neutrino oscillation analysis carried out with SuperKamiokande atmospheric neutrino data in order to obtain information on the mass hierarchy
for the first time. The entire set of oscillation parameters, which include two mass di↵erencesm212 and m232 , three mixing angles ✓12 , ✓23 , ✓13 , and the CP phase parameter CP , andthe mass hierarchies are considered. This is the first study to probe the mass hierarchy withneutrino and anti-neutrino enriched event samples from the Super-Kamiokande atmospheric
neutrino data.
Super-Kamiokande is a 50 kton water Cherenkov detector which started taking data since1996. All the observed data of atmospheric neutrino through running periods SK-I, SK-II, SK-III
and SK-IV collected from 1996 to 2012 are used in this analysis.
Data selection is summarized in this thesis. The separation of neutrino and anti-neutrino
enriched samples with likelihood method is also presented.
Sensitivity study shows that the newly developed neutrino and anti-neutrino enriched samples help to improve the sensitivity on mass hierarchy and other oscillation parameters. The
analysis are carried out for both cases of free ✓13 and fixed ✓13 . In case of fixed ✓13 , inverted
2 (= 2 (N H)
2 (IH)) of 1.2, which ishierarchy is slightly favoured with significancemin
minstill low to draw a conclusion for the mass hierarchy. It is expected that the mass hierarchy canbe constrained with atmospheric neutrinos with longer exposure time or larger detector size infuture experiments.

Study of the neutrino mass hierarchy with theatmospheric neutrino data observed in Super-Kamiokande
by
Lee Ka Pik

October 2012

Acknowledgements
I would like thank my supervisor Prof. Kajita for giving me the chance to study atmospheric
neutrino. Under his continuous guidance, I have learnt a lot of things. I am very grateful that
I am given a chance to join this experiment.
I would like to thank Prof. Masato Shiozawa for his excellent guidance and patience. He
gave me a lot of valuable suggestions throughout the whole process of oscillation analysis.
I would also like to express my gratitude to Prof. Y. Suzuki, for his valuable advice on theanalysis.
I would also like to thank Prof. Y. Hayato and Dr. K. Okumura, for their continuous supportand encouragement.
I would like to thank members of the atmospheric neutrino analysis group especially Prof.
S. Moriyama, Dr. M. Miura, Dr. Y. Obayashi, Dr. J. Kameda, Dr. S. Nakayama, Dr. H. Kaji,
Prof. Y. Itow, Prof. E. Kearns, Prof. C. Walter, Dr. S. Mine, Dr. J.L. Raaf, Dr. R. Wendell,T. Wongirad.
I would also like to thank other ICRR sta↵ members, Prof. M. Nakahata and Dr. Y. Koshio.
I would like to thank my friends who support and encourage me all the time. Dr. C. Ishihara,
Dr. T. Iida, Dr. M. Ikeda, Dr. T. Tanaka, Dr. K. Ueshima, T. F. McLachlan, K. Ueno, T.
Yokozawa, K. Iyogi, M. Dziomba, P. de Perio, K. Choi and T. Irvine.
This analysis is based on the hard work by all members of Super-Kamiokande. I would liketo express my gratitude to all members supporting this experiment.
I would like to specially thank K. Watanabe for her continuous support throughout my stayin Japan.
Lastly, I would like to express my deep gratitude to my family who support me all the time.

iContents
AcknowledgementsiTable of Contentsii

List of Figuresv

List of Tables

xvi

1 Introduction
1.1 Neutrino Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Neutrino Oscillation Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Solar and Reactor Neutrino Observations . . . . . . . . . . . . . . . . . .
1.2.2 Atmospheric Neutrino and Accelerator Neutrino . . . . . . . . . . . . . .
1.2.3 Mixing Angle ✓13 Measurement by Accelerator and Reactor Neutrinos . .
1.2.4 Summary for Neutrino Oscillation Parameters and Remaining Issues . . .
1.3 Sub-Dominant E↵ects of Atmospheric Neutrino Oscillation . . . . . . . . . . . .

1
1
3
3
4
5
7
8

2 The Super-Kamiokande Detector
2.1 Cherenkov Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Water Tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Photomultiplier Tube for the Inner Detector . . . . . . . . . . . . . . . . .
2.2.3 Photomultiplier Tube for the Outer Detector . . . . . . . . . . . . . . . .
2.2.4 Outer Detector (OD) Segmentation . . . . . . . . . . . . . . . . . . . . .
2.3 Water Purification System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Air Purification System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Electronics and Data Acquisition System for SK-I to SK-III Periods . . . . . . .
2.5.1 Inner Detector Electronics and Data Acquisition System . . . . . . . . . .
2.5.2 Outer Detector Electronics and Data Acquisition System . . . . . . . . .
2.5.3 Trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Electronics and Data Acquisition System for SK-IV Period . . . . . . . . . . . .
2.6.1 Trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.2 Improvement by the Electronics and Data Acquisition System in SK-IV
Period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Simulation of Atmospheric Neutrino
3.1 Atmospheric Neutrino Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Neutrino Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Charged Current Quasi-Elastic Scattering . . . . . . . . . . . . . . . . . .
3.2.2 Neutral Current Quasi-Elastic Scattering . . . . . . . . . . . . . . . . . .
3.2.3 Single Meson Production . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.4 Deep Inelastic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.5 Coherent Pion Production . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.6 Nuclear E↵ects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Detector Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Particle Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Cherenkov Photon Tracking in Water . . . . . . . . . . . . . . . . . . . .
3.3.3 Response of PMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Calibration
4.1 Detector Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Absolute Gain Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Relative Gain Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.3 Relative Timing Calibration . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.4 Water Scattering Measurement with a Laser . . . . . . . . . . . . . . . . .
4.1.5 Water Transparency Measurement with Cosmic Ray Muons . . . . . . . .
4.2 Energy Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 High Energy Stopping Muons . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Low Energy Stopping Muons . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.3 Neutrino Induced ⇡ 0 Events . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.4 Decay Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.5 Summary of the Energy Scale Calibration . . . . . . . . . . . . . . . . . .
4.2.6 Time Variation of Energy Scale . . . . . . . . . . . . . . . . . . . . . . . .
4.2.7 Detector Uniformity of Energy Scale . . . . . . . . . . . . . . . . . . . . .

47
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49
50
51
57
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57
61
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5 Data Selection
70
5.1 Reduction for Fully Contained Events . . . . . . . . . . . . . . . . . . . . . . . . 71
5.1.1 First Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.1.2 Second Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.1.3 Third Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.1.4 Fourth Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.1.5 Fifth Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.1.6 FC Reduction Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.2 Reduction for Partially Contained Events . . . . . . . . . . . . . . . . . . . . . . 89
5.2.1 First Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2.2 Second Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2.3 Third Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.2.4 Fourth Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.2.5 Fifth Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.2.6 PC Reduction Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.2.7 PC OD Stop/Through Separation . . . . . . . . . . . . . . . . . . . . . . 104
iii

5.3

5.4

Reduction for Upward-going Muon Events . . . . . . . . . . . . . . . . . . . . . . 107
5.3.1 Charge cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.3.2 Zenith Angle Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.3.3 Scanning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.3.4 UPMU Reduction Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.3.5 Background Estimation for Upward-going Muon . . . . . . . . . . . . . . 110
5.3.6 Upward Through-going Showering Muon . . . . . . . . . . . . . . . . . . . 113
Event Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.4.1 Vertex Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.4.2 Ring Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.4.3 Particle Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.4.4 Precise Vertex Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.4.5 Momentum Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.4.6 Decay Electron Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.4.7 ⇡ 0 Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6 Neutrino Oscillation Analysis
131
6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.2 Data Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.2.1 Selection for Sub-GeV sample . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.2.2 Selection for Multi-GeV Multi-Ring E-like Sample . . . . . . . . . . . . . 135
6.2.3 Separation of CC⌫e and CC⌫ e events in Multi-GeV E-like samples . . . . 138
6.2.4 Data Set For the Oscillation Analysis . . . . . . . . . . . . . . . . . . . . 144
6.3 Oscillation E↵ect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.4 Analysis Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.4.1 Definition of 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.5 Systematic Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.5.1 Systematic Errors in Neutrino flux . . . . . . . . . . . . . . . . . . . . . . 163
6.5.2 Systematic Errors in Neutrino Interactions . . . . . . . . . . . . . . . . . 168
6.5.3 Systematic Errors in Event Selection and Reconstruction . . . . . . . . . 173
6.6 Sensitivity Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
6.7 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
6.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
6.8.1 Event Samples Contributing to the Sensitivity to Mass Hierarchy . . . . . 188
6.8.2 E↵ect of Systematic Errors . . . . . . . . . . . . . . . . . . . . . . . . . . 189
6.8.3 Consistency with Other Experiments . . . . . . . . . . . . . . . . . . . . . 193
7 Conclusion

195

Bibliography

197

iv

List of Figures
1.1
1.2

1.3
1.4

1.5

1.6

2.1
2.2

2.3
2.4
2.5
2.6
2.7
2.8
2.9

Proton and helium spectra from primary cosmic rays. [22] . . . . . . . . . . . . .
90% C.L. allowed regions of oscillation parameters for ⌫µ $ ⌫⌧ oscillation. Horizontal axis is sin2 2✓23 and the vertical axis is m232 .The results are taken from
SK zenith angle analysis (solid line)[36], L/E results (dashed line)[31], MINOS
experiment(dotted line)[34] and K2K(dotted and dashed line)[32]. . . . . . . . .
Three-neutrino mass-squared spectrum, for the two mass hierarchies. . . . . . . .
Left: ⌫µ to ⌫e (or ⌫e to ⌫µ ) transition probability in the Earth; Right: ⌫µ to ⌫e (or⌫e to ⌫µ ) transition probability in vacuum. cos ⇥⌫ = -1 corresponds to upward
direction and cos ⇥⌫ = 0 corresponds to horizontal direction. ( m223 , sin2 ✓23 ,sin2 ✓13 ) = (2.1 ⇥ 10 3 eV2 , 0.5, 0.04) is assumed in the plots [43]. . . . . . . . .
Left: ⌫e transition probability Pex as a function of energy E⌫ and zenith angle,
assuming oscillation parameters ( m212 , m223 , sin2 ✓12 , sin2 ✓23 ) = (7.7 ⇥ 10 50eV2 , 2.1 ⇥ 10 3 eV2 , 0.3, 0.5); Right: Electron flux ratio osc
e / e . There isan excess (a deficit) for sin2 ✓23 in the first (second) octant as shown in the top(bottom) figure. As for sin2 ✓23 = 0.5, both small excess and deficit are seen as
shown in the middle figure. [43]. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Left: Probability of ⌫µ ! ⌫e oscillation as a function of zenith angle and neutrinoenergy, in case of ( m212 , m223 , sin2 ✓12 , sin2 ✓23 , sin2 ✓13 , CP ) = (7.7 ⇥ 10 5 ,2.1 ⇥ 10 3 , 0.3, 0.5, 0.04, 0 ). Right: ⌫e flux ratio osc / 0 assuming the same
oscillation parameters as the left figure. Figures taken from [43]. . . . . . . . . .
Super-Kamiokande detector and its location in Mountain Ikenoyama. . . . . . . .
Visual display of a typical neutrino event in the Super-Kamiokande detector. The
size of the small circles indicates the amount of Cherenkov photons detected ineach PMT. The Cherenkov ring pattern is clearly visible. (Event taken from
SK-IV period.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Supporting frame of PMTs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Schematic view of the 20-inch PMT used in Super-Kamiokande. . . . . . . . . . .
Quantum efficiency distribution of the 20-inch PMT used in the ID against wavelength and the relative Cherenkov spectrum through pure water. [135] . . . . . .
Single photoelectron distribution of the 20-inch PMT. . . . . . . . . . . . . . . .
PMT case used to cover 20-inch PMT in the inner detector since SK-II period. .
A schematic view of the OD region. [43] . . . . . . . . . . . . . . . . . . . . . . .
Left: SK-II OD event display of a partially contained event, without OD segmentation; Right: SK-III OD event displa of a partially contained event, with OD
segmentation. [43] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.10 Block diagram of analog input part of the ATM module. . . . . . . . . . . . . . .
2.11 Schematic view of the ID data acquisition system for SK-I to SK-III periods. . .
2.12 Schematic view of the OD data acquisition system for SK-I to SK-III periods. . .
2.13 Overview of the ID trigger scheme for SK-I to SK-III periods. . . . . . . . . . . .
2.14 Block diagram of the QTC and its surroundings (Data acquisition system for
SK-IV period). [53] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.15 Block diagram of one QTC channel. Each channel has three gain ranges: small,medium and large. (Data acquisition system for SK-IV period)[53] . . . . . . . .
2.16 Timing chart for QTC operation. (Data acquisition system for SK-IV period) [53]Primary cosmic ray proton flux used in Honda flux calculation (line) and theobserved results (points) [55]. Observation data points are taken from Webber [60](crosses), LEAP [61] (upward triangles), MASS1 [62] (open circles), CAPRICE[63] (vertical diamonds), IMAX [64] (downward triangles), BESS98 [65] (circles),AMS [66] (squares), Ryan [67] (horizontal diamonds), JACEE [68] (downward
open triangles), Ivanenko [69] (upward open triangles), Kawamura [70] (opensquares) and Runjob [71] (open diamonds). . . . . . . . . . . . . . . . . . . . . .
3.2 Comparison of the secondary cosmic ray muon flux observed at Tsukuba (left),
Mt. Norikura (middle) with the BESS detector [79] and at ballon altitudes at
Fort Summer [80]. Plots taken from [149]. . . . . . . . . . . . . . . . . . . . . . .
3.3 Schematic view of e↵ective areas of primary cosmic rays interacting with air
molecules for 1-dimentional calculation and 3-dimentional calculation. Solid arrows represent primary cosmic rays and dotted arrows represent neutrinos. The
3-dimentional calculation gives larger area for near-horizontal directions (fromthe view of the SK detector), which induces an enhancement of neutrino flux. [43]3.4 Comparison of 3-dimensional and 1-dimensional calculations of neutrino fluxes
at Super-K. Squares represent ⌫µ , asterisks represents ⌫¯µ , vertical diamonds represent ⌫e , horizontal diamonds represend ⌫¯e for 3-dimensional calculation. Solidlines represent ⌫ fluxes and dotted lines represent ⌫¯ fluxes for 1-dimensional calculation. [56] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Zenith angle dependence of atmospheric neutrino flux for di↵erent energy regions.
The higher lines represent ⌫µ and ⌫ µ and the lower lines represent ⌫e and ⌫ e . [149]
3.6 Comparison of atmospheric neutrino fluxes averaged over all directions (left) andflavor ratio (right). Red solid line represents Honda 11 flux. Dashed line represents the Bartol flux, dotted line represents the FLUKA flux, and blue dashedline represents Honda 06 flux. [149] . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7 Flavor ratio (⌫µ + ⌫¯µ )/(⌫e + ⌫¯e ) as a function of neutrino energy and zenith angle.
Calculation is based on Honda flux. [43] . . . . . . . . . . . . . . . . . . . . . . .
3.8 Charged current quasi-elastic interaction cross section of (a) ⌫µ and (b) ⌫¯µ , withexperimental data from ANL [93], Gargamelle [94] [95], BNL [96], Serpukhov [97]and SKAT [98]. Solid line indicates scattering o↵ a free proton, the dashed line
indicates scattering o↵ bound nucleons in 16 O. . . . . . . . . . . . . . . . . . . .
3.9 Cross section for ⌫µ charged current single pion productions. Solid lines are NEUT
calculations, points are experimental data. [136] . . . . . . . . . . . . . . . . . .
3.10 Cross section for ⌫¯µ charged current single pion productions. Solid lines are NEUT
calculations, points are experimental data. [136] . . . . . . . . . . . . . . . . . .

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3.11 Cross section for neutral current single pion productions. Lines are NEUT calculations, with solid lines for ⌫µ and dashed line for ⌫¯µ . Points are experimental
data taken from [104] (dot) and [105] (square). [136] . . . . . . . . . . . . . . . .
3.12 Cross section for DIS charged current ⌫µ (upper lines and points) and ⌫¯µ (lower
lines and points) interactions. GRV98 is used with Bodek-Yang correction in
NEUT, as shown by the dashed lines. The points represent experimental data.
[136] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.13 Cross section of ⇡ + 12 C scattering as a function of ⇡ + momentum. Solid lines
are NEUT calculation before tuning, dashed lines are NEUT calculation after
tuning. Points with error bars are experimental data. [119] . . . . . . . . . . . .
4.1
4.2
4.3
4.4

41

43

44

The charge distribution of a typical PMT. . . . . . . . . . . . . . . . . . . . . . . 47
Schematic view of the relative gain measurement system. . . . . . . . . . . . . . 48
Schematic view of the timing measurement system using a laser. . . . . . . . . . 49
TQ-map, a measured two-dimensional plot of timing against charge distribution.
Horizontal axis represents charge of each hit and vertical axis represents hit timing. 50
4.5 Positions of the eight laser light injects in the SK tank. ”Top old” refers to theoriginal SK-I light injector which is still being used at the moment. . . . . . . . . 51
4.6 Typical laser light injector event. . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.7 PMT hit time distribution in di↵erent detector regions for data and Monte Carlo(after tuning by scattering and absorption parameters). . . . . . . . . . . . . . . 53
4.8 Water coefficients for December 2009. The top plot shows absorption, the middle
plot shows symmetric scattering and the bottom plot shows asymmetric scattering. From these data, the average and error on the mean are calculated. The
averages for 337 nm, 375 nm, 405 nm and 445 nm are shown by the numbers on
the plots respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.9 A example of fitting. Fitted water coefficient functions for Apr 2009. The solidlines show the fitting in SK4 and the dashed lines show the SK3 functions. Points
are coefficient averages and error bars are the error on the mean. . . . . . . . . . 54
4.10 Schematic view of the attenuation length measurement with vertically penetratingcosmic ray muons. Arrows from the muon track (solid line) show the trajectoriesof Cherenkov photons. li is the flight length of the Cherenkov photons detectedby the i-th PMT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.11 E↵ective observed charge for through-going muons as a function of photon pathlengths (l), where Q is the detected charge and f (✓) is the PMT acceptance. . . 55
4.12 Time variation of water attenuation lengths for SK-I, SK-II, SK-III and SK-IV. . 56
4.13 Upper left panel shows the reconstructed momentum distribution Pp.e. for data(circles with error bars) and MC (solid line). Upper right panel shows the reconstructed opening angle distribution, also for data and MC. Lower two panels
show the correlation between reconstructed Cherenkov opening anlge ✓C and thereconstructed momentum Pp.e. for data (left) and MC (right). All distributions
are made from SK-II. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.14 Left panels show the averaged ratio of momentum derived from charge to thatderived from the opening angle (Pp.e. / P✓ ) as a function of reconstructed momentum P(✓) for data (solid line) and MC (dashed line). Right panels show theratio of MC events to data in the corresponding left panel. . . . . . . . . . . . . . 60
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4.15 Invariant mass distribution of neutrino induced ⇡ 0 events of observed data (dot)and MC (boxes). MC are normalized to the livetime of the observed data. . . . .
4.16 Momentum distribution of decay electrons of data (dot) and MC (solid line) for
each SK period. MC events are normalized by number of observed data. . . . . .
4.17 Summary of absolute energy scale calibration for each SK period. Horizontal axis
shows the momentum range and vertical axis shows the deviation of data fromMC predictions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.18 Time variation of the reconstructed momentum of decay electrons (left) and theaveraged momentum/range of stopping muons (right) as a function of elapsed
days from 1 April 1996 for each SK period. Dotted (dashed) lines show ±1%(±2%) from mean value in each SK period. . . . . . . . . . . . . . . . . . . . . .
4.19 Uniformity of the detector gain as a function of zenith angle for each SK period.
From top to bottom: SK-I, SK-II, SK-III and SK-IV. Y-axis is the ratio of MC
to data of the averaged momentum of decay-electrons. . . . . . . . . . . . . . . .
5.1
5.2
5.3

5.4

5.5

5.6

5.7
5.8
5.9

Schematic view of observed atmospheric neutrino in SK. . . . . . . . . . . . . . .
Expected neutrino spectra of di↵erent event categories. . . . . . . . . . . . . . . .
FC 2nd reduction criterion (1). NHITA800 distributions for SK-IV. (a) shows theobserved data after the FC 1st reduction. (b) shows FC Monte Carlo events after
FC 1st reduction. (c) shows the final samples for both data and Monte Carloevents, where the number of Monte Carlo events is normalized to that of thedata. The arrow at NHITA800 = 30 is the cut in the FC 2nd reduction for SK-IV.
In Figure (c), there are some events remaining at region NHITA800 > 30, they
are events with either PEtotal > 100,000 p.e.s or OD trigger o↵ or both. . . . . .
FC 2nd reduction criterion (2). PEmax /PE300 distributions for SK-IV. (a) shows
the data after the FC 1st reduction. (b) shows FC Monte Carlo events after
the FC 1st reduction. (c) shows the final samples for the data and Monte Carloevents, where the number of Monte Carlo events is normalized to that of the data.
Selection criteria is shown by arrows in the plots. . . . . . . . . . . . . . . . . . .
FC 3rd reduction criterion (1) for through-going muons cut. Scatter plot of thenumber of hit OD PMTs within 8 m of the entering point NHITAin (x-axis) andthe exiting point NHITAout (y-axis) for SK-IV. (a) shows the data after the FC
2nd reduction. (b) shows FC Monte Carlo events after the FC 2nd reduction. (c)shows the final samples for data (bottom left) and Monte Carlo (bottom right)
events. Selection criteria are indicted by solid lines. . . . . . . . . . . . . . . . . .
FC 3rd reduction criterion for stopping muon cut. Number of hit OD PMTs near
the entrance points of muons (NHITAin ) for SK-IV. (a) shows the data after the
FC 2nd reduction. (b) shows FC Monte Carlo events after the FC 2nd reduction.
(c) shows final samples for both data and Monte Carlo. The selection criteria areshown by arrows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Positions of the twelve cable holes. The 4 red circles indicate cable holes installedwith veto counters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Schematic view of a cable hole muon and a veto counter. . . . . . . . . . . . . . .
FC 3rd reduction criterion for flasher event cut. The timing distributions of (i) atypical flasher event and (ii) a typical FC neutrino event. The arrows show thetime window for counting NMIN100 . . . . . . . . . . . . . . . . . . . . . . . . . .
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5.10 FC 3rd reduction criterion for flasher event cut. NMIN100 distributions for SKIV. (a) shows the data after the FC 2nd reduction. (b) shows FC Monte Carloevents after the FC 2nd reduction. (c) shows the final samples for data and theMonte Carlo events, in which the number of Monte Carlo events is normalized tothat of the data. Selection criteria are shown by arrows. . . . . . . . . . . . . . .
5.11 FC 5th reduction invisible muon cut. NHITACearly distribution satisfying thecriteria PEtot < 1000 p.e.s and DISTclust > 500 cm for SK-IV. (a) shows thedata after FC 4th reduction. (b) shows the FC Monte Carlo events after FC 4th
reduction. (c) shows the final samples for data and Monte Carlo events whose
number is normalized to that of the data. The Selection criteria is shown by thearrows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.12 FC 5th reduction coincidence muon cut. PElate distributions for SK-IV. (a) shows
the data after the FC 4th reduction. (b) shows the Monte Carlo events after the
FC 4th reduction. (c) shows the final sample for both data and the Monte Carloevents whose number of events is normalized to that of the data. The selection
criteria is shown by the arrows. In (c), there are some events with PElate > 20,
as they are events with PE500 > 300 p.e.s and hence are not removed by criteria(1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.13 NHITAC distributions for FC and PC final examples for SK-I (top left), SK-II(top right), SK-III (bottom left) and SK-IV (bottom right). Points show theobserved data and error bars, solid lines show the Monte Carlo events assuming
no oscillation (dashed lines) and ⌫⌫ $ ⌫⌧ 2-flavor oscillation with (sin2 2✓, m2 )= (1.00, 2.5 ⇥ 10 3 eV2 ) (solid lines). . . . . . . . . . . . . . . . . . . . . . . . .
5.14 PC 1st reduction criterion (2). Top left and right plots are the scatter plots of
NHITAtop (x-axis) and NHITAbottom (y-axis). Top left figure shows the raw data,
and top right figure shows the PC Monte Carlo events with vertex in the fiducial
volume. Bottom left and right plots show 1-dimentional plot of NHITAtop and
NHITAbottom distributions respectively for the final sample for both data (blackdots) and Monte Carlo (red lines). All plots are for SK-IV. . . . . . . . . . . . .
5.15 PC 1st reduction criterion (3). The upper plots are the scatter plots of NHITAendcap
and NHITAbarrel . The upper left plot shows the raw data and the upper rightplot shows the PC Monte Carlo events with vertex in the fiducial volume. Bottom
left and right plots show 1-dimentional plot of NHITAendcap and NHITAbarrel forfinal samples for both data and Monte Carlo respectively. All the plots are for
SK-IV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.16 PC 1st reduction criterion (4). Top plots are the scatter plots of number of goodOD hits and ODRmean . Top left plot shows the raw data and top right plot shows
the PC Monte Carlo events with vertex in the fiducial volume. The bottom plotshows the 1-dimentional plots of ODRmean of both data (black dots) and Monte
Carlo (red line). All the plots are for SK-IV. . . . . . . . . . . . . . . . . . . . .
5.17 Schematic view of the algorithm to find hit clusters in PC 2nd reduction. Charged
observed in each patch are represented by circles, whose sizes are proportional to
the number of p.e.s. Arrows represent the vector charge gradient, pointing to thehighest charge among the neighbouring patches. . . . . . . . . . . . . . . . . . . .

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5.18 PC 2nd reduction criterion (2) for SK-III. Distributions of number of OD hit
PMTs in the 2nd OD hit cluster. (a) shows the data after PC 1st reduction. (b)shows PC Monte Carlo after PC 1st reduction. (c) shows the final sample forboth data and Monte Carlo. The selection criteria is shown by arrows. . . . . . 94
5.19 PC 2nd reduction criterion (3) for SK-III. Scatter plots of NHITAendcap and
NHITAside . (a) shows the data after PC 1st reduction, (b) shows PC Monte
Carlo after the 1st reduction. (c) and (d) shows the final samples for data andMonte Carlo respectively. The cut criteria are shown by solid lines. . . . . . . . . 95
5.20 PC 4th reduction for SK-IV, criteria (1) to (4) for the final samples. Distributions
for criteria (1) to (4) for data (black dots) and Monte Carlo (red lines) for the
final sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.21 PC 5th reduction Corner clipping muon cut B for SK-IV. The plot shows the
distribution of the distance from the vertex reconstructed by Point-fit to thenearest fringe of ID, for the final sample for both data (black dots) and Monte
Carlo (red line) events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.22 PC 5th reduction Through-going muon cut D for SK-IV. Upper left plot shows
the data after PC 4th reduction while the upper right plot shows the PC Monte
Carlo. The bottom left and right plots show the final samples for data and Monte
Carlo respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.23 Number of background events outside fiducial volume as a function of distance
between reconstructed vertex and the ID wall for SK-III. The solid line shows thefitted exponential curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.24 Distribution of PEanti / PEexp for PC stop / through separation for SK-I (topleft), SK-II (top right), SK-III (bottom left) and SK-IV (bottom right). The
points represent the data with error bar. The shaded regions represent MC oftrue stopping and true through-going events without oscillation and normalized
to data livetime. The selection criteria is shown by the arrow. . . . . . . . . . . . 106
5.25 Distribution of NHITEX for UPMU stop/through separation for SK-I (top left),
SK-II (top right), SK-III (bottom left) and SK-IV (bottom right). Points shows
the observed data, dashed line show MC assuming no oscillation and solid line
shows MC with ⌫µ $ ⌫⌧ 2-flavor oscillation with (sin2 2✓, m2 ) = (1.00, 2.5 ⇥10 3 eV2 ). Selection criteria are shown by arrows. . . . . . . . . . . . . . . . . . 109
5.26 Scatter plots of zenith versus azimuth directions for stopping muons (left) andthrough-going muons (right) for SK-IV. . . . . . . . . . . . . . . . . . . . . . . . 110
5.27 Azimuthal angle distributions for stopping muon (left) and through-going muons(right) for SK-IV. Region (1) has thicker-mountain, while region (2) has thinnermounter. White histograms show the events of downward-going muons, while theblack histograms show the upward-going muons. . . . . . . . . . . . . . . . . . . 111
5.28 Zenith angle distributions for stopping muon (left) and through-going muon (right)
for SK-IV in both regions (1) and (2). Solid curves are fitted functions to estimate
the background contamination. Zenith angle bins are divided equally to 18 bins
between -0.1 < cos ✓ < 0.08. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.29 Energy spectrum of primary neutrino for upward stopping (dotted line), nonshowering (dashed line) and showering (solid line) muon events. . . . . . . . . . 113

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5.30 Distributions for UPMU showering separation for SK-IV. Scatter plot of (Q)against 2 of UPMU through-going events are shown for data (top left) andMC (top right) (normalized to data lifetime). (Q) distribution is shown in thebottom panel for data (points) and MC (red solid line). The cut is shown by reddashed line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.31 Distance between the reconstructed vertex and the true vertex for FC MC MultiGeV multi-ring µ-like sample for SK-IV. Colored region shows 68% of all events. 116
5.32 The basic idea to find ring candidates. By drawing rings around the hit PMTs
with Cherenkov opening angle of 42 from the vertex, center of the actual Cherenkov
ring can be identified. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.33 Ring counting likelihood distribution for SK-I, SK-II, SKIII and SK-IV sub-GeV
samples. Dots are data and solid lines are MC events. . . . . . . . . . . . . . . . 118
5.34 Ring counting likelihood distribution for SK-I, SK-II, SKIII and SK-IV multi-GeV
samples. Dots are data and solid lines are MC events. . . . . . . . . . . . . . . . 119
5.35 Event display of single-ring electron (left) and single-ring muon (right) neutrino
MC event. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.36 PID likelihood distribution for FC sub-GeV 1-ring samples for SK-I, SK-II, SKIII and SK-IV. Points represent the data, red lines represent Monte Carlo, theshaded green regions represent the MC CCQE ⌫µ events. . . . . . . . . . . . . . . 121
5.37 PID likelihood distribution for FC multi-GeV 1-ring samples for SK-I, SK-II, SKIII and SK-IV. Points represent the data, red lines represent Monte Carlo, theshaded green regions represent the MC CCQE ⌫µ events. . . . . . . . . . . . . . . 122
5.38 PID likelihood distribution for FC sub-GeV multi-ring samples for SK-I, SK-II,
SK-III and SK-IV. Points represent the data, red lines represent Monte Carlo,
the shaded green regions represent the MC CCQE ⌫µ events. . . . . . . . . . . . 123
5.39 PID likelihood distribution for FC multi-GeV multi-ring samples for SK-I, SK-II,
SK-III and SK-IV. Points represent the data, red lines represent Monte Carlo,
the shaded green regions represent the MC CCQE ⌫µ events. . . . . . . . . . . . 124
5.40 Distance between the true vertex and the reconstructed vertex of FC MC sample
for SK-IV. Left: Sub-GeV 1-ring elike. Right: Sub-GeV 1-ring µ-like. . . . . . . . 125
5.41 Distance between the true vertex and the reconstructed vertex of FC MC sample
for SK-IV. Left: Multi-GeV 1-ring elike. Right: Multi-GeV 1-ring µ-like. . . . . . 125
5.42 Distance between the true vertex and the reconstructed vertex of PC MC sample
for SK-IV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.43 Angular di↵erence between true direction and reconstructed direction of FC MC
samples for SK-IV. Left: Sub-GeV 1-ring e-like. Right: Sub-GeV 1-ring µ-like. . 126
5.44 Angular di↵erence between true direction and reconstructed direction of FC MC
samples for SK-IV. Left: Mult-GeV 1-ring e-like. Right: Multi-GeV 1-ring µ-like. 127
6.1

Distribution of invariant mass from the ⇡ 0 fitter for CCQE (black solid line) andNC (red dashed line) in SK-IV FC sub-GeV single-ring e-like MC events, in fiveenergy regions (from left to right): (1) Pe < 250 MeV/c, (2) 250 MeV/c  Pe
< 400 MeV/c, (3) 400 MeV/c  Pe < 630 MeV/c, (4) 630 MeV/c  P < 1000
MeV/c, (5) 1000 MeV/c < Pe . The arrows show the events which are taken tobe ⇡ 0 -like. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

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6.3

6.4

6.5

6.6

6.7

6.8
6.9

Distributions used in the ⇡ 0 selection for CCQE (black solid line) and NC (reddashed line) in SK-IV FC sub-GeV single-ring e-like MC events, in four energy
regions (from top to bottom): (1) 250 MeV/c  Pe < 400 MeV/c, (2) 400 MeV/c
 Pe < 630 MeV/c, (3) 630 MeV/c  P < 1000 MeV/c, (4) 1000 MeV/c <Pe . In order to select ⇡ 0 -like and e-like events more efficiently, an additional
likelihood selection is applied for events with momentum above 250 MeV/c. The
distributions of the three likelihood variables used are shown: energy fraction
carried by the second fitted ring (left column), the ⇡ 0 mass (middle column) and-likelihood (right column). All distributions are normalized to the histogram
area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Distributions of the variables used in the multi-GeV multi-ring e-like (MME)
likelihood for SK-IV. From left to write: PID of the most energetic ring, number ofdecay electrons, momentum fraction of the most energetic ring, distance between
the vertex and the decay electron. From bottom to top, each plot represents
an energy bins: (1) 1.33 - 2.5 GeV, (2) 2.5 - 5.0 GeV, (3) 5.0 - 10.0 GeV, (4)10 - 20 GeV, (5) > 20 GeV. Energy bins correspond to the energy of the mostenergetic ring. Black solid line represents CC⌫e +⌫ e and red dashed line representsCC⌫µ + ⌫ µ and NC. All distributions are normalized to the histogram area. . . 137
Distribution of number of decay electrons in both data and MC for SK-IV multiGeV single-ring e-like sample. From bottom to top regions: CC⌫e (red), CC⌫ e(green), CC⌫µ + ⌫ µ (blue) and NC (yellow). Data is represented by black dots.
MC is normalized to data livetime 1097 days. Two-flavor oscillations (sin2 2✓,
3 eV2 ) is assumed. Events on left hand side of the arrow ism ) = (1.0, 2.5⇥10taken as ⌫ e -like sample and those on the right hand side of the arrow is taken as⌫e -like sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Diagram showing the interaction of ⌫e with the nuclei in water, and the Feynman
y = (E’-E)/E where E is the initial energy of the incoming ⌫e and E’ is the energyof the outgoing electron. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Distributions of the variables used in the MME ⌫e and ⌫ e separation likelihood for
SK-IV. From left to right: number of decay electrons, number of rings, momentum
fraction of the most energetic ring. From bottom to top, there are five energy
bins: (1) 1.33 - 2.5 GeV, (2) 2.5 - 5.0 GeV, (3) 5.0 - 10.0 GeV, (4) 10 - 20 GeV, (5)> 20 GeV. Energy bins correspond to the energy of the most energetic ring. Blacksolid line represents CC⌫e and red dashed line represents CC⌫ e . All distributions
are normalized to the historgram area. Area of the histograms are normalized totheir corresponding areas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Distribution of the ⌫e and ⌫ e separation likelihood for SK-IV 500 years MC. Solidline represents CC ⌫e , dashed line represents CC⌫ e and dotted line represents NC
events. The arrow shows the cut which is made at 0. For events having L⌫e /⌫ e
> 0, they are taken as ⌫e -like events, and for events having L⌫e /⌫ e < 0, they aretaken as ⌫ e -like events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Summary of selection steps for the Multi-GeV e-like samples. . . . . . . . . . . . 144
Flux ratio of electron neutrino for oscillations with parameters ( m212 , m232 ,sin2 ✓12 , sin2 ✓23 , sin2 ✓13 , CP ) = (7.6 ⇥ 10 5 , 2.4 ⇥ 10 3 , 0.31, 0.5, 0.025, 220 ).
The flux ratio is the comparison between flux with and without oscillation. . . . 149

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6.10 Zenith angle distribution for multi-GeV single-ring + multi-ring ⌫e -like (top) andmulti-GeV single-ring + multi-ring ⌫ e -like (bottom) samples in SK-IV. The y-axis
shows the number of events with oscillations divided by the number of events
without oscillations. Oscillation paramters are ( m212 , m232 , sin2 ✓12 , sin2 ✓13 ,5 eV2 , 2.4⇥ 10 3 eV2 , 0.31, 0.025, 40 ). Oscillations at di↵erent
CP ) = (7.6 ⇥ 102values of sin ✓23 = 0.4 (green), 0.5 (red), 0.6 (blue) for normal hierarchy (dashedline) and inverted hierarchy (solid line) are shown. . . . . . . . . . . . . . . . . . 151
6.11 Zenith angle distributions of the ratio of number of events with oscilation tothat without oscillation, assuming oscillation parameters ( m212 , m232 , sin2 ✓12 ,sin2 ✓23 , sin2 ✓13 ) = (7.7 ⇥ 10 5 , 2.66 ⇥ 10 3 , 0.30, 0.425, 0.025) in normal hierarchy case for SK-IV. Cases with CP = 80 (blue dotted line), 220 (red solidline) and 300 (green dashed line) are shown. The error bars represent the sizeof the error with SK-I + SK-II + SK-III + SK-IV data livetime. . . . . . . . . . 152
6.12 Definition of the binning used in the neutrino oscillation analysis. FC: 370 bins,
PC: 60 bins, UPMU: 50 bins. White boxes indicate that momentum bin is divided
into 10 zenith angle bins between -1 < cos ✓ < 1 for FC and PC, -1 < cos ✓ < 0for UPMU. Shaded boxes indicate the momentum bin is not further divided into
zenith angle bins. FC single-ring e-like 1-decay, ⇡ 0 -like, µ-like 2-decay samples
and 2-ring ⇡ 0 sample are not divided into zenith angle bins. . . . . . . . . . . . . 155
6.13 Contribution of each error source for atmospheric neutrino flux [78] is shown in
the upper plot. Absolute normalization uncertainty due to only ⇡ (atmosphericmuon data (pion producion)) and(hadronic interaction mode) as a functionof neutrino energy is shown in the lower plot. . . . . . . . . . . . . . . . . . . . . 163
6.14 Flavor ratio (⌫µ + ⌫ µ )/(⌫e + ⌫ e ) of the atmospheric neutrino flux. Vertical axis
stands for the double ratio, FLUKA flux/Honda flux is represented by solid lien,
Bartol flux/Honda flux is represented by dashed line. The kink at 8 GeV in
FLUKA flux due to technical treatment is corrected in the application. . . . . . . 164
6.15 Anti-neutrino / neutrino double ratio for ⌫ e /⌫e (left) and ⌫ µ /⌫µ (right). Solid(dashed) line indicates the double ratio with FLUKA (Bartol) flux. . . . . . . . 165
6.16 Neutrino flight length as a function of zenith angle. Solid curve represents theflight length used in our analysis and dashed curve represents that for the compressed density structure of atmosphere by 10%. . . . . . . . . . . . . . . . . . . 167
6.17 Ratio of calculated neutrino flux in which the fluxes and Bartol (solid line) and
FLUKA (dashed line) are normalized by Honda flux. . . . . . . . . . . . . . . . . 168
6.18 Top left: ratio of CCQE cross section, Nieves model normalized by the Smith
and Monitz model, for ⌫e + ⌫ e (solid line) and ⌫µ + ⌫ µ (dashed line). Top right:(⌫µ + ⌫ µ )/(⌫e + ⌫ e ) ratio. Bottom: (⌫/⌫) ratio. . . . . . . . . . . . . . . . . . . . 169
6.19 Predicted cross section (left) and the ⌫/⌫ ratio (right) of the Hernandez model
normalized by that of the Rein and Sehgal model for ⌫µ p ! µ p⇡ + (solid line),⌫µ n ! µ p⇡ 0 (dashed line) and ⌫µ n !µ p⇡ + (dotted line). . . . . . . . . . . . 170
6.20 ⇡ + scattering on carbon target in: quasi-elastic channel (top left), absorption
channel (top right), total reaction channel (bottom left) and single charge exchange channel (bottom right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
6.21 Pion multiplicity plotted as a function of W2 for NEUT simulation (left) and forthe data from the CHORUS experiment (right). . . . . . . . . . . . . . . . . . . . 174

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6.22 The plots show 2 distributions for sin2 ✓23 (top left), m223 (top right) and CP(bottom) for the sensitivity study carried out with data livetime assuming normal
hierarchy and oscillation parameters of (sin2 ✓12 , m212 , sin2 ✓23 , m232 , sin2 ✓13 ,
53CP ) = (0.304, 7.66 ⇥ 10 , 0.5, 2.66 ⇥ 10 , 0.025, 300 ). Black line represents
the distribution for normal hierarchy, and red line represents the distribution for
inverted hierarchy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
6.23 SK-I + SK-II + SK-III + SK-IV zenith angle distributions for FC events fordata (line with error bar) and MC assuming no oscillation (grey dashed line),
best fit results at normal hierarchy (red solid line) and best fit results at inverted
hierarchy (blue dashed line). From top to the bottom: Sub-GeV 1-ring e-like
0-decay, Sub-GeV µ-like 0-decay, Sub-GeV µ-like 1-decay . . . . . . . . . . . . . 180
6.24 SK-I + SK-II + SK-III + SK-IV zenith angle distributions for FC events fordata (line with error bar) and MC assuming no oscillation (grey dashed line),
best fit results at normal hierarchy (red solid line) and best fit results at inverted
hierarchy (blue dashed line). From top to the bottom: Multi-GeV 1-ring ⌫e -like,
Multi-GeV 1-ring ⌫ e -like, Multi-GeV 1-ring µ-like. . . . . . . . . . . . . . . . . . 181
6.25 SK-I + SK-II + SK-III + SK-IV zenith angle distributions for FC events fordata (line with error bar) and MC assuming no oscillation (grey dashed line),
best fit results at normal hierarchy (red solid line) and best fit results at inverted
hierarchy (blue dashed line). From top to the bottom: Multi-GeV multi-ring⌫e -like, Multi-GeV multi-ring ⌫ e -like, Multi-GeV multi-ring µ-like. . . . . . . . . 182
6.26 SK-I + SK-II + SK-III + SK-IV zenith angle distributions for FC events fordata (line with error bar) and MC assuming no oscillation (grey dashed line),
best fit results at normal hierarchy (red solid line) and best fit results at inverted
hierarchy (blue dashed line). From top to the bottom: PC stop, PC through . . 183
6.27 SK-I + SK-II + SK-III + SK-IV zenith angle distributions for FC events fordata (line with error bar) and MC assuming no oscillation (grey dashed line),
best fit results at normal hierarchy (red solid line) and best fit results at inverted
hierarchy (blue dashed line). From top to the bottom: UPMU stopping, UPMU
non-showering, UPMU showering. . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
2 = 2
2
2 as a function of
6.28
m232 , sin2 ✓23 , CP and
min distributions of2
sin ✓13 (in case of free ✓13 ) for normal hierarchy case. Black line represents fixedsin2 ✓13 = 0.025 and blue line represents free ✓13 . . . . . . . . . . . . . . . . . . . 185
2 = 2
2
6.29
m232 , sin2 ✓23 , CP and sin2 ✓13
min distributions as a function of(in case of free ✓13 ) for inverted hierarchy case. Black line represents fixed sin2 ✓13= 0.025 and blue line represents free ✓13 . . . . . . . . . . . . . . . . . . . . . . . 186
6.30 Contours of m232 and sin2 2✓23 (left) and m232 and sin2 ✓23 (right) for the normal
mass hierarchy, at fixed ✓13 . The three lines indicate allowed regions at 68%
2 = 2.3, 4.6 and 9.2
C.L. 90% C.L. and 99% C.L. respectively, defined as
respectively. The star shows the best fit point. . . . . . . . . . . . . . . . . . . . 187
6.31 Contours of m232 and sin2 2✓23 (left) and m232 and sin2 ✓23 (right) for the inverted
mass hierarchy, at fixed ✓13 . The three lines indicate allowed regions at 68%
2 = 2.3, 4.6 and 9.2
C.L. 90% C.L. and 99% C.L. respectively, defined as
respectively. The star shows the best fit point. . . . . . . . . . . . . . . . . . . . 187

xiv6.32 The numbers on the x-axis indicate the event sample types (18 types in total, with
2 between
SK-I + SK-II + SK-III + SK-IV data). The upper panel shows thenormal hierarchy and inverted hierarchy contributed by each event sample of2
the data at the best fit oscillation parameters. The lower panel shows the
between normal hierarchy and inverted hierarchy expected to be contributed byeach event sample for MC, assuming the ”true” oscillation parameters the sameas the best fit parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
2 as a function of
6.33 Distribution of
CP for normal hierarchy (black line) andinverted hierarchy (red line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
6.34 Contours of m232 and sin2 2✓23 allowed regions for normal hierarchy (left) andinverted hierarchy (right) for free ✓13 . New results from this study is represented
by blue solid line, and the best fit point is represented by the triangle. . . . . . . 194
6.35 Contours of m232 and sin2 2✓23 allowed regions for normal hierarchy (left) andinverted hierarchy (right) for fixed ✓13 . New results from this study is represented
by blue solid line, and the best fit point is represented by the triangle. . . . . . . 194
7.1

Expected sensitivity of mass hierarchy by Hyper Kamiokande (whose fiducial
volume is 25 times of Super Kamiokande) as a function of years of exposure. . . 196

xv

List of Tables
1.1

Summary of global best fit of neutrino oscillation parameters, taken from [39] . .

7

2.1
2.2

Specification of the 20-inch PMT used in Super-Kamiokande. . . . . . . . . . . .
Trigger conditions for each SK period. *All inner PMTs are protected with acrylic
covers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

3.1

List of processes considered in the simulator. . . . . . . . . . . . . . . . . . . . .

45

5.1

Selection efficiencies and number of events for FC sample are summarized foreach SK period. Selection efficiencies are for events whose real vertex is in thefiducial volume for reduction steps 1 to 5, while the selection efficiency for 5th(FV) is for events whose fitted vertex is in the fiducial volume, and number of
OD hits < 16 (10 for SK-I) and visible energy > 30 MeV. Number of events for
MC are normalized without oscillation to real data livetime, 1489.2, 798.6, 518.1
and 1097.0 days for SK-I, SK-II, SK-III and SK-IV respectively. . . . . . . . . . . 87
Estimated contamination by each background source. Sub-GeV means eventswith Evis < 1.3 GeV and Multi-GeV means events with Evis > 1.3 GeV. . . . . . 88
Selection efficiencies and number of events for PC sample are summarized for each
SK period. Selection efficiencies are for events whose real vertex is in the fiducial
volume (after 5th (FV) step, fitted vertex instead of real vertex is considered)and number of OD hits > 15 (9 for SK-I) and total observed charge > 3000 p.e.s.
Number of events for MC are normalized to real data livetime, 1489.2, 798.6,
518.1 and 1097.0 days for SK-I, SK-II, SK-III and SK-IV respectively, 2-flavor
oscillation with (sin2 2✓, m2 ) = (1.00, 2.5 ⇥ 10 3 eV2 ) is assumed. . . . . . . . 104
Estimated contamination by cosmic ray muons for PC. . . . . . . . . . . . . . . . 104
Number of events of UPMU summarized for each SK period. Number of MC(without oscillation) events and event rate are calculated by normalization ofdata livetime, 1645.9, 827.7, 635.6, 1096.7 days for SK-I, SK-II, SK-III and SKIV respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Uncertainties of background events which is the contamination by horizontal
muons to the UPMU sample in the zenith angle -0.1 < cos✓ < 0. . . . . . . . . 111
Vertex resolutions for each SK period. . . . . . . . . . . . . . . . . . . . . . . . . 127
angular resolutions for each SK period. . . . . . . . . . . . . . . . . . . . . . . . . 128

5.2
5.3

5.4
5.5

5.6
5.7
5.8

xvi

25

6.1

Number of FC sub-GeV events and their compositions of neutrino interaction
modes for SK-IV MC scaled to the SK-IV livetime 1097 days with 2-flavor oscillation with (sin2 2✓, m ) = (1.0, 2.5⇥10 3 eV2 ). The upper (lower) table shows
e-like (µ-like) samples. The right column shows the composition before separation
into sub-samples. After the separation, the CCQE purity has increased. . . . . . 133
6.2 Composition of interaction modes in case of with and without MME likelihoodfor SK-IV MC scaled to 1097 days. Two-flavor oscillations (sin2 2✓, m ) = (1.0,
2.5⇥10 3 eV2 ) is assumed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.3 Composition of interaction modes in case of with and without MME likelihoodfor SK-III MC scaled to 518.1 days. Two-flavor oscillations (sin2 2✓, m ) = (1.0,
2.5⇥10 3 eV2 ) is assumed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.4 Composition of interaction modes for multi-GeV single-ring e-like samples for
SK-III and SK-IV respectively. Purity of CC⌫e + ⌫ e is higher for SK-IV due tothe improvement in electronics in the SK-IV period. . . . . . . . . . . . . . . . . 138
6.5 Interaction mode composition of multi-GeV single-ring ⌫e -like and ⌫ e -like samples
for SK-IV. MC is normalized to data livetime 1097 days. Two-flavor oscillations(sin2 2✓, m ) = (1.0, 2.5⇥10 3 eV2 ) is assumed. The separation increases thecomposition of CC⌫e in ⌫e -like sample by 5.2% and that of CC⌫ e in ⌫ e -like sample
by 6.1%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.6 Interaction mode composition of multi-GeV multi-ring ⌫e -like and ⌫ e -like samples
for SK-IV. MC is normalized to data livetime 1097 days. Two-flavor oscillations(sin2 2✓, m ) = (1.0, 2.5⇥10 3 eV2 ) is assumed. The separation increases thecomposition of CC⌫e in ⌫e -like sample by 3.3% and that of CC⌫ e in ⌫ e -like sample
by 1.5%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.7 Summary of number of events for both data and MC in each SK run period.
Number of MC events is normalized by data livetime, assuming 2-flavor mixing
at ( m223 , sin2 ✓23 ) = (2.1 ⇥ 10 3 , 0.5). . . . . . . . . . . . . . . . . . . . . . . . 148
6.8 Systematic errors related to neutrino flux calculation. They (except for solar
activities) are common in all SK periods. The second column shows the best
fit value of the systematic error parameter ✏i and the third column shows theestimated 1 error size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.9 Systematic errors related to neutrino interactions. They are common between all
SK periods. The second column shows the best fit value of the systematic error
parameter ✏i and the third column shows the estimated 1 error size. . . . . . . . 157
6.10 Systematic errors related to event reduction for each SK period. ”Fit” column
shows the best fit value of the systematic error parameter ✏i and ” ” column
shows the estimated 1 error size. . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6.11 Systematic errors related to event reconstruction and selection for SK-I. The
second column shows the best fit value of the systematic error parameter ✏i andthe third column shows the estimated 1 error size. . . . . . . . . . . . . . . . . . 159
6.12 Systematic errors related to event reconstruction and selection for SK-II. The
second column shows the best fit value of the systematic error parameter ✏i andthe third column shows the estimated 1 error size. . . . . . . . . . . . . . . . . . 160
6.13 Systematic errors related to event reconstruction and selection for SK-III. The
second column shows the best fit value of the systematic error parameter ✏i andthe third column shows the estimated 1 error size. . . . . . . . . . . . . . . . . . 161
xvii

6.14 Systematic errors related to event reconstruction and selection for SK-IV. The
second column shows the best fit value of the systematic error parameter ✏i andthe third column shows the estimated 1 error size. . . . . . . . . . . . . . . . . . 162
6.15 Estimated uncertainties of up/down ratio (1 values) for di↵erent event samples.
They are common for SK-I, SK-II, SK-III and SK-IV periods. . . . . . . . . . . . 166
6.16 Estimated uncertainties of horizontal/vertical ratio (1 values) for di↵erent event
samples. They are common for SK-I, SK-II, SK-III and SK-IV periods. . . . . . 166
6.17 1 values for systematic uncertainties of single-ring ⇡ 0 -like events selection for
SK-I, SK-II, SK-III and SK-IV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
2 ( = | 2
2
6.18
min(N H)min(IH) | ) estimated with and without multi-GeV 1-ringand multi-ring e-like samples separated into ⌫e and ⌫ e -like samples. . . . . . . . 176
6.19 Results of the allowed region of each neutrino oscillation parameter at 68% and
90% C.L. for normal hierarchy at fixed ✓13 , where 68% C.L. is defined as 2222
min = 1.0 and 90% C.L. is defined as
min = 2.7. . . . . . . . . . . . . . . 178
6.20 Results of the allowed region of each neutrino oscillation parameter at 68% and
90% C.L. for inverted hierarchy at fixed ✓13 , where 68% C.L. is defined as 2222
min = 1.0 and 90% C.L. is defined as
min = 2.7. . . . . . . . . . . . . . . 179
6.21 Results of the allowed region of each neutrino oscillation parameter at 68% C.L.
and 90% C.L. for normal hierarchy at fixed ✓13 , where 68% C.L. is defined as2222
min = 1.0 and 90% C.L. is defined as
min = 2.7. . . . . . . . . . . . 179
6.22 Results of the allowed region of each neutrino oscillation parameter at 68% C.L.
and 90% C.L. for inverted hierarchy at fixed ✓13 , where 68% C.L. is defined as2222
min = 1.0 and 90% C.L. is defined as
min = 2.7. . . . . . . . . . . . 179
6.23 Best fit values at assumed normal and inverted hierarchies in the original sensitivity study carried out in Section 6.6. . . . . . . . . . . . . . . . . . . . . . . . . 190
2 for normal and inverted hierarchies for data fitted with
6.24 Best fit values and⌫ flux related systematic errors halved. . . . . . . . . . . . . . . . . . . . . . . . . 190
2 for normal and inverted hierarchies for data fitted with
6.25 Best fit values and⌫ flux related systematic errors increased by 50%. . . . . . . . . . . . . . . . . . . 191
2 for normal and inverted hierarchies for data fitted with
6.26 Best fit values and⌫ interactions related systematic errors halved. . . . . . . . . . . . . . . . . . . . 191
2 for normal and inverted hierarchies for data fitted with
6.27 Best fit values and⌫ interactions related systematic errors increased by 50%. . . . . . . . . . . . . . 192
2 for normal and inverted hierarchies for data fitted with
6.28 Best fit values andevent reconstruction related systematic errors halved. . . . . . . . . . . . . . . . 192
2 for normal and inverted hierarchies for data fitted with
6.29 Best fit values andevent reconstruction related systematic errors increased by 50%. . . . . . . . . . 193

xviii

Chapter 1

Introduction
Currently the universe is in matter dominated state, while comparable amounts of matter andanti-matter were created in the Big-Bang. CP violation is believed to contribute to asymmetry
between matter and anti-matter, so that matter remains during the expansion and coolingprocess of the universe [1]. CP invariance is observed to be violated in weak interactions ofquark sector [2], while CP violation in lepton sector is not observed yet. Neutrino oscillation
provides a way to observe the CP violation in lepton sector. Moreover, observations showed thatneutrino mixing angles are very di↵erent from those of quarks, which suggests that the massordering might not be as expected by the knowledge from the quark sector.

1.1

Neutrino Oscillation

Neutrinos are leptons which lack electric charge and are extremely difficult to detect asthey pass through ordinary matter almost without disturbance. There are three flavours ofneutrinos, electron neutrino (⌫e ), muon neutrino (⌫µ ) and tau neutrino (⌫⌧ ). They are assumed
to be massless by the Standard Model. However, neutrino oscillations have been observed in solar[3], atmospheric [4], accelerator[5] and reactor [6] [7] experiments, which implies neutrino flavor
eigenstates are actually made up of superposition of neutrino mass eigenstates [8], suggesting
physics beyond the Standard Model.
Neutrino oscillation was proposed by Maki, Nakagawa, Sakata [9] and Pontecorvo [10]. The
flavor eigenstates can be written as superposition of mass eigenstates:| ⌫↵ i =3
X
i=1

U↵i | ⌫i i

(1.1)

where | ⌫↵ i is the flavor eigenstate ⌫e , ⌫µ or ⌫⌧ , | ⌫i i is the mass eigenstates ⌫1 , ⌫2 and ⌫3 . U
is the Maki-Nakagawa-Sakata-Pontecorvo (MNSP) matrix which is a 3⇥3 matrix describing thetransformation between di↵erent neutrino flavor eigenstates:
0

1
B
U =@ 0010

cos ✓13000C B
cos ✓23 sin ✓23 A ⇥ @0e isin ✓23 cos ✓23
sin ✓130

1

1

sin ✓13C0A
cos ✓13

Chapter 1

2

0

1

0

1

cos ✓12 sin ✓12 0
ei⇢ 0
B
C B
⇥ @ sin ✓12 cos ✓12 0 A ⇥ @ 0 ei
0
0
1
0
0

0
C
0 A (1.2)
1

The time evolution of the four eigenstates is:X| ⌫↵ (t)i =iU↵i exp( iEi t) | ⌫i (t = 0)i

XX

=

i

↵0

(1.3)

U↵i U↵⇤0 i exp( iEi t) | ⌫↵0 (t = 0)i(1.4)

where Ei is the eigenvalue of the mass eigenstate | ⌫i i. The probability for flavor eigenstate ⌫↵at t=0 to change to state ⌫ at time t is:P (⌫↵ ! ⌫ ) = | h⌫ (t) | ⌫↵ (0)i |2
= |

X
i

X

=

i

(1.5)

U i U↵⇤ i exp( iEi t) |2| U↵i U i |2 +

X

(1.6)

U↵i U ⇤ i U↵⇤ j U j exp( i(Ei

Ej )t)

(1.7)

i6=j

The above equations show that the flavor transition phenomenon, neutrino oscillation, occurs
due to the mixing of the mass eigenstate of the non-zero-mass neutrinos.
The mixing matrix U can be simplified into the two-flavor mixing case:cos ✓ sin ✓sin ✓ cos ✓

U=

!

(1.8)

Transition probability shown in equation (1.7) can be simplified into:
P (⌫↵ ! ⌫ ) = sin2 2✓ sin2 (

(Ei

Ej )t
2

).

(1.9)

If neutrinos have non-zero mass, the following approximation is taken:
Ei

Ej

=

q

(m2i + p2i )= pi (1 +⇡ (p +
⇡

m2i 1
)2
p2i

m2i
)
2p

q

(m2j + p2j )(1.10)

m2i 1
)2
p2j

(1.11)

pj (1 +

(p +

m2j
)
2p

m2ij
.
2E

(1.12)(1.13)

where m2ij ⌘ m2i m2j is the mass-squared di↵erence of the neutrino mass eigenstates.
Substituting equation (1.13) into equation (1.9), with neutrino flight length L⌫ and neutrinoenergy E⌫ , the survival probability of ⌫↵ in two-flavor mixing case is:P (⌫↵ ! ⌫ ) = 1= 1

m2ij L⌫)
4E⌫
1.27 m2ij (eV 2 )L⌫ (km)sin2 2✓ sin2 ()
E⌫ (GeV )sin2 2✓ sin2 ((1.14)(1.15)

Chapter 1

3

As shown above, the oscillation probability depends on the mixing angle ✓, the mass squaredi↵erence m2ij , neutrino flight length L⌫ and neutrino energy E⌫ . The oscillation amplitude(km)⇡
is at maximum when EL⌫⌫(GeV) ⇠ 2.54⇥ m2 (eV 2 ) . Therefore, neutrino oscillation experiments canijprobe the neutrino squared mass di↵erence.

1.2

Neutrino Oscillation Experiments

There are many experiments which have observed neutrino oscillation and measured theneutrino oscillation parameters which includes the mixing angle and the squared mass di↵erence.
Some of these experiments are introduced in this section.

1.2.1

Solar and Reactor Neutrino Observations

Electron neutrinos are produced by nuclear reactions 4p !4 He + 2e+ + 2⌫e + in the Sun.
Homestake experiment is the oldest experiment that measured solar neutrinos [3]. Electron
neutrinos were measured by the reaction ⌫e +37 Cl ! e +37 Ar inside the detector whichcontained 520 tons of chlorine, then the produced radioactive Ar atoms which have a half life of
35 days were extracted and counted by measuring its radioactivity. The number of Ar detected
is the number of neutrino interactions in the detector. The count rate was 2.56 SNU (Solar
Neutrino Unit, 1 SNU = 10 36 captures/atom/s), while the Standard Solar Model predicted
a rate of 7.6 SNU [16]. Later experiments SAGE [11] and GALLEX [12] measured electron
neutrinos by the interaction ⌫e +71 Ga ! e +71 Ge and the produced Ge atoms which isalso a radioactive isotope (half life 11.43 days) were counted, deficit was also observed. The
Kamiokande experiment[13] which was a water Cherenkov detector, which detected neutrinos
with di↵erent method (detections of Cherenkov rings emitted by leptons produced by neutrinos
in water), also observed the deficit. This discrepancy in measured and predicted neutrino flux
is referred to as the ”solar neutrino problem”.
The Sudbury Neutrino Observatory (SNO) [14] which is a D2 O Cherenkov detector measures⌫e and total fluxes (⌫e + ⌫µ + ⌫⌧ ) of neutrinos coming from the Sun through the charged current(CC) reactions ⌫e +d ! p+p+e , neutral current (NC) reactions ⌫x +d ! p+n+⌫x and elastic
scattering (ES) reactions ⌫x + e ! ⌫x + e . The total number of neutrinos detected agrees withthe predicted number from the Standard Solar Model, with about one third of the total number
electron neutrinos. On the other hand, Super-Kamiokande (SK) experiment which is a H2 O
Cherenkov detector also made the observation through elastic scattering reactions [15] [17]. As
SK alone cannot indicate non-electron flavor components of neutrinos, only when SK and SNO
results are combined, there was an indication for non-electron flavor neutrinos, so combinationof the results of SNO and SK showed the MSW solution explains the observed flux [18].
The KamLAND experiment confirmed the above results of SNO and SK [19]. KamLAND
is a liquid scintillator type detector, which detects the ⌫¯e from the nuclear power reactor units
with the reaction of inverse -decay ⌫¯e + p ! e+ + n. The prompt signal from e+ together
with the delayed coincidence of the gamma from the neutron capture significantly reduces the
background. The nuclear power reactor units are on average 180 km away from the detector.
The energy of the ⌫¯e from nuclear power reactor is about 3 MeV, from equation (1.15), theexperiment is sensitive to m212 down to 10 6 eV order.
The combined oscillation results of solar neutrino experiments (SNO, SK and Borexino[20])Chapter 1

4

and KamLAND gives[21]:7.4 ⇥ 10 5 <m212<7.8 ⇥ 10 5 eV 2 ,0.30< sin2 ✓12 <0.32

1.2.2(1.16)(1.17)

Atmospheric Neutrino and Accelerator Neutrino

Primary cosmic rays consists of mostly protons and Helium nuclei. The energy spectrum of
primary cosmic rays measured by di↵erent experiments is shown in Figure 1.1. When primarycosmic rays enters the Earth’s atmosphere, hadronic interactions between the protons and theHelium nuclei produce secondary cosmic rays which includes pions and kaons:p + N ! ⇡ + + ⇡ + ....

(1.18)

Figure 1.1: Proton and helium spectra from primary cosmic rays. [22]These pions and kaons further decay into muons, electrons and atmospheric neutrons asshown below:
⇡+ !
⇡ !

µ+

+⌫µ

- e+ + ⌫e + ⌫¯µ ,µ+⌫¯µ- e + ⌫¯e + ⌫µ(1.19)(1.20)(1.21)(1.22)

Atmospheric neutrinos are the ⌫e , ⌫¯e , ⌫µ and ⌫¯µ in the above reactions. Energy spectrum of
atmospheric neutrinos peaks at around 40 MeV and extends up to above TeV while neutrinoChapter 1

5flight length ranges from ⇠10 to ⇠ 104 km, which correspond to the production height fromabout 15 km to the diameter of the Earth (⇠13000 km, neutrinos produced at the other sideof the Earth). Atmospheric neutrino experiments have enough statistics for neutrino energies
above several hundred MeV, hence observation of atmospheric neutrinos gives sensitivity of m2
of 10 1 ⇠ 10 4 eV 2 (equation (1.15)).
The predicted total flux of atmospheric neutrinos has about 10% uncertainty in the GeV
region, which is mainly due to uncertainty of hadronic interaction cross sections. On theother hand, uncertainty of the flavor ratio (⌫µ + ⌫¯µ )/(⌫e + ⌫¯e ) is only about 2% for neutrino energy below 10 GeV [23]. The flavor ratio has been measured by several experiments(Kamiokande [24] [25], IMB-3 [26] [27], Soudan2 [28]) and reported in terms of a double ratioR ⌘ (Nµ /Ne )Data /(Nµ /Ne )M C , where Nµ (Ne ) is the number of muon (electron) events produced by charged current interactions. The measured flavor ratio was roughly one half of what
expected, which was suggested to be due to neutrino oscillation.
Super-Kamiokande (SK) reported in 1998 that the zenith angle distributions of ⌫µ events
have a deficit in the upward direction which gave evidence for neutrino oscillation [4]. Moreover,
MACRO [29] and SK [30] studied upward-going muons produced by energetic ⌫µ in the rocksurrounding the detector through charged current interactions. The observed zenith angle distributions did not agree with the no-oscillation analysis, instead they were consistent with the⌫µ $ ⌫⌧ oscillation hypothesis. Later SK reported observation of a dip in the L/E distribution
which was predicted from the flavor transition probability of neutrino oscillation [31]. Atmospheric neutrino oscillation was then later confirmed by long-baseline experiments using ⌫µ beam
produced by accelerators such as K2K [32] and MINOS [33] which measure ⌫µ disappearance inflight.
The two-flavor neutrino oscillation parameters from MINOS [34] are constrained to be:
2.24 ⇥ 10 3 <m232
2<2.44 ⇥ 10 3 eV 2 ,sin (2✓23 ) >0.90(1.23)(1.24)

at 90% C.L.
The two-flavor neutrino oscillation parameters from the Super-Kamiokande [35] are constrained to be:
1.5 ⇥ 10 3 <m232
2<3.4 ⇥ 10 3 eV 2 ,sin (2✓23 ) >0.92(1.25)(1.26)

at 90% C.L.

1.2.3

Mixing Angle ✓13 Measurement by Accelerator and Reactor Neutrinos

The third mixing angle ✓13 was investigated by the CHOOZ experiment[37] which detectedreactor ⌫¯e by the inverse -decay reaction ⌫¯e + p ! e+ + n. The nuclear reactor was 1km awayfrom the Gadolinium-loaded liquid scintillator detector. The distortion of the energy spectrumof ⌫¯e was investigated. ⌫¯e survival probability is expressed as:P (⌫¯e ! ⌫¯e ) = 1

sin2 2✓13 sin2 (1.27 m213 L)
E⌫(1.27)

Chapter 1

6∆ m232 [10-3 eV2]432
SK(L/E) 90%C.L.
SK(zenith) 90%C.L.
MINOS 90%C.L.
K2K 90%C.L

1

0.6

0.7

0.8

0.9

1

2

sin 2θ23

Figure 1.2: 90% C.L. allowed regions of oscillation parameters for ⌫µ $ ⌫⌧ oscillation. Horizontal axis is sin2 2✓23 and the vertical axis is m232 .The results are taken from SK zenith angle
analysis (solid line)[36], L/E results (dashed line)[31], MINOS experiment(dotted line)[34] andK2K(dotted and dashed line)[32].

The upper limit given by the CHOOZ experiment is[37]:sin2 2✓13 <0.15(1.28)

at 90% C.L. The CHOOZ experiment ended without giving any evidence of non-zero ✓13 .
Later the T2K experiment searches for ⌫e appearance by using a conventional neutrino beam
produced at J-PARC and directed 2.5 o↵-axis to SK at a distance of 295 km. T2K experimentgives indication for non-zero ✓13 [38]:0.03(0.04)< sin2 2✓13 <0.28(0.34)(1.29)

at 90% C.L. (assuming sin2 2✓23 = 1.0, | m223 | = 2.4 ⇥ 10 3 eV2 , CP = 0) for normal(inverted) mass hierarchy.
Recently the Daya Bay Reactor Neutrino Experiment and the Reactor Experiment for Neutrino Oscillation (RENO) experiment both measured non-zero ✓13 .
Daya Bay Reactor Neutrino Experiment detects ⌫¯e from the reactors of the Daya Bay Nuclear
Power Plant and Ling Ao Nuclear Power Plants in China with inverse -decay reaction ⌫¯e + p !
e+ + n in Gadolinium-loaded liquid scintillator [6].

Chapter 1

7

The results from Daya Bay experiment is [6]sin2 2✓13 = 0.092 ± 0.016(stat) ± 0.005(syst).

(1.30)RENO experiment also observes ⌫¯e from the reactors at the Yonggwang Nuclear Power Plant
in Korea through the inverse -decay reactions in hydrocarbon liquid scintillators detectors [7].
Results from RENO is [7]sin2 2✓13 = 0.113 ± 0.013(stat) ± 0.019(syst).

(1.31)As reactor neutrino experiments constrained ✓13 , their combined results are used in thisstudy. By considering the weighed mean x̄,Pni=1x̄ = Pni=1

and the invariance of the weighed mean

xi

2
i

1

(1.32)

2
i

2
x̄ :

1
2
x̄ = Pn1i=1 2(1.33)ithe combined result of Daya Bay experiment and RENO experiment issin2 2✓13 = 0.099 ± 0.014.

(1.34)

Atmospheric neutrino data has also been used to constrain ✓13 in the three-flavor mixing
framework with [44] and without [43] one mass scale dominance approximation , the detailed
analysis will be discussed in section 1.3.

1.2.4

Summary for Neutrino Oscillation Parameters and Remaining Issues

The global best fit results of neutrino oscillation parameters are summarized in Table 1.1.
Parameterm212
sin2 ✓12
sin2 2✓13
sin2 ✓23| m213 |Best fit ± 1 C.L.
5 eV2
7.58 +0.22
0.26 ⇥ 10+0.0180.306 0.015
0.099 ± 0.014
0.42 +0.08
0.03
3 eV2
2.35 0.12⇥
10009

Table 1.1: Summary of global best fit of neutrino oscillation parameters, taken from [39]Neutrino oscillation is well understood and the non-zero mass of neutrinos is confirmed, atthe same time there are several remaining questions about the neutrino oscillation.
Neutrino oscillation experiments are measuring the mass di↵erences ( m212 and m223 ) instead of the absolute mass, therefore the mass ordering is not yet known. Figure 1.2.4 shows

Chapter 1

8normalm3

inverted

m2
m1

2

νe
νµ
ντ

mass

m2
m1

m3

Figure 1.3: Three-neutrino mass-squared spectrum, for the two mass hierarchies.

the conceptual diagram for the two cases of mass ordering. With m3
m2 >m1 , it is defined
as normal hierarchy, while with m3 ⌧ m1 <m2 , it is defined as inverted hierarchy. One of theremaining questions is whether the mass ordering is normal hierarchy or inverted hierarchy.
Another remaining question is the value of the CP-violation phase CP . As T2K and reactor
experiments gives evidence that ✓13 is non-zero, as the CP-violation phase CP appears as a
combination with ✓13 as shown in equation (1.1), it becomes possible to probe the value of CP
and also to address the issue of CP-violation in the lepton sector.
As atmospheric neutrino experiments cover a wide range of L/E⌫ , it is possible to measure
the e↵ect of three-flavor oscillation including ✓13 , 1-2 parameters and CP

1.3

Sub-Dominant E↵ects of Atmospheric Neutrino Oscillation

Atmospheric neutrino data are well fitted to ⌫µ ! ⌫⌧ 2-flavor oscillation. In 3-flavor oscillation, ⌫e oscillation is also considered. ⌫e oscillation is due to the sub-dominant oscillation due to✓13 mixing and/or m212 induced oscillation. Matter e↵ect [40] plays an important role in theseoscillation schemes.
Oscillation of atmospheric ⌫e in vacuum are expressed as (with m12 set to zero):P (⌫e ! ⌫e ) = 1

sin2 2✓13 sin2 (1.27 m223 L)E

P (⌫µ ! ⌫e ) = P (⌫e ! ⌫µ )= sin2 ✓23 sin2 2✓13 sin2 ((1.35)(1.36)

1.27

m223 L
E)(1.37)For the oscillation probability of the neutrinos traversing the Earth, the Earth’s matter
potential due to the forward scattering amplitude of charged current ⌫e and ⌫¯e interactions withelectrons has to be taken into account. The Earth contains a large number of electrons, ⌫einteracts with these electrons via W bosons while ⌫µ and ⌫⌧ do not.
When considering the matter e↵ect in the ⌫µ ! ⌫e oscillation ( m12 ⇠ 0) as shown inequation (1.37), ✓13 and m223 are replaced by the corresponding mixing parameters in matter(✓13,M and m223,M respectively). For constant matter density [41],Chapter 1

9P (⌫µ ! ⌫e ) = P (⌫e ! ⌫µ ) = sin2 ✓23 sin2 2✓13,M sin2 (wherem223,M=sin2 ✓13,M

=

ACC

q

m223 (cos 2✓13

1.27 m223,M L⌫)
E⌫

ACC / m223 )2 + sin2 ✓13(1.38)(1.39)

sin2 2✓13
ACC / m223 )2 + sin2 2✓13

(1.40)(cos 2✓13p= 2 2GF Ne E⌫(1.41)

When | ACC / m223 |= cos 2✓13 , Mikheyev-Smirnov-Wolfenstein (MSW) [40] resonant enhancement occurs to either neutrinos or anti-neutrinos, depending on the mass hierarchy (normal
hierarchy or inverted hierarchy, respectively). Left hand side of figure 1.4 shows the oscillation
probability of ⌫µ ! ⌫e when neutrinos traveling through the Earth as a function of neutrinoenergy and zenith angle with a more realistic matter density, while the right figure shows theoscillation probability in vacuum for comparison. The resonance enhancement is seen in 2-10
GeV region in the upward direction with Chooz limit assumed. The enhancement occurs forneutrinos (antineutrinos) in normal (inverted) hierarchy case.

1

P(νµ → νe)

1

P(νµ → νe)

0.2

0.8

0.18

0.6

0.4

0.6

0.16

0.4

0.35

0.4

0.14

0.2

0.3

0.2

0.12

0

0.25

0

0.1

-0.2

0.2

-0.2

0.08

-0.4

0.15

-0.4

0.06

-0.6

0.1

-0.6

0.04

-0.8

0.05

-0.8

0.02

-1 -1
10

1

10
Eν (GeV)

10

20

cosΘν

0.45

cosΘν

0.8

-1 -1
10

1

10

10

20

Eν (GeV)Figure 1.4: Left: ⌫µ to ⌫e (or ⌫e to ⌫µ ) transition probability in the Earth; Right: ⌫µ to ⌫e (or⌫e to ⌫µ ) transition probability in vacuum. cos ⇥⌫ = -1 corresponds to upward direction andcos ⇥⌫ = 0 corresponds to horizontal direction. ( m223 , sin2 ✓23 , sin2 ✓13 ) = (2.1 ⇥ 10 3 eV2 ,0.5, 0.04) is assumed in the plots [43].
When oscillation driven by LMA-MSW e↵ect obtained from solar neutrino parameters ( m212 ,✓12 ) is considered, oscillation e↵ect can be seen in low energy region even if ✓13 = 0. Assuming✓13 ⇠ 0, ⌫e oscillation probability in constant matter density can be written as [42]:Chapter 1

10

P (⌫e $ ⌫e ) = 1

Pex

P (⌫e $ ⌫µ ) = cos ✓23 Pex

P (⌫µ $ ⌫µ ) = 1(1.42)

2

4cos ✓23 Pex2

sin 2✓23 (1

where

p

(1.43)
1

Pex cos )L
2E⌫
2m21 L⌫ sin2 2✓12Pex (⌫e ! ⌫x ) = sin2 2✓12, M sin2 ()
4E⌫ sin2 2✓12,M
cos 2✓12cos 2✓12,M = q(cos 2✓12 ACC / m212 )2 + sin2 2✓12
⇠ ( m231 + sin2 ✓12 m2 21)(1.44)(1.45)(1.46)(1.47)

Figure 1.5 shows Pex as a function of neutrino energy and zenith angle with a more realistic
matter density. The oscillated atmospheric ⌫e flux is written as:
osc
e

=
=

osc
e
0
e

0
0
e · P (⌫e ! ⌫e ) + µ · P (⌫µ ! ⌫e )02
1))
e (1 + Pex (r cos ✓23= Pex (r cos2 ✓23(1.48)(1.49)

1)(1.50)

where r = Nµ /Ne . r in equation (1.50) is about 2 in low energy region, in case sin2 ✓23 6=
0.5, ⌫e oscillation e↵ect can be seen in either flux excess or deficit as shown in the right sideof Figure 1.5. If cos2 ✓23 is greater than 0.5 (✓23 < 45 ), an enhancement of flux is expected, if
cos2 ✓23 is less than 0.5 (✓23 > 45 ), a deficit of flux is expected. Figure 1.5 shows the flux ratio
for sin2 ✓23 = 0.4, 0.5 and 0.6.
Two sub-dominant oscillation schemes are described above with the approximation of m212 ⇠0 or ✓13 ⇠ 0 respectively. In order to describe the oscillation e↵ect due to the interference between ( m213 , ✓13 ) and ( m212 , ✓12 ), full oscillation parameters have to be taken into accountwithout taking approximation in ✓13 and m212 . Full oscillation parameters include all relevant
oscillation parameters including the CP violation phase CP . When considering all oscillation
parameters, ⌫µ $ ⌫e transition probability in constant matter density is [45]:P (⌫µ $ ⌫e ) = cos2 ✓e13 cos2 ✓23 Pe2+ sin2 ✓e13 cos2 ✓e13 sin2 ✓23 (2

where

Pe2 )
2e
2e
2 sin ✓13 cos ✓13 sin ✓23 cos ✓23 (cosPe2 = sin2 2✓12,M sin2mR2

sin I2 )(1.51)(1.52)2R2 =sin 2✓12,M cos 2✓12,M sin2

I2 =1
sin 2✓12,M sin
2

m

m

2

(1.53)(1.54)

Chapter 1

11Φe / Φe0
0

1.4
1.35
1.3
1.25
1.2
1.15
1.1
1.05
1

2

sin θ23 = 0.4

-0.5

-1
0

2

P(νe → νx)

cosΘν

1
0.8

0.9

0.6

0.8

0.4

0.7

0.2

0.6

0

0.5

-0.2

0.4

-0.4

0.3

-0.6

0.2

-0.8

0.1

-1 -1
10

1

10

10

cosΘν

sin θ23 = 0.5
0.95
-0.5
1.0
1.05
-1
0

1
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0.6

2

sin θ23 = 0.6

-0.5

-1

20

10

-1

1

Eν (GeV)Eν (GeV)Figure 1.5: Left: ⌫e transition probability Pex as a function of energy E⌫ and zenith angle,
assuming oscillation parameters ( m212 , m223 , sin2 ✓12 , sin2 ✓23 ) = (7.7 ⇥ 10 5 eV2 , 2.1 ⇥ 10 320eV2 , 0.3, 0.5); Right: Electron flux ratio osc
e / e . There is an excess (a deficit) for sin ✓23 in2
the first (second) octant as shown in the top (bottom) figure. As for sin ✓23 = 0.5, both small
excess and deficit are seen as shown in the middle figure. [43].
eem is the phase oscillation in matter and ✓13 is the mixing angle in matter, with ✓13 ⇡ ✓13,M .

When full oscillation scheme is considered, ⌫e oscillation occurs with CP violation phase.
Figure 1.6 shows P(⌫µ ! ⌫e ) as a function of zenith angle and neutrino energy with realistic
matter density. Electron neutrino oscillation appears in both sub-GeV and multi-GeV region
as shown in the figure. Sub-GeV (multi-GeV) is defined to be less (greater) than 1.33 GeV invisible energy in a water Cherenkov detector. The ratio between the expected electron events(Ne ) and the non-oscillation electron events (Ne0 ) is:Ne
Ne0= (Pee'1) + rPeµ1 (✓13 ) +2 (✓12 ,(1.55)
m212 ) +3 (✓13 , ✓12 ,

m212 , CP )(1.56)

where r = Nµ /Ne , and r ⇠ 2 to 3.5 in a few MeV to 10 GeV. The term 1 (✓13 ) arises for✓13 > 0 and is independent of m212 . 2 (✓12 , m212 ) arises for m212 > 0 and is independent of✓13 . The last term 3 (✓13 , ✓12 , m212 , CP ) depends on both m212 and ✓13 and it is the only
term which depends on CP phase values. These three terms a↵ect di↵erent energy regions.
2
1 (✓13 ) induces electron excess in the multi-GeV region,2 (✓12 , m12 ) induces excess or deficit

Chapter 1

12

1

P(νµ → νe)

1

0.7

0.8

νe flux ratio Φosc / Φ0

1.4

0.8
0.6

0.6

1.3

0.6
1.2

0.5

0.2

0.4

0

-0.2

0.3

-0.4

0.2

0.4
cosΘν

cosΘν

0.4

-0.6

1.1

0

1

-0.2

0.9

-0.4
0.8
-0.6

0.1

-0.8
-1 -1
10

0.2

1

10

10

20

0.7

-0.8
-1 -1
10

Eν (GeV)

1

10

10

2 0.6

Eν (GeV)Figure 1.6: Left: Probability of ⌫µ ! ⌫e oscillation as a function of zenith angle and neutrinoenergy, in case of ( m212 , m223 , sin2 ✓12 , sin2 ✓23 , sin2 ✓13 , CP ) = (7.7 ⇥ 10 5 , 2.1 ⇥ 10 3 , 0.3,
0.5, 0.04, 0 ). Right: ⌫e flux ratio osc / 0 assuming the same oscillation parameters as the left
figure. Figures taken from [43].
in the sub-GeV region depending on the octant of sin2 ✓23 , 3 (✓13 , ✓12 , m212 , CP ) has e↵ect inboth sub-GeV and multi-GeV regions due to the interference between ( m232 , ✓13 ) and ( m212 ,✓12 ) induced oscillations. By looking at the interference of the sub-dominant e↵ects of theatmospheric neutrino, the CP phase can also be studied.
This study aims to probe the mass hierarchy and to obtain the oscillation parameter values( m212 , m223 , ✓23 , ✓13 , CP phase ( CP )) by studying the full neutrino oscillation. This is thefirst analysis in Super-Kamiokande experiment to probe the mass hierarchy by separating theevent samples into neutrinos and anti-neutrinos. Outline of this thesis is:
Chapter 2 describes the Super-Kamiokande detector.
Chapter 3 describes the simulation for atmospheric neutrinos observed in Super-Kamiokande.
Chapter 4 describes the detector calibration.
Chapter 5 describes the data of atmospheric neutrinos.
Chapter 6 describes the analysis and the results.

Chapter 2

The Super-Kamiokande Detector
Super-Kamiokande is a cylindrical 50 kTon water Cherenkov detector located at Kamioka
Observatory of Institute for Cosmic Ray Research in Gifu Prefecture, Japan. Figure 2.1 shows
the schematic view of the detector and its location in Mountain Ikenoyama. Geographic coordinates of the site is 36 25’N and 137 18’E, the altitude above sea level is 370 m. The detector is
located in a zinc mine under the top of Mountain Ikenoyama which has a mean rock overburdenof ⇠ 1000 m (⇠ 2700 m water equivalent). The rock overburden shields the detector against
cosmic ray muon background, hence the cosmic ray muon flux at the Super-Kamiokande site
is reduced by 5 orders of magnitude when compared with the flux on the Earth surface. The
detector is mainly composed of the water tank and the photomultiplier tubes (PMTs) which
detect Cherenkov radiation due to charged particles in the water.
The main purpose of the Super-Kamiokande experiment are to search for nucleon decays
and the studies of various types of neutrinos which include atmospheric neutrinos, solar neutrinos, neutrinos from supernova and neutrinos from other astrophysical sources. The SuperKamiokande detector is also used as a target for artificial neutrino beam in long-baseline neutrino
oscillation experiment.
The Super-Kamiokande experiment started taking data in April, 1996 and continued theobservation for five years within the SK-I running period till the detector maintenance in July,
2001. During water refilling after the maintenance, an accident occurred in November 2001 inwhich more than half of the PMTs were destroyed. After the accident, the Super-Kamiokande
detector was rebuilt with half of the original PMT density in the inner detector and resumed
observation from October, 2002, which is referred to as SK-II running period. SK-II continued
measurement for three years until October 2005, in which reconstruction work was carried out
to resume the PMT density. Super-Kamiokande restarted observation in June, 2006 with thesame PMT density as SK-I period, this period is referred to as SK-III period. As new electronics
module for the detector was developed, the new electronics was installed in September 2008,
starting the SK-IV running period.
In this thesis, the data observed in all SK periods: SK-I (1996-2001), SK-II (2002-2005),
SK-III (2006-2008) and SK-IV (from 2008 up to 2012 Feb) running periods are used.

13

Chapter 2

14

Figure 2.1: Super-Kamiokande detector and its location in Mountain Ikenoyama.

2.1

Cherenkov Radiation

The Super-Kamiokande detector observes relativistic charged particles in water by detecting
the emitted Cherenkov light. Cherenkov light is emitted when the velocity of a charged particle
exceeds speed of light in the medium:
c
(2.1)
n
where v is the velocity of the charged particle, c is the speed of light in vacuum and n is therefractive index of the medium[46]. Momentum threshold of Cherenkov radiation is determinedby the refractive index of the medium and the particle mass. As the refraction index of water
is about 1.34, the momentum thresholds of Cherenkov radiation for electrons, muons, charged
pions and protons are 0.57, 118, 156 and 1051 MeV/c respectively.
Cherenkov light is emitted in a cone shape with a characteristic half opening angle ✓C along
the direction of travel of the particle. The half opening angle (Cherenkov angle) is determinedby:v

cos ✓C =where= v/c. For particle with1n

⇠ 1 in water, the Cherenkov angle is about 42 .

(2.2)

Chapter 2

15

The number of photons emitted by Cherenkov radiation is written as a function of wavelength
:
d2 N
2⇡↵
= 2 (1
dxd

1
)
n2 2

(2.3)

where x is the path length of the charged particle and ↵ is the fine structure constant. About
340 photons/cm are emitted between the wavelength range of 300 nm to 600 nm, which is thesensitive wavelength region of the PMTs used in the Super-Kamiokande detector.
The Cherenkov light emitted by charged particles are projected as ring images on the wallinside the detector. Super-Kamiokande detects the Cherenkov photons by PMTs arranged on
the wall and the Cherenkov ring patterns can be recognized. Figure 2.2 shows the visual displayof a typical neutrino event observed in the Super-Kamiokande detector.

Figure 2.2: Visual display of a typical neutrino event in the Super-Kamiokande detector. The
size of the small circles indicates the amount of Cherenkov photons detected in each PMT. The
Cherenkov ring pattern is clearly visible. (Event taken from SK-IV period.)

Chapter 2

2.2

Detector

2.2.1

Water Tank

16

The water tank of Super-Kamiokande is in cylindrical shape with 41.4 m height and 39.3 m
diameter. 50 kTon highly pure water is filled in this water tank. The water tank is opticallyseparated into two concentrical cylindrical regions by a PMT support structure and two opaque
sheets. The two regions are referred to as the inner detector and outer detector respectively(Figure 2.3).

Tyvek
8" PMT

Top

Black sheet

Barrel20" PMT

BottomFigure 2.3: Supporting frame of PMTs.
The inner detector (ID) contains 32 kTon water with dimension of 36.2 m height and 33.8 m
diameter. 11,146 inward facing 20-inch PMTs are installed to the supporting frame uniformly at
70 cm intervals in SK-I period. The e↵ective photocathode coverage of the ID was about 40%,and the rest of the surface is covered with black polyethylene terephthalate sheet (called ”black
sheet”). In SK-II period, 5,182 20-inch PMTs (about half of the original PMT density) were
installed to the supporting frame and the photocathode coverage was about 20%. From SK-III

Chapter 2

17period and onwards, the photocathode coverage is resumed to about 40% with 11,129 PMTs.
The outer detector (OD) surrounds the ID with thickness of 2.05 m on the top and bottom,
and 2.2 m along the barrel wall. The OD is monitored by 1,885 outward facing 8-inch PMTs
installed to outer side of the supporting frame. A 60 cm ⇥ 60 cm wavelength shifting plate is
attached to each OD PMT. The wall is covered with reflective material called ”tyvek sheet” toimprove light collection efficiency. The OD is used as a veto counter for cosmic ray muons andto tag outgoing charged particles. The ⇠2 m thick water layer serves as a shield to attenuate
gamma ray and neutrons from surrounding rocks. The inner and outer volumes are separated
by a 55 cm thick dead region in which no PMT is installed. Stainless steel frame, signal cables
and high voltage cables of all the PMTs are contained in this dead region.

2.2.2

Photomultiplier Tube for the Inner Detector

Figure 2.4: Schematic view of the 20-inch PMT used in Super-Kamiokande.
PMTs used in the ID (Hamamatsu R3600) have photocathode with a diameter of 50 cm(20 inch). This 20 inch PMT was developed by Hamamatsu Photonics K.K. in cooperation
with Kamiokande collaborators [47]. Later, the dynode structure and the bleeder circuit weremodified to improve timing response and photo collection efficiency for Super-Kamiokande [48].
Figure {pmt shows a schematic view of the PMT. The PMT specification is listed in table 2.1.
The photocathode of the PMT is coated with bi-alkali (Sb-K-Cs) which has high spectral
sensitivity to Cherenkov light and low thermionic emission. Quantum efficiency (22%) peaks at
the wavelength 360-400 nm region, which also corresponds to the peak of the relative Cherenkov
spectrum through pure water as shown in figure 2.5. The dynode structure and the bleeder
circuit are optimized to achieve high collection efficiency, fast timing response and good energy
resolution. Average collection efficiency at the first dynode is 70% which is uniform within
7% everywhere on the photocathode. Gain of the ID PMT is 107 at a high voltage of 1500 V
to 2000 V. FIgure 2.6 shows the charge distribution for single photoelectron signal, a clear 1

Chapter 2

18

Shape
Photocathode area
Window material
Quantum efficiency
DynodesGain
Dark current
Dark pulse rate
Cathode non-uniformity
Anode non-uniformity
Transit time
Transit time spread
WeightPressure tolerance

Hemispherical
50 cm diamater
Bialkali (Sb-K-Cs)
22% at = 390 nm
11 stage Venetian blind type107 at ⇠ 2000 V
200 nA at 107 gain
3 kHz at 107 gain
<10%
<40%
90 nsec at 107 gain
2.2 nsec (1 ) for 1 p.e. equivalent signals
13 kg
6 kg/cm2 water proof

Table 2.1: Specification of the 20-inch PMT used in Super-Kamiokande.

photoelectron (p.e.) peak can be seen. The transit time spread of the single photoelectron signal
is about 2.2 nsec. The average dark noise rate at 0.25 photoelectrons (p.e.s) threshold used in
SK-I was about 3 kHz.
Magnetic field over 100 mG a↵ects photoelectron trajectories inside the PMT and makes thetiming resolution worse. Geomagnetic field at the detector site is about 450 mG. In order tocompensate the geomagnetic field, 26 sets of horizontal and vertical Hermholtz coils are arranged
around the tank, such that the magnetic field inside the detector is reduced to about 50 mG.
On November 12th, 2001, while the SK tank was being refilled upon the completion of
the maintenance work, one ID PMT at the bottom of the tank imploded, triggering a cascade
implosions of PMTs. About 60% ID and OD PMTs were destroyed in this accident. After this
accident, all the inner PMTs are instrumented with acrylic covers from SK-II period onwards
to avoid this kind of chain reaction of implosions. A clear 12 mm thick UV-transparent acrylic
dome is put over the photocathode area of each PMT and the side of the PMT is protected with
Fiber-Reinforced Plastic (FPR) shield with holes letting water flow in and out of the case freely,
as shown in figure 2.7. Transparency of the acrylic cover for photons with normal incidence inwater is more than 96% above wavelength 350 nm.

2.2.3

Photomultiplier Tube for the Outer Detector

As for the OD, 1,885 8-inch PMTs are installed. The photocathode of the OD PMT is fittedwith 60 cm ⇥ 60 cm ⇥ 1.3 cm wavelength shifter plate, which increases light collection efficiency
by 60%. Even though the timing resolution at single photoelectron slightly degrades from 13
nsec to 15 nsec because of the plate, since the OD is used as a veto counter rather than particle
tracker, poor timing resolution has little consequence while higher light collection efficiency isof more importance.

Chapter 2

19

Figure 2.5: Quantum efficiency distribution of the 20-inch PMT used in the ID against wavelength and the relative Cherenkov spectrum through pure water. [135]2.2.4

Outer Detector (OD) Segmentation

Tyvek sheet was installed to separate top and wall regions, and bottom and wall barrel
regions of the OD before the start of the SK-III period. Figure 2.8 shows a schematic view of
the OD segmentation.
OD aims to distinguish particles (mostly muons) produced in the ID from cosmic ray muons.
Particles (mostly muons) produced inside the ID would leave only one signal in the OD whengoing out of the detector, while cosmic ray muons would leave one signal in the OD when getting
into the detector and then leave another signal in the OD while going out of the detector, leaving
two signals in total in the OD. The installation of the OD segmentation aims to separate thesignal in the ID and the OD more efficiently.
Figure 2.9 shows the OD event display of a simulated partially contained (PC) event which isa particle produced inside the ID and then goes out around the bottom edge of the tank. The left
figure shows the SK-II detector configuration (without the tyvek sheet for the OD segmentation)

20

distribution (arbitrary unit)

Chapter 2

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

0

1

2
3
photo-electron

4

5

Figure 2.6: Single photoelectron distribution of the 20-inch PMT.
and the right figure shows the SK-III detector configuration (with the tyvek sheet for the OD
segmentation between top, bottom and barrel region of the OD installed). Because of the tyvek
sheet, the OD signal in SK-III can be clearly seen at the bottom corner of the detector. The
tyvek sheet improves the efficiency of the event selection process for partially contained (PC)events.

2.3

Water Purification System

Water used in the tank of the Super-Kamiokande experiment is produced from abundant
spring water in the mine. The water in the tank is continuously circulated through the water
purification system at a flow rate of 35 ton/hour in order to keep high attenuation length.
Another important role of the water purification system is to remove radioactive materials in
the water, especially radon (Rn), which is a background source for the solar neutrino observationin the MeV energy region.

2.4

Air Purification System

Clean air with low radon level is essential inside the detector and in the experimental area.
To keep low radon level, radon-free air is produced by the purification system in the mine andis continuously pumped into the space above the water surface inside the tank at a positive
pressure to prevent radon in the SK dorm air from entering the tank and dissolving into thepurified water [49]. Radon concentration in the radon-free air is less than 3 mBq/m3 . The air
purification system is composed of compressors, a bu↵er tank, driers and filters. The air flowrate is about 18 m3 /hour.

Chapter 2

21

Figure 2.7: PMT case used to cover 20-inch PMT in the inner detector since SK-II period.
ODID
OD segmentation

Figure 2.8: A schematic view of the OD region. [43]2.5Electronics and Data Acquisition System for SK-I to SK-III
Periods

Electronics and data acquisition system for SK-I to SK-III periods is described in this section,
as new electronics was installed in September 2008 which corresponds to the SK-IV period, the
new electronics will be described in the next section.

2.5.1

Inner Detector Electronics and Data Acquisition System

During SK-I to SK-III periods, ID PMT signals were processed by custom built electronics
modules called ATM (Analog-Timing-Module) of the TKO standard (TRISTAN KEK Online)[50] [51]. The ATM module records the integrated charge and the arrival timing information of
all PMT signal.

Chapter 2

22

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

Comnt;NUM
78
RUN
999999SUBRUN
500
EVENT
914
DATE 2007-Dec-17
TIME
15:57:41
TOT PE:36346.5
MAX PE:
218.1
NMHIT : 3683
ANT-PE:
372.1
ANT-MX:
24.7
NMHITA:
221

NUM
90
RUN
999999SUBRUN
721
EVENT
1426
DATE 2009-Feb-27
TIME
11: 3:20
TOT PE:
25455.8
MAX PE:
243.3
NMHIT : 3800
ANT-PE:
526.2
ANT-MX:
32.2
NMHITA:
217

RunMODE:MonteCarlo 1990/00/00:NoYet:NoYet
1990/00/00:;R= 0:NoYet
TRG ID :00000011
R :Z : PHI : GOOD
T diff.: 0.00
us 0.00: 0.00: 0.00:0.000
CANG : RTOT : AMOM : MSFEVSK :80000003
nOD YK/LW:79/**
SUB EV : 0/ 1
Dcye: 0( 0/ 0/ 0/ 0)

RunMODE:MonteCarloTRG ID :00000011
T diff.: 0.00
us
FEVSK :80000003
nOD YK/LW:94/**
SUB EV : 0/ 0
Dcye: 0( 0/ 0/ 0/ 0)

Comnt;

Figure 2.9: Left: SK-II OD event display of a partially contained event, without OD segmentation; Right: SK-III OD event displa of a partially contained event, with OD segmentation.
[43]
Figure 2.10 shows the block diagram of the analog input part of the ATM module. Each
input channel of ATM from the PMT has two switching pairs of Charge to Analog Converter(QAC) and Time to Analog Converter (TAC) to minimize electronics dead time during the datataking for two successive events (such as muon followed by its decay electrons). All PMT signal
sent to ATM are amplified 100 times, then they are divided into four signals. One of the four
signals is sent to the discriminator, and when the pulse height of the PMT signal exceeds thethreshold (which is set to 0.23 p.e.s), a 400 ns gate signal for QAC and a start signal for TAC
are generated. At the same time, a rectangular signal called HITSUM (200 ns width and 15
mV pulse height) is sent to the global trigger module. Other two out of the four signals are fed
to QAC. When a global trigger is issued, a stop signal is sent to TAC, then the information in
QAC and TAC are digitized by ADC. The digitized charge and timing information are stored ininternal FIFO memory in ATM. In case no global trigger is issued within 1.3 s, all information
in QAC and TAC would be cleared. ATM has 450 pC dynamic range with a resolution of 0.2
pC, and 1.3 s dynamic range with a resolution of 0.4 ns. The remaining one signal out of thefour signals becomes an output signal of ATM called PMTSUM. Figure 2.11 shows a schematic
view of the ID data acquisition system. Signals from 12 PMTs are fed into 1 ATM board. There
are in total 946 ATM boards installed in 48 TKO crates. The digitized data in the ATM FIFO
memory are then sent to VME memory modules called SMP (Super Memory Partner) for every
16 events. 48 SMP are installed in 8 VME crates, with 1 SMP module handles the data of 20
ATMs. The data in SMP memories are read out by 8 slave computers and then sent to theonline host computer.

2.5.2

Outer Detector Electronics and Data Acquisition System

For the OD data acquisition system, the schematic view is shown in figure 2.12 [52]. The
paddle cards distribute high voltage from the main frame to the OD PMTs. A coaxial cable isused to supply the high voltage to the OD PMT and send signal from the OD PMT, then theseChapter 2

23

to PMTSUMHITOUTDISCRI.
CURRENT SPLITTER&AMPLIFIER

PMT INPUT

W:200ns

to HITSUM

ONE-SHOT HIT
W:300ns

SELF GATESTART/STOP

TAC-A

TAC-A

CLEAR
GATE

QAC-A

QAC-A

to ADCSTART/STOP

TAC-B

TAC-B

CLEAR
GATE

THRESHOLD

QAC-B

QAC-B

DELAY

GATE
TRIGGERPED_START

Figure 2.10: Block diagram of analog input part of the ATM module.
cards pick o↵ the PMT signals through a high voltage capacitor. Signals from the OD PMTs arethen sent to Charge to Time Converter (QTC), which converts the PMT signal to a rectangularpulse whose width is proportional to the input charge. At the same time, a rectangular HITSUM
signal is generated by the QTC and sent to a global trigger module. The threshold of the QTC
is set to 0.25 p.e.. When a global trigger is received, the leading edge and the width of therectangular pulse are converted to timing and charge information respectively by LeCroy 1877
multi-hit TDC module. The dynamic range is set to 16 s which starts from 10 s before theglobal trigger timing.

2.5.3

Trigger

The ID trigger schematic diagram is shown in figure 2.13. When an ID PMT signal exceeds
the threshold, an ATM generates a rectangular HITSUM signal with 15 mV pulse height and
200 ns pulse width. These signals are analog-summed over all PMTs to generate an ID-HITSUM
signal. The pulse height of the ID-HITSUM signal is proportional to the number of hit PMTs
within 200 ns time window. There are three types of trigger signals derived from the IDHITSUM signal. In SK-I , the high energy (HE) trigger is generated when the pulse height of
ID-HITSUM signal exceeds a threshold of -340 mV, which corresponds to 31 hits within a 200
ns time window. The threshold for low energy (LE) trigger is -320 mV, which corresponds to
29 hits. This is equivalent to a signal expected from a 5.7 MeV electron assuming 50 % trigger
efficiency. Trigger rates for HE and LE triggers are ⇠5 Hz and ⇠11 Hz respectively. Super
low energy (SLE) trigger was implemented in May 1997 to lower the solar neutrino analysis
threshold. The threshold level for the SLE trigger was lowered gradually with increasing trigger
CPU power. The threshold was set to -186 mV, which corresponds to 4.6 MeV electron, in July
2000. Its trigger rate is ⇠1 kHz. In SK-II and SK-III, the same trigger scheme were used, withthe trigger conditions listed in table 2.2.

Chapter 2

24

20-inch PMT
SCH
ATM
x 20
x 240

ATM
GONG

20-inch PMTSCHATMinterface
Ultra Sparc

x 20
x 240

SMP

ATM

SMP

GONG

online CPU(slave)

SMP
SMP
SMP
SMP

x6

Super Memory Partner

VME
20-inch PMT
SCHUltra Sparc

ATM

online CPU(slave)

x 20
x 240

Ultra Sparc

ATM
GONG

online CPU(host)

Analog Timing Module

online CPU(slave)
TKO

FDDI

Ultra Sparc

Ultra sparc

online CPU(slave)
Ultra Sparc

online CPU(slave)
Ultra Sparc

FDDI

online CPU(slave)
Ultra Sparc

online CPU(slave)

online CPU(slave)interface
Ultra Sparc
SMP
SMP

Ultra Sparc

online CPU(slave)

SMP
SMP
SMP
SMP

Super Memory Partnerinterface

VME
VME

interrupt reg.
TRG

20-inch PMT
SCH
TRIGGERATM

TRIGGER

x 20
x 240

PROCESSORHIT INFORMATION
ATM
GONG

Analog Timing ModuleTKO

PMT x 11200

ATM x ~1000

SMP x 48

online CPU(slave) x 9

Figure 2.11: Schematic view of the ID data acquisition system for SK-I to SK-III periods.
OD trigger is generated by similar procedure. When an OD PMT signal exceeds a threshold,a QTC module generates a rectangular pulse wit 20 mV height and 200 ns in width. The signals
are also analog-summed to generate an OD-HITSUM signal. Threshold of the OD trigger is 19
hits within 200 ns time window.
Four types of trigger signals (HE, LE, SLE, OD) are fed to a hardware trigger module TRG
which generates a global trigger signal when any of the four trigger signals is issued.

2.6

Electronics and Data Acquisition System for SK-IV Period

In order to achieve zero dead time, SK detector’s front end electronics were replaced in
September 2008. The new electronics [53] are described in this section.
In SK-IV, the former electronics ATM was replaced by new electronics QBEE (QTC-BasedElectronics with Ethernet). It has an integrated 3-channel device for PMT signal readout.

Chapter 2

25

Local Hitsum

QTC
Modulex40/quadx10/quad

TDC
x5
4 LSBsHigh Voltage

16 bit event #TDC

FASTBUSPMT

Paddle Card

StruckSupply~470/quadrant

FSCC

AUX

84 pin data ribbon cables

Ethernet FSCC Control

Figure 2.12: Schematic view of the OD data acquisition system for SK-I to SK-III periods.

SK-I
Periodnumber of ID PMTs
Photocathode coverage
HE trigger threshold
LE trigger threshold
SLE trigger threshold
Other featuresApr 1996 - Jul 2001
11,14640%-340 mV-320 mV-186 mVSK-II
Oct 2002 - Oct 2005
5,182*
20 %-180 mV-152 mV-110 mV

SK-IIIJun 2006 - Sep 2008
11,129*40 %-320 mV-302 mV-186 mV
Tyvek sheet installed
for OD segmentation

Table 2.2: Trigger conditions for each SK period. *All inner PMTs are protected with acrylic
covers.

This application-specific integrated circuit (ASIC) called the high-speed charge-to-time converter (QTC) IWATSU CLC101 was designed in CMOS 0.35 µm technology. It has built-in
discriminators to trigger its integrated circuits by itself and encodes the amount of input charge
to timing signal, where the leading edge represents the timing and the width represents theintegrated charge of the input signal.
Figure 2.14 shows the block diagram of the QTC and its surroundings. The QTC has three
input channels per chip and each channel has three gain ranges (small, medium and large). The
overall charge dynamic range of the QTC is 0.2 - 2500 pC if the gain ratio of the three ranges
1is set to 1: 17 : 49
. Figure 2.15 shows the block diagram of one QTC channel. Input signals
from PMTs are amplified by a low-noise-amplifier (LNA), delayed by a low-pass filter (LPF),
processed by a voltage-to-current (V/I) converter, and integrated by a capacitor. The sum of the
input-signal waveform to the capacitors can be monitored through an output signal designated
PMTSUM.
The QTC is charge sensitive. Charge integration starts when the amplified input signal
crosses the discriminator threshold. The leading edge of the output signal represents the timing
when the charge integration starts. A trigger flag signal (HIT) is generated at the same time.
After a charging period is determined by the timer block, the integrated signal starts to bedischarged by a constant current source. The discharging time is thus proportional to theChapter 2

26

Analog signal
from PMT
ATM threshold

ADC gate
TDC startand stop
HITSUM

400 nsec width

start

stop

200 nsec width

Global triggerATM
Sum of HITSUM
Master threshold
Global trigger
CentralHut

Figure 2.13: Overview of the ID trigger scheme for SK-I to SK-III periods.
integrated charge, which in turn is reflected in the width of the output signal. Hence the QTC
output signal contains both timing and charge information.
In figure 2.15, the timer block contains three timers: a charging timer, a discharging timer
and a VETO timer. The timers generate timing gates by ramps with constant discharge current
and comparators. The time length for these gates are controlled by digital-to-analog converters(DACs) for discharge current and comparator threshold levels.

2.6.1

Trigger

Figure 2.16 shows the timing chart for QTC operation. The charging timer, which is triggeredby the discriminator output signal, operates for ⇠ 400 ns. During the charge gate, the switch
between the V/I converter and charging capacitor closes and input charge accumulates in thecapacitor. Then after the charge gate, the discharging timer operates for ⇠ 350 ns. During
the discharge gate, the switch between the discharging current source and capacitor closes, thearriving input signals are ignored. The trailing edge of the QTC output signal represents thetime when the integrated signal voltage decreases to the comparator threshold level. Hence, theQTC output signal becomes wider when the input signal gets larger. At the end of the dischargegate, reset and VETO signals are issued, all QTC circuits except VETO timer are reinitialized.
Input signals within ⇠ 150 ns after the discharge gate are ignored. the processing time for one
input signal is ⇠ 900 ns in total.

Chapter 2

27

Figure 2.14: Block diagram of the QTC and its surroundings (Data acquisition system for SK-IV
period). [53]2.6.2

Improvement by the Electronics and Data Acquisition System in SK-IV
PeriodThe triggering system during SK-I, SK-II and SK-III period had hardware trigger (by HITSUM), while the new data acquisition (DAQ) readout system used in SK-IV use software trigger
instead. The new system records all hits and apply software triggers,and it is collecting data at
60 kHz frequency and 17 µs TDC window. There is no dead time for the SK-IV DAQ system, sothat all data are taken. The saturation points of the electronic modules also increases from theoriginal ⇠250 photoelectrons during the SK-I to SK-III periods to ⇠1200 photoelectrons in the
new system. Due to these improvement, the decay-electron tagging efficiency becomes higher
and dynamic range also became wider in the SK-IV period.

Chapter 2

28

Figure 2.15: Block diagram of one QTC channel. Each channel has three gain ranges: small,medium and large. (Data acquisition system for SK-IV period)[53]

Figure 2.16: Timing chart for QTC operation. (Data acquisition system for SK-IV period) [53]Chapter 3

Simulation of Atmospheric Neutrino
Atmospheric neutrino events in Super-Kamiokande are simulated by Monte-Carlo method.
The Monte Carlo simulation is composed of three parts: atmospheric neutrino flux, neutrino interactions and particle tracking in the detector. By considering neutrino flux and cross-sections,
expected events observed in the detector are calculated during a certain period. The produced
particles have information such as vertex, direction, momentum and so on. With these information, the detector simulation simulates the particle, Cherenkov photon emission, photon
propagation and response of the detector hardware. The simulated events are reconstructed in
the same way as the real observed data. They are called Monte Carlo (MC) events in this thesis.
This chapter describes the neutrino flux model, NEUT library which is the simulator ofneutrino interaction kinematics and detector simulation.

3.1

Atmospheric Neutrino Flux

There are several atmospheric neutrino flux models calculated by M. Honda et al. [23] [54][55] [149](Honda flux), G. Battistoni et al. [57] (Fluka flux) and G. Barr at al. [58] (Bartol
flux). Honda flux is adopted as the default model in this analysis, with the di↵erence between
Honda flux and the two other flux models are considered as the systematic uncertainties. As
the Honda flux only covers the energy range up to 10 TeV, the Volkava flux [59] is adopted tosimulate events having energy above 10 TeV.
The atmospheric neutrino flux is a convolution of the primary spectrum at the top of theatmosphere with the yield of neutrinos per primary particle. In order to reach the atmosphereand interact, the primary cosmic rays must first pass through the geomagnetic field. Hence, thecalculation of the flux of neutrinos includes flux of primary cosmic rays, filtering e↵ect of thegeomagnetic field and the yield of neutrinos per primary particle [56].
The primary cosmic ray flux model, which is the input of the neutrino flux calculation, isdetermined by experimental measurements. Figure 3.1 shows the model used in the Honda flux
calcution (solid line) and the comparison with di↵erent models [55]. The primary cosmic ray
spectrum has been precisely measured by BESS and AMS experiments up to 100 GeV [65] [66].
The cosmic ray flux changes according to the turbulence of solar activities, which is lower whensolar activities are high (solar maximum) than when the solar activities are low (solar minimum).
The di↵erence of cosmic ray flux between solar minimum and solar maximum is more than afactor of two for 1 GeV cosmic rays, while the di↵erence is about 10 % for 10 GeV cosmic rays.

29

Chapter 3

30

Figure 3.1: Primary cosmic ray proton flux used in Honda flux calculation (line) and the observed
results (points) [55]. Observation data points are taken from Webber [60] (crosses), LEAP[61] (upward triangles), MASS1 [62] (open circles), CAPRICE [63] (vertical diamonds), IMAX[64] (downward triangles), BESS98 [65] (circles), AMS [66] (squares), Ryan [67] (horizontal
diamonds), JACEE [68] (downward open triangles), Ivanenko [69] (upward open triangles),Kawamura [70] (open squares) and Runjob [71] (open diamonds).
In Honda flux, the US Standard Atmosphere model [72] is used for the density structure of
the atmosphere which gives zenith angle dependence of atmospheric neutrinos. Geomagnetic
field e↵ects are calculated based on the IGRF2005 model [73].
Primary cosmic ray protons and nuclei interact with air molecules in the Earth’s atmosphere.
Secondary particles (mostly pions and kaons) are produced through these hadronic interactions.
In the hadronic interactions between the cosmic rays and air molecules, two theoretical models
are used in the calculation: JAM [150] interaction model for primary cosmic ray energies lessthan 32 GeV, and DPMJET-III [75] for energies greater than 32 GeV. Pions and kaons generated
from the hadronic interactions further decay into muons and neutrinos.
In order to make sure that the simulation of the hadronic interactions is correct and to finetune the hadronic interactions in the simulation, the flux of the resulting cosmic ray muons fromthe simulation is compared with external experiment data. The flux of the cosmic ray muons
have been measured by several experiments such as BESS [76] and L3+C [77]. DPMJET-III usedin the Honda flux is modified based on the measurements by BESS and L3+C. JAM used in the
Honda flux for the lower energy hadronic interactions is also modified based on the experimental
studies at Tsukuba [79], Mt. Norikura [81] and at ballon altitudes at Fort Summer [80]. Figure
3.2 shows the comparison of the observed muon flux at di↵erent altitudes and the calculation
with JAM and DPMJET-III.
In the calculation of neutrino flux, interactions and propagation of particles are treated ina 3-dimentional way, considering the curvature of charged particles in geomagnetic field. In
the 3-dimentional calculation, there is an enhancement of neutrino flux in the near-horizontal
direction when compared with the 1-dimentional calculation. This can be explained by thedi↵erence in e↵ective area as shown in Figure 3.3. This e↵ect is significant for low energy
neutrinos (less than 1 GeV), for which transverse momentum of the secondary particles inChapter 3

31

Figure 3.2: Comparison of the secondary cosmic ray muon flux observed at Tsukuba (left), Mt.
Norikura (middle) with the BESS detector [79] and at ballon altitudes at Fort Summer [80].
Plots taken from [149].
hadronic interactions and bending of muons in the geomagnetic field are significant. Figure
3.4 shows the zenith angle distribution of neutrino flux at four di↵erent energy ranges. The
near-horizontal flux enhancement in the lower energy regions (less than 1 GeV) is due to the
3-dimensional e↵ect, while the enhancement in the higher energy regions (above1 GeV) is mostly
due to muons reaching the ground before they decay near vertical.
Figure 3.5 shows the calculated zenith angle distributions of neutrino flux for three di↵erent
neutrino energies (0.32 GeV, 1.0 GeV and 3.2 GeV).
The calculated energy spectrum and flux ratio between ⌫e + ⌫¯e and ⌫µ + ⌫¯µ of atmospheric
neutrinos at the SK site is shown in Figure 3.6 for Honda flux, Fluka flux and Bartol flux, theflux of ⌫e + ⌫¯e and ⌫µ + ⌫¯µ is averaged over all of the direction. The flavor ratio is about 2
for neutrino energies below a few GeV and it becomes larger than 2 as the neutrino energies
increase, as more cosmic ray muons reach the ground before decaying.
The flavor ratio also depends on the zenith angle and the energy of the incoming neutrinos
as shown in Figure 3.7.

Chapter 3

32

Figure 3.3: Schematic view of e↵ective areas of primary cosmic rays interacting with air moleculesfor 1-dimentional calculation and 3-dimentional calculation. Solid arrows represent primary cosmic rays and dotted arrows represent neutrinos. The 3-dimentional calculation gives larger areafor near-horizontal directions (from the view of the SK detector), which induces an enhancementof neutrino flux. [43]Figure 3.4: Comparison of 3-dimensional and 1-dimensional calculations of neutrino fluxes at
Super-K. Squares represent ⌫µ , asterisks represents ⌫¯µ , vertical diamonds represent ⌫e , horizontal
diamonds represend ⌫¯e for 3-dimensional calculation. Solid lines represent ⌫ fluxes and dottedlines represent ⌫¯ fluxes for 1-dimensional calculation. [56]Chapter 3

33

Figure 3.5: Zenith angle dependence of atmospheric neutrino flux for di↵erent energy regions.
The higher lines represent ⌫µ and ⌫ µ and the lower lines represent ⌫e and ⌫ e . [149]Figure 3.6: Comparison of atmospheric neutrino fluxes averaged over all directions (left) andflavor ratio (right). Red solid line represents Honda 11 flux. Dashed line represents the Bartol
flux, dotted line represents the FLUKA flux, and blue dashed line represents Honda 06 flux.
[149]

34

Flux flavor ratio

Chapter 3

35
30
25
20
15
10
5
0
-1
-0.5
0

3

0.5
co
s

1 -1
! 10

1

10

10

2

10 GeV)E "(Figure 3.7: Flavor ratio (⌫µ + ⌫¯µ )/(⌫e + ⌫¯e ) as a function of neutrino energy and zenith angle.
Calculation is based on Honda flux. [43]Chapter 3

3.2

35

Neutrino Interaction

Atmospheric neutrinos interact with nucleons and electrons in water and the rock surrounding the SK detector. In this atmospheric neutrino analysis, the Monte Carlo (MC) simulation
has the neutrino events generated through the simulation of the neutrino interaction with water
nuclei and the rock. Generation volumes include the rock surrounding the SK detector (assumed
to spread spherically up to 4 km from the detector), the OD region and the region inside ID.
A simulation program library NEUT [84] [85] is used for the simulation of neutrino interactions. In the NEUT simulator, the following interactions are considered:
CC/NC (quasi-)elastic scattering ⌫ + N ! l + N 0
CC/NC single meson production ⌫ + N ! l + N 0 + meson
CC/NC deep inelastic interaction ⌫ + N ! l + N 0 + hadrons
CC/NC coherent pion production ⌫ +16 O ! l +16 O + ⇡where N and N 0 are nucleons (proton or neutron) and l is a lepton. These interactions are
explained in the following sections.

3.2.1

Charged Current Quasi-Elastic Scattering

The charged quasi-elastic scattering for free protons (hydrogen atoms) is treated in thissimulator as described by Llewellyn-Smith [86]. For free nucleon, the di↵erential cross sectionis:d ⌫¯
M 2 G2F cos2 ✓C
u
u)2
2
2 s
2 (s
=
[A(q
)
+
B(q
)
+
C(q
)
]
dq 2
8⇡E⌫2
M2
M4

(3.1)

where E⌫ is the neutrino energy, M = 0.938 MeV is the nucleon mass, GF is the Fermi coupling
constant, ✓C is the Cabibo angle, q = p⌫ pl is the four-momentum transfer, s and u are
Mandelstam variables [86]. The factors A, B and C are given as:
A(q 2 ) =B(q 2 ) =C(q 2 ) =m2 q 2
q2
q2
2
[(4
)|F
|
(4
+
)|F 1 |2
A
4M 2
M2
M2 V
q2
q2
4q 2 FV1 ⇠FV2
2 2
|
(1
+
|⇠F
)
V
M2
4M 2
M2
m
((FV1 + ⇠FV2 )2 + |FA |2 )]M
q2(FA (FV1 + ⇠FV2 ))M2
1
q2(|FA |2 + |FV1 |2|⇠FV2 |2 )
4
4M 2

(3.2)
(3.3)
(3.4)

where m is the outgoing lepton mass, ⇠ is defined as anomalous magnetic moment µp µn = 3.71.
Vector form factors FV1 (q 2 ) and FV2 (q 2 ) and axial vector form factor FA (q 2 ) are determinedChapter 3

36experimentally and given by:q2) 1 [GE (q 2 )4M 2
q2⇠FV2 (q 2 ) = (1) 1 [GE (q 2 )
4M 2
q2 2
FA (q 2 ) =1.232(1)
MA2

q2
GM (q 2 )]
4M 2

(3.5)

GM (q 2 )](3.6)

= (1 + ⇠) 1 GM (q 2 ) = (1

q2
) 2
MV2

FV1 (q 2 ) = (1

GE

(3.7)
(3.8)

where GE and GM are the electric and magnetic form vector respectively, MV = 0.84GeV and
MA = 1.21GeV are the vector and axial vector masses respectively with both values taking into
account the experimental results from K2K and MiniBooNE [88] [89] [90].
The same MA value (1.21 GeV) is also assumed for single meson productions. MA valuea↵ects the cross section as it is a function of Q2 . For larger MA , interactions with high Q2 andtherefore larger scattering angles are enhanced. Uncertainty of the MA value in this analysis isestimated to be 10% by the total uncertainty adopted to the K2K and MiniBooNE results.
For scattering o↵ bound nucleons in 16 O, nuclear e↵ects such as Fermi motion of the nucleons
and Pauli exclusion principle are taken into account as described by the Smith and Moniz model[87]. As nucleus are fermions, the outgoing momentum of the nucleons in the interactions has
to be larger than the Fermi surface momentum to allow quasi-elastic scattering to occur. The
Fermi surface momentum is assumed to be 225 MeV/c in this simulator.
Other theoretical models such as Nieves et al. [91] and Nakamura et al. [92] have beendeveloped to account for nuclear e↵ects in neutrino interactions at intermediate energy ranges
besides the Smith and Moniz model. In this analysis, these models are used in the systematic
uncertainty estimation for neutrino interactions and nuclear e↵ects.
Figure 3.8 shows the cross section of quasi-elastic scattering for the calculation from NEUT
and comparison with experimental data.

3.2.2

Neutral Current Quasi-Elastic Scattering

As for the di↵erential cross section of neutral current quasi-elastic scattering, it is of thesame form as equation 3.1, with cos✓C replaced by unity, and nucleon form factors replaced bycorresponding neutral current (NC) nucleon form factors [99]:1= ± FV1
2
1= ± FV2
2

F̃V1,NF̃V2,N(2sin2 ✓W )FN1 ,(2sin2 ✓W )FN2

where
G̃V
FV1 = E

1

FN1 =

G̃N
E
1

q2
G̃V
G̃VM G̃VE2
4M 2 M
,
F
=
2
V
q
q2
1 4M
2
4M 2
q2
N
N
G̃
G̃M G̃N
2
E
4M 2 M
,
F
=
N
q2
q2
1
4M 2
4M 2

(3.9)

37

σ (10

-38

2

cm )

σ (10

-38

2

cm )

Chapter 3

2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

-

(a) νµn→µ p

0

2

4

–

ANL
BNL
SKAT

6
8
10
Eν (GeV)

2

4

12

14

GGM (1977)
GGM (1979)
Serpukhov
SKAT+(b) νµp→µ n0

Serpukhov
GGM (1977)
GGM (1979)

6
8
10
Eν (GeV)12

14

Figure 3.8: Charged current quasi-elastic interaction cross section of (a) ⌫µ and (b) ⌫¯µ , withexperimental data from ANL [93], Gargamelle [94] [95], BNL [96], Serpukhov [97] and SKAT
[98]. Solid line indicates scattering o↵ a free proton, the dashed line indicates scattering o↵bound nucleons in 16 O.
(3.10)
F̃A =1 s
1(FA ± FA ) = ( s ± gA )(1
2
2

F̃p =

2M 2 F̃A
m2⇡ q 2

q2
MA2

) 2
(3.11)

where N = p, n and the upper (lower) sign is for proton (neutron). In NEUT, the weak mixing
angle is set to sin2 ✓W = 0.23117, and the strange quark contribution s is set to 0.

3.2.3

Single Meson Production

Single meson production via baryon resonances is the dominant hadron production process
in the region where the invariant mass of the hadronic system is less than about 2.0 GeV /c2 .
Our simulation for the process is based on Rein and Sehgal’s model [100] . It is used to simulateChapter 3

38

the resonance production of single ⇡, ⌘ and K. It assumes one meson is generated by the decayof intermediate resonances:⌫ + N ! l + N⇤

(3.12)N ⇤ ! N 0 + meson(3.13)

Equations (3.12) is resonance production while (3.13) is resonance decay, in which N and N 0 are
nucleons and N ⇤ is a baryon resonance. Invariant mass W of the intermediate baryon resonancesis restricted to be less than 2 GeV/c2 . For W greater than 2 GeV/c2 , the interactions are
simulated as a part of deep inelastic scattering which will be described in next section.
Rein’s method [101] is used to determine the angular distribution of the pion in the final state
for the P33 (1232) resonance. As for other resonances, the angular distribution of the generated
pion is set to be isotropic in the resonance rest frame. The angular distribution of ⇡ + wasmeasured for ⌫µ p ! µ p⇡ + [102]. The results agree well with the prediction by NEUT.
Pauli exclusion principle in the baryon resonance decay is considered by requiring the nucleon
momentum to be greater than the Fermi surface momentum.
It is known that a baryon resonance in a nucleus can be absorbed without producing a meson[103]. NEUT considers such phenomenon and 20% of the resonance events do not have mesons
generated and only leptons and nucleons are produced.
Figure 3.9, 3.10 and 3.11 show the cross section of single meson production for charged
current ⌫µ , charged current ⌫¯µ and neutral current respectively, from both NEUT calculationand experimental data.

cm2)

νµ p → µ− p π+

−38

1.2

39

σ(ν µ )(10σ(ν µ )(10

−38

cm2)

Chapter 3

1
0.8

1
0.9

νµ n → µ− n π+

0.8
0.7
0.6
0.5

0.6

0.4

0.4

0.3
0.2

0.2

0.1

σ(ν µ )(10

−38

cm2)

0

1
0.9

1

10

0

Eν (GeV)1

10Eν (GeV)νµ n → µ− p π0

0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

1

10

Eν (GeV)Figure 3.9: Cross section for ⌫µ charged current single pion productions. Solid lines are NEUT
calculations, points are experimental data. [136]

cm2)
−38

cm2)

0.8
0.7

1
0.9

0.7
0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

1
0.9

1

10

0
Eν (GeV)νµ n → µ+ n π−

0.8

0.6

0

σ(ν µ )(10νµ p → µ+ p π−

−38

1
0.9

40

σ(ν µ )(10σ(ν µ )(10

−38

cm2)

Chapter 3

1

10Eν (GeV)νµ p → µ+ n π0

0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

1

10

Eν (GeV)Figure 3.10: Cross section for ⌫¯µ charged current single pion productions. Solid lines are NEUT
calculations, points are experimental data. [136]

cm2)
−38

0.35

0.3

0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05
3

4

5

6

7

8

0.4
0.35

0.4

σ(ν µ )(1000.35 νµ(νµ) p → νµ(νµ) p π

0910Eν (GeV)

cm2)

2

−38

1

0.3

0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05
1

2

3

4

5

6

7

8

9
10
Eν (GeV)

1

2

3

4

5

6

7

8

9
10
Eν (GeV)

6

7

8

9
10
Eν (GeV)νµ(νµ) n → νµ(νµ) n π0

0.3

0.25

0

νµ(νµ) p → νµ(νµ) p π−

0.3

0.25

σ(ν µ )(10

cm2)

0.4

σ(ν µ )(10+0.35 νµ(νµ) p → νµ(νµ) n π

0

−38

41

0.4

σ(ν µ )(10

−38

cm2)

Chapter 3

0

1

2

3

4

5

Figure 3.11: Cross section for neutral current single pion productions. Lines are NEUT calculations, with solid lines for ⌫µ and dashed line for ⌫¯µ . Points are experimental data taken from[104] (dot) and [105] (square). [136]Chapter 3

3.2.4

42

Deep Inelastic Scattering

The cross section of charged current deep inelastic scattering in the range of the invariant
mass W > 1.3 GeV/c2 is [106]:d2 ⌫,¯⌫
dxdy

=

C1 =
C2 =

G2F MN E⌫
y2((1 y ++ C1 )F2 (x, q 2 ) ± y(1
⇡
2
yMl2
xyMN
m2l
m2l
2
4MN E⌫ x
2E⌫
4E⌫
2MN E⌫ x
2
ml
4MN E⌫ xy+ C2 )xF3 (x, q 2 ))2

(3.14)

where x = q 2 /(2M (E⌫ El )) and y = (E⌫ El )/E⌫ are Bjorken scaling parameters, MN isnucleon mass, ml is outgoing lepton mass, E⌫ and El are the energy of incoming neutrino andoutgoing lepton in the laboratory frame respectively. The nucleon structure functions F2 andxF3 are taken from the Parton distribution function (PDF) of GRV98 [107]. The correction
given by A. Bodek and U.K. Yang are adopted to use the PDF in lower Q2 region [108].
In NEUT, the cross section of deep inelastic scattering induced by neutral current interactions
are assumed to have relations estimated from experimental results [109] [110].
As the range 1.3<W <2.0 GeV/c2 overlaps with that in single pion production, n⇡
2 isrequired in this W region. Therefore, the kinematics of the hadronic system are simulated by
two di↵erent methods for these two invariant mass ranges:
In the region of 1.3<W <2.0 GeV/c2 , only pions are considered as outgoing mesons. The
average pion multiplicity is estimated from the results of Fermilab 15-foot hydrogen bubble
chamber experiment [111]:hn⇡ i = 0.09 + 1.83 ln W 2(3.15)The number of pions in each event is determined with KNO (Koba-Nielsen-Olsen) scaling.
The forward-backward asymmetry of pion multiplicity in the hadronic center of mass system isincluded with the results from BEBC experiment [112]:nF⇡
0.35 + 0.41 ln W 2=
nB
0.5 + 0.09 ln W 2⇡(3.16)In the region of W >2.0 GeV/c2 , the kinematics of the hadronic system are calculated by
PHYTIA/JETSET [113], which treats ⇡, K, ⌘, and so on.
Cross section of charged current ⌫µ and ⌫¯µ DIS interactions are shown in figure 3.12.

3.2.5

Coherent Pion Production

Coherent pion production is a neutrino interaction with an oxygen nucleus in which one pion
is produced and the pion is of the same charge as the incoming weak current. As only a verysmall amount of momentum is transferred to the oxygen nucleus, the angular distribution of theoutgoing leptons and pions are peaked at the forward direction. The modified Rein and Sehgal
model [114] is adopted to simulate the interactions.

43

σ/Eν (10-38cm2 GeV-1)

Chapter 3

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

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

CCFR 90
CDHSW 87
GGM-PS 79

0

10

CHARM 88
BNL 80
CRS 8020

30 50

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

CCFRR 84
SKAT

100

200

150

250

Eν (GeV)Figure 3.12: Cross section for DIS charged current ⌫µ (upper lines and points) and ⌫¯µ (lower
lines and points) interactions. GRV98 is used with Bodek-Yang correction in NEUT, as shownby the dashed lines. The points represent experimental data. [136]The measurement results by the K2K-SciBar detector set the upper limit of the CC coherent
pion production cross section [115]. The upper limit is significantly lower than that predicted by
the Rein and Sehgal model. Therefore, the modified Rein and Sehgal model [116] which takes
into account the non-vanishing lepton masses in CC interactions is used in NEUT. For ⌫µ , thecross section is suppressed by about 25% at 1.3 GeV due to the interference of the axial vector
and pseudoscalar (pion-exchange) amplitudes by this modification.

3.2.6

Nuclear E↵ects

The secondary interactions of mesons and baryons which are produced in the neutrino interactions with nucleons inside the 16 O nuclei are also simulated in NEUT. All of the mesons
produced within 16 O nuclei are tracked from their production points to either their exiting points
or their absorption point inside the nuclei. This is simulated for ⇡, K and ⌘ by using a cascade
model.
As the cross section of pion production for neutrino is large for E⌫ > 1 GeV and the pionnucleon interaction cross section is also large in this energy region, pion interactions are important. NEUT considers several pion interactions in 16 O nuclei: inelastic scattering, charge
exchange and absorption.
The probabilities for interactions within a nucleus depends on the nuclear density distribution, so the initial pion production point inside the nucleus where neutrino-nucleus interactions
occur, is determined by the Wood-Saxon density distribution [117]. The probabilities for various
interactions are calculated at each step of the pion for low and high energy regions respectively.
In NEUT, the calculation of interaction probabilities for pion momenta p⇡  500 MeV/ccomes from the model of Salcedo et al. [118]. A tuning with absorption and QE interaction
probabilities was done by using ⇡ + 12 C data [119]. As Ashery re-measured the charge exchange
cross section and obtained a larger value than before [120], an important correction was made

Chapter 3

44

σ (mb)

to the Ashery absorption and charge exchange data: the charge exchange data is scaled up andthe absorption data is scaled down accordingly. Only absorption probability and QE probability
are modified, and a good agreement in charge exchange and reactive channels is also obtained.
As for p⇡ > 500 MeV/c, the energy becomes high enough such that nucleons begin to appear
as free particles within the nucleus, ⇡ ± on free proton scattering cross section data are used as
inputs to the calculation of interaction probabilities. Tuning was carried out by using ⇡ ± 12 C
data. The discrepancy in the absorption channel above p⇡ = 500 MeV/c is fixed by the change
in kinematics: more backward scattering which reduces pion momenta to the region where there
is a larger absorption probability. Charge exchange and inelastic scattering data above p⇡ =500 MeV/c are also used to scale the QE probability and charge exchange probability.
Figure 3.13 shows the cross section of ⇡ + 12 C scattering as a function of ⇡ + momentum
before and after tuning.

Reactive (TunedFSI)
Reactive (OldFSI)
Quasi-elastic
Absorption
Single CX
π Production

450
400

π- 12
6 C

350
300
250
200
150
100
50
0
0

200

400

600

800

1000 1200 1400 1600
π- Initial Momentum (MeV/c)

Figure 3.13: Cross section of ⇡ + 12 C scattering as a function of ⇡ + momentum. Solid lines are
NEUT calculation before tuning, dashed lines are NEUT calculation after tuning. Points witherror bars are experimental data. [119]As for kaon interactions, the elastic scattering and charge exchange interaction are consideredwith the results from the cross sections measured by K ± N scattering experiments [121] [122][123]. For ⌘ interactions, the absorption (⌘N ! N ⇤ ! ⇡(⇡)N ) is considered [124].
Nucleon re-scattering inside the oxygen nucleus is also considered with cascade model andthe measurements by Bertini et al. [125] in NEUT. Pion production caused by the decay of
produced deltas is also taken into account as the isobar production model by Lindenbaum et al.
[126].

Chapter 3

3.3

45

Detector Simulation

Produced particles in neutrino interactions are incorporated in a detector simulation code
which simulates (1) tracks of the particles, (2) generation and propagation of Cherenkov photons
in water, (3) PMT response and readout electronics.

3.3.1

Particle Tracking

The detector simulation was developed based on GEANT3 [127]. Table 3.1 lists the processes
which are considered in the simulation program. Hadronic interactions in water are simulatedwith CALOR package [128], which reproduce pion interactions well including low energy region(about 1 GeV/c). For further lower energy region (p⇡  500 MeV/c), the NEUT cascade modelis used.

e±µ±

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

Table 3.1: List of processes considered in the simulator.

3.3.2

Cherenkov Photon Tracking in Water

Dispersion of the refractive index is taken in account in the simulation. Group velocity
vg =

c
n( )

@n( )
@

(3.17)

Chapter 3

46

where c is the speed of light in vacuum, is light wavelength, is used as the light velocity inwater. The generated Cherenkov light is simulated to be scattered and absorbed in water.

3.3.3

Response of PMT

Charge and timing response of PMTs in considered in the simulation. Detected charge
value of each hit PMT is simulated by random numbers distributed as the measured photon
distribution in Figure 2.6. The timing distribution depends on the charge. The relation between
timing and charge is measured using laster calibration (see next chapter). The timing of each
PMT is a random variable distribution as Gaussian with one sigma.

Chapter 4

Calibration
4.1

Detector Calibration

Calibrations of the detector charge and timing information are performed in order to understand and keep high quality of the detector performance. Water transparency is also monitoredcontinuously for the SK detector.

4.1.1

Absolute Gain Calibration

Absolute gain calibration is necessary in order to convert the charge detected by a PMT inpico Coulomb (pC) into the number of photoelectrons (p.e.s). The absolute gain is determinedby using the charge distribution of single photoelectron signals.

mean

2.055pC

5

10

3

2

1

0

0

pC

Figure 4.1: The charge distribution of a typical PMT.

47

Chapter 4

48

The absolute gain was measured by using a gamma-emitting nickel source which gives about
100 photons per gamma emission. The source was placed near the center of the SK tank withcoordinates (x = 35.3 cm, y = -70.7 cm, z = 0 cm) with the origin defined as the center of
the tank. The nickel source consists of a neutron source put inside nickel material. The nickel
source releases gamma rays of energy about 9 MeV which is produced when neutrons are releasedby spontaneous fission of 252 Cf and captured on nickel. The charge distribution of a typical
PMT is shown in figure 4.1. The sharp peak near zero is caused by electrons emitted from thephotocathode but missing the first dynode, and the peak around 2 pC corresponds to single
photoelectrons. The mean value of the 1 p.e. charge distribution is used as a constant toconvert PMT charge from pico Coulomb to the number of p.e.s. The constant used for SK-I,
SK-II, SK-III and SK-IV are 2.044 pC/p.e., 2.297 pC/p.e., 2.243 pC/p.e. and 2.658 pC/p.e.
respectively.

4.1.2

Relative Gain Calibration

The high voltage of each PMT is set to work at approximately uniform gain for all the PMTs
in the detector. The uniformity of the gain of all PMTs in the detector is important in terms ofdetermination of momentum without systematic di↵erences.

Figure 4.2: Schematic view of the relative gain measurement system.
Figure 4.2 shows the schematic view of the relative gain calibration system. Light from theXe lamp is passed through an ultraviolet filter and neutral density filter and is injected intoa scintillation ball in order to produce uniform di↵used light. The other outputs from the Xelamp are used to monitor the intensity of the Xe light and to make a calibration trigger.
The high voltage of the PMTs are adjusted so that they give the same gains. The relative
gain Gi of the i-th PMT is given by:

Gi =

Qi
li
⇥ li2 ⇥ exp
Q0 f (✓)
L(4.1)

Chapter 449

Qi is the charge detected by the i-th PMT, li is the distance from the light source to the PMT,f (✓) is the PMT acceptance as a function of the photon incidence angle ✓, L is the attenuation
length of water and Q0 is the normalization factor. This measurement is performed with thedi↵user ball at several di↵erent positions.
The relative gain spread for SK-I is about 7% . From SK-II, the relative gain calibration
is improved by using standard PMTs whose gains are adjusted within a few per cents spread
relative to one another in advance. The position dependence of the light from the Xe lamp canbe reduced by standard PMTs. The relative gain uncertainty was adjusted to 2 % for SK-II,
SK-III and SK-IV.

4.1.3

Relative Timing Calibration

The timing of all PMTs must be calibrated so that the event vertices and directions can bereconstructed accurately. The timing response of the PMTs depends on length of cable between
each PMT, the process time of electronics and ”time walk” (which means di↵erent light intensity
makes di↵erent triggered timing). ”Time walk” exists because PMTs with larger charge hits
exceed TDC discriminator threshold earlier than those with less charge even if both get photons
at the same timing. Hence the timing calibration is to make a correction table for the time walkand the overall process time for each PMT in the detector.

0.CUGT

PORWNUGYKFVJPUGE&[G.CUGTPORWNUGYKFVJPUGE64)/QP2/6

5-+&

8CTKCDNGHKNVGT&KHHWUGTDCNN
Figure 4.3: Schematic view of the timing measurement system using a laser.
Figure 4.3 shows the schematic view of the timing calibration system. The nitrogen laser
pulse is divided into two, one goes to a trigger PMT which makes an external trigger for timing

Chapter 4

50

calibration event, the other goes to dye laser module to produce a dye laser pulse. The dye laser
pulse which has a wavelength of 398 nm and pulse width 0.2 nsec goes through variable filter
which is used to adjust the laser intensity. The dye laser pulse travels through a 70 m opticalfiber and then its di↵used light is emitted from a di↵user ball. The PMT timing response ismeasured with various light intensity from 1 p.e. to 250 p.e. by using an adjustable attenuation
filter. The result is shown as a scatter plot of timing and charged (TQ-map) in figure 4.4. Eachdot represents one measurement. The TQ-map is made for all ID PMTs and is used to correct
the timing information.

Figure 4.4: TQ-map, a measured two-dimensional plot of timing against charge distribution.
Horizontal axis represents charge of each hit and vertical axis represents hit timing.

4.1.4

Water Scattering Measurement with a Laser

Water transparency is important to determine the number of photons arriving PMTs throughwater in the detector. Water transparency is a↵ected by the combined e↵ect of absorption
and scattering of the light. Water attenuation length in water can be described as L = (↵abs+ ↵scat.sym + ↵scat.asym ) 1 , where ↵abs , ↵scat.sym and ↵scat.asym are absorption, symmetricscattering and asymmetric scattering coefficients respectively. These coefficients are measured
with a N2 laser and dye laser system which can be tuned to di↵erent wavelengths. During SK-III,
the wavelengths are 337 nm, 365 nm (371 nm in SK-II), 400 nm and 420 nm. Each laser fires
every 1 sec during normal data taking (which is faster than 6 sec of SK-I). In SK-I, there is only
one injector, but since SK-II, seven more injectors were installed and there are eight injectors
in total now, their positions are shown in figure 4.5.
In SK-IV, the nitrogen/dye laser is replaced with laser diode (LD), as nitrogen laser isunstable and intensity gradually decreases, while LD can be used permanently and the intensity
is stable. The nitrogen/dye laser was replaced with LD with wavelengths of 375 nm, 405 nm

Chapter 4

51

Figure 4.5: Positions of the eight laser light injects in the SK tank. ”Top old” refers to theoriginal SK-I light injector which is still being used at the moment.
and 445 nm.
Figure 4.6 shows a typical top laser light injector event. A cluster of hits at the bottom of
the tank is due to the unscattered photons from the laser. Other hits at the barrel or top of
the tank is due to the dark current or photons scattered in water or reflection by PMTs andblack sheet. The number of scattered hits are related to water quality and proportional to lightintensity. Figure 4.7 shows the PMT hit timing distributions in each region of the detector forboth data and Monte Carlo. The first peaks are characterized by absorption and scattering
coefficients while the second peaks are due to the photons reflected by PMTs or black sheets.
In SK-IV, monthly fitting of parameters is carried out. The water coefficient averages anderrors on the mean are collected for 3-5 wavelengths as shown in figure 4.8. The coefficients are
obtained by fitting shown in figure 4.9.

4.1.5

Water Transparency Measurement with Cosmic Ray Muons

Water transparency is also measured by Cherenkov light from cosmic ray muons passing
through the detector. Cosmic ray muons are used as a calibration source because the energy
deposit of an energetic muon is almost constant (about 2 MeV/cm). Only vertical downwardChapter 4

52

Figure 4.6: Typical laser light injector event.
muons are selected for this measurement. The muon track is reconstructed by connecting theentrance and exit points in the ID. Assuming the light detected by each PMT is not scattered,the detected charge is expressed by:

Q = Q0 ⇥

f (✓)
⇥ exp
l

l
L

(4.2)

where Q0 is a constant, f (✓) is the PMT acceptance, l is the photon path length and L is theattenuation length. Figure 4.10 shows the schematic view of the measurement. Figure 4.11
shows log(Q ⇥ l/f (✓)) as a function of photon pathlengths. The attenuation length is estimatedto be 95 m for this data set.
As cosmic ray muons are measured during normal data taking, continuous check of watertransparency is possible with this method. Figure 4.12 shows the time variation of the attenuation length of water in the entire period from SK-I to SK-IV (till 20 June 2012).

Chapter 4

53

Figure 4.7: PMT hit time distribution in di↵erent detector regions for data and Monte Carlo(after tuning by scattering and absorption parameters).

0.01
0.008
0.006
0.004
0.002
0.020
12
0.016
0.012
0.008
0.004-3
x 10
0.60
12
0.48
0.36
0.24
0.12
0
12

Figure 4.8: Water coefficients for December 2009. The top plot shows absorption, the middle
plot shows symmetric scattering and the bottom plot shows asymmetric scattering. From these
data, the average and error on the mean are calculated. The averages for 337 nm, 375 nm, 405
nm and 445 nm are shown by the numbers on the plots respectively.

Chapter 4

54

Figure 4.9: A example of fitting. Fitted water coefficient functions for Apr 2009. The solid lines
show the fitting in SK4 and the dashed lines show the SK3 functions. Points are coefficient
averages and error bars are the error on the mean.

Chapter 4

55: Cherenkov
Photons

Inner Tank

20-inch
PMT

42o

li

Θi

i’th PMT
Cosmic Muonlog(Ql/f( θ))Figure 4.10: Schematic view of the attenuation length measurement with vertically penetratingcosmic ray muons. Arrows from the muon track (solid line) show the trajectories of Cherenkov
photons. li is the flight length of the Cherenkov photons detected by the i-th PMT.

500

RUN 3106

P1
P2

5.987
-0.9487E-04

1000

2000

3000

0.4609E-03
0.4444E-06

400
300
200
0

4000

5000

Figure 4.11: E↵ective observed charge for through-going muons as a function of photon pathlengths (l), where Q is the detected charge and f (✓) is the PMT acceptance.

Chapter 456

Figure 4.12: Time variation of water attenuation lengths for SK-I, SK-II, SK-III and SK-IV.

Chapter 4

4.2

57

Energy Calibration

Momentum of a particle is determined based on the charge information of PMTs. The
accuracy of energy scale is important for atmospheric neutrino analysis. Four independentcalibration sources are used for the energy scale calibration in momentum range from a few tensof MeV/c to about 10 GeV/c:• Track range of high energy stopping muons (1 ⇠ 10 GeV/c)• Cherenkov angle of low energy stopping muons (200 ⇠ 500 MeV/c)• Invariant mass of ⇡ 0 produced by neutrino interactions (⇠ 130 MeV/c)• Momentum of decay electron (⇠ 50 MeV/c)
The accuracy of the absolute energy scale is checked by comparing data and Monte Carlosimulation for all of the above calibration source. Time variation check of energy scale, detector
uniformity of energy scale and uncertainty on energy scale are also estimated.

4.2.1

High Energy Stopping Muons

Momentum of high energy muons can be determined by the track lengths which are approximately proportional to the momentum. The stopping cosmic ray muon track lengths can be
estimated by the distance between the entering position at the detector and the vertex positionof the subsequent decay electron. Resolutions for stopping muon entering position and decay
electron vertexes are better than 50 cm. Selection criteria for stopping muons are listed in thefollowing:
1. Entering position is on the top wall of the detector
2. Direction of the stopping muon is downward (cos✓ > 0.94)
3. One decay electron event is detected
4. Reconstructed range of the muon track is 7 < L < 30 m.
Criteria (1) and (2) require down-going muon event entering the detector from the top of
the detector. Momentum loss per cm is about 2.3 MeV/c. Data and MC agree within 0.7%,
1.1%, 2.0% and 2.2% for SK-I, SK-II, SK-III and SK-IV respectively.

4.2.2

Low Energy Stopping Muons

For low energy stopping muons (< 400 MeV/c), momentum can be estimated by Cherenkov
angle which depends on momentum as shown below. The Cherenkov angle of charged particles
can be expressed as following:1
1
cos ✓C =
=
n
n

s

1+

m2
p2

(4.3)

Chapter 4

58

where ✓C , n, , m and p are Cherenkov angle, refraction index of water, v/c, mass and momentum respectively.
Selection criteria for low energy stopping muons are:
1. Entering point is at the top wall of the detector
2. Direction of the stopping muon is downward (cos ✓ > 0.9)
3. One decay electron event is detected
4. Total number of p.e.s in ID is less than 1500 p.e.s (750 p.e.s for SK-II)

500

200

number of events

number of events

250

150

100

400

300

200

100

50

0
100

150

200

250

300

350

400

450

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500

24

26

28

32

34

36

38

40

42

44

500

reconstracted momemtum (MeV/c)

500

reconstracted momemtum (MeV/c)

30

opening angle (deg.)

reconstracted momentum (MeV/c)

DATA

450

400

350

300

250

Monte Carlo

450

400

350

300

250

200

200
30

32

34

36

38

40

opening angle (deg.)

42

44

30

32

34

36

38

40

42

44

opening angle (deg.)

Figure 4.13: Upper left panel shows the reconstructed momentum distribution Pp.e. for data(circles with error bars) and MC (solid line). Upper right panel shows the reconstructed opening angle distribution, also for data and MC. Lower two panels show the correlation between
reconstructed Cherenkov opening anlge ✓C and the reconstructed momentum Pp.e. for data (left)
and MC (right). All distributions are made from SK-II.
Criterion (4) selects low momentum stopping muons with momentum < 380 MeV/c. In
figure 4.13, the upper left panel shows the reconstructed momentum distribution, upper rightChapter 459

panel shows the reconstructed opening angle ✓C distribution. In the lower two panels, the scatterplots of reconstructed momentum and the opening angle ✓C is shown for both data and MC.
The momentum and opening angle correlation can be seen in these two scatter plots. Figure
4.14 shows the averaged Pp.e. /P✓ (where P✓ is the momentum estimated with ✓C ) and the ratioof MC to data as a function of P(✓). They agree within 0.7%, 1.3%, 2.1% and 2.1% for SK-I,
SK-II, SK-III and SK-IV respectively.

Chapter 4

60

SK-I

1.4

SK-I

1.05
1.04

1.3

1.03
1.2

MC/DATAP(p.e.)/P(θ)

1.02
1.1
1
0.9

1.01
1
0.99
0.98

0.8

DATA

0.97

MC

0.7

0.96

0.6
150

200

250

300

350

400

450

0.95
150

500

expected momentum P(θ) (MeV/c)

SK-II

1.4

200

250

300

350

400

450

500

expected momentum P(θ) (MeV/c)

SK-II

1.05
1.04

1.3

1.03
1.2

MC/DATAP(p.e.)/P(θ)

1.02
1.1
1
0.9

1.01
1
0.99
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0.8

DATA

0.97

MC

0.7

0.96

0.6
150

200

250

300

350

400

450

0.95
150

500

expected momentum P(θ) (MeV/c)

SK-III

1.4

200

250

300

350

400

450

500

expected momentum P(θ) (MeV/c)

SK-III

1.05
1.04

1.3

1.03
1.2

MC/DATAP(p.e.)/P(θ)

1.02
1.1
1
0.9

1.01
1
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0.8

DATA

0.97

MC

0.7

0.96

0.6
150

200

250

300

350

400

450

500

0.95
150

expected momentum P(θ) (MeV/c)

200

250

300

350

400

450

500

expected momentum P(θ) (MeV/c)

SK-IV

SK-IV
1.05

1.4

1.04

1.3

1.03
1.2

MC/DATAP(p.e.)/P(θ)

1.02
1.1
1

1.01
1

0.99

0.9

0.98
0.8

Data

0.97

MC

0.7
0.6
150

200

250

0.96
300

350

400

450

500

expected momentum P(θ) (MeV/c)

0.95
150

200 250 300 350 400 450
expected momentum P(θ) (MeV/c)

500

Figure 4.14: Left panels show the averaged ratio of momentum derived from charge to thatderived from the opening angle (Pp.e. / P✓ ) as a function of reconstructed momentum P(✓) fordata (solid line) and MC (dashed line). Right panels show the ratio of MC events to data in thecorresponding left panel.

Chapter 4

4.2.3

61

Neutrino Induced ⇡ 0 Events⇡ 0 events are produced by interactions of atmospheric neutrinos in the detector. As a
⇡ 0 decays into two rays immediately, the invariant mass of ⇡ 0 can be calculated with thereconstructed momentum of the two rays (P 1 and P 2 ):M⇡ 0 =

q

2P 1 P 2 (1cos ✓)(4.4)

where ✓ is the opening angle between the two rays. Neutral current ⇡ 0 events are selectedfrom atmospheric neutrino event sample by the following criteria:
1. Two Cherenkov rings are recognized and both are identified as electron-like
2. Electron from muon decay is not detected
3. Vertex position is reconstructed within the fiducial volume (> 2 m away from the ID wall)
Criterion (2) rejects contamination by ⇡ + ⇡ 0 or µ± ⇡ 0 events. Figure 4.15 shows the invariant
mass distribution of ⇡ 0 events compared with the prediction by MC. The peak positions are
fitted by Gaussian distribution. The peak positions of data and MC agree within 0.7%, 1.3%,
0.3% and 1.7% for SK-I, SK-II, SK-III and SK-IV respectively.

Chapter 4

62Super Kamiokande I 1489.2 days

Super Kamiokande II 798.6 days
70

140
120

60

100

50

80

40

60

30

40

20

20

10

0
0

50

100

150

200

250

300

Mass(2 e-like rings) MeV/c2

00Super Kamiokande III 518.1 days

100

150

200

250

300

Mass(2 e-like rings) MeV/c2

Super Kamiokande IV 1096.7 days : Monitoring

50

100

40

80

30

60

20

40

10

20

0
0

50

50

100

150

200

250

300

Mass(2 e-like rings) MeV/c2

0
0

50

100

150

200

250

300

Mass(2 e-like rings) MeV/c2

Figure 4.15: Invariant mass distribution of neutrino induced ⇡ 0 events of observed data (dot)and MC (boxes). MC are normalized to the livetime of the observed data.

Chapter 4

4.2.4

63

Decay Electrons

Large number of electron events are produced in the detector from the decay of stopping
cosmic ray muons. Decay electrons’ energy spectrum is well understood and the spectrum
spreads around 50 MeV. Selection criteria for decay electrons is listed in the following:
1. Time interval from a stopping muon is 2.0 sec to 8.0 sec
2. Number of PMT hits in a 50 nsec window is > 60 (30 for SK-II)
3. Goodness of vertex fit is greater than 0.5
4. Vertex position is reconstructed within the fiducial volume (> 2 m away from the ID wall)
Criterion (1) is for efficient timing for decay electron tagging. Criterion (2) rejects ⇠6 MeV
rays from µ capture on nucleons. In simulation, µ+ /µ ratio of 1.37 [132] and the e↵ectof µ capture by oxygen nuclei are considered. The vertex fitter developed for low energy
neutrino observations (such as solar and supernova neutrinos) is used [133]. Figure 4.16 shows
the overlaid momentum spectra for decay electrons for data and MC. The mean values of data
spectrum agree with MC within 0.6%, 1.6%, 0.8% and 1.6% for SK-I, SK-II, SK-III and SK-IV
respectively.

Chapter 4

64

SK-I

SK-II
500

200

DATA

175

MC

DATA

400
number of events

number of events

225

150
125
100
75
50

MC

300

200

100

25
0

0

10 20 30 40 50 60 70
reconstracted momentum (MeV/c)

0

80

0

SK-III

10 20 30 40 50 60 70
reconstracted momentum (MeV/c)

80

SK-IV

700
4000

DATA

600

DATA

MC

number of events

number of events

MC

500
400
300
200

3000

2000

1000

100
0

0

10 20 30 40 50 60 70
reconstracted momentum (MeV/c)

80

0
0

10 20 30 40 50 60 70
reconstructed momentum (MeV/c)

80

Figure 4.16: Momentum distribution of decay electrons of data (dot) and MC (solid line) for
each SK period. MC events are normalized by number of observed data.

Chapter 4

4.2.5

65

Summary of the Energy Scale Calibration

The absolute energy scale is checked by various methods for di↵erent momentum ranges(from a few tens of MeV/c to about 10 GeV/c). Figure 4.17 summarizes the absolute energy
calibrations.The uncertainty of absolute energy scale is estimated to be less than 0.88%, 0.55%,
1.79% and 2.19% for SK-I, SK-II, SK-III and SK-IV respectively.

4.2.6

Time Variation of Energy Scale

Stability of the energy scale is confirmed with stopping muons and decay electrons. During
the reconstruction process, particle momentum is corrected with water transparency. Figure
4.18 shows the time variation of the mean value of momentum/range for stopping muons andthe mean value of momentum for decay electrons. As water quality is worse and not stable in
SK-III, the stability of energy scale is slightly worse than other SK periods. The fluctuationof absolute energy scale is estimated to be less than 0.74%, 1.60%, 2.08% and 0.39% for SK-I,
SK-II, SK-III and SK-IV respectively. SK-IV has smallest time variation among all other SK
periods. The main reasons are the treatment of scattered light calculation as a function of timein momentum reconstruction is improved and the measurement of PMT gain correction factor
as a function of time in momentum reconstruction is improved.
By adding the energy scale uncertainties discussed in the previous section to the time variation uncertainties in quadrature, the final energy scale uncertainties for SK-I, SK-II, SK-III and
SK-IV are estimated to be 1.1%, 1.7%, 2.7% and 2.3% respectively.

66(MC - DATA) / DATA (%)

Chapter 4

SK-I
8
6
4
2
0
-2
-4
-6
-8

(MC - DATA) / DATA (%)

10

2

3

4

10
10
10
momentum range (MeV/c)

10

5

SK-II
8
6
4
2
0
-2
-4
-6
-8
10

(MC - DATA) / DATA (%)µ rangeµ angle0π massdecay eµ rangeµ angle0π massdecay e
2

3

4

10
10
10
momentum range (MeV/c)

10

5

SK-III
8
6
4
2
0
-2
-4
-6
-8
10

µ rangeµ angle0π massdecay e
2

3

4

10
10
10
momentum range (MeV/c)

105Figure 4.17: Summary of absolute energy scale calibration for each SK period. Horizontal axis
shows the momentum range and vertical axis shows the deviation of data from MC predictions.

Chapter 4

67

±1%

0

e momentum(MeV/c)

43
42
41
40
39
38
37
36
2400

250

500

µ momentum/track(MeV/c/cm)

SK-I

750 1000 1250 1500 1750 2000
days from 1996/Apr./1

2.6
2.55
2.5
2.45
2.4
2.35
2.3
2.25
2.2

SK-II
µ momentum/track(MeV/c/cm)

e momentum(MeV/c)

45
44
43
42
41
40
39
38

2600

2800

3000
3200
3400
days from 1996/Apr./1

SK-III

±1%

0

2.6
2.55
2.5
2.45
2.4
2.35
2.3
2.25
2.2
2400

250

500

750 1000 1250 1500 1750 2000
days from 1996/Apr./1

SK-II

2600

2800

2.6
2.55
2.5
2.45
2.4
2.35
2.3
2.25
2.2
3800 3900 4000 4100 4200 4300 4400 4500
days from 1996/Apr./1µ momentum/track (MeV/c/cm)

e momentum (MeV/c)

SK-IV
42

40

38

36
4500

5000

5500days from 1996/Apr./1

3000
3200
3400
days from 1996/Apr./1

SK-IIIµ momentum/track(MeV/c/cm)

e momentum(MeV/c)

45
44
43
42
41
40
39
38
3800 3900 4000 4100 4200 4300 4400 4500
days from 1996/Apr./1

SK-I

SK-IV

2.6

2.5

2.4

2.3

2.2
4500

5000

5500days from 1996/Apr./1

Figure 4.18: Time variation of the reconstructed momentum of decay electrons (left) and theaveraged momentum/range of stopping muons (right) as a function of elapsed days from 1 April
1996 for each SK period. Dotted (dashed) lines show ±1% (±2%) from mean value in each SK
period.

Chapter 4

4.2.7

68

Detector Uniformity of Energy Scale

Uniformity of the detector is measured with decay electrons from cosmic ray muons. Decay
electron vertexes distribute uniformly in the fiducial volume and the momentum distribution isalmost uniform, making them a good calibration source to check the detector uniformity. In
order to take into account the muon polarization, only electrons whose direction is perpendicular
to the parent muon direction is used. This condition is -0.25 < cos ⇥e$µ < 0.25, where cos⇥e$µis the opening angle between the decay electron and the muon directions. Figure 4.19 shows theratio of averaged momentum of decay electrons between MC and data. These figures show thatthe detector gains are uniform within ±0.6%, ±0.6%, ±1.3% and ±1.3% for SK-I, SK-II, SK-III
and SK-IV respectively.

electron momentum

69

1.05
1.04
1.03
1.02
1.01
1
0.99
0.98
0.97
0.96
0.95

electron momentum

Chapter 4

1.05
1.04
1.03
1.02
1.01
1
0.99
0.98
0.97
0.96
0.95

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
cosΘ

-1

-0.8 -0.6 -0.4 -0.20

0.2 0.4

0.6

0.8

1

cos Θ

Figure 4.19: Uniformity of the detector gain as a function of zenith angle for each SK period.
From top to bottom: SK-I, SK-II, SK-III and SK-IV. Y-axis is the ratio of MC to data of theaveraged momentum of decay-electrons.

Chapter 5

Data Selection
About 106 high energy and low energy trigger events are collected per day in the SK detector.
Most events are background events such as cosmic rays and gamma rays from radioactivity. An
efficient data selection is essential to select neutrino events from such a large amount of data.
Atmospheric neutrino events observed in SK are categorized to 3 types:• Fully contained (FC)• Partially contained (PC)• Upward-going muon (UPMU)For FC and PC event types, the neutrino interacts in the fiducial volume of the detector(at least 2 m away from the ID wall). If all the energy of the generated (charged) particles
is deposited inside the ID, the event is classified as FC. If high energy muon exits the ID anddeposits its energy in the OD, the event is classified as PC.
UPMU events are high energy muons produced by neutrino interactions with the rock surrounding the detector. As these events cannot be distinguished from cosmic rays which travelin downward direction, only muons traveling in the upward direction through the detector are
selected. UPMU events are divided into two types: for events which come to rest in the detector,
they are categorized as upward stopping muons, and those which traverse through the entire
detector volume, they are categorized as upward through-going muons.

Fully contained

Partially contained

Upward stoppingmuon

Upward through-going
muon

Figure 5.1: Schematic view of observed atmospheric neutrino in SK.

70

Chapter 5

71

Figure 5.1 shows the schematic view of the above three event types. The neutrino meanenergies are ⇠1 GeV for FC, ⇠10 GeV for PC and stopping UPMU, ⇠100 GeV through-going
UPMU, as shown in Figure 5.2.
1000
FC νe
FC νµ
PC

800

Events / 1000 days

600
400
200
0
160
140
120
100
80
60
40
20
0 -1
10

Upward stopping µ
Upward through-going µ

1

10
10
Eν (GeV)

2

10

3

10

4

Figure 5.2: Expected neutrino spectra of di↵erent event categories.
The event selection process consists of reduction and reconstruction processes.
Reduction is to select neutrino events by removing background events in the collected data.
There are several background observed in SK:• Cosmic ray through-going muons which pass through the detector• Cosmic ray stopping muons which enter and stop inside the detector• Cosmic ray corner clipping muons which pass through the corner of the detector• Flasher event which is caused by PMT emitting light through internal discharge• Low energy events such as gamma ray from radioactivities
The above background events can be removed during the reduction processes.

5.1

Reduction for Fully Contained Events

FC events are distinguished from PC events by using the number of OD hit PMTs in thehighest charge cluster. The data reduction process consists of five steps for FC.
As the number of ID PMTs in SK-II period was only about half of that in other SK periods,
the SK-II selection criteria related to the number of hits and observed charge of ID PMTs isdi↵erent from that in other SK periods.

Chapter 5

5.1.1

72

First Reduction

An event passes the first reduction if it fulfils the following criteria:
1. PEtotal > 200 p.e.s (100 p.e.s for SK-II)
where PEtotal is the number of total p.e.s observed by the ID PMTs
2. NHITA800  50 or OD trigger is o↵where NHITA800 is the number of hit OD PMTs in a fixed 800 nsec time window from
500 nsec to 1300 nsec before and after the trigger timing.
Criterion (1) rejects low energy background events due to radioactivities. 200 p.e.s (100 p.e.s
for SK-II) corresponds to 22 MeV/c of electron momentum. As events which have visible energy
below 30 MeV are not used in the analysis, this cut can be applied without loss of neutrino
events. Criterion (2) removes cosmic ray muon events.
FC 1st reduction reduces the number of events from 106 events/day to ⇠3000 events/day.

5.1.2

Second Reduction

Selection criteria for the second reduction are:
1. NHITA800  25 (30 for SK-IV) or PEtotal > 100,000 p.e.s (50,000 p.e.s for SK-II) or OD
trigger is o↵
2. PEmax /PE300 < 0.5
where PEmax is the maximum number of p.e.s observed by an ID PMT. PE300 is the
maximum number of total p.e.s observed by ID PMTs in a sliding 300 nsec time window.
Note that the cut criteria for NHITA800 is changed in SK-IV, as the electronics was changed
to QBEE and the high voltage gain has changed. The cut criteria is tuned so that it agrees with
SK-III.
Criterion (1) rejects cosmic ray muons by a tighter threshold than the first reduction. Figure
5.3 shows the NHITA800 distributions for data, FC Monte Carlo simulation events whose vertices
are 2 m away from the ID wall, and the FC final sample events.
Criterion (2) rejects low energy events and electrical noise events, which have one larger hit
signal from one single PMT. Figure 5.4 shows the PEmax /PE300 distributions.
Event rate becomes ⇠200 events/day after the FC 2nd reduction.

Chapter 5

73

10

5

(a) data

rejected

104

10

3

Number of events

0

10

20

30

40

rejected
1050(b) MC

3

102

10

10

0

10

20

30

3

40

50

(c) final sample
MC

102

data
10
1
10-1
0

10

20

30

40

50

NHITA800

Figure 5.3: FC 2nd reduction criterion (1). NHITA800 distributions for SK-IV. (a) shows theobserved data after the FC 1st reduction. (b) shows FC Monte Carlo events after FC 1st
reduction. (c) shows the final samples for both data and Monte Carlo events, where the numberof Monte Carlo events is normalized to that of the data. The arrow at NHITA800 = 30 is the
cut in the FC 2nd reduction for SK-IV. In Figure (c), there are some events remaining at region
NHITA800 > 30, they are events with either PEtotal > 100,000 p.e.s or OD trigger o↵ or both.

Chapter 5

74

10

6

10

5

rejected(a) data

104

10

3

102
0

0.2

0.4

0.6

0.8

1

FC2 PEmax/PE300

Mumber of events

104

rejected
103(b) MC

102

10

10

0.2

0.4

0.6

0.8

1

FC2 PEmax/PE300
104
10

3

10

2

(c) final sample
MC
data

10
1
10-1
0

0.2

0.4

0.6

0.8

1

FC2 PEmax/PE300

Figure 5.4: FC 2nd reduction criterion (2). PEmax /PE300 distributions for SK-IV. (a) shows thedata after the FC 1st reduction. (b) shows FC Monte Carlo events after the FC 1st reduction.
(c) shows the final samples for the data and Monte Carlo events, where the number of Monte
Carlo events is normalized to that of the data. Selection criteria is shown by arrows in the plots.

Chapter 5

5.1.3

75

Third Reduction

Remaining background events, which consists of mostly noise events and cosmic ray muons
which have a small number of OD hits, are rejected by the following cuts.
Through-going Muon Cut
Through-going muons are very energetic and deposit a lot of charge in the ID. In order
to eliminate these events, a through-going muon fitter is applied when there are more than
1000 PMTs detecting more than 230 photoelectrons (p.e.s) (for each of them). This fitter finds
the through-going muon entering point at the ID by using the earliest hit PMT with someneighbouring hit PMTs. The exiting point at which the through-going muon goes out of the ID
is defined as the centre of the saturated ID PMTs. The goodness of through-going muon fit isdefined as:

goodness = P

1
i

1 ⇥
2
i

X 1

2 exp

i

i

(ti Ti )22(1.5 ⇥ i )2

(5.1)

where ti and i are the observed hit time of the i-th PMT and its resolution, Ti is the hit time
estimated from the muon entering time and its track.
The cut criteria is:
1. goodness of through-going muon fit > 0.75
2. NHITAin 10 or NHITAout 10
where NHITAin (NHITAout ) is the number of OD hit PMTs located within 8 m from the entrance(exit) point in a fixed 800 nsec time window.
Events that satisfy the above criteria are rejected. Figure 5.5 shows the scatter plots of
NHITAin and NHITAout satisfying criterion (1) for data and Monte Carlo after the 2nd reduction
and the final samples.
Stopping Muon Cut
For the rejection of stopping muons, a stopping muon fitter is applied to find the entrance
point with similar way as the through-going muon fit. The goodness is defined by equation (5.1).
Events fulfilling the following criterion are rejected as stopping muons:
1. NHITAin 10 if goodness of stopping muon fit 0
where NHITAin is the number of OD hit PMTs located within 8 m from the entrance
point in a fixed 800 nsec time window.
or NHITAin

5 if goodness of stopping muon fit > 0.5 (additional criterion for SK-I)

The direction of muon is reconstructed so that the total number of p.e.s inside the cone
within half opening angle 42 is maximized. Figure 5.6 shows NHITAin distributions for thedata and Monte Carlo after the 2nd reduction and their final samples.

Chapter 5

76

30

30

(b) MC20

NHITAout

NHITAout(a) Data

rejected
10

0
0

10

20

20

rejected
10

0
0

30

FC3 NHITAin
30

20

30

30

(c) final sample (data)(c) final sample (MC)20

NHITAout

NHITAout

10

FC3 NHITAin

10

0
0

10

20

FC3 NHITAin

30

20

10

0
0

10

20

30

FC3 NHITAin

Figure 5.5: FC 3rd reduction criterion (1) for through-going muons cut. Scatter plot of thenumber of hit OD PMTs within 8 m of the entering point NHITAin (x-axis) and the exiting
point NHITAout (y-axis) for SK-IV. (a) shows the data after the FC 2nd reduction. (b) shows
FC Monte Carlo events after the FC 2nd reduction. (c) shows the final samples for data (bottom
left) and Monte Carlo (bottom right) events. Selection criteria are indicted by solid lines.
Cable Hole Muons
On top of the SK detector tank, there are twelve cable holes for the signal and high voltage
supply cables to pass through. There is no OD PMT installed at the positions of these holes
as shown in Figure 5.7. When cosmic ray muons enter through these holes, they enter the
ID region without hitting the OD PMTs, and these events may be misidentified as contained
neutrino events because there is no OD signal.
In order to avoid such mis-identification, veto counters ( 2 m ⇥ 2.5 m plastic scintillation
counters) were installed in April, 1997 for four out of the twelve cable holes. A schematic viewof the cable hole and the veto counter is shown in figure 5.8. The rejection criteria with the vetocounters are:
1. One veto counter hit

Chapter 577

2. Lveto < 4 mwhere Lveto is the distance from the cable hole to the reconstructed vertex.
Beside the above veto counter cut, a new cut is introduced in SK-IV reduction. For cableholes which do not have veto counters, the following cut is added:
1. Goodness of stopping muon fit

0.4

2. PEtotal > 4000
where PEtotal is the total number of p.e.s observed by the ID PMTs
3. travel in downward direction (cos✓ < -0.6)
4. Lveto < 2.5 m
where Lveto is the distance from the cable hole to the reconstructed vertex.
The inefficiency (of wrongly rejecting neutrino events) is less than 0.01%.

Chapter 578

rejected

10410(a) data

3

102
10
10

10

20

30

40

50

40

50

FC3 NHITAent

Number of events

104

rejected
103(b) MC

102

10

10

10

20

30

FC3 NHITAent
104(c) final sample
10

3

MC
data

102

10
0

10

20

30

40

50

FC3 NHITAent

Figure 5.6: FC 3rd reduction criterion for stopping muon cut. Number of hit OD PMTs near theentrance points of muons (NHITAin ) for SK-IV. (a) shows the data after the FC 2nd reduction.
(b) shows FC Monte Carlo events after the FC 2nd reduction. (c) shows final samples for bothdata and Monte Carlo. The selection criteria are shown by arrows.

Chapter 5

79

Figure 5.7: Positions of the twelve cable holes. The 4 red circles indicate cable holes installedwith veto counters.

Figure 5.8: Schematic view of a cable hole muon and a veto counter.

Chapter 5

80

Flasher Event Cut
Flasher events usually have a broader hit timing distribution when compared with that ofneutrino events. The following cut criteria is applied by using this feature:
For SK-I:
NMIN100 15
or NMIN100 10 if the number of hit ID PMTs  800
For SK-II to SK-IV:
NMIN100 20
where NMIN100 is the minimum number of hit ID PMTs in a sliding 100 nsec time window
from +300 nsec to +800 nsec after the trigger.

Figure 5.9: FC 3rd reduction criterion for flasher event cut. The timing distributions of (i) atypical flasher event and (ii) a typical FC neutrino event. The arrows show the time window forcounting NMIN100 .
Figure 5.9 shows the timing distributions of (i) a typical flasher event and (ii) a typical FC
neutrino event. Figure 5.10 shows NMIN100 distributions for data and FC Monte Carlo after
the 2nd reduction and their final samples.
Accidental Coincidence Events Cut
Accidental coincidence occurs when a low every event form a trigger and a cosmic ray muon
events follows in the same trigger gate. These events are not rejected in the former reduction
steps because there is no OD activities on the trigger timing and there is a large number of total
p.e.s in the ID caused by muons. Accidental coincidence event are removed by the following
cuts:Chapter 5

81

1. NHITAof f
20
where NHITAof f is the number of hit OD PMTs in a fixed 500 nsec o↵-timing window
from +400 nsec to +900 nsec after the trigger timing.
2. PEof f > 5000 p.e.s (2500 p.e.s for SK-II)
where PEof f is the number of p.e.s observed by ID PMTs in a fixed 500 nsec o↵-timing
window from +400 nsec to +900 nsec.
Low Energy Events Cut
This is the rejection of the remaining low energy events such as the decay of radioisotopes
and electrical noise. Events satisfying the following criteria are removed:
1. NHIT50 < 50 (25 for SK-II)
where NHIT50 is the number of hit ID PMTs in a sliding 50 nsec time window.
NHIT50 is counted after subtracting the time of flight (TOF) of each observed photon
assuming all photons are generated at the same point. The vertex is defined as the position
at which the timing residual distribution peaks. NHIT50 = 50 corresponds to visible energyof 9 MeV, and this cut is low enough to keep efficiency for contained neutrino events which
is Evis > 30 MeV.
After the FC third reduction, the event rate is ⇠45 events/day.

Chapter 582

rejected(a) data

104

10

3

102

10

Number of events

0

10

20

30

rejected
10

40

50

(b) MC

3

102
10
1
10-1
0

10

10

20

30

40

50

(c) final sample

3

MC

102

data

10

1
0

10

20

30

40

50

NMIN100

Figure 5.10: FC 3rd reduction criterion for flasher event cut. NMIN100 distributions for SK-IV.
(a) shows the data after the FC 2nd reduction. (b) shows FC Monte Carlo events after the FC
2nd reduction. (c) shows the final samples for data and the Monte Carlo events, in which thenumber of Monte Carlo events is normalized to that of the data. Selection criteria are shown byarrows.

Chapter 5

5.1.4

83

Fourth Reduction

In the FC fourth reduction, a pattern matching algorithm is used to remove the remaining
flasher events. Flasher events usually repeat with similar hit patterns in the detector, which are
not likely to be caused by neutrinos.
Algorithm of pattern matching:
1. Divide the ID wall into 1450 patches for 2m ⇥ 2m squares.
2. Computer the correlation factor by comparing the total charge in each patch of two events(A and B). The correlation is defined as follows:r=

1 X (QAi
N i

hQA i) ⇥ (QBi
A⇥ B

hQB i)(5.2)

where N is the number of patches , hQA(B) i and A(B) are the average charge and its
standard deviation for event A and event B respectively.
3. Calculate the distance (DISTmax ) between the PMTs which have maximum pulse heights
in the two events respectively.
4. If DISTmax < 75 cm, an o↵set value is added to r by r = r + 0.15.
5. If r exceeds the threshold (rth )m events A and B are determined to be matched events,
where rth is defined as:
AB
r>rth = 0.168 ⇥ log10 ((P Etot
+ P Etot)/2.) + 0.130(5.3)

where PEtot is the total number of p.e.s observed in the ID.
6. Repeat the above calculation over 10,000 events around the target event and count the
number of matched events.
7. Remove events with large correlation factor r with several events.
The event rate after FC fourth reduction is ⇠18 events/day.

5.1.5

Fifth Reduction

The final rejection cut is applied by several criteria specialized for each background source.
Invisible Muon Cut
If a cosmic ray muon has momentum less than the Cherenkov threshold and there is subsequent decay electron observed, the event may be misidentified as a neutrino event. This kindof events is characterized by a low energy signal from a decay electron in the ID and a signal
in the OD before the trigger timing. Events fulfilling the following cut are rejected as invisible
muons:

Chapter 5

84

1. PEtot <1000 p.e.s (500 p.e.s for SK-II)
where PEtot is the total number of p.e.s observed in the ID.
2. NHITACearly + NHITAC500 10 if DISTclust < 500 cm
NHITACearly >9 otherwise
where NHITACearly is the maximum number of hit PMTs in the OD hit cluster in a sliding
200 nsec time window from 8800 nsec to 100 nsec,
NHITAC500 is the number of hit PMTs in the OD hit cluster in a fixed 500 nsec time
window from 100 nsec to +400 nsec,
DISTclust is the distance between two OD hit clusters which are used for NHITACearlyand NHITAC500

104

rejected
10(a) data

3

102

10

1
0

10

20

30

40

Number of events

FC5 NHITACearly

103rejected(b) MC

102
10
1
10-1
0

10

20

30

40

FC5 NHITACearly103(c) final sample
102

MC
10

data
1
10-1
0

10

20

30

40

FC5 NHITACearly

Figure 5.11: FC 5th reduction invisible muon cut. NHITACearly distribution satisfying thecriteria PEtot < 1000 p.e.s and DISTclust > 500 cm for SK-IV. (a) shows the data after FC 4th
reduction. (b) shows the FC Monte Carlo events after FC 4th reduction. (c) shows the final
samples for data and Monte Carlo events whose number is normalized to that of the data. The
Selection criteria is shown by the arrows.

Chapter 585

Figure 5.11 shows the NHITACearly distribution, satisfying criterion (1) and DISTclust >500 cm, for the data and Monte Carlo after the FC 4th reduction and the final samples.
Coincidence Muon Cut
The remaining accident coincidence muon events which are not rejected in the third reduction
accident coincidence events cut are removed by the following cuts:
1. PE500 < 300 p.e.s (150 p.e.s for SK-II)
where PE500 is the total number of p.e.s observed in the ID in a fixed 500 nsec time window
from 100 nsec to +400 nsec.
2. PElate 20 p.e.s
where PElate is the maximum number of hit OD PMTs in a 200 nsec sliding time window
from +400 nsec to +1600 nsec.
Figure 5.12 shows the PElate distribution satisfying criterion (2) for data and Monte Carloafter the 4th reduction and their final samples.
Long-tail Flasher Cut
This flasher cut is stricter than the one in FC 3rd reduction.
Events satisfying the following criteria are removed as flasher events:
1. NMIN100 6 if goodness of Point-fit < 0.4
where NMIN100 is the minimum number of hit ID PMTs in a sliding 100 nsec time window
from +300 nsec to +800 nsec.
From SK-II onwards, additional cuts are added:
2. NMIN100  5 if the goodness of Point-fit < 0.3(Point-fit would be explained in Section 5.4.1)
After FC fifth reduction, FC event rate becomes ⇠16 events/day.

Chapter 5

86

104

10

(a) data

rejected

3

102

10

1
0

20

40

60

80

100

FC5 PElate

Number of events

103

rejected(b) MC

102
10

1
10-1
0

20

40

60

80

100

FC5 PElate

10

3

10

2

(c) final sample

MC

10

data

1
10-1
0

20

40

60

80

100

FC5 PElate

Figure 5.12: FC 5th reduction coincidence muon cut. PElate distributions for SK-IV. (a) shows
the data after the FC 4th reduction. (b) shows the Monte Carlo events after the FC 4th
reduction. (c) shows the final sample for both data and the Monte Carlo events whose numberof events is normalized to that of the data. The selection criteria is shown by the arrows. In (c),there are some events with PElate > 20, as they are events with PE500 > 300 p.e.s and hence
are not removed by criteria (1).

Chapter 5

5.1.6

87

FC Reduction Summary

Finally FC neutrino events are selected by:
1. Vertex of neutrino interaction is inside the fiducial volume (2 m away from the ID wall).
2. Number of hit PMTs in the highest charge OD cluster (NHITAC) is less than 16 (10 forSK-I).
3. Visible energy (Evis ) > 30 MeV.
Figure 5.13 shows the NHITAC distributions for FC and PC final sample for SK-I to SK-IV
periods. Detection efficiencies in each reduction step are estimated by atmospheric neutrino MC
as shown in table 5.1. The number of events in the final sample and the event rate are also
summarized in table 5.1. Event rates are consistent with one another for all SK periods.
Uncertainty of FC reduction efficiency is estimated by comparing the distributions of eachcut variable of data and Monte Carlo. Uncertainties for SK-I, SK-II, SK-III and SK-IV are
estimated to be 0.2%, 0.2%, 0.8% and 0.3% respectively.
FC
reduction step
1st
2nd
3rd
4th
5th
5th (FV)
Monte Carlo
Data
Data/MC
event rate/day

SK-I

SK-II
SK-III
selection efficiency (%)
100.0
99.97
100.0
100.0
99.92
99.98
99.93
99.78
99.81
99.29
99.38
99.30
99.26
99.30
99.24
99.25
99.95
99.62
number of events
14375.77772.75085.8
12299
6610
4355
0.86 ±0.01 0.85±0.01 0.86±0.02
8.26±0.07 8.28±0.10 8.41±0.13

SK-IV
100.0
99.99
99.82
99.00
98.95
99.19
10669.58929
0.84±0.01
8.14±0.09

Table 5.1: Selection efficiencies and number of events for FC sample are summarized for each
SK period. Selection efficiencies are for events whose real vertex is in the fiducial volume for
reduction steps 1 to 5, while the selection efficiency for 5th (FV) is for events whose fitted vertex
is in the fiducial volume, and number of OD hits < 16 (10 for SK-I) and visible energy > 30
MeV. Number of events for MC are normalized without oscillation to real data livetime, 1489.2,
798.6, 518.1 and 1097.0 days for SK-I, SK-II, SK-III and SK-IV respectively.
The background events contamination in the final FC samples are mainly cosmic ray muonsand flasher events. Flasher events is a background source for e-like sample, while cosmic raymuons is a background source for µ-like sample. These background events are largely rejectedby the fiducial volume cut (reconstructed vertex position > 2 m from the ID wall). Background
events are checked by eye-scan with a visual event display. Flasher events is a background sourcefor e-like sample and cosmic ray muons is a background source for µ-like sample, contaminations
by background for each sample are summarized in table 5.2.

Chapter 5

88

SK-I
cosmic ray flasher
0.2%
0.1%
0.3%
0.5%

BG
sub-GeVmulti-GeV

SK-II
cosmic ray flasher
0.2%
0.4%
0.2%
0.2%

SK-III
cosmic ray flasher
0.3%
0.1%
0.3%
0.2%

SK-IV
cosmic ray flasher
0.1%
0.1%
0.1%
0.1%

Table 5.2: Estimated contamination by each background source. Sub-GeV means events with
Evis < 1.3 GeV and Multi-GeV means events with Evis > 1.3 GeV.

105

105
FC

PC

104

104

103

103

102

102

10

10

1
0

20

40

60

80

105

FC

1
0

20

FC

60

80

103

102

102

10

10

20

40

60

80

PC

104

PC

103

1
0

40

105
FC

104

PC

40

60

80

1
0

20

Figure 5.13: NHITAC distributions for FC and PC final examples for SK-I (top left), SK-II(top right), SK-III (bottom left) and SK-IV (bottom right). Points show the observed data anderror bars, solid lines show the Monte Carlo events assuming no oscillation (dashed lines) and⌫⌫ $ ⌫⌧ 2-flavor oscillation with (sin2 2✓, m2 ) = (1.00, 2.5 ⇥ 10 3 eV2 ) (solid lines).

Chapter 5

5.2

89

Reduction for Partially Contained Events

PC events have OD activities which FC events do not have, hence PC reduction process isdi↵erent from that of the FC. As OD segmentation was installed in SK-III period, the reduction process is substantially modified to adjust for SK-III onwards. There are 5 steps for PC
reduction.

5.2.1

First Reduction

For SK-I and SK-II, the 1st reduction rejects the through-going cosmic ray muons and low
energy eventss. Events satisfying the following criteria is selected:
For SK-I:
1. PEtot 1000 p.e.s
where PEtot is the number of p.e.s observed in the ID.
2. TWIDA  260 nsec
where TWIDA is the width of the hit timing distribution in the OD PMTs.
3. NCLSTA  1
where NCLSTA is the number of the hit clusters in the OD.
For SK-II:
1. PEtot

500 p.e.s

2. TWIDA  170 nec.
In criterion (1), PEtot 1000 p.e.s (500 p.e.s for SK-II) corresponds to 310 MeV/c for muons.
As exiting particles in PC samples (mostly muons) must have at least 2 m track length in the
ID which corresponds to momentum loss of 500 MeV/c for muons, this criterion rejects events
which do not have 2 m track length in the ID. Criterion (2) rejects through-going muons whichhave broad hit timing distribution and two hit clusters around the entrance and exit point in
the OD. As the reflected photons in the OD increased and the quantum efficiency of OD PMTs
was increased in SK-II, TWIDA cut criterion is tuned for SK-II.
For SK-III onwards, reduction of through-going muons is more efficient with the OD segmentation. Events satisfying the following criteria would be selected:
For SK-III onwards:
1. PEtot

1000 p.e.s

2. NHITAtop < 10 or NHITAbottom < 10where NHITAtop (NHITAbottom ) is the number of OD hits at the top (bottom) part of OD
3. NHITAendcap < 25 or NHITAside < 70
where NHITAendcap (NHITAside ) is the number of OD hits in the top and bottom (side)
region of the OD.

Chapter 590

4. ODRmean < 2100 cm if OD hits < 20 in 500 nsec time window
where ODRmean is the average distance between all hit pairs which is defined as follows:
ODRmean =1

N
X1

Npair i=1|!
xi!
xj | .

(5.4)

Criterion (2) rejects through-going muons which pass through both top and bottom parts ofthe detector. Figure 5.14 upper plots show the scatter plots of NHITAtop and NHITAbottom of
observed data (upper left) and PC Monte Carlo (upper right) events, while bottom plots show
the distributions for each of the variables for the final sample.

200

300

180

104

3

10

250

140
120

102

100
80
60

10

OD bottom cap hits

OD bottom cap hits

160

103

200
150

102

100
10

40

50

20
0
0

20 40 60 80 100 120 140 160 180 200

10
0

OD top cap hits

102

10

10

1

1

20 40 60 80 100 120 140 160 180 200
Number of OD hits in Bottom Cap

100

150

200

250

300

1

OD top cap hits

102

0

50

0

20 40 60 80 100 120 140 160 180 200
Number of OD hits in Top Cap

Figure 5.14: PC 1st reduction criterion (2). Top left and right plots are the scatter plots of
NHITAtop (x-axis) and NHITAbottom (y-axis). Top left figure shows the raw data, and top rightfigure shows the PC Monte Carlo events with vertex in the fiducial volume. Bottom left andright plots show 1-dimentional plot of NHITAtop and NHITAbottom distributions respectively forthe final sample for both data (black dots) and Monte Carlo (red lines). All plots are for SK-IV.
Criterion (3) rejects corner clipping muons. Figure 5.15 shows the scatter plots of NHITAendcap
and NHITAside .
In criterion (4), ODRmean value for through-going muons is expected to be larger than thatof PC events. Figure 5.16 shows the scatter plots of ODRmean and the number of OD hits forboth data and Monte Carlo.
PC 1st reduction reduces number of events to ⇠2 ⇥ 104 events/day.

Chapter 5

91

250

250

500

200

400

150

300

100

200

50

100

35
30
25

150

20
100

15

OD end cap hits

OD end cap hits

200

10
50
5
0
0

50

100

150

200

250

300

0

0
0

50

100

150

200

250

300

0

OD barrel cap hits

OD barrel hits

102
102

10
10

1

1
0

20 40 60 80 100 120 140 160 180 200
Number of OD hits in Barrel

0

20 40 60 80 100 120 140 160 180 200
Number of OD hits in End Cap

Figure 5.15: PC 1st reduction criterion (3). The upper plots are the scatter plots of NHITAendcap
and NHITAbarrel . The upper left plot shows the raw data and the upper right plot shows the
PC Monte Carlo events with vertex in the fiducial volume. Bottom left and right plots show
1-dimentional plot of NHITAendcap and NHITAbarrel for final samples for both data and Monte
Carlo respectively. All the plots are for SK-IV.

Chapter 5

92

300

140

40
35

200

30
25

150

20
100

15
10

50

250
Number of good OD hits

250
Number of good OD hits

300

45

120
200

100
80

150

60

100

40
50

20

5
0
0

500

1000

1500

2000

2500

ODRMEAN: ave. hit-hit distance in OD (cm)

0

0
0

500

1000

1500

2000

2500

0

ODRMEAN: ave. hit-hit distance in OD (cm)

102

10

1

0

500

1000 1500 2000 2500 3000

Average OD hit-hit distance

Figure 5.16: PC 1st reduction criterion (4). Top plots are the scatter plots of number of goodOD hits and ODRmean . Top left plot shows the raw data and top right plot shows the PC Monte
Carlo events with vertex in the fiducial volume. The bottom plot shows the 1-dimentional plotsof ODRmean of both data (black dots) and Monte Carlo (red line). All the plots are for SK-IV.

Chapter 5

5.2.2

93

Second Reduction

In PC 2nd reduction, a clustering algorithm of OD hits is used to reject through-going
muons and stopping muons. OD (ID) walls are divided into 11 ⇥ 11 (21 ⇥ 21) patches and thecharge observed in each patch is counted. Clusters are formed by looking for charge gradient toneighbouring patches. The algorithm is illustrated in Figure 5.17.

more charge

Cluster 1

less charge

Cluster 2

Cluster 3

Figure 5.17: Schematic view of the algorithm to find hit clusters in PC 2nd reduction. Charged
observed in each patch are represented by circles, whose sizes are proportional to the numberof p.e.s. Arrows represent the vector charge gradient, pointing to the highest charge among theneighbouring patches.
Events satisfying the following criteria are selected:
For both SK-I and SK-II:
1. NCLSTA2  1
where NCLSTA2 is the number of OD hit clusters with more than 6 hit PMTs.
For SK-I:
2. NCLSTA2(2)  1
where NCLSTA2(2) is the number of the 2nd OD hit clusters with more than 6 hit PMTs.
3. NHITACmin < 7
where NHITACmin is the minimum number of hit PMTs among top (or bottom) and side
regions in the OD hit cluster.
4. PE200 > 1000 p.e.s if NCLSTA2 = 1
where PE200 is the number of observed p.e.s within 200 cm from the highest charge PMT
in the ID hit cluster closest to the OD hit cluster.
For SK-II:
2. NCLSTA2(2)  1
where NCLSTA2(2) is the number of the 2nd OD hit clusters with more than 10 hit PMTs.

Chapter 5

94

3. NHITAendcap < 20 or NHITAendcap < MAX(NHITAside )where MAX(NHITAside ) is defined as follows:
MAX(NHITAside ) = exp(5.8 - 0.023 ⇥ NHITAside ) if NHITAside < 75
MAX(NHITAside ) = exp(4.675 - 0.008 ⇥ NHITAside ) if NHITAside 75
where NHITAendcap is the number of OD hit PMTs in the top and bottom region of the
OD
and NHITAside is the number of hit OD PMTs in the side region of the OD.
4. NHITAC2 < 12 + 0.085 ⇥ PE200
where NHITAC2 is the number of OD hit PMTs in the 2nd cluster.
Criteria (3) rejects corner clipping muons, which leave hit PMTs in both top (or bottom)
and side region of the detector.
Criteria (4) rejects stopping muons by using ID hit cluster located behind the OD hit cluster.
(a) DATA

number of events

rejected
10

6

10

5

10

4 (b) MC

10

3

10

2

rejected

101(c) final sample
10

2

data
MC

10
1
0

5

10

15

20

25

30

PC2 NCLSTA2

Figure 5.18: PC 2nd reduction criterion (2) for SK-III. Distributions of number of OD hit PMTs
in the 2nd OD hit cluster. (a) shows the data after PC 1st reduction. (b) shows PC Monte
Carlo after PC 1st reduction. (c) shows the final sample for both data and Monte Carlo. The
selection criteria is shown by arrows.
For SK-III onwards, as 1st reduction with the OD segmentation is very efficient, the 2nd
reduction only consists of criteria (2) and (3) of SK-II. Figure 5.18 shows the number of OD
hit PMTs in the 2nd OD hit cluster for both data and Monte Carlo after PC 1st reduction andtheir final samples. Criterion (3) is a more strict cut than the criterion (3) in PC 1st reduction.
Figure5.19 shows the scatter plots of NHITAendcap and NHITAside for both data and PC Monte
Carlo after PC 1st reduction and their final samples.
Event rate after PC 2nd reducction is ⇠8000 events/day.

Chapter 595(a) DATA

250

300

NHITA(endcap)

NHITA(endcap)

300

rejected

200
150
100
50
0

0

100

200

250

rejected

200
150
100
50
0

300

(b) MC0NHITA(side)(c) final sample (DATA)

250
200
150
100
50
0

300

NHITA(endcap)

NHITA(endcap)

300

0

100

200

NHITA(side)

100

200

300

NHITA(side)

300(d) final sample (MC)

250
200
150
100
50
0

0

100

200

300

NHITA(side)

Figure 5.19: PC 2nd reduction criterion (3) for SK-III. Scatter plots of NHITAendcap and
NHITAside . (a) shows the data after PC 1st reduction, (b) shows PC Monte Carlo after the 1st
reduction. (c) and (d) shows the final samples for data and Monte Carlo respectively. The cut
criteria are shown by solid lines.

5.2.3

Third Reduction

PC 3rd reduction rejects flasher events in the same way as that in the FC 3rd reduction, in
which the broad timing distribution of flasher events is used for the cut. Events which satisfy
the following criteria are rejected as flasher events:
1. NMIN100 >14
or100 > 9 if number of hit ID PMTs < 800
where NMIN100 is the minimum number of hit ID PMTs in a sliding 100 nsec time window
from +300 nsec to +800 nsec.
PC 3rd reduction also rejects stopping muons by using the number of hit OD PMTs nearentrance point. In order to find the entrance point, vertex and direction are fitted by Point-fit.
The entrance position is estimated by backward extrapolation from the reconstructed vertex.
Events satisfying the following criteria are rejected as stopping muons for SK-I and SK-II:
2. NHITAin > 10
where NHITAin is the number of hit OD PMTs located within 8 m from the entrance
point in a fixed 500 nsec time window.
For SK-III onwards, this cut is removed at this stage and this cut variable is used in the 5th
reduction.
Event rate after PC 3rd reduction is ⇠100 events/day.

Chapter 5

5.2.4

96

Fourth Reduction

For SK-I and SK-II
PC 4th reduction rejects cosmic ray muons which have relatively less OD activities. Twotypes of event reconstruction tools are used here: Point-fit and through-going muon fit. Throughgoing muon fit defines the entrance point as the position of the earliest hit cluster in the ID.
For PC events, the entrance point cannot not determined correctly by through-going muon fit
because they are generated inside the ID, the goodness tends to be worse when compared withthrough-going muon events. On the other hand, Point-fit can determine the vertex position andthe direction reasonably well for both PC and cosmic ray muons. Events fulfilling the following
criteria are selected:!
!
1. d pf it • d P M T > -0.8!
!
where d pf it is the reconstructed direction by Point-fit, d P M T is the reconstructed vertex
to the earliest saturated PMT.
and2. DCORN > 150 cm
where DCORN is the distance from the reconstructed vertex by Point-fit to the nearest
fringe of the ID.
3. TLMU > 30 m if goodness of through-going muon fit > 0.85
where TLMU is the track length of muons from the entrance point to the exit point
estimated with through-going muon fit.
Criterion (1) rejects the cosmic ray stopping muons which have entrance points in the opposite direction to the reconstructed direction by Point-fit. Criterion (2) rejects corner clippingmuons. Criterion (3) rejects through-going muons which have long track lengths.
For SK-III onwards
Event reconstruction tool is used, which classifies each event as stopping muons, throughgoing muons, multiple muons or corner clipping muons. 97%s background events are categorized
as stopping or through-giong muons, 96% PC events are categorized as other muon types. 5
selections based on the results of the fitter are:
1. anglemuon < 90
where anglemuon is the angle between the fitted direction and the vector between the
Point-fit vertex and the center of the highest charge OD cluster.
2. dotprodmuon > -0.8
where dotprodmuon the angle between the fitted direction and the vector between the
Point-fit vertex and the earliest saturated ID PMT.
3. lengthmuon < 1750 cm
where lengthmuon is the length of the fitted muon track. Cosmic ray muons tend to have
longer track lengths.
4. goodnessmuon < 0.52
where goodnessmuon is the goodness from muon fitter.

Chapter 597

5. cornermuon 300 cm
where cornermuon is the distance between fitted entrance point and the corner of the track.
If an event is classified as a through-going muon, it has to pass 4 of the above 5 criteria inorder to be selected. If an event is classified as a stopping muon, it has to pass 4 of the above
5 criteria including dotprodmuon pass in order to be selected, then it has to pass the criteria
goodnessmuon < 0.5 or ehitmuon < where ehitmuon is the number of OD hits located within 8 mfrom the fitted entrance point in a fixed 500 nsec time window. For a event classified as othertypes, it only has to pass 2 of the above 5 criteria in order to be selected.

160
90
140

80

120

70

100

60
50

80

40

60

30
40

20

20
0
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10
20

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60

80 100 120 140 160 180

0-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

PC4 Angle (degrees)1

PC4 Dot Prodcut

200

250

180
160

200

140
120

150

100
80

100

60
40

50

20
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0

1000

2000

3000

PC4 Dot Prodcut

4000

5000

0
0

0.2

0.4

0.6

0.8

1

PC4 Muboy Goodness

Figure 5.20: PC 4th reduction for SK-IV, criteria (1) to (4) for the final samples. Distributions
for criteria (1) to (4) for data (black dots) and Monte Carlo (red lines) for the final sample.
Figure 5.20 shows the distributions for criteria (1) to (4) for both data and MC events forthe final sample.

Chapter 5

5.2.5

98

Fifth Reduction

Fifth reduction is done by some elaborate criteria specialized for each remaining backgroundsource.
For SK-I and SK-II
Low Energy Event Cut
Events satisfying the following criteria are removed as low energy background events:
1. PEtot < 3000 p.e.s (1500 p.e.s for SK-II)
where PEtot is the total number of p.e.s observed in the ID
Through-going Muon Cut A
Remaining through-going muons are removed by looking for the existence of two OD hit
clusters and the existence of hit OD PMTs near the entrance and exit points. It uses the OD
hit cluster information obtained by the algorithm used in PC 2nd reduction. Events satisfying
the following criteria are removed as through-going muons:
1. DISTclust > 20 m
where DISTclust is the distance between the highest charge OD hit cluster and the 2nd
highest charge one.
2. PEAC2nd 10 p.e.s
where PEAC2nd is the number of p.e.s detected in the 2nd highest charge OD hit cluster.
3. NCLSTA5 2
where NCLSTA5 is the number of OD hit clusters which contain more than 9 hit PMTs.
Criterion (3) uses the same algorithm as that in PC 2nd reduction (only the parameters used
are di↵erent). The OD wall is divided into 6 ⇥ 6 (instead of 11 ⇥ 11) in PC 5th reduction to
avoid boundary e↵ects of patches.
Event passing through the above two cuts are reconstructed, then the final backgroundrejections are performed as described in the following after the reconstruction.
Through-going Muon Cut B
Some cosmic ray muons enter the detector from the edge on the top part of the detector
then exist from the edge of the bottom of the detector. These through-going muons tend to passthrough the previous reduction cuts as the light collection efficiency is not good around the edgeof the OD and the event reconstruction is relatively not so accurate for these events. In orderto reject these events, the number of hit OD PMTs and observed p.e.s in 8 m radius spheres
centered at the top and bottom edges are studied. Events satisfying the following criteria arerejected as through-going muons:
1. NHITAtop 7 and NHITAbottom 7
where NHITAtop (NHITAbottom ) is the maximum number of hit OD PMTs in a 8 m radius
sphere centered at the top (bottom) edge.

Chapter 5

99

2. PEAtop 10 p.e.s and PEAbottom 10 p.e.s
where PEAtop (PEAbottom ) is the number of p.e.s in OD detected in the same sphere as
defined for NHITAtop (NHITAbottom )3. 0.75 < TDIFFA ⇥ c/40 m < 1.5
TDIFFA is the time interval between the averaged hit timing in the to and bottom spheres.
Through-going Muon Cut C
Remaining through-going muons are removed by the number of hit OD PMTs near theentrance and exit points. Their vertex positions and ring directions are reconstructed by aprecise fit (MS-fit) which use Cherenkov ring patter. The entrance and exit points on thedetector wall are estimated by extrapolation. Cut criteria are listed as follows:
1. NHITAin 5 and NHITAout 5
where NHITAin (NHITAout ) is the number of OD PMTs within 8 m from the entrance(exit) points.
2. 0.75 < TDIFFA ⇥ c/TRACK < 1.5
where TRACK is the distance between entrance and exit points estimated with vertex
position and ring direction reconstructed by MS-fit.
Stopping Muon Cut A
Entrance positions of stopping muons are estimated by reconstructing the vertex positionand direction by using MS-fit and extrapolation. The cut is made with counting the number of
hit OD PMTs near the entrance point. Events satisfying the following criteria are rejected as
stopping muons:
1. NHITAin 10
where NHITAin is the number of OD hit PMTs within 8 m from the reconstructed entrance.
Stopping Muon Cut B
Opening angles between the OD hit cluster and the ring estimated by two di↵erent fitters(TDC-fit and MS-fit) are compared. For stopping muon events, the opening angles are relatively
larger than PC events. Events satisfying the following criteria are rejected as stopping muons:
1. ⇥T DC f it > 90 or ⇥M S f it > 90
where ⇥T DC f it (⇥M S f it ) is the opening angle between the direction to the OD hit
cluster and the ring direction reconstructed by TDC-fit (MS-fit).
Stopping Muon Cut C
The charge inside a 42 cone in the ID is used. Vertex and direction are reconstructed bystopping muon fit which defines the entrance point as the position of the earliest hit cluster in
the ID. The entrance position cannot be reconstructed correctly for PC events as the vertexes
are not on the wall. Events satisfying the following criteria are rejected as stopping muons:Chapter 5100

1. goodness of stopping muon fit > 0
2. PEcone /PEtot 0.6
where PEcone is the number of p.e.s observed by ID PMTs located inside a 42 cone, andPEtot is the total number of p.e.s observed in ID.
3. NHITAin > 6
where NHITAin is the number of hit OD PMTs within 8 m from the entrance point.
Cable Hole Muon Cut
Veto scintillation counters are placed over four cable holes (out of twelve) on top of thedetector. Events satisfying the following criteria are rejected as cable hole muons:
1. One veto counter hit.
!
!
2. d ring • d veto vertex > -0.8!
!
where d ring is the reconstructed ring direction by TDC-fit, d vertex veto is the direction
from the hit veto counter to the reconstructed vertex.
Corner Clipping Muon Cut A
Corner clipping muon events have a small hit cluster in the ID, and the vertexes are sometimes
mis-reconstructed in the ID. Therefore, the tracklength from vertex to exit point reconstructedby MS-fit is large, while the tracklength estimated with visible energy by using the energy lossof muons ⇠2 MeV/cm is small. The cut criteria for rejecting corner clipping muons is:
1. Evis /2 (MeV/cm) < TRACK-1500 if TRACK > 15 m
where Evis /2 is the estimated tracklength by visible energy and TRACK is the tracklengthfrom vertex to the exit point estimated with vertex point and direction by MS-fit.
SK-III onwards
There are two types of cuts: Hard cuts and Soft cuts. PC events are required to pass all
Hard cuts, and it is allowed to fail Soft cuts once.
Hard cuts include:
1. Through-going muon cut A
2. Through-going muon cut B
3. Stopping muon cut B
4. Cable hole muon cut
5. Corner clippping muon cut B
where criteria (1), (2), (3) and (4) are described in the above reductions for SK-I and
SK-II.

Chapter 5

101

Corner Clipping Muon Cut B
This cut rejects corner clipping muons:
1. DCORN > 150 cm
where DCORN is the distance from the vertex reconstructed by Point-fit to the nearest
fringe of ID.

90
80
70
60
50
40
30
20
10
0
0

500

1000

1500

2000

2500

Distance To Nearest Corner

Figure 5.21: PC 5th reduction Corner clipping muon cut B for SK-IV. The plot shows the
distribution of the distance from the vertex reconstructed by Point-fit to the nearest fringe ofID, for the final sample for both data (black dots) and Monte Carlo (red line) events.
Figure 5.21 shows the DCORN distributions for both data and PC Monte Carlo and their
final samples.
Soft cuts include:
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 cut
where criteria (1), (3), (4) and (7) are described in the above reductions for SK-I and
SK-II.

Chapter 5

102

Through-going Muon Cut D
Remaining through-going muons are rejected by using the number of hit OD PMTs in the
1st and 2nd highest charge clusters. The cut criterion is:
1. NCLSTA5(1) 10 and NCLSTA5(2) 17
where NCLSTA5(1) (NCLSTA5(2)) is the number of hit OD PMTs in the 1st (2nd) highest
charge cluster.

50

50
500

40

400

35
30

300

25
20

200

15
10

100

5
0
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OD Hits in Largest Cluster

50

50
12

40

10

35
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30
25

6

20
4

15
10

2

5

45
OD Hits in 2nd Largest Cluster45
OD Hits in 2nd Largest Cluster

1600

40

OD Hits in Largest Cluster

0
0

1800

45
OD Hits in 2nd Largest Cluster

OD Hits in 2nd Largest Cluster

45

6

40
5

35
30

4

25
3

20
15

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10
1

5
50

100

150

200

OD Hits in Largest Cluster

250

0

0
0

50

100

150

200

250

0

OD Hits in Largest Cluster

Figure 5.22: PC 5th reduction Through-going muon cut D for SK-IV. Upper left plot shows thedata after PC 4th reduction while the upper right plot shows the PC Monte Carlo. The bottom
left and right plots show the final samples for data and Monte Carlo respectively.
Figure 5.22 shows scatter plots of NCLSTA5(1) and NCLSTA5(2) for data and PC Monte
Carlo and their final samples.
Stopping Muon Cut D
Stopping muons are rejected by using the number of OD hits within 8 m of the reconstructedentry point (NHITAin ) which is the same as that for SK-I and SK-II in PC 3rd reduction criterion(2), in which the reconstructed entry point is based on Point-fit. If the entry points reconstructedby Point-fit and MS-fit agree, this stopping muon rejection is applied:Chapter 5

103

!
!
1. NHITAin > 6 if | P point P M S | < 1500 cm
where NHITAin is the number of hit OD PMTs within 8 m from the entrance point. Ppoint!
( P M S ) is the vertex position reconstructed by Point-fit (MS-fit).
Stopping Muon Cut E
The same criterion in PC 4th reduction for SK-III criterion (1) is applied, as the former cut
was looser for events which are classified as stopping or through-going muons.
1. anglemuon < 90
where anglemuon is the angle between the fitted direction and the vector between the
Point-fit vertex and the center of the highest charge OD cluster.
Decay electron Cut
High energy neutrinos mostly interact with nucleons in water through DIS interactions. They
produce hadrons including charged pions. The pions decay into muons which in turn decay into
electrons. These decay-electrons are tagged by Decay Electron Search.
For high cosmic ray muon decays in the detector, there is no decay electron as cosmic raymuons decay in flight.
Therefore events having high energy (Evis > 25 GeV) but having no decay electron are
rejected as cosmic ray muons.

5.2.6

PC Reduction Summary

Finally, PC neutrino events are selected by the following criteria:
1. Vertex of neutrino interactions is inside the fiducial volume (2 m from the wall of detector).
2. Number of hit PMTs in the highest charge OD cluster (NHITAC) is larger than 15 (9 forSK-I).
3. Visible energy (Evis ) > 350 MeV(Total observed charge in ID > 3000 p.e.s (1500 p.e.s for SK-II)
Detection efficiency in each reduction step is estimated by MC events as shown in table 5.3.
Detection efficiency in final events are estimated to be 81%, 74.8%, 88.8% and 86.3% for SK-I,
SK-II, SK-III and SK-IV respectively.
Background events in final PC sample are mainly due to cosmic ray muons. These background events are checked by eye-scan. Most background events are rejected by the fiducial
volume cut, so only few background events contaminate the PC final sample. Contamination
due to background events is estimated by extrapolating the distribution of background events
outside the fiducial volume. Figure 5.23 shows the number of background events outside thefiducial volume for SK-II. Background events inside the fiducial volume may lead to contamination as shown by the fitted exponential curve in Figure 5.23. Table 5.4 summarizes the estimatedcontamination due to cosmic ray background for each SK period.

Chapter 5

104

PC
reduction step
1st
2nd
3rd
4th
5th
5th (FV)
Monte Carlo
Data
Data/MC
event rate/day

SK-I

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
number of events
913.8
448.6
356.0
744.9
902
427
344
735
0.99±0.05 0.95±0.06 0.97±0.07 0.97±0.05
0.66±0.02 0.65±0.04 0.62±0.03 0.66±0.02

Table 5.3: Selection efficiencies and number of events for PC sample are summarized for each
SK period. Selection efficiencies are for events whose real vertex is in the fiducial volume (after
5th (FV) step, fitted vertex instead of real vertex is considered) and number of OD hits > 15
(9 for SK-I) and total observed charge > 3000 p.e.s. Number of events for MC are normalized
to real data livetime, 1489.2, 798.6, 518.1 and 1097.0 days for SK-I, SK-II, SK-III and SK-IV
respectively, 2-flavor oscillation with (sin2 2✓, m2 ) = (1.00, 2.5 ⇥ 10 3 eV2 ) is assumed.

cosmic ray BG

SK-I
0.2%

SK-II
0.7%

SK-III
1.8%

SK-IV
0.6%

Table 5.4: Estimated contamination by cosmic ray muons for PC.

5.2.7

PC OD Stop/Through Separation

OD stopping events have better neutrino energy resolution than through-going events, therefore PC sample is further separated into OD stopping and OD through-going, by considering theenergy deposited into the OD region. OD stopping events stop in the OD, while OD throughgoing events travel through the OD into the rocks surrounding the detector. OD stopping
events have less charge in the OD compared to OD through-going events as they have shorter
pathlengths through OD. Selection criteria for OD stopping events are listed as follows:
1. Maximum number of p.e.s observed in OD in a sliding 500 nsec time window from 400
nsec to +600 nsec (PEanti ) is less than PEexp /1.5.
where PEexp is the expected number of p.e.s in the OD from the tracklength in the OD,
PEanti is the observed charge for that event.
2. The most energetic ring or the second energy ring is identified to be µ-like.
The ratio PEanti /PEexp is position dependent as the PMT charge response changes along
the height of the detector tank.

Chapter 5

105

Figure 5.23: Number of background events outside fiducial volume as a function of distance between reconstructed vertex and the ID wall for SK-III. The solid line shows the fitted exponential
curve.
Figure 5.24 shows the PEanti / PEexp distributions for data and PC MC events.

Chapter 5

160

Stopping

106100

Through-going

Stopping

Through-going

DATA (1489 days)

DATA (798 days)

140

80
MC w/ osc.: Total

120
100

MC w/ osc.: Total

60

MC w/ osc.: True Stopping

MC w/ osc.: True Stopping80
MC w/ osc.: True Through-going

MC w/ osc.: True Through-going

40

60
40

20

20
0
0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0
0

0.5

1

1.5

2

2.5

3

3.5

OD Charge Ratio

4

4.5

5

OD Charge Ratio

160
70 Stopping

Through-going

Stopping

60

DATA (1069 days)120
MC w/ osc.: Total

50

MC w/ osc.: Total

100
MC w/ osc.: True Stopping40

MC w/ osc.: True Stopping80
MC w/ osc.: True Through-going

30

40

10

20
0.5

1

1.5

2

2.5

3

3.5

4

4.5

MC w/ osc.: True Through-going

60

20

0
0

Through-going

140

DATA (518 days)

5

OD Charge Ratio

0
0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

OD Charge Ratio

Figure 5.24: Distribution of PEanti / PEexp for PC stop / through separation for SK-I (top left),
SK-II (top right), SK-III (bottom left) and SK-IV (bottom right). The points represent thedata with error bar. The shaded regions represent MC of true stopping and true through-going
events without oscillation and normalized to data livetime. The selection criteria is shown by
the arrow.

Chapter 5

5.3

107

Reduction for Upward-going Muon Events

The reduction for upward-going muon (UPMU) events are described in this section.

5.3.1

Charge cut

Low energy and extremely high energy events are rejected in the 1st reduction:
1. PEtot > 6000 p.e.s (3000 p.e.s for SK-II)
2. PEtot < 1,750,000 p.e.s (800,000 p.e.s for SK-II)
where PEtot is the total number of p.e.s observed in the ID.
PEtot corresponds to muon momentum of 1 GeV/c and tracklength of 3.5 m (requirement for
final sample is tracklength > 7 m). At very high ID charge ⇠1,750,000 p.e.s, the ID electronics
is saturated and muon fitter does not work.

5.3.2

Zenith Angle Cut

In order to reject the the downward-going muon cosmic-ray muons, seven di↵erent fittersspecialized to fit stopping muons, through-going muons and muons with Bremsstrahlung areused. Algorithm of the zenith angle cut is listed as follows:
1. Apply a muon fitter
2. If the event is classified as upward and goodness of fit is above the threshold, the event issaved.
3. If the event is classified as downward and goodness of fit is above the threshold, the eventis rejected.
4. If the event is classifed as horizontal with goodness of fit above the threshold, or if goodnessof fit is below the threshold, the judgement is suspended.
5. Go to (1) and apply the next muon fitter.
This algorithm continues until the event has passed through all the fitters or has beenclassified. If no fitter gives a goodness above the threshold, the event is rejected. If at least one
fitter classifies the event as horizontal, the event is saved. Detailed description about the sevenmuon fitters and definition of goodness can be found in [134].
All events from the output of the upward-going muon reduction are passed through theprecise fitter. The fitter results are used to select upward-going events by the criterion of zenith
angle cos⇥ < 0.

5.3.3

Scanning

In order to eliminate the remaining misfitted background events, the selected events arescanned using a visual display. Upward-going muon events are selected one by one. All eventsare checked by two scanners so that no neutrino events is missed. Efficiency of scanning isestimated to be almost 100%. About half of the events remaining after all the automatedreduction steps are rejected by this final scan.

Chapter 5

5.3.4

108

UPMU Reduction Summary

The final selection criteria for UPMU stopping events are:
1. Fitter classification is stopping event
2. Fitted momentum

1.6 GeV/c (which correspond to tracklength = 7 m)

3. Number of hit OD PMTs within 8 m from the exit point (NHITEX) < 10 (16 for SK-II)
Selection criteria for UPMU through-going events are:
1. Fitter classification is through-going event
2. Distance from the ID entrance point to the ID exit point

7m

3. Number of hit OD PMTs within 8 m from the exit point (NHITEX)

10 (16 for SK-II)

Figure 5.25 shows the distribution of NHITEX. Detection efficiency for the final events is
estimated to be 98.0% (99.4%) for SK-I stopping muons (through-going muons), 97.0% (98.1%)
for SK-II stopping muons (through-going muons), 98.2% (99.4%) for SK-III stopping muons(through-going muons) and 98.0% (99.4%) for SK-IV stopping muons (through-going muons).
Number of events in the final sample are summarized in 5.5. The final sample is made by
subtracting the expected number of background events.
Note that the disagreement between data and Monte Carlo for SK-III in figure 5.25 is dueto the problem in the OD tuning. However, as the cut for stopping muons and through-going
muons is far away from the disagreement, the disagreement does not a↵ect the efficiency ofseparation of UPMU stopping and through-going events.

final sample (MC)final sample (data)
BG subtracted
BG subtracted/MC
event rate (per day)

SK-I
stop
thru
729.6 1892.4
462
1866
429
1835.60.59
0.97±0.04 ±0.03
0.26
1.12
±0.01 ±0.03

SK-II
stop
thru
364.6 926.2
223
871
210
833.4
0.58
0.90
±0.10 ±0.04
0.25
1.01
±0.02 ±0.03

SK-III
stop
thru
286.0 725.1
210
735
193
722.9
0.67
1.00
±0.06 ±0.05
0.30
1.14
±0.02 ±0.04

SK-IV
stop
thru
474.0 1257.6
306
1345
284
1330.90.60
1.06±0.04 ±0.04
0.26
1.21
±0.02 ±0.03

Table 5.5: Number of events of UPMU summarized for each SK period. Number of MC (without
oscillation) events and event rate are calculated by normalization of data livetime, 1645.9, 827.7,
635.6, 1096.7 days for SK-I, SK-II, SK-III and SK-IV respectively.

Chapter 5

109

Stop

Thru

103

103

Stop

Thru

102

102
10

10
1
1
10-1

0

20
40
60
Number of hits within 8 m of exit

103 Stop Thru

10-1
0

103

20
40
60
Number of hits within 8 m of exit

Stop

Thru

102
102
10
10
1
1
10-1

0

20
40
60
Number of hits within 8 m of exit

10-1
0

20
40
60
Number of hits within 8 m of exit

Figure 5.25: Distribution of NHITEX for UPMU stop/through separation for SK-I (top left),
SK-II (top right), SK-III (bottom left) and SK-IV (bottom right). Points shows the observed
data, dashed line show MC assuming no oscillation and solid line shows MC with ⌫µ $ ⌫⌧2-flavor oscillation with (sin2 2✓, m2 ) = (1.00, 2.5 ⇥ 10 3 eV2 ). Selection criteria are shownby arrows.

Chapter 5

0

cos θ

Background Estimation for Upward-going Muon

cos θ

5.3.5

110

-0.2

-0.2

-0.4

-0.4

-0.6

-0.6

-0.8

-0.8

-1

0

60

120

180

240

300
360
φ (degrees)

0

-1

0

60

120

180

240

300
360
φ (degrees)

Figure 5.26: Scatter plots of zenith versus azimuth directions for stopping muons (left) andthrough-going muons (right) for SK-IV.
Background events are mostly cosmic-ray muons, which are reconstructed as upward-going
due to multiple Coulumb scattering of the muons or slight mis-reconstruction of near horizontal going muons. Hence they are mostly distributed in the horizontal bin (-0.1 < cos✓ < 0).
Contamination by the background is estimated by extrapolating the distribution of downward
events in 0< cos✓. Figure 5.26 shows the zenith versus azimuth directions for upward going
muon samples. Clusters of cosmic ray downward muons are seen in the regions zenith anglecos⇥ > 0 (downward) and azimuthal angle around = 120 , 180 and 270 . These clusters arecaused by thin rock over the detector. When contamination of background is estimated, the
UPMU samples are divided into two azimuth angle regions. Figure 5.27 shows the azimuthal
angle distributions for upward going muon samples, in which region (2) represents the region
with thinner mountain (which is defined as 60 < < 310 ), while region (1) represents theregion with thicker mountain.
Figure 5.28 shows the zenith angle distributions for upward-going muon samples, in which
region (1) and (2) are normalized by the coverage of azimuthal angle. The zenith angle distribution is almost flat for upward going events (cos⇥ < 0), but the number of events in region (2)exponentially increase with cosine of zenith angle for downward-going events (cos⇥ > 0). Contamination by cosmic ray muons in the upward-going direction are estimated by extrapolatingfrom the downward direction in region (2) with (exponential + constant) function. In figure ??,there are best-fitted curves which is defined as f (cos ✓) = p0 + exp p1 + p2(cos ✓). p0 is fixed to
the value determined by Rthe average of region (1) events. Contaminations of background events0are estimated by Nbg = cos✓x (exp p1 + p2(cos ✓)). The uncertainties on the estimated numbersof background events are evaluated by the maximum change on the number of background events
when fit parameters (p0, p1 and p2) are shifted to increase or decrease the background events
within the parameters’ allowed regions. They are summarized in table 5.6.

Events/day

111

Events/day

Chapter 5

All thru-going muons (cos θ < 0.1)
Upward thru-going muons (cos θ < 0)

10

All thru-going muons (cos θ < 0.1)
Upward thru-going muons (cos θ < 0)
3

102

(1)

(2)

(1)

(1)

(2)

(1)

102

10

0

60

120

180

240

300
360
φ (degrees)

0

60

120

180

240

300
360
φ (degrees)

Figure 5.27: Azimuthal angle distributions for stopping muon (left) and through-going muons(right) for SK-IV. Region (1) has thicker-mountain, while region (2) has thinner-mounter. White
histograms show the events of downward-going muons, while the black histograms show the
upward-going muons.

stop µthru µ (non-shower)
thru µ (shower)

SK-I
16%
11%
18%

SK-II
21%
15$
14%

SK-III
20%
19%
24%

SK-IV
17%
14%
20%

Table 5.6: Uncertainties of background events which is the contamination by horizontal muons
to the UPMU sample in the zenith angle -0.1 < cos✓ < 0.

102

Events/day/sr

112

Events/day/sr

Chapter 5

χ2 / ndf
17.18 / 16Prob0.3739p0
0.07993 ± 0.00000
p1-1.57 ± 0.12
p2
44.23 ± 1.89
Region (1) φ<60°, φ>310°
Region (2) 60°< φ <310°

103

102

χ2 / ndf
Prob
p0
p1
p2

27.05 / 16
0.04094
0.256 ± 0.000-2.285 ± 0.112
76.46 ± 1.63

Region (1) φ<60°, φ>310°
Region (2) 60°< φ <310°

10
10
1
1
10-1
10-1
10-2
-0.1 -0.08 -0.06 -0.04 -0.02

00.02 0.04 0.06 0.08
cos θ

-0.1 -0.08 -0.06 -0.04 -0.02

00.02 0.04 0.06 0.08
cos θ

Figure 5.28: Zenith angle distributions for stopping muon (left) and through-going muon (right)
for SK-IV in both regions (1) and (2). Solid curves are fitted functions to estimate the background contamination. Zenith angle bins are divided equally to 18 bins between -0.1 < cos ✓ <0.08.

Chapter 5

5.3.6

113

Upward Through-going Showering Muon

0.7

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

0.6

Stopping

0.5
0.4
0.3
0.2
0.1
0
1

10

2

10
10
Eυ (GeV)

3

10

4

10

5

Figure 5.29: Energy spectrum of primary neutrino for upward stopping (dotted line), nonshowering (dashed line) and showering (solid line) muon events.
Upward through-going muons are further separated into showering muon sample and nonshowering muon sample. This is because some upward through-going muon events are accompanied with electromagnetic shower. Showering muons are observed as high energy muons whichlose energy through radiative processes such as Bremstrahlung radiation, e+ e pair production
and photo-nuclear interactions. The energy of the parent neutrino is approximately 1 TeV.
Figure 5.29 shows energy spectrum of primary neutrino for the UPMU samples.
Selection of showering muons uses 2 test based on the observed charge and the expectedcharge of non-showering muons [134]. Di↵erence between corrected observed charge and expectedcharge of non-showering muon is defined as (Q). Criteria of showering muon selection are:
1. Di↵erence between the corrected charge and the expected charge for non-showering muon
( (Q) = <Q> ql ) > 0.5 when 2 variable for showering selection 50.
2.

(Q)> 4.5 - 0.08

2 when2 < 50.

Figure 5.30 shows scatter plot of (Q) and 2 ,and
events which are classified as through-going muons.

(Q) distributions for data and MCChapter 5

114

15

15

MC

Data
10

∆(Q)

∆(Q)

10

5

0
0

5

0
50

100

150

200

χ2showering

0

50

100

150

200

χ2showering

100

80

60

40

20

0
-5

0

5

10

15

∆(Q)

Figure 5.30: Distributions for UPMU showering separation for SK-IV. Scatter plot of (Q)against 2 of UPMU through-going events are shown for data (top left) and MC (top right)(normalized to data lifetime). (Q) distribution is shown in the bottom panel for data (points)
and MC (red solid line). The cut is shown by red dashed line.

Chapter 5

5.4

115

Event Reconstruction

Selected neutrino events are reconstructed to determine their neutrino flavor type, directions,
momentum etc. For FC and PC, the reconstruction process consists of the following steps:
1. Vertex fitting
Vertex position is defined as the point at which the timing residual distribution of hit
PMTs has the sharpest peak. The direction and the outer edge of the dominant ring isdetermined.
2. Ring counting
Other possible rings are searched for by using the vertex position and the direction of
the dominant ring. Ring candidates are tested whether they are true rings or not with alikelihood, then the number of rings is determined.
3. Particle identification
Each ring is classified as two types: showering type (e± , ) as e-like and non-showering
type (µ± , ⇡ ± ) as µ-like, based on a likelihood using Cherenkov ring pattern and Cherenkov
opening angle.
4. Precise vertex fitting (MS-fit) (only for single-ring events)
Vertex position is determined more accurately by using Cherenkov ring pattering assuming
the obtained particle type.
5. Momentum determination
Momentum of each ring is determined by the detected charge inside a Cherenkov cone.
Conversion from the charge to momentum is based on a Monte Carlo simulation anddetector calibration.
6. Decay electron search
Decay electrons produced by primary events are searched.
7. Ring number correction (only for FC multi-ring events)
Rings which are fitted mistakenly are discarded by using precise reconstructed information.
8. ⇡ 0 fitting
Candidate ⇡ 0 events are searched for among the events which are recognized as single-ringe-like by the above reconstruction processes.
UPMU events are also reconstructed by using some of the above processes. PC and UPMU
events are basically treated as single-ring µ-like events, information such as PID and number ofrings are not used in the analysis for PC and UPMU.
Reconstruction methods are described briefly in the following, detailed explanation of reconstruction can be found in [135]. Distribution and numbers of Monte Carlo events in the following
sections are assumed to have undergone 2-flavor oscillation with (sin2 ✓, m2 ) = (1.00, 2.5 ⇥10 3 eV2 ).

Chapter 5

5.4.1

116

Vertex Fitting

Vertex position is reconstructed by using the timing information of hit PMTs in three steps.
In the first step, a simple fit called Point-fit is applied. Vertex is estimated assuming thatall photons are emitted at the same time from a point source. After subtracting the time offlight from a tested vertex, a distribution of residual PMT hit time is constructed for that tested
vertex. The vertex is estimated as the point where the timing residual distribution is peaked.
In the second step, the direction and the outer edge of the dominant ring is determined. A
pair of direction and opening angle is tested by the parameter defined as:Q(✓edge ) =R ✓edge0(✓edge ✓exp )2
P E(✓)d✓dP E(✓)⇥(|✓=✓edge )2 ⇥ exp()
sin ✓edge
d✓
2 ✓2

(5.5)

where ✓exp and ✓ are the Cherenkov opening angle expected from the charge within the
Cherenkov cone and its resolution respectively.
P E(✓) is the angular distribution of the observed charge as a function of opening angle ✓.
The observed charge is corrected for the e↵ect from water transparency and PMT acceptance.
The direction and the ring edge which maximize Q(✓edge ) are taken as the results of fit by
Point-fit.

Figure 5.31: Distance between the reconstructed vertex and the true vertex for FC MC MultiGeV multi-ring µ-like sample for SK-IV. Colored region shows 68% of all events.
In the final step, the vertex position is determined more precisely with TDC-fit. Unlike
Point-fit, photons are assumed to be generated along the track of the particle. Then the time
residual for PMTs inside the Cherenkov ring is calculated with this e↵ect also considered. For
photons outside the ring, the time residual is also computed assuming all photons originatedfrom the vertex. The goodness-of-fit is determined including the e↵ect of scattered photons.
The vertex is determined to be the position which maximize the goodness parameter. The
vertex position is used for multi-ring events. Figure 5.31 shows the estimated vertex position
resolutions for SK-IV multi-GeV multi-ring µ-like sample.

Chapter 5

5.4.2

117

Ring Counting

42 deg. ringhit PMT(possible center)

center(most probable)

Cherenkov ring

Figure 5.32: The basic idea to find ring candidates. By drawing rings around the hit PMTs
with Cherenkov opening angle of 42 from the vertex, center of the actual Cherenkov ring canbe identified.
After the initial Cherenkov ring and the vertex is determined, other ring candidates aresearched for by a Hough transform method [137], which is a technique to extract a particularshape from an image. Figure 5.32 illustrates the method to find other possible rings, in which a
spherical coordinate centred on the vertex position is considered. The shaded circle in the figure
represents the Cherenkov ring image projected to a plane perpendicular to the ring direction.
Hit PMTs are picked up and virtual circles (dashed lines) centred at the position of the hit
PMTs are drawn with 42 half angle. The direction of the Cherenkov ring is identified as theintersection point of these dashed line circles. The second ring candidate is searched by choosing
possible ring directions based on this method. Likelihood technique is used to determine if thecandidate ring is consistent with a ring. If the second ring is determined to exist, the sameprocedure is repeated to find other possible rings up to 5 rings.
Figure 5.33 and figure 5.34 show the ring counting likelihood distributions between singlering and multi-ring for both Data and MC. The cut threshold is at 0 to separate single-ringand multi-ring events. The di↵erence between data and MC is taken as a source of systematic
uncertainty in event selection.

Chapter 5

118

Super Kamiokande I 1489.2 days
450

single ring

Super Kamiokande II 798.6 days
250

multi ring400

single ring

multi ring

CCQE

CCQE

CC Single π

350

CC Single π

200

CC DISCC DIS

CC Coherent π

300

CC Coherent π

NC

NC

150

250
200

100

150
100

50

50
0
-15

-10

-5

0

5

10

15

Ring counting likelihood sub-GeV(FC), stack0-15

Super Kamiokande III 518.1 days
220

-10

-5

0

5

10

15

Ring counting likelihood sub-GeV(FC), stack

Super Kamiokande IV 1096.7 days : Monitoring

450
single ring

multi ring200

single ring

multi ring

400

CCQE

CCQE

CC Single π

180

CC Single π

350

CC DIS

160

CC DIS

CC Coherent π

CC Coherent π

300

NC

NC

140
250

120
100

200

80

150

60

100

40

50

20
0
-15

-10

-5

0

5

10

15

Ring counting likelihood sub-GeV(FC), stack

0
-15

-10

-5

0

5

10

15

Ring counting likelihood sub-GeV(FC), stack

Figure 5.33: Ring counting likelihood distribution for SK-I, SK-II, SKIII and SK-IV sub-GeV
samples. Dots are data and solid lines are MC events.

Chapter 5

119Super Kamiokande I 1489.2 days

Super Kamiokande II 798.6 days

400

220
single ring

multi ring

200

350
CCQE

CC DIS

CCQE
CC Single π
CC DIS

160

CC Coherent πNC

250

multi ring180

CC Single π

300

single ring

CC Coherent π
NC

140
120

200

100
150

80

100

60
40

50
0
-15

20
-10

-5

0

5

10

15

Ring counting likelihood multi-GeV(FC), stack0-15

Super Kamiokande III 518.1 days
140

single ring

-5

0

5

350

single ring

CCQE

CC Single π

CC Single π

CC DISCC DIS

250

CC Coherent π

CC Coherent π

NC

NC

80

200

60

150

40

100

20

50

0
-15

-10

-5

0

5

15

multi ring

300

CCQE

100

10

Ring counting likelihood multi-GeV(FC), stack

Super Kamiokande IV 1096.7 days : Monitoring

multi ring

120

-10

10

15

Ring counting likelihood multi-GeV(FC), stack

0
-15

-10

-5

0

5

10

15

Ring counting likelihood multi-GeV(FC), stack

Figure 5.34: Ring counting likelihood distribution for SK-I, SK-II, SKIII and SK-IV multi-GeV
samples. Dots are data and solid lines are MC events.

Chapter 5

5.4.3

120

Particle Identification

Figure 5.35: Event display of single-ring electron (left) and single-ring muon (right) neutrino
MC event.
The detected Cherekov rings are identified as e-like or µ-like by Particle Identification (PID).
Figure 5.35 shows the event display for single-ring electron (left) and muon (right) neutrino MC
events. An electron (and a gamma ray) produces di↵used ring patterns as electrons produce
electromagnetic shower and low energy electrons undergo multiple scattering. Muon (and acharged pion) produces ring with sharp edge. Moreover, Cherenkov rings from muons or charged
pions have smaller Cherenkov angles if they are not highly relativistic ( = v/c< 1) Thesedi↵erences are taken into account when events are classified into two types by PID algorithm.
Figure 5.36, figure 5.37 (single-ring) figure 5.38 and figure 5.39 (multi-ring) show the PID
likelihood for FC samples. Separation of particle type in multi-ring events is not as good as thatfor single-ring events because Cherenkov photons from multiple rings overlap. The di↵erence
between data and MC is taken as a source of systematic uncertainty in event selection.

Chapter 5

121Super Kamiokande I 1489.2 days

Super Kamiokande II 798.6 days

300
e-like 3469

180

muon-like 3184

e-like 1856

muon-like 1684

160

250

140
200

120
CCQE muon

CCQE muon
100

150

80
100

60
40

50
20
0
-10

-8

-6

-4

-2

0

2

4

6

8

10

PID likelihood sub-GeV 1ring (FC)0
-10Super Kamiokande III 518.1 days
120

e-like 1268

-8

-6

-4

-2

0

2

4

6

8

10

PID likelihood sub-GeV 1ring (FC)Super Kamiokande IV 1096.7 days : Monitoring200

muon-like 1139

e-like 2457

muon-like 2412

180
100

160
140

80
CCQE muon

CCQE muon

120
100

60

80
40

60
40

20

20
0
-10

-8

-6

-4

-2

0

2

4

6

8

10

PID likelihood sub-GeV 1ring (FC)

0
-10

-8

-6

-4

-2

0

2

4

6

8

10

PID likelihood sub-GeV 1ring (FC)

Figure 5.36: PID likelihood distribution for FC sub-GeV 1-ring samples for SK-I, SK-II, SK-III
and SK-IV. Points represent the data, red lines represent Monte Carlo, the shaded green regions
represent the MC CCQE ⌫µ events.

Chapter 5

122Super Kamiokande I 1489.2 days
e-like 856

Super Kamiokande II 798.6 days

muon-like 712

100

120
100

e-like 396

muon-like 400

80
CCQE muon

80

CCQE muon

60

60
40
40
20

20
0
-30

-20

-10

0

10

20

30

PID likelihood multi-GeV 1ring (FC)0-30Super Kamiokande III 518.1 days

-20

-10

0

10

20

30

PID likelihood multi-GeV 1ring (FC)Super Kamiokande IV 1096.7 days : Monitoring

120

45
e-like 274

muon-like 238

e-like 579

40

muon-like 520

100

35
80

30
CCQE muon

25

CCQE muon
60

20
40

15
10

20
5
0
-30

-20

-10

0

10

20

30

PID likelihood multi-GeV 1ring (FC)

0
-30

-20

-10

0

10

20

30

PID likelihood multi-GeV 1ring (FC)

Figure 5.37: PID likelihood distribution for FC multi-GeV 1-ring samples for SK-I, SK-II, SK-III
and SK-IV. Points represent the data, red lines represent Monte Carlo, the shaded green regions
represent the MC CCQE ⌫µ events.

Chapter 5

123

Super Kamiokande I 1489.2 days
e-likeSuper Kamiokande II 798.6 days
100

muon-likee-likemuon-like

100
80
80
nu_mu CC

nu_mu CC

60

60

40

40

20

20

0
-300

-200

-100

0

100

200

300

PID likelihood for Pmax ring in sub-GeV multi-ring (FC)0-300Super Kamiokande III 518.1 days
e-like

40

-200

-100

0

100

200

300

PID likelihood for Pmax ring in sub-GeV multi-ring (FC)Super Kamiokande IV 1096.7 days : Monitoringmuon-likee-like60

muon-like

35
50
30
40

nu_mu CC25
20

nu_mu CC

30

15

20

10
10

5
0
-300

-200

-100

0

100

200

300

PID likelihood for Pmax ring in sub-GeV multi-ring (FC)

0
-300

-200

-100

0

100

200

300

PID likelihood for Pmax ring in sub-GeV multi-ring (FC)

Figure 5.38: PID likelihood distribution for FC sub-GeV multi-ring samples for SK-I, SK-II,
SK-III and SK-IV. Points represent the data, red lines represent Monte Carlo, the shaded green
regions represent the MC CCQE ⌫µ events.

Chapter 5

124Super Kamiokande I 1489.2 days

Super Kamiokande II 798.6 days

120
e-like180

muon-likee-likemuon-like

160

100

140
80

120
nu_mu CC

nu_mu CC
100

60

80
40

60
40

20
20
0
-600

-400

-200

0

200

400

600

PID likelihood for Pmax ring in multi-GeV multi-ring (FC)

0
-600

-400

-200

0

200

400

600

PID likelihood for Pmax ring in multi-GeV multi-ring (FC)Super Kamiokande III 518.1 days

Super Kamiokande IV 1096.7 days : Monitoring

45

100
e-likemuon-likee-likemuon-like

40
80

35
30
nu_mu CC25

nu_mu CC

60

20
40
15
10

20

5
0
-600

-400

-200

0

200

400

600

PID likelihood for Pmax ring in multi-GeV multi-ring (FC)

0
-600

-400

-200

0

200

400

600

PID likelihood for Pmax ring in multi-GeV multi-ring (FC)

Figure 5.39: PID likelihood distribution for FC multi-GeV multi-ring samples for SK-I, SK-II,
SK-III and SK-IV. Points represent the data, red lines represent Monte Carlo, the shaded green
regions represent the MC CCQE ⌫µ events.

Chapter 5

5.4.4

125

Precise Vertex Fitting

Since Point-fit only uses timing information, FC/PC single-ring events are re-fitted to optimize their vertex position and direction by using expected light pattern as e-like or µ-like events.
This precise vertex fitter is called MS-fit. UPMU events are also fitted by MS-fit assuming thatthe particle is a single-ring muon and the vertex is at the ID surface.
MS-fit adjusts the vertex position parallel to the particle direction obtained by Point-fit
with timing information. The process is iterated until the changes in the vertex position andthe particle direction is less than 5 cm and 0.5 degrees.

Figure 5.40: Distance between the true vertex and the reconstructed vertex of FC MC sample
for SK-IV. Left: Sub-GeV 1-ring elike. Right: Sub-GeV 1-ring µ-like.

Figure 5.41: Distance between the true vertex and the reconstructed vertex of FC MC sample
for SK-IV. Left: Multi-GeV 1-ring elike. Right: Multi-GeV 1-ring µ-like.

Chapter 5

126

Figure 5.42: Distance between the true vertex and the reconstructed vertex of PC MC sample
for SK-IV.
Performance of MS-fit for single-ring events is represented by the resolution which is estimated by using MC. Figure 5.40, figure 5.41 and figure 5.42 show the distance between the true
vertex and the reconstructed vertex for single-ring samples. Figure 5.43 and figure 5.44 show
the angular di↵erence between true direction and the reconstructed direction.
Table 5.7 and table 5.8 summarize the estimated resolutions for di↵erent samples in each SK
period.

Figure 5.43: Angular di↵erence between true direction and reconstructed direction of FC MC
samples for SK-IV. Left: Sub-GeV 1-ring e-like. Right: Sub-GeV 1-ring µ-like.

Chapter 5

127

Figure 5.44: Angular di↵erence between true direction and reconstructed direction of FC MC
samples for SK-IV. Left: Mult-GeV 1-ring e-like. Right: Multi-GeV 1-ring µ-like.

Vertex resolution (cm)
SK-I SK-II SK-III
FC sub-GeV
single-ringe-likeµ-likesingle-ringe-likeµ-likemulti-ringµ-like
PC

SK-IV

31.2
35.6
31.1
23.8
30.3
23.9
FC multi-GeV

31.5
23.5

33.5
24.8

34.3
26.9

33.4
25.1

33.5
23.7

67.4
53.6

111.8
62.8

73.8
52.6

87.8
67

Table 5.7: Vertex resolutions for each SK period.

Chapter 5

128

Angular resolution (degree)
SK-I SK-II SK-III SK-IV
FC sub-GeV
single-ringe-like
3.1
3.2
3.0
3.0
µ-like
1.9
2.1
1.9
1.9
FC multi-GeV
single-ringe-like
1.2
1.3
1.2
1.2
µ-like
0.8
1.0
0.8
0.8
PC
1.0
1.2
0.9
0.9
Table 5.8: angular resolutions for each SK period.

Chapter 5

5.4.5

129

Momentum Reconstruction

Momentum for each particle is estimated by the observed p.e.s inside a Cherenkov cone withhalf opening angle of 70 (in order to include all the hits). In order to determine the momentum
for individual rings, the observed p.e.s in hit PMTs are separated into contribution from eachring. Separation of the observed p.e.s is carried out based on the expected p.e. distribution
from each ring as a function of the opening angle ✓, assuming uniformity in azimuthal angle .
The observed p.e.s in the i-th PMT from the n-th ring is estimated as:
exp
qi,n
obs
qi,n
= qiobs ⇥ P exp
n0 qi,n0(5.6)

obs is the fractional p.e.s from the n-th ring in the i-th PMT,
where qi,nobs
qi is the observed p.e.s in the i-PMT
exp
qi,nis the expected p.e.s.
To calculate the total number of p.e.s inside the 70 cone, the number of p.e.s in eac PMT
is corrected considering the light attenuation and scattering in water, light reflection correction
and the acceptance of PMTs as:

RT OTn =

GM C
[↵ ⇥
Gdata
✓

i,n

X

ri
cos⇥i
(qiobs ⇥ exp( ) ⇥)L
f (⇥i )
<70 and 50nsec<t <250nsec
iX✓i,n <70

Si

X✓i,n <70

where↵ = normalization factor
Gdata , GM C = relative PMT gain parameters for data and MC✓i,n = opening angle between the n-th ring direction and the i-th PMT directionti = photon time-of-flight subtracted hit timing of i-th PMT
L = light attenuation length in waterri = distance from the vertex position to i-th PMT
f (⇥i ) = correction function for the PMT acceptance as a function of photon incidence angle ⇥i
Si = expected p.e.s for i-th PMT from scattered photons
Ri = correction for the reflected photons from black sheets in the detector
The summation is restricted to the time window -50 nsec to +250 nsec around the peakof the photon time-of-flight subtracted hit timing distribution to reject the e↵ect from muon
decay electrons. Attenuation length in water L is measured continuously by using cosmic ray
through-going muons. The absolute energy scale is calibrated by using the four independentcalibration sources as described in Chapter 4.

5.4.6

Decay Electron SearchElectrons may be produced from the decay of muons which stopped in the detector. ⇡ +produced by a neutrino interaction may decay into µ± which in turn decays into e+ . Therefore,
for example, if a muon decay is observed for a single-ring e-like event, the event is likely to bea CC-non QE ⌫e event that produces ⇡ + along with an electron. Hence the information fromdecay-electrons is useful to determine the type of neutrino interactions. Decay-electrons are

Ri ](5.7)

Chapter 5130

observed in a later independent time window (900 nsec) from that of the primary event or in
the same time window of the primary event. If the decay occurs around 900 nsec, the electron
event may be observed separately in the primary trigger or subsequent event trigger.
Detection efficiency of decay electron is 80% for µ+ and 63% for µ , as approximately 20%
of µ is captured by 16 O nuclei [138] for SK-I, SK-II and SK-III. As the detection efficiency of
decay electrons is improved in SK-IV, the tagging efficiency is 96% for decay electrons from µ+
and 83% for decay electrons from µ .
Note that detection efficiency of decay electron also includes low energy de-excitation gammas
from µ captures. There is a cut from SK-I to SK-III to remove low energy events (in whichmost of them are de-excitation gammas from µ captures), and this cut is removed for SKIV in order to improve the decay electron efficiency. This explains why the ratio of detection
efficiencies of decay electrons from µ+ to that from µ is larger for SK-IV when compared to
SK-I to SK-III.

5.4.7⇡ 0 Reconstruction⇡ 0 reconstruction is applied to only single-ring e-like samples. In FC sub-GeV single-ringe-like event sample, the contamination by NC events is estimated to be about 9%. These NC
events are mostly from NC single ⇡ 0 production. ⇡ 0 immediately decays into two gamma rays
which produce two e-like Cherenkov rings. In case one of these two gammas is not detected due
to too low energy or overlapping rings, the event can be misidentified to be single-ring e-like. In
order to find these ⇡ 0 events, ⇡ 0 fitter reconstructs the second gamma ray on the assumptionof existence of two Cherenkov rings by comparing the observed photoelectron distribution withthe expected photoelectron distribution of two gamma rays. Then likelihood technique is used
to determine the best-fit configuration of two gamma rays.

Chapter 6

Neutrino Oscillation Analysis
6.1

Overview

Atmospheric neutrino oscillation is studied with the information of detected neutrino momenta and directions. Observed data is compared with MC expectation by using a 2 method.
In this chapter, data samples for this analysis and expected e↵ects from the full oscillation onthese samples are explained. Analysis method, systematic errors and results will be presented
in this chapter.

6.2

Data Set

All data samples including FC, PC and UPMU are used for this analysis.
In order to separate CCQE, CC non-QE and NC events from 1-ring sample, FC sub-GeV
1-ring samples are further separated to sub-samples by considering number of muon decayelectrons.
In order to obtain higher purity of charged current e-like events in Multi-GeV multi-ringe-like sample, the sample is processed with likelihood method to reject NC events and µ-likeevents.
Multi-GeV e-like samples are further divided into ⌫e and anti-⌫e -like samples, so as to improve
the sensitivity to mass hierarchy.
These additional selection methods are described in the following sections.

6.2.1

Selection for Sub-GeV sample

FC sub-GeV single-ring e-like sample contains background events which are mainly NC ⇡ 0
events producing two gamma rays. In case one out of the two gamma rays is missed by theevent reconstruction, the NC ⇡ 0 event would be identified as a single-ring e-like event, as theelectromagnetic shower from gamma gives similar light pattern to that of an electron. In orderto reduce this kind of background, a specialized ⇡ 0 fitter is used. This fitter reconstructs thesecond gamma-ray based on the first fitted ring. It predicts the light pattern resulting from two
gamma rays propagating through the tank with the vertex of the fitted ring and the best fitted
direction.
Figure 6.1 shows the invariant mass distributions from the ⇡ 0 fitter for CCQE (black solidlines) and NC (red dashed line) in FC sub-GeV single-ring e-like sample MC in five energy131

Chapter 6

132

0.04
0.015
0.04

0.02

0.02

0.03
0.03

0.01
0.02

0.02

0.01

0.01

0.005
0.01

0
0

50

100

150

200

0
0

50

100

150

200

0
0

0.01

50

100

150

200

0
0

50

100

150

200

0
0

50

100

150

200

Figure 6.1: Distribution of invariant mass from the ⇡ 0 fitter for CCQE (black solid line) andNC (red dashed line) in SK-IV FC sub-GeV single-ring e-like MC events, in five energy regions(from left to right): (1) Pe < 250 MeV/c, (2) 250 MeV/c  Pe < 400 MeV/c, (3) 400 MeV/c 
Pe < 630 MeV/c, (4) 630 MeV/c  P < 1000 MeV/c, (5) 1000 MeV/c < Pe . The arrows show
the events which are taken to be ⇡ 0 -like.
regions. NC events tend to form a peak near the ⇡ 0 mass while CCQE events do not. For eventswith electron momentum below 250 MeV/c, a cut at 100 MeV/c2 of ⇡ 0 mass is used to createa ⇡ 0 sample. This cut is not efficient for events at higher electron momentum, so a likelihood
selection is done by using three variables: fraction of the reconstructed momentum of the second
ring, ⇡ 0 invariant mass and the di↵erence of two likelihood variables which results from a ⇡ 0 -fitand electron fit. Distribution of these three variables are shown in Figure 6.2 for each energy
region starting from 250 MeV/c. The ⇡ 0 -like selection likelihood functions are defined as:L⇡ 0 =

3
X

log( Si (xi ))log( Bi (xi ))

(6.1)

i=1

where Si (xi ) and Bi (xi ) are CCQE (NC) events’ probability distribution function (PDF) forth
the i variable with observable x.
After separating the ⇡ 0 -like sample, the remaining e-like events are divided into two categories: 0-decay with no decay electrons and 1-decay with one or more decay electrons. As ⌫e
CCQE events are not expected to produce decay electrons, there is a high fraction of CCQE
interactions in the 0-decay sample.
As for FC sub-GeV single-ring µ-like sample, there are three categories according to thenumber of decay electrons reconstructed: 0-decay, 1-decay and 2-decay (2 or more decay electrons). As these events produce one muon, so they are expected to have one decay electron.
The fraction of CCQE events is high in 1-decay µ-like sample.
Table 6.1 shows the interaction mode compositions for FC sub-GeV samples.

Chapter 6

133

MC events
Q.E.
CC
single meson⌫e + ⌫ emulti ⇡
coherent ⇡CC ⌫µ + ⌫ µNCMC events
Q.E.
CC
single meson⌫µ + ⌫ µmulti ⇡
coherent ⇡CC ⌫e + ⌫ e
NCFC sub-GeV single-ring e-like
0-decay 1-decay⇡ 0 -like
1932.9
196.1
95.9
81.5%
3.7%
13.0%
12.5%
56.0%
6.0%
0.9%
10.4%
1.0%
1.0%
8.4%
0.4%
0.2%
12.1%
1.3%
3.9%
9.3%
78.2%
FC sub-GeV single-ring µ-like
0-decay 1-decay
2-decay
365.8
1651.9
130.2
63%
79.6%
5.4%
11.7%
15.1%
69.3%
0.9%
1.3%
14.8%
0.4%
1.1%
7.5%
5.4%
0.1%
0.05%
18.5%
2.8%
3.0%

FC sub-GeV
single-ring e-like
2224.9
72.2%
16.2%
1.0%
1.6%
1.3%
7.6%
FC sub-GeV
single-ring µ-like
2147.9
72.3%
17.8%
2.0%
1.4%
1.0%
5.5%

Table 6.1: Number of FC sub-GeV events and their compositions of neutrino interaction modes
for SK-IV MC scaled to the SK-IV livetime 1097 days with 2-flavor oscillation with (sin2 2✓,
3 eV2 ). The upper (lower) table shows e-like (µ-like) samples. The rightm ) = (1.0, 2.5⇥10column shows the composition before separation into sub-samples. After the separation, the
CCQE purity has increased.

Chapter 6

134

e2/(e1+e2) (signal)
0.2

0.3

0.15
0.15
0.2

0.1
0.1

0.1

0.05
0.05

0
0

0.2

0.4

0.6

0
100

120

140

160

180

200

0
-200

0

200

400

0

200

400

0

200

400

0

200

400

0.15
0.15
0.2
0.1
0.1
0.1

0.05
0.05

0
0

0.2

0.4

0.6

0
100

0.2

0.1

0.15

0.08

0.1

0.06

120

140

160

180

200

0
-200

0.1

0.05
0.04
0.05
0.02
0
0

0.2

0.4

0.6

100

120

140

160

180

200

0
-200

0.08
0.15
0.1
0.06
0.1
0.05

0.04
0.05
0.02
0
0

0.2

0.4

0.6

100

120

140

160

180

200

0
-200

Figure 6.2: Distributions used in the ⇡ 0 selection for CCQE (black solid line) and NC (reddashed line) in SK-IV FC sub-GeV single-ring e-like MC events, in four energy regions (fromtop to bottom): (1) 250 MeV/c  Pe < 400 MeV/c, (2) 400 MeV/c  Pe < 630 MeV/c, (3) 630
MeV/c  P < 1000 MeV/c, (4) 1000 MeV/c < Pe . In order to select ⇡ 0 -like and e-like events
more efficiently, an additional likelihood selection is applied for events with momentum above
250 MeV/c. The distributions of the three likelihood variables used are shown: energy fraction
carried by the second fitted ring (left column), the ⇡ 0 mass (middle column) and -likelihood(right column). All distributions are normalized to the histogram area.

Chapter 6

6.2.2

135

Selection for Multi-GeV Multi-Ring E-like Sample

In order to make CC ⌫e + ⌫ e -enriched Multi-GeV Mult-ring e-like (MME) sample, a selection
is applied based on likelihood method. Four variables are used in this likelihood:
1. PID likelihood
2. momentum fraction of the most energetic ring
3. number of decay electrons
4. the distance between the neutrino vertex and any muon decay electrons.
Distributions of these variables are shown in Figure 6.3 for each energy region. Likelihood
function is defined as:

LM M E =4
X

log( Si (xi ))log( Bi (xi ))

(6.2)

i=1

where Si xi ( B
i xi ) represents the CC⌫e and the background (CC⌫µ and NC) events’ PDF forththe i variable with observable xi .
This MME likelihood selection increases CC⌫e purity and decrease background events (CC⌫µ events and NC events) significantly. Table 6.2 shows the interaction mode compositions forthe cases without and with the MME likelihood.
Interaction mode composition
CC ⌫e + ⌫ e
CC ⌫µ + ⌫ µNCTotal number of events

Without LM M E
53.8%
21.5%
24.7%
654.3

With LM M E
75.5%
7.2%
17.3%
341.2

Table 6.2: Composition of interaction modes in case of with and without MME likelihood for
SK-IV MC scaled to 1097 days. Two-flavor oscillations (sin2 2✓, m ) = (1.0, 2.5⇥10 3 eV2 ) is
assumed.
Due to the improved electronics in SK-IV, decay-electron tagging efficiency is improved andthe charge dynamic range becomes wider in SK-IV, hence purity of CC⌫e + ⌫ e becomes higher
in multi-GeV multi-ring e-like samples, as shown in Table 6.2 and 6.3 for SK-IV and SK-III
respectively.

Chapter 6

136

Interaction mode composition
CC ⌫e + ⌫ e
CC ⌫µ + ⌫ µNCTotal number of events

Without LM M E
53.0 %
22.0%
25.0%
298.9

With LM M E
74.1%
8.9%
17.1%
154.3

Table 6.3: Composition of interaction modes in case of with and without MME likelihood for
SK-III MC scaled to 518.1 days. Two-flavor oscillations (sin2 2✓, m ) = (1.0, 2.5⇥10 3 eV2 ) is
assumed.

Chapter 6

137

0.25

0.03

0.2

10-1

> 20.0 GeV

0.15
0.15
0.02

0.1

10-2

0.1
0.01

0.05
0.05
10-3

10.0 - 20.0 GeV

0
-200

-150

-100

-50

0

0

0.08

0.4

0.06

0.3

0.04

0.2

0.02

0.1

5

10

0

0

0.5

1

0

0.5

1

0

0.5

1

0

0.5

1

0

0.5

1

0

0.5

1

10-1

0.1

10-2
0.05
10-3

0
-200

-150

-100

-50

0

0

0

5

10

0

0.5

1

0.15

0.1

5.0 - 10.0 GeV

0

0.08

0.4

0.06

0.3

0.04

0.2

10-1
0.1
10-2
0.05
10-3
0.1

0.02

0
-80

-60

-40

-20

0

0

0

5

10

0

0

0.5

1

10-4

0.06

1
0.1

2.5 - 5.0 GeV

10

-1

0.4

0.04

10-2
0.05
0.2

0.02

10-3

10-4
0
-40

-30

-20

-10

0

0

5

10

0

0

0.5

1

0.15

0.04

1.33 - 2.5 GeV

0

1

0.6

10-1

0.03
0.1
0.4

10-2
0.02
0.05

0.2

10-3

0.01
10-4
0
-20

-15

-10

PID of MER

-5

0

0

0

5

10

Number of decay electrons

0

0

0.5

momentum fractino of MER

1

distance from vertex to decay-e

Figure 6.3: Distributions of the variables used in the multi-GeV multi-ring e-like (MME) likelihood for SK-IV. From left to write: PID of the most energetic ring, number of decay electrons,
momentum fraction of the most energetic ring, distance between the vertex and the decay electron. From bottom to top, each plot represents an energy bins: (1) 1.33 - 2.5 GeV, (2) 2.5 - 5.0
GeV, (3) 5.0 - 10.0 GeV, (4) 10 - 20 GeV, (5) > 20 GeV. Energy bins correspond to the energyof the most energetic ring. Black solid line represents CC⌫e + ⌫ e and red dashed line representsCC⌫µ + ⌫ µ and NC. All distributions are normalized to the histogram area.

Chapter 6

6.2.3

138

Separation of CC⌫e and CC⌫ e events in Multi-GeV E-like samples

As the enhancement of electron neutrinos due to matter e↵ect (as shown in figure 1.4)
occurs for neutrinos in normal hierarchy and anti-neutrinos in inverted hierarchy, the sensitivity
to mass hierarchy can be improved by further dividing multi-GeV e-like samples into ⌫e and⌫ e -like samples.
Multi-GeV Single-Ring E-like Sample
Due to the improved electronics in SK-IV, the charge dynamic range becomes wider in SKIV, which helps to improve the PID, hence the purity of CC⌫e + ⌫ e becomes slightly higher in
SK-IV when compared to previous SK periods, as shown in Table 6.4.
Interaction mode composition
CC ⌫e + ⌫ e
CC ⌫µ + ⌫ µNCTotal number of events

SK-III
87.2%
3.4%
9.4%
286.5

With SK-IV
88.1%
3.1%
8.8%
568.7

Table 6.4: Composition of interaction modes for multi-GeV single-ring e-like samples for SK-III
and SK-IV respectively. Purity of CC⌫e + ⌫ e is higher for SK-IV due to the improvement inelectronics in the SK-IV period.
In order to improve the sensitivity to mass hierarchy, this multi-GeV single-ring e-like sample
is further divided into ⌫e -like and ⌫ e -like samples by considering the di↵erence in interactions of⌫e and ⌫ e in water:
For CC ⌫e :⌫e + n ! e + N 0 + pions (Total charge for the N’ and pion system is +1)⌫e + p ! e + N 0 + pions (Total charge for the N’ and pion system is +2)
where the pion further decays into
⇡+ !

µ+

+⌫ + µ- e+ + ⌫ e + ⌫ µ

producing decay electrons.
For CC⌫ e :⌫¯e + p ! e+ + N 0 + pions (Total charge for the N’ and pion system is 0)⌫¯e + n ! e+ + N 0 + pions (Total charge for the N’ and pion system is -1)
where ⇡ is more likely to be absorbed by water nuclei and hence no decay electrons are emitted.
Therefore, by considering the number of decay electrons, multi-GeV single-ring e-like sample isfurther separated as follows:
Number of decay electrons > 0 ! ⌫e -like
Number of decay electrons = 0 ! ⌫ e -like

Chapter 6

139

Figure 6.4 shows the number of decay electrons distribution for multi-GeV single-ring elike sample in SK-IV. The arrows represent the cut described above. Table 6.5 shows thecompositions of interaction modes in the separated multi-GeV single-ring ⌫e -like sample and⌫ e -like sample. The separation increases the composition of CC⌫e in ⌫e -like sample by 5.2% andthat of CC⌫ e in ⌫ e -like sample by 6.1%.

500

400

300

200

100

0
0

2

4

6

8

10

Number of decay electrons

Figure 6.4: Distribution of number of decay electrons in both data and MC for SK-IV multi-GeV
single-ring e-like sample. From bottom to top regions: CC⌫e (red), CC⌫ e (green), CC⌫µ + ⌫ µ(blue) and NC (yellow). Data is represented by black dots. MC is normalized to data livetime
1097 days. Two-flavor oscillations (sin2 2✓, m ) = (1.0, 2.5⇥10 3 eV2 ) is assumed. Events onleft hand side of the arrow is taken as ⌫ e -like sample and those on the right hand side of thearrow is taken as ⌫e -like sample.

Interaction modeCC ⌫eCC ⌫ eCC⌫µ + ⌫ µ
NC
Total

Without separation
57.6%
30.6%
3.2%
8.8%
568.7

⌫e -like
62.8%
10.8%
10.6%
15.7%
135.9

⌫ e -like
55.9%
36.7%
0.8%
6.6%
432.8

Table 6.5: Interaction mode composition of multi-GeV single-ring ⌫e -like and ⌫ e -like samples
for SK-IV. MC is normalized to data livetime 1097 days. Two-flavor oscillations (sin2 2✓, m )= (1.0, 2.5⇥10 3 eV2 ) is assumed. The separation increases the composition of CC⌫e in ⌫e -likesample by 5.2% and that of CC⌫ e in ⌫ e -like sample by 6.1%.

Multi-GeV Multi-Ring E-like Sample
Multi-GeV multi-ring e-like (MME) sample, after passing through the MME likelihood described in subsection 6.2.2, the sample is passed through another likelihood to separate into

Chapter 6

140⌫e -like and ⌫¯e -like samples. Unlike the multi-GeV single-ring e-like sample which is separated
into ⌫e -like and ⌫¯e -like samples with a simple cut depending on number of decay electrons, a likelihood method is used, the MME sample is passed through another likelihood for the separationof ⌫e -like and ⌫¯e -like events.
Figure 6.5 shows the interaction of an incoming ⌫e with initial energy E with water nuclei
and the outgoing electron carries energy E’. Feynman y of the interaction is defined as:
y=

E0

E
E

(6.3)

As Feynman y distribution is larger for CC⌫e than for CC⌫ e , the outgoing electrons from CC⌫einteraction with water nuclei tend to have smaller energy. This implies:• CC⌫e has smaller momentum fraction for the most energetic ring than CC⌫ e• CC⌫e has more number of rings than CC⌫ e• CC⌫e has larger transverse momentum than CC⌫ e• CC⌫e has more decay electrons than CC⌫ e

Figure 6.5: Diagram showing the interaction of ⌫e with the nuclei in water, and the Feynman y= (E’-E)/E where E is the initial energy of the incoming ⌫e and E’ is the energy of the outgoing
electron.
By using the above properties, a likelihood is newly made for the separation into ⌫e -like and⌫¯e -like samples. In the ⌫e and ⌫¯e separation, the following parameters are used.
1. number of decay electrons
2. number of rings
3. transverse momentumChapter 6

141

Figure 6.6 shows the distributions of the variables used in the ⌫e and ⌫ e separation likelihoodfor SK-IV in each energy region. Likelihood functions are defined as:L⌫e /⌫ e =3
Xlog( Si (xi ))log( Bi (xi ))

(6.4)

i=1

th
where Si (xi ) ( Bi (xi ))representsCC⌫e (CC⌫ e ) events’ PDF for the i variable with observable
xi .
Figure 6.7 shows the likelihood distribution for SK-IV MC events, in which the separation ismade at L⌫e /⌫ e = 0. Table 6.6 shows the compositions of interaction modes in the separated
multi-GeV multi-ring ⌫e -like sample and ⌫ e -like sample. The separation increases the composition of CC⌫e in ⌫e -like sample by 3.3% and that of CC⌫ e in ⌫ e -like sample by 1.5%.

Interaction modeCC ⌫eCC ⌫ eCC⌫µ + ⌫ µ
NC
Total

Without separation
56.1%
19.5%
7.2%
17.3%
341

⌫e -like
59.4%
17.9%
8.3%
14.4%
161.9

⌫ e -like
53.1%
21.0%
6.1%
19.8%
168.1

Table 6.6: Interaction mode composition of multi-GeV multi-ring ⌫e -like and ⌫ e -like samples for
SK-IV. MC is normalized to data livetime 1097 days. Two-flavor oscillations (sin2 2✓, m ) =(1.0, 2.5⇥10 3 eV2 ) is assumed. The separation increases the composition of CC⌫e in ⌫e -likesample by 3.3% and that of CC⌫ e in ⌫ e -like sample by 1.5%.

Chapter 6

142

0.6
0.4

> 20.0 GeV

0.3
0.3
0.4
0.2
0.2
0.2
0.1

0

0.1

0

5

10

0

0

2

4

6

0

0

0.5

1

0

0.5

1

0

0.5

1

0

0.5

1

0.5

1

0.3

10.0 - 20.0 GeV

0.4
0.4
0.3

0.2

0.2
0.2
0.1
0.1

0

0

5

10

0

0

2

4

6

0

0.3

5.0 - 10.0 GeV

0.4

0.4
0.2

0.2

0

0.2

0

5

10

0

0.1

0

2

4

6

0

0.8
0.2

2.5 - 5.0 GeV

0.6
0.4

0.15

0.4
0.1
0.2
0.2

0

0.05

0

5

10

0

0

2

4

6

0

0.6
0.6

1.33 - 2.5 GeV

0.15
0.4
0.4

0.1

0.2

0.2

0

0.05

0

5

10

number of decay electrons

0

0

2

4

Number of rings

6

0

0

Transverse momentum

Figure 6.6: Distributions of the variables used in the MME ⌫e and ⌫ e separation likelihood for
SK-IV. From left to right: number of decay electrons, number of rings, momentum fraction of
the most energetic ring. From bottom to top, there are five energy bins: (1) 1.33 - 2.5 GeV,
(2) 2.5 - 5.0 GeV, (3) 5.0 - 10.0 GeV, (4) 10 - 20 GeV, (5) > 20 GeV. Energy bins correspondto the energy of the most energetic ring. Black solid line represents CC⌫e and red dashed line
represents CC⌫ e . All distributions are normalized to the historgram area. Area of the histograms
are normalized to their corresponding areas.

Chapter 6

143

0.3

0.2

0.1

0

-0.4

-0.2

0

0.2

0.4

Likelihood

Figure 6.7: Distribution of the ⌫e and ⌫ e separation likelihood for SK-IV 500 years MC. Solidline represents CC ⌫e , dashed line represents CC⌫ e and dotted line represents NC events. The
arrow shows the cut which is made at 0. For events having L⌫e /⌫ e > 0, they are taken as ⌫e -likeevents, and for events having L⌫e /⌫ e < 0, they are taken as ⌫ e -like events.

Chapter 6

144

Figure 6.8: Summary of selection steps for the Multi-GeV e-like samples.
Summary for Multi-GeV E-like Samples Selection
Figure 6.8 shows the schematic diagram which summarizes the steps for selection of MultiGeV e-like samples.

6.2.4

Data Set For the Oscillation Analysis

In summary, data sets for the analysis are listed as follows:
1. FC sub-GeV single-ring e-like 0-decay electron sample
single-ring identified as e-likepe > 100 MeV/c
Evis < 1.33 GeV
without any decay electrons
2. FC sub-GeV single-ring e-like 1-decay electron sample
single-ring identified as e-likepe > 100 MeV/c
Evis < 1.33 GeV
with one decay electron
3. ⇡ 0 -like sample
single-ring identified as e-likepe > 100 MeV/c
Evis < 1.33 GeV
fitted as ⇡ 0 -like by ⇡ 0 -fitter and ⇡ 0 likelihood

4. FC sub-GeV single-ring µ-like 0-decay electron sample
single-ring identified as µ-like

Chapter 6

pµ > 200 MeV/c
Evis < 1.33 GeV
without any decay electrons
5. FC sub-GeV single-ring µ-like 1-decay electron sample
single-ring identified as µ-likepµ > 200 MeV/c
Evis < 1.33 GeV
with one decay electron
6. FC sub-GeV single-ring µ-like 2-decay electron sample
single-ring identified as µ-likepµ > 200 MeV/c
Evis < 1.33 GeV
with two or more decay electrons

7. FC sub-GeV 2-ring ⇡ 0 -like sample
Two rings are recognized and both of them are identified as e-likeNo decay electrons

8. FC multi-GeV single-ring ⌫e -like sample
single ring identified as e-like
Evis > 1.33 GeV
Number of decay electrons > 0
9. FC multi-GeV single-ring ⌫ e -like sample
single ring identified as e-like
Evis > 1.33 GeV
Number of decay electrons = 0

10. FC multi-GeV single-ring µ-like sample
single ring idenfied as µ-like
Evis > 1.33 GeV

11. FC multi-GeV multi-ring ⌫e -like sample
Most energetic ring identified as e-like
Evis > 1.33 GeV
Events that passed through the MME likelihood cut
Events that have ⌫e and ⌫ e separation likelihood > 0145

Chapter 6

146

12. FC multi-GeV multi-ring ⌫ e -like sample
Most energetic ring identified as e-like
Evis > 1.33 GeV
Events that passed through the MME likelihood cut
Events that have ⌫e and ⌫ e separation likelihood < 0

13. FC multi-GeV multi-ring µ-like sample
Most energetic ring identified as µ-like
Momentum pµ > 600 MeV/c
Evis > 600 MeV

14. PC OD stopping sample
Maximum number of p.e.s observed in OD in a sliding 500 nsec time window from
nsec to +600 nsec (PEanti ) is less than PEexp /1.5.
The most energetic ring or the second one is identified as µ-like.

400

15. PC OD through-going sample
Remaining of the PC events after the OD stopping events are selected in the above step.

16. Upward stopping muon sample
Classified as stopping event by fitter
Number of hit OD PMTs within 8 m from the exit point (NHITEX) < 10 for SK-I, SK-III
and SK-IV (16 for SK-II)
Fitted momentum 1.6 GeV/c
17. Upward muon through-going showering sample
Classified as through-going event by fitter
Number of hit OD PMTs within 8 m from the exit point (NHITEX) 10 for SK-I, SK-III
and SK-IV (16 for SK-II)
Distance from the entrance point to the exit point 7 m.
Di↵erence between average corrected charge and the expected charge for non-showeringmuon ( (Q) = <Q> ql ) > 0.5 when 2 valuable for showering selection 50(Q) > 4.5 - 0.08 2 when 2 < 50.
18. Upward muon through-going non-showering sample
Classified as through-going event by fitter
Number of hit OD PMTs within 8 m from the exit point (NHITEX)and SK-IV (16 for SK-II)
Distance from the entrance point to the exit point 7 m.
Remaining of the UPMU events after the showering event selection.
In this analysis, FC, PC and UPMU data during:

10 for SK-I, SK-III

Chapter 6147• SK-I, with livetime of 1489.2 days for FC and PC, 1645.9 days for UPMU
• SK-II, with livetime of 798.6 days for FC and PC, 827.7 days for UPMU
• SK-III, with livetime of 518.1 days for FC and PC, 635.6 days for UPMU
• SK-IV with livetime of 1097 days for FC and PC, 1096.7 days for UPMU
The livetime is determined by the requirement of good detector condition. As the reconstructionof long pathlength muon for UPMU is less a↵ected by detector condition, the livetime is in
general longer than that for FC and PC. The amount of simulated MC atmospheric neutrino
events are equivalent to SK exposure of 500 years for FC and PC, 200 years for UPMU for each
SK run period. Table 6.7 summarizes the number of events in each categorized event sample forboth data and MC in each SK period.

Chapter 6

FC sub-GeV
single-ringe-like0-decay
1-decay⇡ 0 -likeµ-like0-decay
1-decay
2-decay
2-ring ⇡ 0 -likeFC multi-GeV
single-ring⌫e -like⌫ e -likeµ-likemulti-ring⌫e -like⌫ e -likeµ-likePC
OD stopping
OD through-going
UPMU
stopping
non-showering
showering

148

SK-I
Data
MC

SK-II
Data
MC

SK-III
Data MC

SK-IV
Data
MC

2992
301
176

2705.4
248.1
160.0

1573
172
111

1445.4
138.9
96.3

1092
118
58

945.3
85.3
53.8

2098
243
116

1934.9
198.4
96.2

1025
2012
147
524

893.7
1883.0
130.4
492.8

561
1037
86
266

501.9
1006.7
71.3
259.8

336
742
61
182

311.8
664.1
46.6
172.2

405
1833
174
380

366.3
1654.1
132.2
355.9

191
665
712

152.8
656.2
775.3

79
317
400

78.4
349.5
415.7

68
206
238

54.9
231.6
266.4

156
423
420

135.9
432.8
554.8

216
227
603

224.7
219.7
640.1

143
134
337

121.9
121.1
337.0

65
80
228

81.8
72.4
231.4

175
212
479

161.9
179.1
499.0

143
759

141.4
772.4

77
350

71.3
377.3

54
290

52.0
304.0

109
626

115.0
629.9

435.9
1564.4271.6

388.3
1306.3
285.1

241.2
725.3
108.1

210.1
687.7
106.9

193.7
612.9
110

210
486.3
123.2

286.7
1047.6
282.9

248.7
847.6
f 209.2

Table 6.7: Summary of number of events for both data and MC in each SK run period. Numberof MC events is normalized by data livetime, assuming 2-flavor mixing at ( m223 , sin2 ✓23 ) =(2.1 ⇥ 10 3 , 0.5).

Chapter 6

149

Figure 6.9: Flux ratio of electron neutrino for oscillations with parameters ( m212 , m232 , sin2 ✓12 ,sin2 ✓23 , sin2 ✓13 , CP ) = (7.6 ⇥ 10 5 , 2.4 ⇥ 10 3 , 0.31, 0.5, 0.025, 220 ). The flux ratio is thecomparison between flux with and without oscillation.

6.3

Oscillation E↵ect

As discussed in Section 1.3, electron neutrino oscillation is observable by sub-dominant oscillation. Figure 6.9 shows the expected flux ratio at the detector for electron neutrino as afunction of zenith angle (y-axis) and energy (x-axis). The expected events are calculated withboth disappearance and appearance oscillation probabilities. Figure 6.9 shows flux ratio forelectron neutrino at ( m212 , m232 , sin2 ✓12 , sin2 ✓23 , sin2 ✓13 , CP ) = (7.7 ⇥ 10 5 , 2.4 ⇥ 10 3 ,0.31, 0.5, 0.025, 220 ). For electron neutrino, the oscillation e↵ect due to ✓13 is dominant froma few GeV to 10 GeV.
In multi-GeV region, resonance enhancement of ⌫µ ! ⌫e (⌫ µ ! ⌫ e ) oscillation due to nonzero ✓13 is expected for normal hierarchy (inverted hierarchy) in the upward direction, while
there is no enhancement for inverted hierarchy (normal hierarchy). Therefore, by separation
the multi-GeV e-like samples into ⌫e -like and ⌫ e -like samples (as described in section 6.2.3), theenhancement is expected to be seen in ⌫e -like samples in case of normal hierarchy, while theenhancement is expected to be seen in ⌫ e -like samples in case of normal hierarchy. Hence bytagging the increase in number of upward events in these samples, mass hierarchy can be probed.
Figure 6.10 shows the zenith angle distribution of ratio of number of events with oscillation (at( m12 , m232 , sin2 ✓12 , sin2 ✓13 , CP ) = (7.6 ⇥ 10 5 eV2 , 2.4⇥ 10 3 eV2 , 0.31, 0.025, 40 )) toChapter 6150

that without oscillation. The enhancement can be seen in the upward direction (-1 < cos ✓ <0), and a sizable di↵erence can be seen between the normal hierarchy (dashed line) and theinverted hierarchy (solid line). Moreover, as the absolute cross section and d /dy (where y is
Feynman y = (E⌫ Elepton )/E⌫ ) are di↵erent between neutrinos and anti-neutrinos, upward
electron neutrino appearance observed in the detector is larger in normal hierarchy case thanthat in the inverted hierarchy case as shown by the dashed lines (normal hierarchy) being above
the solid line (inverted hierarchy) in Figure 6.10.
CP phase also change the sub-dominant e↵ects. Figure 6.11 shows the zenith angle distributions of ratio of number of events with to that without oscillations at three di↵erent CP
phase values. The plots show that the enhancement is dependent on the value of CP . Hence bystudying the excess in upward direction in sub-GeV e-like 0-decaye and multi-GeV elike samples,
CP may also be probed.
In this analysis, an algebraical computation for neutrino oscillation analysis described in[139] is used. Earth density is mesaured by seismic velocity observation. Density described in[140] is used.

Chapter 6

151

1.2
s223=0.4, inverted hierarchys223=0.5, inverted hierarchys223=0.6, inverted hierarchys223=0.4, normal hierarchys223=0.5, normal hierarchys223=0.6, normal hierarchy

N e/Ne0

1.1Multi-GeV ν e-like

1

0.9
-1

-0.5

0

0.5

1

cos Θ
1.2
s223=0.4, inverted hierarchys223=0.5, inverted hierarchys223=0.6, inverted hierarchys223=0.4, normal hierarchys223=0.5, normal hierarchys223=0.6, normal hierarchy

N e/Ne0

1.1Multi-GeV anti-ν e-like

1

0.9
-1

-0.5

0

0.5

1

cos Θ

Figure 6.10: Zenith angle distribution for multi-GeV single-ring + multi-ring ⌫e -like (top) andmulti-GeV single-ring + multi-ring ⌫ e -like (bottom) samples in SK-IV. The y-axis shows thenumber of events with oscillations divided by the number of events without oscillations. Oscillation paramters are ( m212 , m232 , sin2 ✓12 , sin2 ✓13 , CP ) = (7.6 ⇥ 10 5 eV2 , 2.4⇥ 10 3 eV2 ,0.31, 0.025, 40 ). Oscillations at di↵erent values of sin2 ✓23 = 0.4 (green), 0.5 (red), 0.6 (blue)
for normal hierarchy (dashed line) and inverted hierarchy (solid line) are shown.

Chapter 6

152

1.1

1.1

1.15
Multi-GeV single-ring nue

Sub-GeV e-like 0-decaye

Multi-GeV single-ring anti-nue

1.1
1.05
1.05
1.05
1
1
1
0.95
0.95

0.9
-1

-0.5

0

0.5

1

0.9
-1

-0.5

0

0.5

1

cos θ
1.1

-0.5

0

0.5

1

cos θ

1.1
Multi-GeV multi-ring anti-nue

Multi-GeV multi-ring nue
1.05

1.05

1

1

0.95

0.95

0.9
-1

0.95
-1

-0.5

0

0.5

1

0.9
-1

-0.5

0

0.5

1

Figure 6.11: Zenith angle distributions of the ratio of number of events with oscilation to thatwithout oscillation, assuming oscillation parameters ( m212 , m232 , sin2 ✓12 , sin2 ✓23 , sin2 ✓13 ) =(7.7 ⇥ 10 5 , 2.66 ⇥ 10 3 , 0.30, 0.425, 0.025) in normal hierarchy case for SK-IV. Cases with
CP = 80 (blue dotted line), 220 (red solid line) and 300 (green dashed line) are shown. The
error bars represent the size of the error with SK-I + SK-II + SK-III + SK-IV data livetime.

Chapter 6

6.4

153

Analysis Method

In order to examine the neutrino oscillation parameters, data are compared with MC expectation by using a pull 2 method based on Poisson probability distribution.

6.4.1

Definition of2

Assume that there are n bins for data, the likelihood can be defined as [136]:
exp

L(N

exp

,N

obs

)=

n
Y
e Ni

i=1

Niexp Ni
Niobs !

obs

(6.5)

where
Niobs is the number of observed events in the i-th bin
Niexp is the expected number of events (MC) in the i-th bin.
2 is given by the log likelihood ratio:
2

⌘

n
X
L(N exp , N obs )2 ln= 2 (Niexp
obs
obs
L(N , N )i=1

Niobs + Niobs ln

Niobs
Niexp

(6.6)

The e↵ect of systematic uncertainties are included into the bin with ”pull approach” [151].
The number of expected events in a bin is allowed to vary according to the systematic uncertainties, so Niexp is replaced as:
Niexp ! Niexp (1 +mXfji ✏j )

(6.7)

j

where
✏j is a variation of j-th systematic uncertainty.
fji is the fractional change of the event rate in the i-th bin due to the j-th systematic uncertainty.
m is the total number of systematic uncertainties.
For each uncertainty, j is determined during the fitting to 2 and the quantity ✏j / jsys follow
Gaussian distribution, where jsys is the 1 value of the j-th systematic uncertainty. In order toconstrain the range of j , an additional term
m
X
j

(

✏j

sys )
j

2

(6.8)

is added to equation 6.6.
Therefore, 2 is defined as:

2

⌘2

X
i

[(Niexp (1 +Xjfji ✏j )Niobs ) + Niobs ln(

X ✏j
Niobs
P i )] +( sys )2exp
Ni (1 + j fj ✏j )
j
j

(6.9)

Chapter 6

154

Explanation about the systematic errors would be presented in the next section.
At the minimum 2 location for each combination of oscillation parameters, each of theparameter ✏j is estimated according to @ 2 /@✏j = 0. Hence the minimization of 2 in equation
6.9 is equivalent to solving the following equation for every ✏j [141]:
X

(

(
j=1

X

X

X

1
Njobs fji fki )✏j =((1 + (
fji ✏j )2
sys 2 jk +
)
j
i=1
i=1
j=1

...)Niobs

Niexp )fki

(6.10)In order to ensure a proper function in equation 6.9, the binning is defined such that there
are at least 5 expected MC events for each bin after scaling to data livetime. The binningis defined by zenith angle and momentum information as shown in figure 6.12. All sub-GeV
samples are divided into 5 momentum bins. Sub-GeV e-like 0-decay, µ-like 0-decay and 1-decay
samples are divided into 10 zenith angle bins. All multi-GeV samples (FC, PC and UPMU)
are all divided into 10 zenith angle bins. The total number of bins is 480 for each SK-I, SK-II,
SK-III and SK-IV period.
SK-I, SK-II, SK-III and SK-IV bins are merged after evaluating the systematic error parameters by equation 6.10:

Niexp (1 +Xj

Niobs !
fji ✏j )!

X

obs
Ni,SKniXnexp
Ni,SKn(1 +X

fji ✏j )jSystematic error coefficients for each SK period are computed and their e↵ects specific toeach SK period are properly taken into account.

6.5

Systematic Uncertainties

Details of systematic error terms and best fit results of the error parameter ✏j is describedin this section.
151 systematic error parameters are used in this analysis, among which 19 are errors related
to neutrino flux model, 20 are related to neutrino interactions, 17 x 4 = 68 are related to event
reduction and 11 x 4 = 44 are related to event reconstruction and selection.
The best fit systematic uncertainty parameters ✏j for the fit in normal hierarchy case aresummarized with the estimated 1 value ( jsys ) for each systematic error term from tables 6.8
to 6.14.

Chapter 6

155

Upmu
shower
FC
FC
FC
FC
FC
(1x10 bins)
1ring
Multi-ring Multi-ring Multi-ring
1ring
1ring
PC
PC
Upmu
µ-likeνe-likeanti-νe-likeνe-likeanti-νe-likeµ-likestop
throughstop(4x10 bins) (4x10 bins) (2x10 bins) (3x10 bins) (3x10 bins) (4x10 bins)(2x10 bins) (4x10 bins) (3x10 bins)

Log10 P (MeV/c)

5.0

Upmu
non-shower(1x10 bins)

4.0
3.7
3.4
3.0

3.2
3.0
2.8
2.6
2.4
2.0

FC Sub-GeV
1-ring e-like

FC Sub-GeV
1-ring π0-like

0-decay
1-decay(5x10 bins) (5x1 bins)(5x1 bins)

FC Sub-GeV 1-ring µ-like
0-decay
1-decay
2-decay(5x10 bins) (5x10 bins) (5x1 bins)
3.2
3.0
2.8
2.6
2.4
2.2

FC Sub-GeV
2-ring π0-like

3.2

2-decay(5x1 bins)

2.8
2.6
2.4
2.2
2.0

Figure 6.12: Definition of the binning used in the neutrino oscillation analysis. FC: 370 bins,
PC: 60 bins, UPMU: 50 bins. White boxes indicate that momentum bin is divided into 10 zenith
angle bins between -1 < cos ✓ < 1 for FC and PC, -1 < cos ✓ < 0 for UPMU. Shaded boxes
indicate the momentum bin is not further divided into zenith angle bins. FC single-ring e-like
1-decay, ⇡ 0 -like, µ-like 2-decay samples and 2-ring ⇡ 0 sample are not divided into zenith anglebins.

Chapter 6

Systematic error related to neutrino flux
Flux normalization
E⌫ < 1 GeV
E⌫ > 1 GeV(⌫µ + ⌫µ )/(⌫e + ⌫e )E⌫ < 1 GeV
1 < E⌫ < 10 GeV
E⌫ > 10 GeV⌫e /⌫eE⌫ < 1 GeV
1 < E⌫ < 10 GeV
E⌫ > 10 GeV⌫µ /⌫µE⌫ < 1 GeV
1 < E⌫ < 10 GeV
E⌫ > 10 GeV
K/⇡ ratio in flux calculation
Neutrino path length
Sample-by-sample normalization FC Multi-GeV
PC + UPMU stop
Matter e↵ect
Solar Activity
SK-I
SK-II
SK-III
SK-IV

156

Fit value (%)
32.50
18.33

(%)
25a
15b

-0.43
-1.46
0.83

2
3
5c

3.0
1.62
-0.36

5
5
8d

0.06
0.64
-0.08
-10.41
0.15
-6.92
-6.68
-0.17
2.23
19.28
3.50
-0.42

2
6
15e
10f
10
5
5
1.0g
20
50
20
10

a

Uncertainty linearly decreases with log E⌫ from 25% (0.1 GeV) to 7% (1 GeV).
Uncertainty is 7% upto 10 GeV, linearly increases with log E⌫ from 7% (10 GeV) to 12% (100 GeV) and thenfrom 12% (100 GeV) to 20% (1 TeV).
c
Uncertainty linearly increases with log E⌫ from 5%(30 GeV) to 30%(1 TeV).
dUncertainty linearly increases with log E⌫ from 8%(100 GeV) to 20%(1 TeV).
e
Uncertainty linearly increases with log E⌫ from 6% (50 GeV) to 40% (1 TeV).
f
Uncertainty increases linearly from 5% to 20% between 100 GeV and 1 TeV.
g
Di↵erence of electron density in the core region from heavy element case is set to 1.0.
b

Table 6.8: Systematic errors related to neutrino flux calculation. They (except for solar activities) are common in all SK periods. The second column shows the best fit value of the systematic
error parameter ✏i and the third column shows the estimated 1 error size.

Chapter 6

Systematic error in neutrino interaction
CCQE cross section
CCQE ⌫ i /⌫i (i = e, µ) ratio
CCQE ⌫µ / ⌫e ratio
MA in CCQE and single meson production
Single meson production cross section
Single ⇡ production, ⇡ 0 /⇡ ± ratio
Single ⇡ production, ⌫i /⌫i (i=e,µ) ratio
DIS cross section (E⌫ < 10 GeV)DIS cross section
Q2 spectrum in DIS (W < 1.3 GeV/c2 )Q2 spectrum in DIS (W > 1.3 GeV/c2 )Coherent ⇡ production
NC/CC ratio
Nuclear e↵ect in 1 6O nucleus⌫⌧ contamination
NC in FC µ-like (hadron simulation)
Decay electron taggig (⇡ + interaction)f

157

Fit value(%)
11.1
14.54
2.26
-0.35
-2.80
-42.39
-4.22
-4.47
5.11
3.33
0.00
-30.6
14.3
-2.32
-5.5
2.0
-4.79

(%)
1.0a
1.0a
1.0a
10
20
40
1.0b
1.0c
5
1.0d
1.0d
100
20
1.0e
30
10
10

a

Di↵erence from the Nieves model is set to 1.0.
Di↵erence from the Hernandez model is set to 1.0.
cDi↵erence from CKMT parametrization is set to 1.0.
dDi↵erence from original GRV98 model is set to 1.0.
eUncertainty depends on the di↵erence between NEUT model and external data.
f
They are assumed to be identical and fully correlated for each SK periodbTable 6.9: Systematic errors related to neutrino interactions. They are common between all SK
periods. The second column shows the best fit value of the systematic error parameter ✏i andthe third column shows the estimated 1 error size.

Chapter 6

Systematic error in event reduction
FC/PC separation
FC reduction
PC reductionNon-⌫ BG (e-like)
Sub-GeV e-like
Multi-GeV e-likeNon-⌫ BG (µ-like)
Sub-GeV µ-like
Multi-GeV µ-likePC
PC-stop/throughTop
separation
BarrelBottomFiducial volume
UPMU reduction
stopping
Through-going
UPMU stop/through separation
UPMU through-going showering separation
Energy cut for UPMU stop
Pathlength cut for upward through-going µ
BG subtraction for UPMU

158

SK-I
fit
-0.06
0.6
0.008 0.2
-0.55
2.4
0.03
0.5
0.018 0.1
0.018 0.1
0.018 0.1
0.036 0.2
31.6
46.1

SK-II
fit
0.06
0.5
0.007 0.2
-2.46
4.8
0.004 0.2
0.01
0.1
0.01
0.10
0.01
0.10
0.07
0.7
-7.0
19.37

SK-III
fit
-0.09
0.9
-0.03
0.8
-0.02
0.5
-0.004 0.1
0.008 0.4
0.002 0.1
0.004 0.2
0.036 1.8
-19.22 86.6

SK-IV
fit
0.00
0.38
0.006 0.3
-0.11
1.0
-0.003 0.1-0.007 0.1-0.007 0.1-0.056 0.8
-0.03
0.49
2.48
40.44

-1.17
-27.7
0.37
-0.091-0.065
0.006
3.0
-0.03
-0.44
5.83

-3.90
2.41
-0.04
-0.028
-0.02
0.002
0.14
-0.03
-0.33
-8.84

-6.61
-0.47
-0.57
0.084
0.06
0.03
1.88
0.75
-0.21
8.79

-0.95
-1.02
-0.18
0.065
0.039
-0.06
0.59
-0.03
1.0
9,17

7.01
22.7
2
0.7
0.5
0.4
3.4
-0.9
-1.5
20

9.44
12.9
2
0.7
0.5
0.6
4.4
-1.3
-2.3
21

28.7
12.1
2
0.7
0.5
0.4
2.4
2.0
-2.8
20

Table 6.10: Systematic errors related to event reduction for each SK period. ”Fit” column shows
the best fit value of the systematic error parameter ✏i and ” ” column shows the estimated 1
error size.

8.54
6.8
2
0.5
0.3
0.6
3.0
1.7
-1.5
20

Chapter 6(SK-I) Systematic error in event reconstruction and selection
Ring separation< 400 MeV
e-likeµ-like> 400 MeV
e-likeµ-likeMulti-GeV
e-likeµ-likeMulti-ring sub-GeVµ-likeMulti-ring multi-GeV e-like
Multi-ring multi-GeV µ-likePID (single-ring) sub-GeV
e-likeµ-likeMulti-GeV
e-likeµ-likePID (multi-ring) sub-GeVµ-likeMulti-GeV
e-likeµ-likeEnergy calibration
Up/down asymmetry energy calibration
Multi-GeV single-ring e-like non CC⌫e BG
Multi-GeV multi-ring elike non CC⌫e BG
Multi-GeV multi-ring ⌫e and ⌫ e separation (likelihood)
Sub-GeV 2-ring ⇡ 0
Decay-e tagging (µ-decay)

159

Fit value(%)
1.51
0.46
0.26
0.46
2.42
1.12
-2.95
-2.03
-2.69
-0.01
0.01
0.02
-0.02
0.49
-0.21
0.37
-0.55
0.13
1.97
2.55
-4.17
-0.24
-3.43

(%)
2.3
0.7
0.4
0.7
3.7
1.7
-4.5
-3.1
-4.1
0.1
-0.1
-0.2
0.2
-3.9
1.7
-2.9
1.1
0.6
13.2
12.1
7.16
2.0
10

Table 6.11: Systematic errors related to event reconstruction and selection for SK-I. The second
column shows the best fit value of the systematic error parameter ✏i and the third column showsthe estimated 1 error size.

Chapter 6(SK-II) Systematic error in event reconstruction and selection
Ring separation< 400 MeV
e-likeµ-like> 400 MeV
e-likeµ-likeMulti-GeV
e-likeµ-likeMulti-ring sub-GeVµ-likeMulti-ring multi-GeV e-likeµ-likePID (single-ring) Sub-GeV
e-likeµ-likeMulti-GeV
e-likeµ-likePID (multi-ring) Sub-GeVµ-likeMulti-GeV
e-ilkeµ-likeEnergy calibration
Up/down asymmetry energy calibration
Multi-GeV single-ring e-like non CC⌫e BG
Multi-GeV multi-ring elike non CC⌫e BG
Multi-GeV multi-ring ⌫e and ⌫ e separation (likelihood)
Sub-GeV 2-ring ⇡ 0
Decay-e tagging (µ-decay)

160

Fit value(%)
-1.2
-2.12
-1.57
-0.65
-2.4
-1.57
7.57
1.75
0.74
0.075
-0.06
0.015
-0.015
0.80
0.65
-1.2
-0.42
-0.08
-7.26
-0.74
0.55
-0.71
-1.18

(%)
1.3
2.3
1.7
0.7
2.6
1.7
-8.2
-1.9
-0.8
0.5
-0.4
0.1
-0.1
2.2
1.8
-3.4
1.7
0.6
38.1
11.1
7.91
2.0
10

Table 6.12: Systematic errors related to event reconstruction and selection for SK-II. The second
column shows the best fit value of the systematic error parameter ✏i and the third column showsthe estimated 1 error size.

Chapter 6(SK-III) Systematic error in event reconstruction and selection
Ring separation< 400
e-likeµ-like> 400
e-likeµ-likeMulti-GeV
e-likeµ-likeMulti-ring sub-GeVµ-likeMulti-ring multi-GeV e-likeµ-likePID (single-ring) Sub-GeV
e-likeµ-likeMulti-GeV
e-likeµ-likePID (multi-ring) Sub-GeVµ-likeMulti-GeV
e-likeµ-likeEnergy calibration
Up/down asymmetry energy calibration
Multi-GeV single-ring e-like non CC⌫e BG
Multi-GeV multi-ring elike non CC⌫e BG
Multi-GeV multi-ring ⌫e and ⌫ e separation (likelihood)
Sub-GeV 2-ring ⇡ 0
Decay-e tagging (µ-decay)

161

Fit value(%)
0.54
0.9
0.39
0.18
0.39
0.3
0.78
-0.33
-0.63
0.03
-0.04
0.06
-0.06
-0.13
-0.10
-0.18
-0.34
0.18
1.65
-0.22
-6.86
-0.14
2.25

(%)
1.8
3.0
1.3
0.6
1.3
1.0
-2.6
-1.1
-2.1
0.3
-0.4
0.5
-0.5
3.1
2.5
4.5
2.7
1.3
26.7
11.4
7.68
1.0
10

Table 6.13: Systematic errors related to event reconstruction and selection for SK-III. The second
column shows the best fit value of the systematic error parameter ✏i and the third column showsthe estimated 1 error size.

Chapter 6(SK-IV) Systematic error in event reconstruction and selection
Ring separation< 400 MeV
e-likeµ-like> 400 MeV
e-likeµ-likeMulti-GeV
e-likeµ-likeMulti-ring sub-GeVµ-likeMulti-ring multi-GeV e-likeµ-likePID (single-ring)
Sub-GeV
e-likeµ-likeMulti-GeV
e-likeµ-likePID (multi-ring)
Sub-GeVµ-likeMulti-GeV
e-likeµ-likeEnergy calibration
Up/down asymmetry energy calibration
Multi-GeV single-ring e-like non CC⌫e BG
Multi-GeV multi-ring elike non CC⌫e BG
Multi-GeV multi-ring ⌫e and ⌫ e separation (likelihood)
Sub-GeV 2-ring ⇡ 0
Decay-e tagging (µ-decay)

162

Fit value(%)
-1.0
-1.89
0.63
0.38
0.63
-0.76
-1.45
-0.57
1.51
-0.09
0.09
-0.09
0.09
0.83
1.29
2.57
-0.23
0.02
-0.23
-2.25
-4.24
0.75
4.98

(%)
-1.6
-3.0
1.0
0.6
1.0
-1.2
-2.3
-0.9
2.4
0.4
-0.4
0.4
-0.4
2.2
3.4
6.8
2.3
0.3
17.6
11.6
6.82
1.8
10

Table 6.14: Systematic errors related to event reconstruction and selection for SK-IV. The second
column shows the best fit value of the systematic error parameter ✏i and the third column showsthe estimated 1 error size.

Chapter 6

6.5.1

163

Systematic Errors in Neutrino flux

Absolute Normalization

Relative Uncertainty

Uncertainty in the absolute atmospheric neutrino flux is calculated by Honda et al. [78],taking into account the atmospheric muon data ( ⇡ pion production in the hadronic interaction
model), Kaon production ( K ), hadronic interaction cross section ( ) and the atmosphericdensity profile ( air ) as shown in the upper plot of Figure 6.13.
The sum of dominant uncertainties which are ⇡ andis shown in the lower plot of Figure
6.13. It is taken as the absolute normalization uncertainty. The systematic error is divided into
two terms for two energy regions: E⌫ < 1 GeV and E⌫ > 1 GeV.
0.30

δ tot
δπ
δK
δσ
δ air

0.25
0.20
0.15
0.10
0.05
0

0

Absolute normalization (%)

10

10

1

10

2

10
E ν (GeV)

3

35
30
25
20
15
10
5
0
10−1

1

10

102

103
104
Eν (GeV)Figure 6.13: Contribution of each error source for atmospheric neutrino flux [78] is shown in theupper plot. Absolute normalization uncertainty due to only ⇡ (atmospheric muon data (pionproducion)) and(hadronic interaction mode) as a function of neutrino energy is shown in
the lower plot, which is considered to be the absolute normalization error. The kaon production
and air density uncertainties are separately taken into account in the systematic errors.

Flavor Ratio
Systematic uncertainty in the flavor ratio (⌫µ +⌫ µ )/(⌫e +⌫ e ) of the atmospheric neutrino flux
is estimated by comparing the Honda flux [78] with FLUKA [57] and Bartol [58] flux models.
Figure 6.14 shows the flavor ratio normalized by Honda flux model. Uncertainty estimated is 2%
for E⌫ < 1 GeV, 3% for 1 < E⌫ < 10 GeV, and 5% for 10 < E⌫ < 30 GeV. Above 30 GeV, theuncertainty increases as a function of logE⌫ from 5% to 30% up to 1 TeV. As the error sources

Chapter 6164(νµ+νµ)/(νe+νe) double ratio

in each energy region are di↵erent, the uncertainty is divided into three terms according to theenergy ranges: E⌫ < 1 GeV, 1 < E⌫ < 10 GeV , 10 < E⌫ .
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6 −1
10

1

10

102

103
Eν(GeV)

Figure 6.14: Flavor ratio (⌫µ +⌫ µ )/(⌫e +⌫ e ) of the atmospheric neutrino flux. Vertical axis stands
for the double ratio, FLUKA flux/Honda flux is represented by solid lien, Bartol flux/Honda
flux is represented by dashed line. The kink at 8 GeV in FLUKA flux due to technical treatment
is corrected in the application.

Chapter 6

165

Anti-Neutrino / Neutrino Ratio

1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5 −110νµ/ νµ double ratioνe/ νe double ratio

The systematic uncertainty in anti-neutrino to neutrino ratio comes from ⇡ + / ⇡ andK + /K ratio in hadronic interaction of the flux calculation. The uncertainty is estimated bycomparing Honda flux with FLUKA and Bartol flux models [78] [57] [58]. Figure 6.15 shows
anti-neutrino to neutrino ratio normalized by Honda flux for ⌫e and ⌫µ .
For ⌫ e / ⌫e , the estimated uncertainty is 5% in E⌫ < 1 GeV, 5% for 1 < E⌫ < 10 GeV, 8%
for 10 < E⌫ < 100 GeV, for above 100 GeV, the error increases with logE⌫ upto 30% (1 TeV).
For ⌫ µ / ⌫µ , the estimated uncertainty is 2% for E⌫ < 1 GeV, 6% for 1 <E⌫ < 10 GeV, 6%
for 10 < E⌫ < 50 GeV. For energy above 50 GeV, the error increases with log E⌫ upto 60 % (1
TeV).

1

10

102

103
Eν(GeV)

1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5 −1
10

1

10

102

103
Eν(GeV)

Figure 6.15: Anti-neutrino / neutrino double ratio for ⌫ e /⌫e (left) and ⌫ µ /⌫µ (right). Solid(dashed) line indicates the double ratio with FLUKA (Bartol) flux.

Up/Down Ratio
Neutrino flux has up/down asymmetry due to the geomagnetic field, but the uncertainty
in the low energy neutrino flux does not a↵ect the zenith angle distribution, as the up/down
asymmetry in low energy region is largely smeared by the poor angular correlation in neutrino
interactions. The systematic uncertainties of up/down ratio are estimated by comparison of
the zenith angle distributions with the three flux models [78] [57] [58] for each neutrino event
sample, as shown in table 6.15.
Horizontal/Vertical Ratio
The systematic uncertainty of horizontal/vertical ratio is caused mainly by the di↵erence the3-dimentional calculation method in each flux calculation model [78] [57] [58] below neutrinoenergy of 3 GeV. For neutrino energy above 3 GeV, the uncertainty is caused by the di↵erence inthe predicted K/⇡ ratio in hadronic interactions in the atmosphere. The systematic uncertainties
are estimated by comparing the zenith angle distributions using the three flux models for eachneutrino sample, as summarized in table 6.16.

Chapter 6166< 400 MeV

> 400 MeV

Multi-GeV
Multi-ring sub-GeV
Multi-ring multi-GeV

e-like 0-decayµ-like 0-decayµ-like 1-decay
e-like 0-decayµ-like 0-decayµ-like 1-decay
e-likeµ-likeµ-likee-likeµ-likePC

fit value (%)
-0.02
-0.2
-0.07
-0.2
-0.4
-0.1
-0.2
-0.04
-0.04
-0.07
-0.04
-0.04

1 (%)
0.1
1.1
0.3
0.8
1.7
0.5
0.7
0.2
0.2
0.3
0.2
0.2

Table 6.15: Estimated uncertainties of up/down ratio (1 values) for di↵erent event samples.
They are common for SK-I, SK-II, SK-III and SK-IV periods.

< 400 MeV

> 400 MeV

Multi-GeV
Multi-ring sub-GeV
Multi-ring multi-GeV
PC

e-like 0-decayµ-like 0-decayµ-like 1-decay
e-like 0-decayµ-like 0-decayµ-like 1-decay
e-likeµ-likeµ-likee-likeµ-likefit value (%)
0.02
0.07
0.02
0.3
0.3
0.4
0.7
0.5
0.3
0.5
0.3
0.4

1 (%)
0.1
0.3
0.1
1.4
1.4
1.9
3.2
2.3
1.3
2.8
1.5
1.7

Table 6.16: Estimated uncertainties of horizontal/vertical ratio (1 values) for di↵erent event
samples. They are common for SK-I, SK-II, SK-III and SK-IV periods.

167

path length (km)

Chapter 6

10

4

10

3

10

2

10

1

-0.2

0

0.2

0.4

0.6

0.8

1

Zenith angle (cosΘ)

Figure 6.16: Neutrino flight length as a function of zenith angle. Solid curve represents theflight length used in our analysis and dashed curve represents that for the compressed density
structure of atmosphere by 10%.
K/⇡ Ratio
Source of atmospheric neutrino is mainly ⇡ + and ⇡ in the neutrino energy range below
10 GeV. K’s contribute to neutrino production above a few tens of GeV. The uncertainty ofK/⇡ production ratio in cosmic ray interactions in atmosphere is taken as the systematic error.
According to the SPY experiment [142], the measurement of the K/⇡ ratio is obtained withan accuracy of 3%, where K and ⇡ momentum ranges from 7 GeV/c to 135 GeV/c. By using
the SPY experiment results and the correspondence between handron momentum and neutrinoenergy, the uncertainty of K/⇡ ratio is estimated to be 5% below E⌫ = 100 GeV and 20% aboveE⌫ = 1 TeV. The uncertainty increases linearly from 5% to 20% between 100 GeV and 1 TeV.
Neutrino Path Length
As atmospheric neutrino flight length is used in the calculation of neutrino oscillation probabilities, the uncertainty in neutrino production height has to be taken into account, especiallyfor downward going and horizontal going neutrinos. The uncertainty of neutrino production
height is related to the structure of atmosphere. The systematic uncertainty is estimated byusing the flux calculation and by changing the density structure of atmosphere by 10%. This
10% number is given by comparison between US-standard ’76 and MSISE90 [143]. Figure 6.16
shows the comparison of flight path lengths in two di↵erent density structures. The di↵erence ofthe comparison of neutrino flux due to the change of path length is taken to be the systematic
uncertainty.
Solar Activity
Primary cosmic ray flux is a↵ected by the solar activity in an 11 year cycle. ±1 year of
uncertainty is assigned to the modulation of solar activity, which corresponds to uncertainty inthe solar activity of 20% for SK-I, 50% for SK-II, 20% for SK-III and 10% for SK-IV.

168

Flux ratio

Chapter 6

1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6 −1
10

1

10

102

103
Eν(GeV)

Figure 6.17: Ratio of calculated neutrino flux in which the fluxes and Bartol (solid line) and
FLUKA (dashed line) are normalized by Honda flux.
Sample-by-sample Normalization
Energy dependence of the absolute normalization is di↵erent for each flux model, and thedi↵erence cannot be represented by a simple function, as shown in figure 6.17. 5% is assigned
as the relative normalization uncertainty for FC multi-GeV, PC and UPMU-stop samples.
Matter E↵ect
Electron density in the Earth a↵ects the calculation of neutrino oscillation probabilities.
Matter density is well measured by [140], but electron density depends on the chemical composition of the Earth. As the core region is expected to consist of mostly heavy elements (iron),the electron density is less than that of the light elements by about 6.8%. The di↵erence of
number of oscillated neutrino events from reduced electron density is taken as the uncertaintyof the matter e↵ect.

6.5.2

Systematic Errors in Neutrino Interactions

CCQE Cross Section
Cross section of CCQE for bound nucleon is calculated based on the Smith and Monitz
Fermi-gas model. The di↵erence from Nieves et al. model [91] is set to be the 1 uncertainty
for CCQE cross section.
The uncertainties in CCQE cross section (⌫/⌫) ratio and flavor ratio (⌫µ + ⌫ µ )/ (⌫e +⌫ e ) arealso set by the di↵erence from the Nieves model.
Figure 6.18 shows the calculated CCQE cross section ratio of Nieves model to Smith andMonitz model, as a function of neutrino energy for total cross section, flavor ratio and ⌫/⌫.

169

2

µ/e double ratio(Nieves/Smith-Moniz)

Ratio(Nieves/Smith-Moniz)

Chapter 6

1.8
1.6
1.4
1.2
1
0.8
0.6
0.4

1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6

0.2

! / ! double ratio(Nieves/Smith-Moniz)

1.5

0

50

100

150

200

250

300

350

400

450
500
E! (MeV)

0

50

100

150

200

250

300

350

400

450
500
E! (MeV)

0.5
0

50

100

150

200

250

300

350

400

450
500
E! (MeV)

1.4
1.2
1
0.8
0.6
0.4
0.2

Figure 6.18: Top left: ratio of CCQE cross section, Nieves model normalized by the Smith andMonitz model, for ⌫e + ⌫ e (solid line) and ⌫µ + ⌫ µ (dashed line). Top right: (⌫µ + ⌫ µ )/(⌫e + ⌫ e )ratio. Bottom: (⌫/⌫) ratio.
MA in CCQE and Single Meson Production
For MA in CCQE and single meson production, the di↵erence of Q2 spectrum between MA= 1.2 and MA = 1.1 is set to be the systematic uncertainty in order to take into account the
10% uncertainty of MA parameter.
Single Meson Production Cross Section
Uncertainty in the cross section of single meson production is estimated to be 20% by investigating the di↵erence between the expectation and experimental data of ⌫µ p ! µ p⇡ +observation. ⌫µ n ! µ p⇡ 0 interaction is less understood as there are only few experiment results. Hence the additional uncertainty is estimated by comparison with the Hernandez model[144]. The cross section ratio of ⌫µ n ! µ p⇡ 0 is shown the dashed line in the left panel of
figure 6.19. It is considered that ⇡ 0 production interaction has 40% uncertainty relative to the⇡ ± emitting interactions.
Uncertainty in the single meson production cross section of ⌫/⌫ ratio is also estimated byusing the di↵erence from the Hernandez model. Right panel of figure 6.19 shows the ⌫/⌫ ratioof CC cross section nomalized by Rein and Sehgal model.

170

2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

CC ν/ ν double ratio

CC νµ cross section ratio

Chapter 6

0.5

1

1.5

2

2.5

3

3.5

4

4.5 5
Eν (GeV)

2.2
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0

0.5

1

1.5

2

2.5

3

3.5

4

4.5 5
Eν (GeV)Figure 6.19: Predicted cross section (left) and the ⌫/⌫ ratio (right) of the Hernandez model
normalized by that of the Rein and Sehgal model for ⌫µ p ! µ p⇡ + (solid line), ⌫µ n ! µ p⇡ 0(dashed line) and ⌫µ n !µ p⇡ + (dotted line).

Chapter 6

171

Deep Inelastic Scattering Cross section
Calculation of cross section for DIS agrees with experimental measurement within 5% forenergies above 10 GeV, but the uncertainty is larger in lower energy region (< 10 GeV), theadditional error is estimated by the di↵erence from CKMT model [48].
Q2 Spectrum in Deep Inelastic Scattering
The uncertainty of Q2 spectrum is taken into account by comparing with the original GRV98
model. This uncertainty is separated into 2 terms for W < 1.3 GeV/c2 and W > 1.3 GeV/c2 .
Coherent Pion Production Cross Section
This uncertainty is obtained as 50% by the di↵erence between the Rein and Sehgal modeland experimental measurement for NC and CC ⌫e interactions. As CC ⌫µ coherent pion has noexperimental measurement, the uncertainty is taken to be 100%.
NC/CC ratio
This uncertainty is estimated to be 20%.
Nuclear E↵ect in 16 O Nucleus
The cross section of pion final state interactions inside nuclei are compared between simulation by NEUT and external experimental data [145]. The cross sections of these final stateinteractions in the simulation are varied with weighing method so that the di↵erence with theexperimental data is 1 as shown in figure 6.20. Then di↵erence between the minimum weightand maximum weight on each simulated event is used to compute the uncertainty on the nucleare↵ect in 16 O nucleus.

300

σAbsorption (mb)172

σQuasi-elastic (mb)

Chapter 6

qe1.0_ab1.1_cx1.0
qe0.6_ab0.7_cx0.5
qe0.6_ab0.7_cx1.6
qe0.7_ab1.6_cx0.4
qe0.7_ab1.6_cx1.6
qe1.3_ab0.6_cx0.6
qe1.3_ab0.7_cx1.6
qe1.5_ab1.5_cx0.4
qe1.6_ab1.6_cx1.6

200

200
150
100

100
50
0
0

200
400
600
π+ Initial Momentum (MeV/c)σSingle CX (mb)σReactive (mb)

0
0

400

200
400
600
π+ Initial Momentum (MeV/c)

100

50
200

0
0

200
400
600
π+ Initial Momentum (MeV/c)

0
0

200
400
600
π+ Initial Momentum (MeV/c)

Figure 6.20: ⇡ + scattering on carbon target in: quasi-elastic channel (top left), absorption
channel (top right), total reaction channel (bottom left) and single charge exchange channel(bottom right).

Chapter 6

6.5.3

173

Systematic Errors in Event Selection and Reconstruction

FC/PC separation
FC and PC events are separated by number of OD hit PMT clusters (NHITAC). The systematic uncertainty is estimated to be 0.6% for SK-I, 0.5% for SK-II, 0.9% for SK-III and 0.38%
for SK-IV by using the di↵erence between the observed data and the MC NHITAC distribution(Figure 5.13).
FC Reduction Related
The uncertainty in FC reduction efficiency is estimated to be 0.2%, for SK-I, 0.2% for SK-II,
0.8% for SK-III and 0.3% for SK-IV by comparing the distributions of each cut variable forobserved data and MC. Contamination of the background is estimated in Chapter 5.
PC Reduction Related
The uncertainty in PC reduction efficiency is estimated to be 2.5% for SK-I, 4.8% for SK-II, 0.5% for SK-III and 1.0% for SK-IV by comparing the distributions of each cut variables forobserved data and MC. The contamination of background is estimated in Chapter 5.
PC Stop/Through Separation
PC OD stopping and through-going events are separated by using the energy deposit in the
OD region (PEanti / PEexp ). The systematic uncertainty is estimated by the di↵erence between
the observed data and MC PEanti / PEexp distributions. The estimation is separately down for
OD bottom, barrel and top, as the OD calibration has di↵erence depending on the OD position.
UPMU Reduction Related
The uncertainty in UPMU reduction effiiciency is estimated by comparing the distributions ofeach cut variables for the observed data and MC. For stopping muons, the estimated uncertainty
are 0.7% for SK-I, 0.7% for SK-II, 0.7% for SK-III and 0.5% for SK-IV. For through-going muons,the estimated uncertainty are 0.5% for SK-I, 0.5% for SK-II, 0.5% for SK-III and 0.3% for SK-IV.
These uncertainties for stopping and through-going muons are assumed to be correlated.
The stopping and through-going muons are separated by using the number of hit OD PMTs
within 8 m from the projected exit point (NHITEX). The separation uncertainty are estimatedto be 0.4% for SK-I, 0.6% for SK-II, 0.4% for SK-III and 0.6% for SK-IV, by using the di↵erence
between observed data and MC NHITEX distributions (Figure 5.25).
UPMU Non-Showering/Showering Separation
UPMU non-showering and showering events are separated by using the di↵erence between the
average corrected charge and expected charge for non-showering events ( (Q)). The systematic
uncertainty is estimated by comparing the (Q) distribution (figure 5.29) for data and MC. The
estimated uncertainty are 3.4% for SK-I, 4.4% for SK-II , 2.4% for SK-III and 3.0% for SK-IV.

Chapter 6

174

0.4

0.35
10

0.3

Fraction

10

nch

0.5

Fraction

CHORUS (Column normalized)

nch

NEUT (Column normalized)

0.25
0.3
5

0.2

0.2
5

0.15
0.1

0.1
0

50

0

100

W2

0.05
0

50

0

100

W2

Figure 6.21: Pion multiplicity plotted as a function of W2 for NEUT simulation (left) and forthe data from the CHORUS experiment (right).
Ring Separation
Ring separation is done based on the ring-counting algorithm. The uncertainties are estimated by the di↵erence of likelihood distribution between observed data and MC for eachsample.
Particle Identification
Uncertainty in particle identification are estimated by the di↵erence of PID likelihood distribution between observed data and MC for each sample.
Energy Calibration
Energy scale uncertainties are estimated in Chapter 4. The systematic uncertainty for absolute energy scale is estimated to be 1.1% for SK-I, 1.7% for SK-II , 2.7% for SK-III and 2.3%
for SK-IV.
Multi-GeV Multi-Ring ⌫e and ⌫ e Separation
Multi-GeV multi-ring ⌫e and ⌫ e separation likelihood depends on number of decay electrons,
number of rings, etc, which depend on the number of pions which in turn depends on the crosssection of the final state interactions of pions and pion multiplicity.
In order to take into account the e↵ect of pion final state interaction cross sections, the MC
is compared with external experimental data, then the MC events are weighed so that the pion
final state interaction cross sections are 1 from the experimental data [145].
As for pion multiplicity, the NEUT model MC is compared with the experimental data fromthe CHORUS experiment [146]. The left and the right plots in figure 6.21 are compared andeach bin is weighed with the ratio of CHORUS/NEUT.
Each MC event is weighed with both final state interaction cross section weight and pion
multiplicity weight respectively, then the di↵erence in purities of CC⌫e events in the weighed
MC likelihood and the original likelihood are taken as the systematic uncertainty.
The estimated systematic uncertainties are 7.16% for SK-I, 7.91% for SK-II, 7.68% for SK-III
and 6.82% for SK-IV.

Chapter 6

175

The non-CC⌫e background uncertainty for multi-GeV multi-ring e-like events is also evaluated by the same method.
Single-Ring ⇡ 0 -Like Sample Selection
The uncertainty in the selection efficiency of FC sub-GeV single-ring ⇡ 0 -like events are estimated by comparing observed data with MC for the ⇡ 0 mass and likelihood distributions.
As the likelihood distributions are di↵erent for each momentum, the systematic uncertainties
are estimated for each momentum region. Part of the uncertainty in the selection efficiency
also comes from the absolute energy scale uncertainty. The combined uncertainties from thedi↵erence of distribution and shift of energy scale, are summarized in table 6.17.

100 < Pe < 250 MeV/c
250 < Pe < 400 MeV/c
400 < Pe < 630 MeV/c
630 < Pe < 1000 MeV/c1000 < Pe < 1330 MeV/c

SK-I
fit value
(%)
1.14
11.2
1.17
11.5
2.39
23.4
1.95
19.1
1.33
13.0

SK-II
fit value
(%)
1.91
7.5
2.26
8.9
4.45
17.5
2.72
10.7
2.82
11.1

SK-III
fit value
(%)
1.73
7.7
5.94
26.4
2.81
12.5
6.01
26.7
6.01
26.7

SK-IV
fit value
(%)
6.48
10.8
6.96
11.6
6.06
10.1
9.06
15.1
8.7
14.5

Table 6.17: 1 values for systematic uncertainties of single-ring ⇡ 0 -like events selection for SK-I,
SK-II, SK-III and SK-IV.

Two-Ring ⇡ 0 Selection
The uncertainty in selection of two-ring ⇡ events comes from ring-counting and PID. The
estimated uncertainties are 2.0% for SK-I, 2.0% for SK-II, 1.0% for SK-III and 1.8% for SK-IV.
Decay Electron Tagging
Systematic error in decay electron detection efficiency is estimated to be 1.5% by using
cosmic ray muons [147], which is assumed to be identical for each SK period. In addition, ⇡ +decay probability uncertainty is considered, as ⇡ + interacts in water depending on its energy,
the decay probability are between 3% to 13% depending on the ⇡ + momentum, hence thedecay probability also a↵ects the uncertainty for decay electron tagging by 0.3% to 0.6% for FC
sub-GeV single-ring e-like 0-decay sample.

Chapter 6

6.6

176

Sensitivity Study

Sensitivity study was carried out to study the e↵ect of the new binning of multi-GeV ⌫e and⌫ e -like samples.
The sensitivity study was carried out with SK-I, SK-II, SK-III and SK-IV MC corresponding
to data livetime, assuming the following set of parameters:m223
sin2 ✓23CP

2.66 ⇥ 10 3 eV20.5300

In case of assumed true normal hierarchy, the2 between normal and inverted hierarchies

is
2

=

2

(IH)

2

(N H) = 1.06

(6.11)

Figure 6.22 shows the 2 distribution for oscillation parameters sin2 ✓23 , m223 and CP for
both normal and inverted hierarchies for the sensitivity study.
The same study was carried out with assumed true inverted hierarchy, assuming the same
oscillation parameters as in Table 6.6.
2 between normal and inverted hierarchies
In case of assumed true inverted hierarchy, the
is
2

=

2

(IH)

2

(N H) = 0.90

(6.12)

The above sensitivity study on mass hierarchy was repeated for combined binnings of multiGeV ⌫e and ⌫ e -like samples, in which Multi-GeV single-ring ⌫e -like and ⌫¯e -like samples are
combined as Multi-GeV single-ring e-like sample, and Multi-GeV multi-ring ⌫e -like and ⌫¯e -likesamples are combined as Multi-GeV multi-ring e-like sample. The results are listed in Table
6.18. The results shows that the introduction of ⌫e -like and ⌫¯e -like samples in Multi-GeV e-likeevents improves the sensitivity to the mass hierarchy.

Combined ⌫e /⌫ e bins
Separated ⌫e /⌫ e bins

Assumed true inverted hierarchy
2 = 0.68
2 = 0.90

Assumed true normal hierarchy
2 = 0.81
2 = 1.06

2 ( = | 2
2
Table 6.18:
min(N H)min(IH) | ) estimated with and without multi-GeV 1-ringand multi-ring e-like samples separated into ⌫e and ⌫ e -like samples.

Chapter 6

177

Figure 6.22: The plots show 2 distributions for sin2 ✓23 (top left), m223 (top right) and CP(bottom) for the sensitivity study carried out with data livetime assuming normal hierarchy andoscillation parameters of (sin2 ✓12 , m212 , sin2 ✓23 , m232 , sin2 ✓13 , CP ) = (0.304, 7.66 ⇥ 10 5 ,0.5, 2.66 ⇥ 10 3 , 0.025, 300 ). Black line represents the distribution for normal hierarchy, andred line represents the distribution for inverted hierarchy.

Chapter 6

6.7

178

Results

In this analysis, the zenith angle and momentum distributions of di↵erent event samples are
fitted with the simulated distributions, considering all oscillation parameters, which includes
two mass di↵erences m212 , m232 , three mixing angles ✓12 , ✓23 , ✓13 , CP phase CP and mass
hierarchy cases: normal and inverted mass hierarchies.
2 is calculated in the fitting process by four dimensional oscillation parameter spaces of
2
m32 , ✓23 , ✓13 and CP . Parameters m212 and sin2 ✓12 are fixed to the values determined bysolar neutrino analyses, as they are already well constrained.
The examined oscillation parameter spaces are:• ( m212 , sin2 ✓12 ) = (7.65 ⇥ 10 5 , 0.304) (fixed)•

m232 = 1.33 ⇥ 10 3 to 10 ⇥ 10 3 (36 points)• sin2 ✓23 = 0.3 to 0.7 (17 points)• sin2 ✓13 = 0.0 to 0.2 (65 points) (2 studies: free and fixed)•

CP = 0

to 340 (18 points)

for normal hierarchy and inverted hierarchy.
As the mixing angle ✓13 is measured and well constrained by reactor experiments [6] [7], twostudies: one with ✓13 free as usual and another one with ✓13 fixed at the value obtained by thereactor experiments are carried out.
With Free ✓13With free ✓13 , in the normal hierarchy case, the best-fit is found at ( m232 , sin2 ✓23 , sin2 ✓13 ,
32CP ) = (2.51 ⇥ 10 , 0.55, 0.003, 260 ) with min /dof = 555.33 / 476. In the inverted hierarchy
case, the best-fit is found at ( m232 , sin2 ✓23 , sin2 ✓13 , CP ) = (2.51 ⇥ 10 3 , 0.55, 0.006, 240 )with 2min /dof = 554.91 / 476. Tables 6.19 and 6.20 summarize the 68% C.L. and 90% C.L.
allowed region for ( 32 , sin2 ✓23 , sin2 ✓13 , CP ).

m232 (eV2 )sin2 ✓23
sin2 ✓13
CP

Normal hierarchy
68% C.L.
90% C.L.
(2.22 < m232 < 2.81) ⇥ 10 3 (2.06 < m232 < 3.04) ⇥ 10 30.420 < sin2 ✓23 < 0.601
0.391 < sin2 ✓23 < 0.619
20.00037 < sin ✓13 < 0.015
sin2 ✓13 < 0.0655
all values allowed
all values allowed

Table 6.19: Results of the allowed region of each neutrino oscillation parameter at 68% and 90%
2
C.L. for normal hierarchy at fixed ✓13 , where 68% C.L. is defined as 2
min = 1.0 and 90%
22
C.L. is defined as
=
2.7.
min

Chapter 6179

m232 (eV2 )sin2 ✓23
sin2 ✓13
CPInverted hierarchy
68% C.L.
90% C.L.
2
3(2.32 < m32 < 2.79) ⇥ 10
(2.14 < m232 < 3.04) ⇥ 10 320.430 < sin ✓23 < 0.605
0.393 < sin2 ✓23 < 0.630
0.00067 < sin2 ✓13 < 0.034
sin2 ✓13 < 0.0944
all values allowed
all values allowed

Table 6.20: Results of the allowed region of each neutrino oscillation parameter at 68% and 90%
2
C.L. for inverted hierarchy at fixed ✓13 , where 68% C.L. is defined as 2
min = 1.0 and 90%
22
C.L. is defined as
min = 2.7.
With Fixed ✓13
With fixed ✓13 at sin2 ✓13 = 0.025 [6] [7], in the normal hierarchy case, the best-fit is found at( 23 , sin2 ✓32 , CP ) = (2.66 ⇥ 10 3 , 0.425, 300 ) with 2min /dof = 556.70 / 477. In the inverted
hierarchy case, the best-fit is found at ( 32 , sin2 ✓23 , sin2 ✓13 , CP ) = (2.66 ⇥ 10 3 , 0.575, 260 )with 2min /dof = 555.50 / 477. Summary of 68% C.L. and 90% C.L. allowed regions of each
oscillation parameter is summarized in Tables 6.21 and 6.22.
Zenith angle distributions of data and MC at best fit (both normal and inverted hierarchies)
with SK-I, SK-II, SK-III and SK-IV combined are shown from Figure 6.23 to figure 6.27.

m232 (eV2 )sin2 ✓23
CP

Normal hierarchy
68% C.L.
90% C.L.
2
3(2.26 m32 < 2.81) ⇥ 10
(2.11 m232 < 3.09) ⇥ 10 3
20.398 < sin ✓23 < 0.511
0.380 < sin2 ✓23 < 0.628
180 < CP < 340
all values allowed

Table 6.21: Results of the allowed region of each neutrino oscillation parameter at 68% C.L. and2
90% C.L. for normal hierarchy at fixed ✓13 , where 68% C.L. is defined as 2
min = 1.0 and2
2
90% C.L. is defined as
=
2.7.
min

m232 (eV2 )sin2 ✓23
CP

Inverted hierarchy
68% C.L.
90% C.L.
2
3(2.43 m32 < 2.83) ⇥ 10
(2.21 < m232 < 3.11) ⇥ 10 320.416 < sin ✓23 < 0.612
0.388 < sin2 ✓23 < 0.630
162 < CP < 330
all values allowed

Table 6.22: Results of the allowed region of each neutrino oscillation parameter at 68% C.L. and2
90% C.L. for inverted hierarchy at fixed ✓13 , where 68% C.L. is defined as 2
min = 1.0 and2
2
90% C.L. is defined as
=
2.7.
min

Chapter 6

180

300

300

300

300

300

250-400

100-250 (MeV/c)

400-6301000 < P

630-1000

200

200

200

200

200

100

100

100

100

100

0

0

-1 -0.5

0

0.5

1

0

-1 -0.5200-250 (MeV/c)

0

0.5

1

250-40000-1 -0.50

0.5

1-1 -0.5

400-63000.5

1-1 -0.5

100

100

100

100

50

50

50

50

50

0

-1 -0.5

0

0.5

1

0

-1 -0.50

0.5

1-1 -0.500250-400

200-250 (MeV/c)

0.5

1

0

0.5

1

-1 -0.5

630-1000

200

200

200

200

100

100

100

100

100

-1 -0.5

0

0

0

0.5

1

-1 -0.50

0.5

1-1 -0.5

0

0

0.5

1

-1 -0.5

0

0.5

1

0.5

1

1000 < P

200

0

1

0

-1 -0.5

400-6300.51000 < P

630-1000

100

0

0

0

0

0.5

1

-1 -0.5

0Figure 6.23: SK-I + SK-II + SK-III + SK-IV zenith angle distributions for FC events for data(line with error bar) and MC assuming no oscillation (grey dashed line), best fit results at normal
hierarchy (red solid line) and best fit results at inverted hierarchy (blue dashed line). From top
to the bottom: Sub-GeV 1-ring e-like 0-decay, Sub-GeV µ-like 0-decay, Sub-GeV µ-like 1-decay

Chapter 6

181

1-2.5(GeV/c)

2.5-5

5-10

10 < P

20

20

20

20

10

10

10

10

0

0

0

0

-1 -0.50

0.5

1-1 -0.50

0.5

1-1 -0.5

2.5 - 5

1-2.5 (GeV/c)

0

0.5

1

-1 -0.5

5 - 10

100

100

100

50

50

50

50

0

-1 -0.5

0

0.5

1

0

-1 -0.5

0

0.5

1

0

0.5

200

150

150

100

100

50

50

0

0.5

1

-1 -0.50

0.5

1

2.5 < P

200

-0.5

1

0

-1 -0.5

1 - 2.5 (GeV/c)

0
-1

0.5

10 < P

100

0

0

1

0
-1

-0.5

0

0.5

1

Figure 6.24: SK-I + SK-II + SK-III + SK-IV zenith angle distributions for FC events for data(line with error bar) and MC assuming no oscillation (grey dashed line), best fit results at normal
hierarchy (red solid line) and best fit results at inverted hierarchy (blue dashed line). From top
to the bottom: Multi-GeV 1-ring ⌫e -like, Multi-GeV 1-ring ⌫ e -like, Multi-GeV 1-ring µ-like.

Chapter 6

182

40 1-2.5 (GeV/c)

40

30

30

30

20

20

20

10

10

10

0
-1

-0.5

0

0.5

1

2.5-5

0
-1

-0.5

0

1-2.5 (GeV/c)

0.5

0
-1

1

40

20

20

20

0

0

100

0.5

P < 1 (GeV/c)

100

50

0

1 - 2.5

0.5

1-1 -0.5

0.5

100

1

-1 -0.5

2.5 - 5

1

0.5

1

0.5

1

-1 -0.5

5<P

50

0

0

0

100

50

0

0

0.5

0

-1 -0.5500-1 -0.5

1

0

5<P

40

0

-0.5

2.5 - 540

-1 -0.5

5<P

40

0

0

0.5

1

-1 -0.5

0

0.5

1Figure 6.25: SK-I + SK-II + SK-III + SK-IV zenith angle distributions for FC events for data(line with error bar) and MC assuming no oscillation (grey dashed line), best fit results at
normal hierarchy (red solid line) and best fit results at inverted hierarchy (blue dashed line).
From top to the bottom: Multi-GeV multi-ring ⌫e -like, Multi-GeV multi-ring ⌫ e -like, Multi-GeV
multi-ring µ-like.

Chapter 6

183

P < 2.5 GeV/c

P > 2.5

30

30

20

20

10

10

0
-1

-0.5

0

200

0.5

0
-1

1

200

P < 1 GeV/c

-0.5

0

200

1 - 2.5

0.5

200

5<P

2.5 - 5

150

150

150

150

100

100

100

100

50

50

50

50

0

0

0

0

-1 -0.50

0.5

1-1 -0.5

0

0.5

1

1

-1 -0.50

0.5

1-1 -0.5

0

0.5

1Figure 6.26: SK-I + SK-II + SK-III + SK-IV zenith angle distributions for FC events for data(line with error bar) and MC assuming no oscillation (grey dashed line), best fit results at normal
hierarchy (red solid line) and best fit results at inverted hierarchy (blue dashed line). From top
to the bottom: PC stop, PC throughChapter 6

184

150

150

150

2.5 -5

P < 2.5 GeV/c

5<P

100

100

100

50

50

50

0
-1

-0.8

-0.6

-0.4

-0.2

0

0
-1

-0.8

-0.6

-0.4

-0.2

0

0
-1

-0.8

-0.6

-0.4

-0.2

0

600

400

200

0
-1

-0.8

-0.6

-0.4

-0.2

0

-0.8

-0.6

-0.4

-0.2

0

150

100

50

0
-1

Figure 6.27: SK-I + SK-II + SK-III + SK-IV zenith angle distributions for FC events for data(line with error bar) and MC assuming no oscillation (grey dashed line), best fit results at normal
hierarchy (red solid line) and best fit results at inverted hierarchy (blue dashed line). From top
to the bottom: UPMU stopping, UPMU non-showering, UPMU showering.

Chapter 6

185

2
Figures 6.28 and 6.29 show the 2
min distributions as a function of oscillation parameters.
2
When finding the minimum
against one oscillation parameter, the other parameters are chosen
so that 2 is minimized. 90% C.L. is at 2 = 2min +2.7.

99% C.L.

99% C.L.

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

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

∗θ13 free

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

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

2 = 2
2Figure 6.28:
m232 , sin2 ✓23 , CP and sin2 ✓13 (inmin distributions as a function ofcase of free ✓13 ) for normal hierarchy case. Black line represents fixed sin2 ✓13 = 0.025 and blueline represents free ✓13 .

Figures 6.30 and 6.31 show the allowed regions drawn in two dimensions at 68%, 90% and
99% C.L..
2
2
2
min ( =
min(N H)min(IH) ) is found to be 556.7 (normal hierarchy) - 555.50 (inverted
hierarchy) = 1.2, with inverted hierarchy slightly preferred. However, the significance is still too
low to draw a conclusion on mass hierarchy. Figure 6.33 shows the distribution of 2 as afunction of CP for both normal and inverted hierarchies.
As for CP , with ✓13 fixed, the best fit values are 300 and 260 , and 180 < CP < 340 and
162 < CP < 330 at 68% C.L., for normal and inverted hierarchies respectively. However, the2 is still low to draw a conclusion on the value of
CP . More statistics is needed to improve
the sensitivity on CP .

Chapter 6

186

99% C.L.

99% C.L.

90% C.L.

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

68% C.L.

*θ13 free
90% C.L.
99% C.L.
68% C.L.
90% C.L.
68% C.L.

2 = 2
2Figure 6.29:
m232 , sin2 ✓23 , CP and sin2 ✓13 (inmin distributions as a function ofcase of free ✓13 ) for inverted hierarchy case. Black line represents fixed sin2 ✓13 = 0.025 and blueline represents free ✓13 .

Chapter 6

187

Normal hierarchy
0.01

0.008

0.008

∆ m232∆ m232

Normal hierarchy
0.01

0.006

0.004

0.006

0.004

0.002

0.002
0.85

0.9

0.95

1

0.3

0.4

2

0.5

0.6

0.7

2

sin 2θ23sin θ23

0.01

0.01

0.008

0.008

∆ m232∆ m232

Figure 6.30: Contours of m232 and sin2 2✓23 (left) and m232 and sin2 ✓23 (right) for the normal
mass hierarchy, at fixed ✓13 . The three lines indicate allowed regions at 68% C.L. 90% C.L. and
2 = 2.3, 4.6 and 9.2 respectively. The star shows the best
99% C.L. respectively, defined as
fit point.

0.006

0.006

0.004

0.004

0.002

0.002
0.85

0.9

0.95

sin2 2θ23

1

0.3

0.4

0.5

0.6

0.7

sin2 θ23

Figure 6.31: Contours of m232 and sin2 2✓23 (left) and m232 and sin2 ✓23 (right) for the inverted
mass hierarchy, at fixed ✓13 . The three lines indicate allowed regions at 68% C.L. 90% C.L. and
2 = 2.3, 4.6 and 9.2 respectively. The star shows the best
99% C.L. respectively, defined as
fit point.

Chapter 6

188

6.8

Discussion

6.8.1

Event Samples Contributing to the Sensitivity to Mass Hierarchy

This full neutrino oscillation analysis is the first attempt to use ⌫e and ⌫ e enriched samplesof the atmospheric neutrino data observed in the Super-Kamiokande to increase the sensitivityof mass hierarchy.
The sensitivity of mass hierarchy was carried out with fixed ✓13 (at sin2 ✓13 = 0.025), byassuming the ”true” mass hierarchy as the inverted hierarchy case, oscillation parameters ( 23 ,sin2 ✓32 , CP ) are assumed to be at (2.66 10 3 , 0.425, 300 ) (best fit of normal hierarchy with
fixed ✓13 ).

MGeV(1R(
elike(nuebar(

Prefer(NH(χ 2 best( IH ) − χ 2 best( NH )SK1+2+3+4
χ2NH0χ2IH=(1.2(

MGeV(MR(elike(
Prefer(IH(€

Sub$GeV((
mu$like(1(dcye(

MGeV(1R(
mulike(
Sub$GeV(

Mul+$GeV(

upmu(

Sensi8vity(study((Assumed(true(IH)(χ2NH0χ2IH=(0.90(€

Prefer(IH(χ2 best(NH ) − χ2 best(IH )MGeV(( MGeV((
1R(
MR(
elike( elike(

Figure 6.32: The numbers on the x-axis indicate the event sample types (18 types in total,
2 between norwith SK-I + SK-II + SK-III + SK-IV data). The upper panel shows themal hierarchy and inverted hierarchy contributed by each event sample of the data at the best
2 between normal hierarchy and infit oscillation parameters. The lower panel shows theverted hierarchy expected to be contributed by each event sample for MC, assuming the ”true”
oscillation parameters the same as the best fit parameters.

Chapter 6

189

Normal hierarchy
Inverted hierarchy

Figure 6.33: Distribution of
inverted hierarchy (red line).

2 as a function ofCP

for normal hierarchy (black line) and

2 is calculated by fitting the assumed inverted hierarchy case with normal hierarchy MC.
2 =2min(IH)2
min(N H) is 1.06 when normal hierarchy is assumed to be true, and 0.90

when inverted is assumed to be true (Table 6.18). Contribution from each event sample in thesensitivity study is shown in the lower panel of Figure 6.32. It can be seen that the event samples
2 are multi-GeV 1-ring ⌫ and ⌫ -like samples andwhich give the largest contributions ofeemulti-GeV multi-ring ⌫e and ⌫ e -like samples.
2 distribution for observed data
From the upper panel of Figure 6.32 which shows thefitted with MC, it can be seen that multi-GeV 1-ring ⌫ e like sample prefers the normal hierarchy,
while multi-GeV 1-ring µ-like sample prefers the inverted hierarchy. As mention in the previous
section, with ✓13 fixed (sin2 ✓13 = 0.025), 2min for normal hierarchy is 556.70, while 2min for
2 (=
2
2
2
inverted hierarchy is 555.50. Hencemin(N H)min(IH) ) = 1.2. As min is smaller
for the inverted hierarchy, it is slightly favoured, but the significance is still low to draw aconclusion. The distribution of 2 of both normal and inverted hierarchies as a function of CP
is shown in Figure 6.33, the di↵erence between the 2 of normal and inverted hierarchies atdi↵erent values of CP can be seen from the plot. It is expected that with more statistics, andfurther improved purity of CC ⌫e and CC ⌫ e in e-like samples, the sensitivity to mass hierarchy
could be further improved and probed with atmospheric neutrino analysis.

6.8.2

E↵ect of Systematic Errors

A study was carried out by taking a particular category of systematic errors as ±50% of
their original values. The assumed true values of the oscillation parameters are the same as thatin Section 6.6.
The best fit oscillation parameters at assumed true normal and inverted hierarchies in thesensitivity study in Section 6.6 are listed in the Table 6.23:

Chapter 6

m232
sin2 ✓23
CP
2

190

Best fit for assumed normal hierarchy
2.82 ⇥ 10 3 eV2
0.5
300
1.06

Best fit for assumed inverted hierarchy
2.51 ⇥ 10 3 eV2
0.475
300
0.90

Table 6.23: Best fit values at assumed normal and inverted hierarchies in the original sensitivity
study carried out in Section 6.6.

⌫ flux systematic errors
Systematic errors related to ⌫ flux were set to half of their original values.
2 = 2 (NH)
2 (IH) is 0.93, increased by
In the case of assumed inverted hierarchy,
0.03 when compared with the original sensitivity study, with the best fit point the same as theoriginal sensitivity study best fit point.
2 = 2 (NH)
2 (IH) is 1.08, which increased by
In case of assumed normal hierarchy,
0.02, with the best fit point same as the original sensitivity study best fit point.
Data was also fitted with systematic errors related to ⌫ flux halved. The best fit points andthe 2 is listed in Table 6.24 for both normal and inverted hierarchies. Both best fit points
2 between normal and inverted hierarchies
shifted when compared with the original fitting.
becomes 0.8 (decreased by 0.4 when compared with the original fit), with inverted hierarchyslightly preferred.

m232
sin2 ✓23
CP
2

Normal hierarchy
2.37 ⇥ 10 3 eV2
0.425
260
569.1 / 477 dof

Inverted hierarchy
2.51 ⇥ 10 3 eV 20.45
260568.3 / 477 dof

2 for normal and inverted hierarchies for data fitted with ⌫Table 6.24: Best fit values andflux related systematic errors halved.

2 = 2 (NH)
2 (IH) is studied.
The errors are also increased by 50% and the e↵ect
2 = 2 (NH)
2 (IH) is 0.88, decreased by
In the case of assumed inverted hierarchy,
0.02 when compared with the original sensitivity study, with the best fit point the same as theoriginal sensitivity best fit point.
2 = 2 (NH)
2 (IH) becomes 1.05, decreased by
In case of assumed normal hierarchy,
0.01 when compared with the original value, with the best fit point the same as the original
sensitivity study.
The study with ⌫ flux systematic errors increased by 50% was also done for fitting withdata. The best fit point the 2 for both normal and inverted hierarchies are listed in Table 6.25.
2 between normal andBoth best fit points changed when compared with the original fitting.
inverted hierarchies becomes 1.6 (increased by 0.4), with inverted hierarchy slightly preferred.

Chapter 6

191

m232
sin2 ✓23
CP
2

Normal hierarchy
2.66 ⇥ 10 3 eV2
0.425
300
550.4 / 477 dof

Inverted hierarchy
2.66 ⇥ 10 3 eV 20.6
260548.8 / 477 dof

2 for normal and inverted hierarchies for data fitted with ⌫Table 6.25: Best fit values andflux related systematic errors increased by 50%.

⌫ interactions systematic errors
Systematic errors related to ⌫ interactions were set to half of their original values.
2 = 2 (NH)
2 (IH) is 0.96, increased by
In the case of assumed inverted hierarchy,
0.06 when compared with the original sensitivity study, with the best fit point the same as theoriginal sensitivity study best fit point.
2 = 2 (NH)
2 (IH) is 1.13, increased by 0.07
In the case of assumed normal hierarchy,
when compared with the original sensitivity study, with the best fit point the same as the original
sensitivity study best fit point.
Data fitting was also performed with ⌫ interaction systematic errors halved. The best fit
2 between
points and 2 for both normal and inverted hierarchies are listed in Table 6.26.
normal and inverted hierarchies becomes 2.0 (increased by 0.8) with inverted hierarchy slightly
preferred. Best fit point for sin2 ✓23 and CP for normal hierarchy and CP for inverted hierarchychanged when compared with the original data fitting.

m232
sin2 ✓23
CP
2

Normal hierarchy
2.66 ⇥ 10 3 eV20.6
260569.4 / 477 dof

Inverted hierarchy
2.66 ⇥ 10 3 eV 20.6
240567.4 / 477 dof

2 for normal and inverted hierarchies for data fitted with ⌫Table 6.26: Best fit values andinteractions related systematic errors halved.

2 = 2 (NH)
2 (IH)The ⌫ interactions errors are also increased by 50% to see howchanges.
2 = 2 (NH)
2 (IH) is 0.86, decreased by
In the case of assumed inverted hierarchy,
0.04 when compared with the original sensitivity study, with the best fit point the same as theoriginal one.
2 = 2 (NH)
2 (IH) becomes 1.01, decreased
In the case of assumed normal hierarchy,
by 0.05 when compared with the original value, with the best fit point also the same as that in
the original sensitivity study.
Data fitted was also carried with ⌫ interaction related errors increased by 50%. The best fit
points for both normal and inverted hierarchies are listed in Table 6.27. The best fit point forChapter 6

192

for normal hierarchy and sin2 ✓23 for inverted hierarchy changed when compared with the
2 between normal and inverted hierarchy becomes 0.3 (decreased by 0.9),
original fitting.
with inverted hierarchy very slightly preferred.
CP

m232
sin2 ✓23
CP
2

Normal hierarchy
2.66 ⇥ 10 3 eV2
0.425
280
549.7 / 477 dof

Inverted hierarchy
2.66 ⇥ 10 3 eV 2
0.425
280
549.4 / 477 dof

2 for normal and inverted hierarchies for data fitted with ⌫Table 6.27: Best fit values andinteractions related systematic errors increased by 50%.

Event reconstruction systematic errors
Systematic errors related to event reconstruction were set to half of their original values.
2 = 2 (NH)
2 (IH) is 0.91, increased by
In the case of assumed inverted hierarchy,
0.01 when compared with the original sensitivity study, with the best fit point the same as theoriginal sensitivity study best fit point.
2 = 2 (NH)
2 (IH) is 1.07, increased by 0.01
In the case of assumed normal hierarchy,
when compared with the original sensitivity study, with the best fit point the same as the original
sensitivity study best fit point.
Data fitting was also performed with event reconstruction related errors halved. The best fit
points for both normal and inverted hierarchies are listed in Table 6.28. The best fit points form232 and CP for normal hierarchy and m232 , sin2 ✓23 and CP for inverted hierarchy changed
2 between normal and inverted hierarchy becomes 0.9
when compared with the original fit.
with inverted hierarchy slightly preferred.

m232
sin2 ✓23
CP
2

Normal hierarchy
2.37 ⇥ 10 3 eV2
0.425
280
561.0 / 477 dof

Inverted hierarchy
2.66 ⇥ 10 3 eV 2
0.575
260
560.1 / 477 dof

2 for normal and inverted hierarchies for data fitted with
Table 6.28: Best fit values andevent reconstruction related systematic errors halved.

2 = 2 (NH)
2 (IH) changes.
This category of errors are also increased by 50% to see how222
In the case of assumed inverted hierarchy,= (NH)(IH) is 0.89, decreased by 0.01
when compared with the original sensitivity study, with the best fit point the same as the one
in the original sensitivity study.
2 = 2 (NH)
2 (IH) becomes 1.06, no change
In the case of assumed normal hierarchy,
when compared with the original sensitivity study, with the best fit point the same as the original

Chapter 6

193

sensitivity study.
Data was also fitted with event reconstruction related systematic errors increased by 50%.
2 between normal and inverted hierarchies
The best fit points are listed in Table 6.29.
becomes 1.3 (increased by 0.1) with inverted hierarchy slightly preferred, but the best fit points
for both normal and inverted hierarchy changed significantly when compared with the original
fit.

m232
sin2 ✓23
CP
2

Normal hierarchy
2.66 ⇥ 10 3 eV20.45
300552.4 / 477 dof

Inverted hierarchy
2.66 ⇥ 10 3 eV 2
0.575
240
551.1 / 477 dof

2 for normal and inverted hierarchies for data fitted with
Table 6.29: Best fit values andevent reconstruction related systematic errors increased by 50%.

In conclusion, the errors related to ⌫ interactions have the largest e↵ect on the sensitivity on
mass hierarchy, but the e↵ect is still not very significant. Increasing or decreasing each of thethree categories of systematic errors does not have significant e↵ect on the sensitivity of mass
hierarchy.

6.8.3

Consistency with Other Experiments

The new results of m232 and sin2 ✓23 is compared with other experiments are shown in Figure6.34 for free ✓13 and Figure 6.35 for fixed ✓13 . The results is shown to be consistent with resultsfrom other experiments.

194

0.004

0.004

0.003

0.003

| ∆ m232 | (eV2)| ∆ m232 | (eV2)

Chapter 6

0.002

0.001
0.8

90% C.L.
T2K 2νMINOS 2νSK 1+2+3+4 L/E 2ν
SK 1+2+3+4 Zenith 2ν
SK 1+2+3+4 Zenith 3ν ( sin2 2θ13 = free , NH)

0.85

0.9

0.95

0.002

0.001
1

0.8

90% C.L.
T2K 2νMINOS 2νSK 1+2+3+4 L/E 2ν
SK 1+2+3+4 Zenith 2ν
SK 1+2+3+4 Zenith 3ν ( sin2 2θ13 = free , IH)

0.85

sin2 2 θ23

0.9

0.95

1

sin2 2 θ23

0.004

0.004

0.003

0.003

| ∆ m232 | (eV2)| ∆ m232 | (eV2)

Figure 6.34: Contours of m232 and sin2 2✓23 allowed regions for normal hierarchy (left) andinverted hierarchy (right) for free ✓13 . New results from this study is represented by blue solidline, and the best fit point is represented by the triangle.

0.002

0.001
0.8

90% C.L.
T2K 2νMINOS 2νSK 1+2+3+4 L/E 2ν
SK 1+2+3+4 Zenith 2ν
SK 1+2+3+4 Zenith 3ν ( sin2 2θ13 = 0.099 , NH)

0.85

0.9

sin2 2 θ23

0.95

0.002

0.001
1

0.8

90% C.L.
T2K 2νMINOS 2νSK 1+2+3+4 L/E 2ν
SK 1+2+3+4 Zenith 2ν
SK 1+2+3+4 Zenith 3ν ( sin2 2θ13 = 0.099 , IH)

0.85

0.9

0.95

1

sin2 2 θ23

Figure 6.35: Contours of m232 and sin2 2✓23 allowed regions for normal hierarchy (left) andinverted hierarchy (right) for fixed ✓13 . New results from this study is represented by blue solidline, and the best fit point is represented by the triangle.

Chapter 7

Conclusion
Full three flavor oscillation analysis was carried out with Super-Kamiokande atmospheric
neutrino data in order to obtain information on the mass hierarchy for the first time. It is thefirst time to separate multi-GeV e-like samples into ⌫e and ⌫ e enriched samples so as to increase
the sensitivity to the mass hierarchy. All data observed in SK-I (1489.2 days exposure for FC
and PC, 1645.9 days for UPMU), SK-II (798.6 days exposure for FC and PC, 827.7 days for
UPMU), SK-III (518.1 days exposure for FC and PC, 635.6 days for UPMU) and SK-IV (1097days exposure for FC and PC, 1096.7 days for UPMU) period through Apr. 1996 to Feb 2012were used.
For both free and fixed ✓13 cases, the allowed regions of the m232 and sin2 ✓23 are consistent
with other experiments (Figures 6.34 and 6.34).
As for the constraint on mass hierarchy, it is probed with ⌫e and ⌫ e -like samples. The
2 = 1.2. The significance is still low to draw ainverted hierarchy is slightly favoured withconclusion. However the sensitivity study shows that the newly developed ⌫e and ⌫ e enrichedsamples improve the sensitivity on the mass hierarchy. This study demonstrated the contribution
by the separation of ⌫e and ⌫ e samples.
2 between the normal and inverted mass hierarchies is expected to be proportional toAs
the exposure as shown in Figure 7.1, it is expected that the mass hierarchy could be constrainedwith atmospheric neutrino analysis in the future. Longer exposure time or larger detector size
(such as Hyper-Kamiokande experiment [148]) which both give higher statistics would help toconstrain the mass hierarchy.

195

Chapter 7

196

Figure 7.1: Expected sensitivity of mass hierarchy by Hyper Kamiokande (whose fiducial volume
is 25 times of Super Kamiokande) as a function of years of exposure.

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