Kyoto University

Doctor Thesis

Study of the neutrino mass hierarchy
with the atmospheric neutrino datacollected in Super-Kamiokande IV

Miao Jiang

A thesis submitted in fulfilment of the requirements
for the degree of Doctor of Sciencein theHigh Energy Physics Group
Department of Physics

March 2019

Abstract
Super-Kamiokande (SK) is a large underground water Cherenkov detector located in the
Kamioka mine in Mt. Ikenoyama, Gifu Prefecture, Japan. The detector consists of awater tank holding 50 kt of ultra-pure water, and more than 11,000 photomultiplier tubes(PMT) for the Cherenkov light’s detection. A new algorithm, named fiTQun, employs amaximum likelihood method using the charge and time information that is observed by
the PMTs to reconstruct particle types and their kinematics in the detector. Comparing
with the conventional reconstruction algorithm, fiTQun shows better performance.
An analysis of the neutrino mass hierarchy with the atmospheric neutrino data reconstructed by fiTQun in SK-IV with a livetime of 3118.5 days is presented in this thesis.
Due to the improved performance of fiTQun, the fiducial volume (FV) is expanded from
22.5 kt to 29.7 kt by changing the FV cut from Dwall 1 > 200 cm to Dwall > 50cm
which provides higher statistics in this analysis. This analysis is performed for both thecases of free θ13 and θ13 constrained based on the result of reactor experiments. The
best-fit results of the oscillation parameters for the analysis with constrained θ13 are
+0.22
+2.67
2
−3
2
sin2 θ23 = 0.0425+0.046−0.037 , |∆m32,31 | = 2.53−0.12 × 10 eV , δCP = 3.14−1.35 . The fit result

also shows a weak preference for the normal hierarchy (NH) disfavoring the inverted
hierarchy (IH) at 74% confidence level with a ∆χ2 ≡ χ2NH − χ2IH = −2.45 assuming
neutrino oscillations at the best fit of the analysis. The median expected sensitivity to
the mass hierarchy is expected to over ∆χ2 = 3 if the data collected in SK-I, II and III
are also included. This sensitivity can be further improved to over ∆χ2 = 5 if the datafrom the T2K experiment are analyzed simultaneously.

1

The distance from the event vertex to the nearest wall of detector.

Acknowledgements
I sincerely appreciate the amazing people I met in this long journey. First and foremost,
I would like to express my sincere gratitude to my advisor Prof. Tsuyoshi Nakaya. WhenI expressed the interests on neutrino physics, Prof. Tsuyoshi Nakaya kindly acceptedme as a member of this excellent laboratory in Kyoto University and helped me a lot on
the scholarship application. He also gave me enough freedom to select topics and manysage advice during research.
My work has benefited immensely from Roger Wendell, Michael Wilking, Shunichi Mineand Prof. Masato Shiozawa. Roger Wendell is both pragmatic and knowledgeable, andmore importantly, always willing to go out of his way to help. He is delightful to have aconversation with on just about anything. Michael Wilking gave many insightful ideas
from unique viewpoint about the fiducial volume expansion, which is essential part inthis thesis. I learned many things about the energy scale and calibration from Shunichi
Mine. He also showed me his academic rigor and research altitude, which inspired mea lot during my research. Prof. Shiozawa’s comments always make me think about myanalyses critically. My study always becomes more comprehensive during the discussion
with him.
I appreciate all my friends from Kyoto University. My days at Kyoto University started
with Tatsuya Hayashino, Keigo Nakamura, Ichinori Kamiji, Kota Nakagiri and Takuto
Kunigo, who made me feel welcome in the new country and new environment. It was
Tatsuya Hayashino who guided me through the first years on the life in Kyoto University
and endured all my unsophisticated questions. Keigo Nakamura is always smiling andenthusiastic about helping others. I’m happy to have a mutual hobby with Ichinori
Kamiji, who is kind and a lot of fun to talk with. The most friendly person in ourlaboratory is Kota Nakagiri, he often gives helpful advises to the juniors when theymeet with trouble. Takuto Kunigo is a pretty cool guy with extensive observation andbroad knowledge, I can always learn new things during conversation with him.
I would also thank senpai (senior students) who helped me a lot in Kamioka Observatory.
Seiko Hirota is like a elder sister, she picked me up many times when I was lost in the
Kamioka mountain or problems of physics. Kunxian Huang is a reliable brother, who
led me on the road to python and pyROOT and taught me lot of things about dataprocessing. Many thanks go to Shimpei Tobayama and Yusuke Suda, who gave me
many insightful suggestions when I studied the new reconstruction algorithm, fiTQun,
which is a crucial element of this thesis.
I appreciate the members of the ATMPD group who have endured my talks with increasing length over the years and gave me many helpful feedback. The work about theiii

iv
retuning SK-I in this thesis is also benefited from the calibration and low-e group forthe analysis of Nickel data and the simulation group for their help on SKDETIM.
It has been an honor to be member of Super-Kamiokande collaboration. Although I
can’t thank everyone whose path has crossed with the mine, your work ethic have beenreally inspiring. Your kindness has made the time in Kamioka Observatory one of mymost treasured memories.
I appreciate the excellent laboratory of high energy physics at Kyoto University. Not
only neutrino members, but also members from KOTO, ATLAS and CMB gave me
stimulus of particle physics, and I benefit a lot from the discussions with them: Atsuko
Ichikawa, Osamu Tajima, Masaya Ishino, Hajime Nanjo, Akihiro Minamino, Toshi Sumida, Kiseki Nakamura, Syunsuke Andachi, Shuhei Obara, Shunsuke Honda, Takahiko
Masuda, Shota Takahashi, Naoki Kawasaki, Tatsuya Kikawa, Kento Suzuki, Daichi
Naito, Yosuke Maeda, Shigeto Seki, Takuya Tashiro, Naoyuki Kamo, Shinichi Akiyama,
Takaaki Hineno, Yuki Ishiyama, Kento Yoshida, Keisuke Kondo, Matoshi Shinohara,
Saori Yanagita, Sei Ban, Mika Yamamoto, Ryutaro Monden, Kento Haneda, Ayami
Hiramoto, Yohei Noguchi, Yoshie Nakanishi, Shunsuke Tanaka, Yosuke Ashida, Shunichi Akatsuka, Masamitsu Mori, Kazuhiro Nakamura, Yuuta Okazaki, Masashi Yoshida,
Koichiro Kuniyoshi, Junta Komine, Kenji Yasutome, Tomofumi Abe, Takuji Ikemitsu,
Takahiro Odagawa, Soichiro Kuribayashi, Masanori Tajima, Masayuki Hatano, Yuya
Mino.
Also, I appreciate the financial support from Ministry of Education, Culture, Sports,
Science and Technology of Japan (MEXT).
Finally, I give greatest thanks to my wife and my parents for their continuous support
over the years and trust.

Contents
Abstractii

Acknowledgements

iii

Contentsv

List of Figuresix

List of Tables

xxvii

1 Introduction
1
1.1 Neutrino . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Neutrino Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Neutrino Oscillation Experiments . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.1 Atmospheric Neutrino and Accelerator Neutrino . . . . . . . . . . 5
1.3.2 Solar Neutrino and Reactor Neutrino . . . . . . . . . . . . . . . . . 7
1.3.3 Mixing Angle θ13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Unresolved Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 Sub-Dominant Effects of Atmospheric Neutrino Oscillation . . . . . . . . 12
1.6 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 The Super-Kamiokande Detector
17
2.1 Cherenkov Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.1 Water Tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.2 Photomultiplier Tube . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.3 Outer Detector (OD) . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 Water Purification and Air Purification System . . . . . . . . . . . . . . . 24
2.4 Electronics and Data Acquisition System for SK-IV . . . . . . . . . . . . . 24
3 Simulation of Atmospheric Neutrino
29
3.1 Atmospheric Neutrino Flux . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2 Neutrino Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.1 (Quasi-)Elastic Scattering . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.2 Single Meson Production . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.3 Coherent Pion Production . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.4 Deep Inelastic Scattering . . . . . . . . . . . . . . . . . . . . . . . 35v

vi

Contents

3.3

3.2.5 Nuclear Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Detector Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4 Calibration
39
4.1 Detector Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.1.1 Relative Gain Calibration . . . . . . . . . . . . . . . . . . . . . . . 39
4.1.2 Absolute Gain Calibration . . . . . . . . . . . . . . . . . . . . . . . 40
4.1.3 Quantum Efficiency Calibration . . . . . . . . . . . . . . . . . . . . 41
4.1.4 Relative Timing Calibration . . . . . . . . . . . . . . . . . . . . . . 42
4.1.5 Water Property Calibration . . . . . . . . . . . . . . . . . . . . . . 43
4.2 Energy Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2.1 High Energy Stopping Muons . . . . . . . . . . . . . . . . . . . . . 46
4.2.2 Low Energy Stopping Muons . . . . . . . . . . . . . . . . . . . . . 49
4.2.3 Neutrino Induced π 0 Events . . . . . . . . . . . . . . . . . . . . . . 51
4.2.4 Decay Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2.5 Time Variation of Energy Scale . . . . . . . . . . . . . . . . . . . . 53
4.2.6 Detector Uniformity of Energy Scale . . . . . . . . . . . . . . . . . 53
4.2.7 Summary of the Energy Scale Error . . . . . . . . . . . . . . . . . 55
5 Neutrino Event Selection
59
5.1 Reduction for Fully Contained Events . . . . . . . . . . . . . . . . . . . . 61
5.1.1 First Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.1.2 Second Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.1.3 Third Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.1.4 Fourth Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.1.5 Fifth Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.1.6 FC Reduction Summary . . . . . . . . . . . . . . . . . . . . . . . . 69
5.2 Reduction for Partially Contained Events and Up-µ Event . . . . . . . . . 70
6 Event Reconstruction
71
6.1 Conventional Event Reconstruction . . . . . . . . . . . . . . . . . . . . . . 71
6.2 New FC Event Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . 72
6.2.1 Likelihood Function . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.2.2 Fitting Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.2.3 FiTQun in SK-I to SK-III . . . . . . . . . . . . . . . . . . . . . . . 86
7 Neutrino Oscillation Analysis
89
7.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
7.1.1 FC Event Selection and Binning . . . . . . . . . . . . . . . . . . . 90
7.1.2 Selection for Multi-GeV e-like Sample . . . . . . . . . . . . . . . . 92
7.2 Oscillation Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.3 Fiducial Volume Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.3.1 MC Study for Fiducial Volume Expansion . . . . . . . . . . . . . . 100
7.3.2 OD Hits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.3.3 Data/MC Comparison and Background Estimation . . . . . . . . . 109
7.4 Analysis Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
7.5 Systematic Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.5.1 Energy Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Contents

vii

7.5.2

7.6

Ring Counting, Particle Identification and Multi-GeV Multi-ringe-like Event Separation . . . . . . . . . . . . . . . . . . . . . . . . 117
7.5.3 Other Systematic Error related to Event Reconstruction . . . . . . 120
Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.6.1 Fit Results and Discussion with Free θ13 . . . . . . . . . . . . . . . 123
7.6.2 Fit Results and Discussion with Constrained θ13 . . . . . . . . . . 128

8 Conclusions and Outlook

145

A SK-I retuning155

B Performance check for fiducial volume expansion

163

C Distribution for systematic error evaluation

181

D Statistical Study for Oscillation Analysis Result Validation

187

E Improvement of decay electron detection for fiTQun

195

Bibliography

201

List of Figures
1.1

1.2

1.3

1.4

1.5

Zenith angle distributions for electron-like (top) and muon- like (bottom)
events in sub-GeV (left) and multi-GeV (right) energy region, observed bySuper-Kamiokande. Upward- and downward-going events are represented
by negative and positive value of cos θ, respectively. The expectations
without oscillations are represented by the hatched bands, while the blacklines denote the best-fit to data with the oscillation between νµ and ντ .
Taken from [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
8 B solar neutrino flux measured by SNO with different channels. The
horizontal axis shows the νe flux and the vertical axis shows the sum ofthe νµ and ντ flux. The allowed regions at 68% from the measurementsof CC, NC and ES channels are represented as the red, blue and green
bands respectively. The gray band denotes the ES measurement result
from Super-Kamiokande [2]. The dashed lines represents the predictionof the total neutrino flux based on Standard Solar Model. The point andthe contours show the best estimate and the allowed regions at 68, 95 and
99% confidence levels by combining the results from CC and NC. Taken
from [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Two possible mass hierarchies for the three neutrino mass eigenstates.
The varying width of the shadings shows the variation with respect tocosine of the CP violating phase δ. Taken from [4]. . . . . . . . . . . . . . 10
Effective Majorana neutrino mass hmββ i as a function of the lightest
neutrino mass mlightest . The dark shaded region shows the expectations
based on the best-fit values of neutrino oscillation parameters for bothnormal hierarchy (NH) and inverted hierarchy (IH), respectively. The
light shaded regions considers the 3σ uncertainties of those parameters.
The cyan horizontal bands indicate 90% C.L. upper limits on hmββ i with
1 36Xe from KamLAND-Zen. Right panel shows the corresponding limits
for each nucleus as a function of the mass number. Taken from [5]. . . . . 11
Oscillation probabilities for neutrinos (upper panels) and antineutrinos(lower panels) as a function of energy and zenith angle assuming a normal mass hierarchy. Matter effects in the Earth distorted the neutrino
oscillation probability between 2 and 10 GeV, while there is no such distortion in the antineutrino figures. For an inverted hierarchy, the distortion
caused by matter effects appear in the antineutrino figures. The discontinuities near cosθzenith = −0.5 and -0.8 are due to the matter densitychange between crust, mantle and core. Here the oscillation parameters
are taken to be ∆m232 = 2.5 × 10−3 eV2 , sin2 θ23 = 0.5, sin2 θ13 = 0.0219,
and δCP = 0. Taken from [6] . . . . . . . . . . . . . . . . . . . . . . . . . 14ix

x

List of Figures
2.1

Super-Kamiokande detector and its location in Mountain Ikenoyama.
Take from [7] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 A example of neutrino event observed by Super-K detector. Each point
denotes a PMT hit, whose charge is represented by the color. The
Cherenkov ring pattern can be seen clearly. . . . . . . . . . . . . . . . . .
2.3 Supporting structure of PMTs. Taken from [7]. . . . . . . . . . . . . . . .
2.4 Schematic view of the 20-inch ID PMT for Super-Kamiokande experiment. Taken from [7]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Single photoelectron distribution of the 20-inch PMT. . . . . . . . . . . .
2.6 PMT case for 20-inch ID PMT since SK-II period. . . . . . . . . . . . . .
2.7 A schematic view of outer detector (OD). Taken from [8] . . . . . . . . . .
2.8 OD event display for a simulated partially contained event, without (with)tyvek sheet for left (right) figure. The distribution of OD hit is shown inthe center of each figure, while the distribution of ID hits is shown on the
top left corner. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.9 Block diagram and surroundings of QTC. Taken from [9]. . . . . . . . . .
2.10 Block diagram of one QTC channel. There are three gain ranges for eachchannel. Taken from [9]. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.11 Timing chart for QTC operation. Taken from [9]. . . . . . . . . . . . . . .
3.1

3.2

3.3

3.4

3.5

3.6

Dependence of atmospheric neutrino flux on zenith angle for different
energy regions. The atmospheric neutrino is vertically downward goingwhen cosθ = 1 and it is upward going when cosθ = -1. Taken from [10]. .
Atmospheric neutrino fluxes (left) and flavor ratio (right) after averaging
over all directions. The prediction based on Honda 11 flux is shown by
red solid line, while the result of the Bartol flux and the FLUKA flux are
represented by dashed line and dotted line. Taken from [10]. . . . . . . . .
Axial mass MA extractions. Left panel: From (quasi)elastic neutrino
and antineutrino scattering experiments. The weighted average is MA =(1.026 ± 0.021) GeV. Right panel: From charged pion electroproduction
experiments. The weighted average is MA = (1.069 ± 0.016) GeV. Taken
from [11]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cross section of Charged current quasi-elastic interaction of νµ (left) andν̄µ (right), with experimental data from ANL [12], Gargamelle [13] [14],BNL [15], Serpukhov [16] and SKAT [17]. Solid line represents the result
for scattering off a free proton, the dashed line represents the result for
scattering off bound nucleons in 16 O. Taken from [18] . . . . . . . . . . .
Cross section for νµ charged current single pion productions. Solid lines
represents the calculation result of NEUT, points represent data fromexperiments shown in bottom right figure. Taken from [18]. . . . . . . . .
Cross section of charged current DIS for νµ (upper lines and points) andν¯µ (lower lines and points) interactions. Dashed lines show the predictionof NEUT, which uses GRV98 with Bodek-Yang correction. The experimental data is represented by the points. Taken from [18]. . . . . . . . . .

17

19
20
21
22
23
23

24
25
26
27

31

31

34

34

36

37

List of Figures
3.7

xi

Cross section of π + −12 C scattering as a function of the momentum of π + .
Solid lines represent the result of NEUT calculation before tuning, while
the result after tuning are represented by dashed lines. Experimental data
are shown as the points with error bars. Taken from [19]. Right figure is
the update for the absorption (ABS) and charge exchange (CX) based on
new scattering data [20] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.1

Distribution of the observed charge for single p.e. signals obtained fromthe calibration data with nickel source. Taken from [21]. . . . . . . . . . .
4.2 Time variation of the peak of the charge distribution for the PMT darkhits. Different color denote the different production year of PMT. A
correction on PMT gain is applied based on this measurement. . . . . . .
4.3 Schematic view of the timing measurement system using a laser. Takefrom [21]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 A typical TQ distribution for ID PMT. The horizontal axis and the vertical axis show the charge and TOF-corrected hit time, respectively. The
scale of horizontal axis is linear up to 10 pC and then goes to log scale.
A larger T value corresponds to earlier hit in this plot. Taken from [21]. .
4.5 The laser injector system for the water property measurement. Taken
from [21]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 PMT hit time distribution in different detector regions for data (open circle) and Monte Carlo (red line) after tuning by scattering and absorption
parameters. Taken from [21]. . . . . . . . . . . . . . . . . . . . . . . . . .
4.7 An example of water parameter fitting. The data used here is taken in Apr
2009. Points represent the average value of coefficients. The errors on the
average value are represented as error bars. The vertical axis represent
the coefficient α in Equation 4.4. Taken from [21] . . . . . . . . . . . . . .
4.8 Time variation of the light absorption and scattering coefficients measured
at various wavelengths. The coefficients for absorption αabs , asymmetricscattering αasym and symmetric scattering αsym for various wavelength
are shown in the blue (top), black (middle) and purple (bottom), respectively. In 2008, the wavelength was changed from 365 nm to 375 nm, from
400 nm to 405 nm, and from 420 nm to 445 nm. The time of this change
is indicated by the black vertical bars. Taken from [21] . . . . . . . . . . .
4.9 Momentum over the range of the stopping muon events with various track
length for the data (black dots) and MC (blue line). From the top left
plot to the bottom right plot, the track length is 5-10 m, 10-15 m, 15-20
m, 20-25 m, 25-30 m and 30-35 m, respectively. The fiducial volume forthe decay electron used to determine the track length is defined as Dwall >50 cm. MC are normalized to data by area. . . . . . . . . . . . . . . . . .
4.10 Distribution of p(p.e.)/p(θ) at each p(θ) bin for data (black dots) andMC (blue line). From the top, the p(θ) range is 200-280 MeV/c, 280-360
MeV/c and 360-440 MeV/c, respectively. MC are normalized to data by
area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.11 Invariant mass of neutral pions for the data (black dots) and MC (bluelines). From the top, the Dwall range is > 200 cm, 50 ∼ 200 cm and > 50
cm. MC are normalized to data by livetime. . . . . . . . . . . . . . . . . .

41

41
42

43
44

45

46

47

48

50

52

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List of Figures
4.12 Momentum of decay electrons for the data (black dots) and MC (blueline). From the top, the Dwall range is > 200 cm, 50 ∼ 200 cm and > 50
cm. MC are normalized to data by area. . . . . . . . . . . . . . . . . . . .
4.13 Time variation plots of the p/range of muon (top figures) and the momentum of decay electron (bottom three figures) for fiTQun. The left two
figures show the result for Dwall > 200 cm, while the right two figures arefor Dwall > 50 cm. The bottom figure shows the time variation of the
momentum of decay electron with 200cm > Dwall > 50 cm. The dashedline shows the range of average value with ±1% error. . . . . . . . . . . .
4.14 Uniformity of the detector as a function of zenith angle of decay electrons.
Y axis is the ratio of MC to data of the average momentum of decay
electrons. X axis is direction of the electron on z axis, a positive value
represents a downward direction. From the top, the Dwall range is > 200
cm, 50 ∼ 200 cm and > 50 cm. The dashed line shows the ±1% error
range. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.15 Absolute energy scale error with Dwall > 200 cm (top) and Dwall > 50
cm (bottom). The statistical uncertainty is denoted by the vertical error
bars. The momentum range of each sample used to evaluate energy scaleerror is shown by the horizontal error bar. The final absolute energy scaleerror 2.09% (1.92%) for Dwall > 200(50) cm. . . . . . . . . . . . . . . . .
5.1

5.2
5.3

5.4

6.1

6.2

Schematic view of observed atmospheric neutrino in SK. Dashed lines
represent the track of atmospheric neutrinos, while the solid lines show
the lepton’s track. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Expected neutrino energy spectra of different event categories. Taken
from [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Two flasher event displays in SK-IV. Each circle represents a hit on PMT.
A larger circle denotes a higher charge signal for the PMT. The flash
light for both events come from the same PMT on the left part of plots,
therefore similar hit pattern observed. . . . . . . . . . . . . . . . . . . . .
Final likelihood for flasher event rejection. Points show the distributionof data, while the black (red) solid line shows the distribution of nominal
MC (fitted MC). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

55

56

57

60
60

65

66

Event display of single-ring electron (right) and single-ring muon (left)
neutrino MC event. Each circle denotes a PMT hit, whose observed
charge is represented by the size of the circle. . . . . . . . . . . . . . . . . 72
An example of the goodness function with respect to hit time. A muon
event with its decay electron is shown here. The goodness function withrespect to hit time is shown as the black line, while the vertex is fixed
to the pre-fit vertex. Thresholds for identifying candidate peaks: F (x)and 0.6 × F (x) are denoted by blue (dashed) and green (dotted) curves.
Candidates found by the hit clustering algorithm are marked by red triangles and their associated time windows are indicated by gray verticallines. Overlapped time windows are merged into one. . . . . . . . . . . . . 77

List of Figures
PID likelihood distribution of FC single-ring events. Left figure is for
sub-GeV events and right figure is for multi-GeV events. Distribution forneutrino data is denoted by points and that for atmospheric neutrino MC
after oscillation is shown as histograms. The true hierarchy is assumed to
be normal hierarchy. The oscillation parameters is taken to be sin2 θ13 =0.0210, sin2 θ23 = 0.5, ∆m223 = 2.4 × 10−3 eV2 , and δCP = 0. Sameparameters are applied below. Charged-current νµ interaction component
is shown as the shaded histograms. The statistical error are denoted byerror bars. The reconstructed event vertex to the nearest ID wall (Dwall )is required to be larger than 200 cm. . . . . . . . . . . . . . . . . . . . . .
6.4 Single-ring electron(left) and muon(right) vertex resolution for FC truefiducial CCQE events in atmospheric neutrino MC, compared between
APFit(dashed line) and fiTQun(solid line). The resolution is defined as
the 68 percentile of the respective distributions, which is shown by corresponding vertical line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5 Vertex resolution of FC single-ring charged current quasi-elastic (CCQE)
event as a function of visible energy. Left figure is for CC νe events andright figure is for CC νµ events. The performance of fiTQun is indicatedby the full triangles, while that for APFit is indicated by the open circles.
6.6 Single-ring electron (left) and muon (right) direction resolution for FC
CCQE events in atmospheric neutrino MC, compared between APFit(dashed line) and fiTQun (solid line). The resolution is defined as the 68
percentile of the respective distributions which is shown by corresponding
vertical line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.7 Direction resolution of single-ring electron (left) and muon (right) eventsin the FC CCQE event sample in the atmospheric neutrino MC, plotted
as a function of visible energy. The full triangles indicate the performanceof fiTQun and the open circles are for APFit. . . . . . . . . . . . . . . . .
6.8 Single-ring electron (left) and muon (right) momentum resolution for FC
CCQE events in atmospheric neutrino MC, compared between APFit(dashed line) and fiTQun (solid line). The bias (resolution) is defined as
the mean (RMS) value of the ratio distribution between the reconstructed
momentum and true momentum. . . . . . . . . . . . . . . . . . . . . . . .
6.9 Momentum resolution and bias as a function of visible energy. Top two
figures show the momentum resolution and bottom ones show the momentum bias. Left two figures are for FC CCQE νe events in the atmospheric
neutrino MC while right ones are for νµ events. The performance offiTQun is indicated by the full triangles, while that for APFit is indicated by the open circles. . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.10 Mis-identification rate as a function of visible energy. Left figure shows theresult of FC CCQE νe events in the atmospheric neutrino MC, while rightfigure shows the result of CCQE νµ events. The performance of fiTQun
is indicated by the full triangles, while that for APFit is indicated by theopen circles. The statistical error are denoted by error bars. . . . . . . . .

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6.3

79

79

80

80

80

81

81

82

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List of Figures
6.11 The distribution of the likelihood ratio between the best-fit result of
single-ring hypothesis and multi-ring hypothesis of FC atmospheric neutrino events. The most energetic ring is required as e-like ring. Left figure
shows the result of sub-GeV events, while right figure shows multi-GeV
events’ result. Distribution for atmospheric neutrino data is denoted by
the points and the MC prediction including neutrino oscillations is denoted by the histogram. The events with single-ring final state is shownby the shaded histogram. The statistical error are denoted by error bars.
The terms “1R” and “MR” in the figures represent single-ring and multiring, respectively. The reconstructed event vertex to the nearest ID wall
(Dwall ) is required to be larger than 200 cm. . . . . . . . . . . . . . . . . .
6.12 PID likelihood distributions of the most energetic ring for fully contained
multi-ring events. Left figure shows the results of sub-GeV events whileright figure are for multi-GeV events. Distribution for atmospheric neutrino data is denoted by the points and the MC prediction including
neutrino oscillations is denoted by the histogram. The component of νµcharged-current interactions is shown by the shaded histogram. The statistical error are denoted by error bars. The reconstructed event vertex
to the nearest ID wall (Dwall ) is required to be larger than 200 cm. . . .
6.13 Reconstructed invariant mass calculated based the second and the third
rings in order of energy for events with three e-like rings in the sub-GeV
atmospheric neutrino MC sample. Left figure shows the events reconstructed by fiTQun, and right figure is for the events reconstructed by
APFit. True CCνe 1π 0 events are denoted by shaded histograms. The invariant mass range used for the event rate and purity calculation is shownby arrows. The reconstructed event vertex to the nearest ID wall (Dwall )is required to be larger than 200 cm. . . . . . . . . . . . . . . . . . . . . .
6.14 PID Likelihood distribution of SK-I for sub-GeV single-ring events. Error
bars represent the statistical uncertainty of the data. Events with negative(positive) likelihood values are designated µ-like (e-like). Color denotes
the true flavor interaction component. . . . . . . . . . . . . . . . . . . . .
6.15 Time variation of the light attenuation length in water. Different color
denote different Super-K phases, SK-I, II, III and IV are shown in black,green, blue and red, respectively. . . . . . . . . . . . . . . . . . . . . . . .
7.1
7.2

7.3

84

84

85

87

88

Summary of event categories for FC samples. The texts in the dashedline boxes shows the result classified by the variables which is on the boxes. 92
Definition of the binning used in the neutrino oscillation analysis. FC:
405 bins, PC: 60 bins, Up-µ: 50 bins. The samples which is divided into
10 zenith angle bins between -1 < cos θ< 1 for FC and PC, -1 < cos θ<
0 for UPMU are denoted by white boxes, while no zenith angle bins are
divided for the samples represented by shaded boxes. . . . . . . . . . . . . 93
Distribution of number of decay electrons in both data and MC for SK-IV
multi-GeV single-ring e-like sample. Cyan (magenta) shaded histogram
represent the distribution of CCνe (CCν̄e ) events, red solid line represents
total atmospheric neutrino MC and open circle indicates the distributionof data. Error bars show the statistical uncertainty of data. . . . . . . . . 94

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7.4

xv

Distributions of the variables used in the multi-GeV multi-ring e-like(MME) likelihood for SK-IV. Shaded histogram represents CCνe + ν e ,red solid line represents total atmospheric neutrino MC and open circle
indicates the distribution of data. Error bars represent the statisticaluncertainty of the data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7.5 Likelihood distribution for the first stage of SK-IV multi-GeV multi-ringe-like events separation. The statistical error are denoted by error bars.
The cut is determined at -0.25 according to the result of MC study. . . . 97
7.6 Distributions of the variables used in the MME νe and ν e separation
likelihood for SK-IV passed the first selection. Cyan (magenta) shaded
histogram represent the distribution of CCνe (CCν̄e ) events, red solid line
represents total atmospheric neutrino MC and open circle indicates the
distribution of data. Error bars show the statistical uncertainty of data. . 98
7.7 Likelihood distribution for the second stage of SK-IV multi-GeV multiring e-like events separation. The statistical error are denoted by error
bars. Events with likelihood values larger (less) than 0 are selected asν̄e -like (νe -like). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.8 Sensitivity to reject wrong mass hierarchy (MH) of each sample for atmospheric neutrino. The vertical axis shows the ∆χ2 value between the true
oscillation point at normal hierarchy and the best fit point at inverted hierarchy. The true oscillation paramters are (∆m212 , ∆m232 , sin2 θ12 , sin2 θ13 ,δCP ) = (7.49 × 10−5 eV2 , 2.5× 10−3 eV2 , 0.309, 0.021, 200◦ ). True MH isassumed to be normal hierarchy. Oscillations at different values of sin2 θ23= 0.4 (red), 0.5 (blue), 0.6 (magenta) are shown. . . . . . . . . . . . . . . 100
7.9 Oscillation effect on zenith angle distribution for multi-GeV samples. The
vertical axis shows the ratio between the number of events with oscillations and the ones without oscillations. Oscillation parameters are (∆m212 ,∆m232 , sin2 θ12 , sin2 θ13 , δCP ) = (7.49 × 10−5 eV2 , 2.5× 10−3 eV2 , 0.309,
0.021, 200◦ ). Oscillations at different values of sin2 θ23 = 0.4 (red), 0.5(blue), 0.6 (green) for normal hierarchy are shown. . . . . . . . . . . . . 101
7.10 Sensitivity to δCP of each sample for atmospheric neutrino. The y-axis
shows the ∆χ2 value between the true oscillation point and the best fit
point. The true oscillation parameters are (∆m212 , ∆m232 , sin2 θ12 , sin2 θ13 ,sin2 θ23 , δCP ) = (7.49 × 10−5 eV2 , 2.5× 10−3 eV2 , 0.309, 0.021,0.425,200◦ ). True MH is assumed to be normal hierarchy. Oscillations at different values of δCP = 0◦ (red), 80◦ (blue), 300◦ (magenta) are shown.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7.11 Zenith angle distributions of the ratio of number of events with oscillation to that without oscillation, assuming oscillation parameters (∆m212 ,∆m232 , sin2 θ12 , sin2 θ23 ) = (7.49 × 10−5 , 2.5 × 10−3 , 0.309, 0.5) in normal
hierarchy case for SK-IV. Cases with δCP = 80◦ (red), 220◦ (blue) and300◦ (green) are shown. The error bars represents statistical uncertainty. . 103
7.12 The composition of the FC sub-GeV µ-like sample without decay electron.
Horizontal axis shows the distance between the reconstructed vertex andthe nearest ID wall. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

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7.13 Sensitivity for incorrect mass hierarchy rejection. The exposure time is3118.5 days. Horizontal axis shows different FV cut position. The truehierarchy is assumed to be normal hierarchy. The true oscillation parameters are taken as: sin2 θ13 = 0.0210, sin2 θ23 = 0.5, ∆m223 = 2.4 × 10−3 eV2 ,and δCP = 0. The systematic error for the analysis with a FV cut at
200 cm is used for all sensitivity calculation with different FV cut. . . . . 106
7.14 Defination of “Dwall ” and “Towall”. . . . . . . . . . . . . . . . . . . . . . 106
7.15 Purity of each component for sub-GeV single ring e-like 1 decay electron
as function of towall value. A Dwall cut at 50 cm has been applied. . . . 107
7.16 Sensitivity to reject wrong mass hierarchy. True mass hierarchy is NH.
The left figure shows the contribution from sub-GeV single-ring e-like
1 decay sample, while the right figure shows the total sensitivity fromall samples. The open circle at (200,200) in right figure represents thesensitivity at the FV cut used in previous APFit analysis. . . . . . . . . 108
7.17 Towall distribution for multi-Gev e-like events. The statistics with small
towall value is low. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
7.18 Distribution of the number of OD hits in cluster. Left figure shows the
distribution for data with in conventional FV (Dwall > 200 cm). Right
figure shows the result in new region (50 cm < Dwall < 200 cm). The
peak around 20 in right figure is due to cosmic ray muon events, whichis confirmed by eye-scanning. The threshold line to select FC events isdenoted by the red dashed line. . . . . . . . . . . . . . . . . . . . . . . . . 109
7.19 Number of OD bad channels. . . . . . . . . . . . . . . . . . . . . . . . . . 109
7.20 Number of events as the function of the distance between the reconstructed event vertex and the nearest ID wall for FC samples. Points
represent the result of data and solid line is for MC. The ratio between
MC and data is shown in bottom figure. . . . . . . . . . . . . . . . . . . . 111
7.21 The distribution of the distance from the reconstructed vertex to thenearest ID wall of µ-like samples. Left figure shows the distribution of
single-ring events while right figure shows the one of multi-ring events.
Both of the independent cosmic ray muon sample (solid line) and eyescanned background events in FC sample (points) are shown. Histograms
are normalized by the total number of backgrounds events seen in the FC
sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7.22 The distribution of the momentum and zenith angle of single-ring (top)and multi-ring (bottom) µ-like samples. The left two figures show the
2-d distribution, where points represent the background events identified
by eye scanning in FC and color box indicate the distribution of cosmic
ray muon sample. In right four figures, solid line represents the resultof cosmic ray muon sample and the cross shows the distribution of theeye-scanned background events in FC. Histograms are normalized by the
total number of events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7.23 Momentum distribution of single ring event in different detector region.
Samples are divided based on the reconstructed particle ID. Upper twoplots show the events within conventional FV, Dwall > 200 cm, while thelower two plots show the events within new region, 50 cm < Dwall <200 cm. Good agreement between data and MC can be observed. . . . . . 113

List of Figuresxvii

7.24 Comparisons between SK-IV data and MC within expanded FV (Dwall >50 cm). The sample has been divided into 18 categories. The reconstructed momentum distributions for the samples with only one zenith
angle bin are shown in the first column. The other samples are shown aszenith angle distributions from second to fifth column. The best-fit MC
assuming the normal hierarchy is denoted by solid lines. The ratio between MC to data is shown in the narrow panels below each distribution.
The error bars in each panels represent the statistical error. . . . . . . . . 114
7.25 Final event rates variation with respect to running time since SK-IV
launch. The error bars represent the statistical error. The fully contained
event rate is denoted by circle and the partially contained (upward-goingmuon) event rates are denoted by upward-facing (downward-facing) triangles. The fiducial volume cut value for FC events are shown in theparentheses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.26 Ring counting likelihood distribution for multi-GeV e-like events in the
new region (Dwall from 50 cm to 200 cm). The original p.d.f constructedfrom MC (blue line), the fitted p.d.f. (red line) and the data (points)
are shown. The dashed lines show the contribution from different components: true single-ring subsample and true multi-ring sample. The true
number of rings is counted using only particles with energy more than 30
MeV above Cherenkov threshold in the final states. Error bars represent
the statistical error. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7.27 Ring counting likelihood distribution for multi-GeV e-like events in the
new region. (dwall from 50 cm to 200 cm.) Point shows the data, andsolid (dashed) line shows fitted (nominal) distribution. The dotted lineis the cut line for single-ring/multi-ring event separation. The shaded
histograms show the after fitting distribution for true single-ring eventsin final state. The term “1R” and “MR” represent “single-ring” and“multi-ring” respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.28 Fractional change of the single ring component distribution of multi-GeV
e-like ring counting likelihood systematic error evaluation for conventional
FV (Dwall > 200 cm, black) and new region (50 cm < Dwall < 200 cm,blue). The dashed lines show the final systematic error value, which isdefined as the 68.7% percentile points with larger absolute value. The
systematic error in new region is larger than the one in conventional FV. . 119
7.29 Likelihood distribution of ring counting (top) and PID (bottom) for both
first ring (left two plots) and second ring (right two plots). Fiducial
volume cut is Dwall > 50 cm. The shaded histogram show the distributionof CC 1π 0 events. The solid red (dashed blue) line denote the fitted MC(nominal) while the points are for data. Error bars show the statisticaluncertainty of data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
7.30 Expected median sensitivity of the atmospheric neutrino sample to rejecta zero value of sin2 θ13 with a livetime of 3118.5 days. Horizontal axis
shows the assumed true value of sin2 θ23 . The result of the events reconstructed by fiTQun within the conventional FV and expanded FV are
denoted by grey and blue lines, respectively. The sensitivity with eventsreconstructed by APFit and conventional FV is denoted by orange line. . 123

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7.31 Difference of χ2 value between global best fit result in this analysis andthe result with the oscillation parameter shown in the horizontal axis.
In these plots, no external constraints are applied and the events withinconventional FV (Dwall > 200 cm) are used. The normal hierarchy result
is denoted by blue solid lines. The result of inverted hierarchy is denotedby orange dashed lines and has been offset from the normal hierarchy
result by the difference of minimum χ2 values between them. . . . . . . . 125
7.32 Difference of χ2 value between global best fit result in this analysis andthe result with the oscillation parameter shown in the horizontal axis.
In these plots, no external constraints are applied and the events within
expanded FV (Dwall > 50 cm) are used. The normal hierarchy result isdenoted by blue solid lines. The result of inverted hierarchy is denoted byorange dashed lines and has been offset from the normal hierarchy result
by the difference of minimum χ2 values between them. . . . . . . . . . . 126
7.33 Upward- (cosθ < −0.4) to downward-going (cosθ > 0.4) event ratio fordifferent momentum within expanded FV. The momentum for multi-ring
event is defined as the sum of the momentum of each ring. Statisticaluncertainties are denoted by the error bars. The cyan line denotes thebest fit from the normal hierarchy hypothesis, and the orange line thebest fit from the inverted hierarchy hypothesis. . . . . . . . . . . . . . . . 127
7.34 The preference for MH for each event types within expanded FV. The
y-axis shows the difference of χ2 between normal hierarchy (NH) andinverted hierarchy (IH) contributed by each event sample at the bestfit oscillation parameters. The definition of the vertical axis is ∆χ2 =χ2NH − χ2IH , which means that the negative value represent the preference
for the normal hierarchy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.35 Expected median sensitivity with a livetime of 3118.5 days to mass hierarchy. Top figure shows the sensitivity to normal mass hierarchy andbottom figure shows the sensitivity to inverted hierarchy. Horizontal axis
shows the assumed true value of sin2 θ23 . Here the value of sin2 θ13 is
fixed to 0.0210. The result of the events reconstructed by fiTQun within
the conventional FV and expanded FV are denoted by the gray and blue
bands. The sensitivity with events reconstructed by APFit algorithm andthe conventional FV is denoted by the orange band. The bands represents
the effect of the uncertainty from δCP . . . . . . . . . . . . . . . . . . . . 129
7.36 Difference of χ2 value between global best fit result in this analysis andthe result with the oscillation parameter shown in the horizontal axis. In
these plots, no external constraints are applied. Top three figures show
the result within conventional FV (Dwall > 200 cm) and bottom threefigures show the result within expanded FV (Dwall > 50 cm). The normal
hierarchy result is denoted by blue solid lines. The result of inverted
hierarchy is denoted by orange dashed lines and has been offset from thenormal hierarchy result by the difference of minimum χ2 values between
them. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

List of Figuresxix

7.37 Two-dimensional atmospheric mixing contour for the preferred regions at
90% C.L.. The left (right) figure shows 90% C.L. for the normal (inverted)hierarchy. The solid line denotes the fit result within expanded FV while
the dashed line is for the result within conventional FV. The full (open)
star denotes the best-fit value for expanded FV (conventional FV). In
each contour sin2 θ13 is fixed to 0.0210. Both contours have been drawn
with respect to the global best-fit for each FV. . . . . . . . . . . . . . . . 131
7.38 Constraints on neutrino oscillation contours at the 90% C.L. from analyses
assuming the normal mass hierarchy. The contour from Super-K IV with
fiTQun (red) is taken from the analysis with sin2 θ13 constrained to be
0.0210 ± 0.0011. Contours from the Super-K I-IV with APFit (dashedcyan) [6], T2K (dotted yellow) [22], NOvA (dashed green) [23], IceCube(dashed black) [24] and MINOS+ (dashed blue) [25] experiments are also
shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.39 Distributions of the ∆χ2 for octant preference (χ2sin2 θ =0.4 − χ2sin2 θ =0.6 )23
23
for each sample. Blue boxes represents the fit result with sin θ13 free,
whose the best-fit value is 0.0075, while the red triangles show the fit
result with sin θ13 fixed to 0.02. . . . . . . . . . . . . . . . . . . . . . . . . 132
7.40 Upward- (cosθ < −0.4) to downward-going (cosθ > 0.4) event ratio as afunction of energy for expanded FV. Statistical error are denoted by theerror bars. For the multi-ring samples the momentum is defined as thesum of each ring’s momentum. The cyan line denotes the best fit whensin2 θ23 = 0.4, and the orange line the best fit when sin2 θ23 = 0.6. Top
five plots show the result when sin2 θ13 = 0.0075, which is the best-fit
value in the sin2 θ13 -free analysis, while the bottom ones show the result
when sin2 θ13 = 0.02, which is the global best-fit value. . . . . . . . . . . . 133
7.41 Distributions of the best-fit χ2 values fits to pseudo data sets. The pseudodata have been generated with the best-fit value in NH expanded FV θ13
constrained fit shown in Table 7.13. Shaded portions of the histograms
denote the fraction of pseudo data sets with more extreme values thanthat observed in the data, χ2N H = 576.5. . . . . . . . . . . . . . . . . . . . 134
7.42 Distributions of the best-fit χ2 of observed data (red triangle) and pseudodata sets (blue box). The error bar of pseudo data sets result represents
the 1σ uncertainty from statistical fluctuations and systematic errors. The
pseudo data have been generated with the best-fit value in NH expanded
FV θ13 constrained fit shown in Table 7.13. The variation shows the effectof the statistical fluctuations of the new region (50 cm < Dwall < 200 cm). 139
7.43 Zenith angle distributions for of data (open circle) and best-fit MC for
both NH (blue solid line) and IH (orange dashed line). Sub-GeV single
ring e-like 0-decay electron events are shown here. Left figures are forconventional FV (Dwall > 200) cm, while right figures are for new region(50 cm < Dwall < 200 cm). The momentum range are 100 MeV ∼ 200
MeV, 200 MeV ∼ 300 MeV, 300 MeV ∼ 600 MeV, 600 MeV ∼ 1000 MeV,
1000 MeV ∼ 1330 MeV from most top figures, respectively. . . . . . . . . 140

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7.44 Zenith angle distributions for of data (open circle) and best-fit MC for
both NH (blue solid line) and IH (orange dashed line). Multi-GeV single
ring νe -like events are shown here. Left figures are for conventional FV(Dwall > 200) cm, while right figures are for new region (50 cm < Dwall <200 cm). The momentum range are 1000 MeV ∼ 2500 MeV, 2500 MeV ∼5000 MeV, 5000 MeV ∼ 10000 MeV, >10000 MeV from most top figures,
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
7.45 Distributions of the difference in best-fit χ2 values assuming the firstoctant (sin2 θ23 = 0.4) and second octant (sin2 θ23 = 0.6). The value ofsin2 θ13 is fixed to 0.0210 and expanded FV are used. The result of pseudodata generated assuming sin2 θ23 = 0.4 (sin2 θ23 = 0.6) is shown as thecyan (orange) histogram. The left figure shows the true NH case, while
the right figure show the true IH case. The calculation of p-value to obtaina value of ∆χ2 more extreme than the observed value, ∆χ2data = −0.73,
is according to the shaded histogram. . . . . . . . . . . . . . . . . . . . . . 142
7.46 Distributions of the difference in best-fit χ2 values assuming the normal
hierarchy and inverted hierarchy. The value of sin2 θ13 is fixed to 0.0210
and expanded FV are used. The result of pseudo data generated assuming
normal (inverted) hierarchy is shown as the cyan (orange) histogram. The
results with different assumed true sin2 θ23 are shown. The calculation of
p-value to obtain a value of ∆χ2 more extreme than the observed value,∆χ2data = −2.45, is according to the shaded histogram. . . . . . . . . . . . 142
8.1

8.2

8.3

8.4

Median expected sensitivity to mass hierarchy for different livetime. Normal hierarchy is assumed to be true and the true value of sin2 θ23 isassumed to be 0.5. Here the value of sin2 θ13 is fixed to 0.0210. The
result of the events reconstructed by fiTQun within the conventional FV
and expanded FV are denoted by the gray and blue lines. The sensitivity
with events reconstructed by APFit algorithm and the conventional FV
is denoted by the orange line. . . . . . . . . . . . . . . . . . . . . . . . . 146
A schematic of a neutrino’s journey from the neutrino beamline at JPARC, through the near detectors (green dot) which are used to determine the properties of the neutrino beam, and then 295 km underneaththe main island of Japan to Super-Kamiokande. Taken from [26] . . . . . 147
Sub-binning structure for the T2K systematic uncertainty parameterization. The binnings shown on the left are based on reconstructed information, while the ones shown on the right are based on true information.
En,0 represents the number of events of nominal MC prediction in n-th
bin based on reconstructed information, while En,m,0 denotes the numberof events in the m-th sub-bin for n-th bin. . . . . . . . . . . . . . . . . . . 148
Sensitivity to mass hierarchy as function of true sin2 θ23 . The solid lines(dotted lines) are the sensitivity with atmospheric neutrino only (accelerator neutrino only), while the dashed lines represent the result of joint
analysis. The result shown in left figure assumes true δCP = 0◦ , while
the right figure is for true δCP = 270◦ . The orange, purple, magenta andgreen lines represent the result at the end of year 2026, 2023, 2020 and2017, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

List of Figuresxxi

Sensitivity to CP violation (sin δCP 6= 0) as function of true sin δCP . The
true mass hierarchy of left figure is assumed to be normal hierarchy, while
for right figure it is assumed to be inverted hierarchy. The green, blue
and red lines represent the sensitivity with atmospheric neutrino only,
accelerator neutrino only and combined analysis at the end of year 2026,
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
8.6 Sensitivity to sin2 θ23 octant ( > 0.5 or < 0.5) as function of true sin2 θ23 .
The green, blue and red lines represent the sensitivity with atmospheric
neutrino only, accelerator neutrino only and combined analysis at the endof year 2026, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
8.7 Sensitivity to the normal mass hierarchy for components of a combined
measurement of beam and atmospheric neutrinos for a 10 year exposure.
Here JD refers to a single Hyper-K detector in Kamioka, Japan, and JD
×2 refers two to such detectors operating simultaneously. The horizontal
axis shows the assumed value of sin2 θ23 and the width of the bands shows
the variation in sensitivity with δCP . . . . . . . . . . . . . . . . . . . . . . 152
8.8 Sensitivity to δCP = 0 for components of a combined measurement ofbeam and atmospheric neutrinos for a 10 year exposure. Here JD refers
to a single Hyper-K detector in Kamioka, Japan. . . . . . . . . . . . . . . 153
8.9 Sensitivity to CP violation (sin δCP 6= 0) for components of a combined
measurement of beam and atmospheric neutrinos for a 10 year exposure.
Here JD refers to a single Hyper-K detector in Kamioka, Japan, and JD
× 2 refers two to such detectors operating simultaneously. The horizontal
axis shows the assumed true value of δCP . . . . . . . . . . . . . . . . . . . 153
8.10 Sensitivity as a function of year to the mass hierarchy (left) and the fraction of δCP phase space for which CP violation (sin δCP 6= 0) can be
determined at 3σ or better. Red lines show a combined beam and atmospheric neutrino measurement with one Hyper-K detector in Kamioka,
Japan (JD) and one at the Mt. Bisul site in Korea. Cyan lines show thesame analysis assuming two detectors in Kamioka (JD × 2) and grey lines
assume only one detector in Kamioka (JD × 1). Different symbols show
the assumed value of sin2 θ23 . . . . . . . . . . . . . . . . . . . . . . . . . . 154
8.5

A.1 PID likelihood distribution of sub-GeV single-ring events for SK-I. Left(right) figure shows the distribution reconstructed by fiTQun (APFit). . . 155
A.2 Charge profile with respect to the reconstructed vertex and direction of
sub-GeV single-ring events for SK-I (left) and SK-IV (right). Points denote the data and solid line is for MC. . . . . . . . . . . . . . . . . . . . . 156
A.3 Hit profile with respect to the reconstructed vertex and direction for subGeV single-ring events for SK-I (left) and SK-IV (right). Points denotethe data and solid line is for MC. . . . . . . . . . . . . . . . . . . . . . . . 156
A.4 Ratio between data and MC of charge profile (left) and hit profile (right)
for sub-GeV single-ring events for SK-I (black) and SK-IV (red). . . . . . 156
A.5 Charge distribution around the Cherenkov angle (42◦ ) for sub-GeV singlering events of SK-I (left) and SK-IV (right). Points denote the distributionof data, while the solid line is for MC. . . . . . . . . . . . . . . . . . . . . 157
A.6 Charge distribution around the Cherenkov angle (42◦ ) for Michel electronfrom cosmic ray muon of SK-I (left) and SK-IV (right). Points denotethe distribution of data, while the solid line is for MC. . . . . . . . . . . . 157

xxii

List of Figures

A.7 Hit time distribution of each PMT for Nickel source after event selection.
The on-time region and off-time region are marked with blue line. . . . . 158
A.8 Charge distribution of each PMT for Nickel source. Black (red) line denote the distribution from on-time (off-time) region. . . . . . . . . . . . . 159
A.9 Charge distribution for single p.e. with (left) and without (right) the
trigger effect. The points on the right shows the data from Nickel source
taken on 10, August, 1996 for calibration. . . . . . . . . . . . . . . . . . . 159
A.10 Charge profile (left) and hit profile (right) for sub-GeV single-ring eventsfor SK-I (black) and SK-IV (red). . . . . . . . . . . . . . . . . . . . . . . . 159
A.11 PID likelihood distribution for SK-I Michel electron from cosmic ray muon
before (left) and after (right) retuning. . . . . . . . . . . . . . . . . . . . . 160
A.12 Momentum distribution for SK-I Michel electron from cosmic ray muon
before (left) and after (right) retuning. . . . . . . . . . . . . . . . . . . . . 160
A.13 PID likelihood distribution of sub-GeV single-ring events for SK-I after
retuning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
A.14 Charge distribution around the Cherenkov angle (42◦ ) for sub-GeV singlering events of. Points denote the distribution of data, while the solid line
is for MC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
B.1 Basic distribution for SubGeV single-ring e-like 0 decay-e sample reconstructed and selected by fiTQun. Top four figures shows the event rate,component fraction, direction resolution and momentum bias as the function of the distance to nearest wall (dwall), respectively. The error bar
shows the statistical uncertainty. Bottom six figures shows the energy distribution and zenith angle distribution in different detector region. Sameplots are shown for next 12 figures. . . . . . . . . . . . . . . . . . . . . . 165
B.2 Basic distribution for SubGeV single-ring e-like with 1 or more decay-esample reconstructed and selected by fiTQun. . . . . . . . . . . . . . . . . 166
B.3 Basic distribution for SubGeV single-ring µ-like 0 decay-e sample reconstructed and selected by fiTQun. . . . . . . . . . . . . . . . . . . . . . . . 167
B.4 Basic distribution for SubGeV single-ring µ-like 1 decay-e sample reconstructed and selected by fiTQun. . . . . . . . . . . . . . . . . . . . . . . . 168
B.5 Basic distribution for SubGeV single-ring µ-like with 2 or more decay-esample reconstructed and selected by fiTQun. . . . . . . . . . . . . . . . . 169
B.6 Basic distribution for SubGeV two-ring π 0 sample reconstructed and selected by fiTQun. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
B.7 Basic distribution for MultiGeV single-ring νe -like sample reconstructed
and selected by fiTQun. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
B.8 Basic distribution for MultiGeV single-ring ν̄e -like sample reconstructed
and selected by fiTQun. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
B.9 Basic distribution for MultiGeV single-ring νµ -like sample reconstructed
and selected by fiTQun. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
B.10 Basic distribution for MultiGeV multi-ring νe -like sample reconstructed
and selected by fiTQun. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
B.11 Basic distribution for MultiGeV multi-ring ν̄e -like sample reconstructed
and selected by fiTQun. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
B.12 Basic distribution for MultiGeV multi-ring µ-like sample reconstructed
and selected by fiTQun. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

List of Figuresxxiii

B.13 Basic distribution for MultiGeV multi-ring e-like other sample reconstructed and selected by fiTQun. . . . . . . . . . . . . . . . . . . . . . . . 177
C.1 Ring counting likelihood distribution of sub-GeV e-like events with momentum less than 400 MeV. Points show the distribution from data, while
the red solid line (blue dashed line) represents the distribution after (before) fitting. The shaded histogram show events with only single-ring in
the final states. Left figure shows the distribution for conventional FV(Dwall > 200 cm), while right figure shows the one for new region (200cm > Dwall > 50 cm). All following figures in this section have the same
configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
C.2 Ring counting likelihood distribution of sub-GeV e-like events with momentum larger than 400 MeV. . . . . . . . . . . . . . . . . . . . . . . . . 182
C.3 Ring counting likelihood distribution of multi-GeV e-like events. . . . . . 182
C.4 Ring counting likelihood distribution of sub-GeV µ-like events with momentum less than 400 MeV. . . . . . . . . . . . . . . . . . . . . . . . . . . 182
C.5 Ring counting likelihood distribution of sub-GeV µ-like events with momentum larger than 400 MeV. . . . . . . . . . . . . . . . . . . . . . . . . 183
C.6 Ring counting likelihood distribution of multi-GeV µ-like events. . . . . . 183
C.7 Ring counting likelihood distribution of µ-like events with Evis > 600
MeV. This category has overlap with Figure C.5 and Figure C.6. . . . . 183
C.8 PID likelihood distribution for sub-GeV single-ring events. The shaded
histogram show νµ charged current interactions. . . . . . . . . . . . . . . . 184
C.9 PID likelihood distribution for multi-GeV single-ring events. . . . . . . . . 184
C.10 PID likelihood distribution for sub-GeV multi-ring events. . . . . . . . . 184
C.11 PID likelihood distribution for sub-GeV single-ring events. . . . . . . . . 185
C.12 Likelihood distribution used to separate SK-IV Multi-ring events withe-like most energetic ring into the e-like and unclassified samples. The
shaded histogram show νe and ν̄e charged current interactions. . . . . . . 185
C.13 Likelihood distribution used to separate SK-IV Multi-ring e-like events
into the neutrino-like and antineutrino-like samples. The shaded histogram show ν̄e charged current interactions. . . . . . . . . . . . . . . . . 185
D.1 Constraints on neutrino oscillation contours from SK-IV atmospheric neutrino data with expanded FV (Dwall > 50 cm) assuming sin2 θ13 = 0.0210±
0.0011 . Only normal hierarchy fit result is shown here. The solid line
represents the result with data until Apr. 2016, while the dashed line
show the result with data until Apr. 2018. Both of them are offset with
their respective minimal χ2 value. . . . . . . . . . . . . . . . . . . . . . . 187
D.2 Distributions of the best fit χ2 values fits to pseudo data sets. The pseudodata have been generated with the best fit value in NH expanded FV θ13
constrained fit shown in Table 7.13. Shaded portions of the histograms
denote the fraction of pseudo data sets with more extreme values thanthat observed in the data, χ2N H = 576.5. . . . . . . . . . . . . . . . . . . . 188
D.3 Distributions of the best fit χ2 of observed data (red triangle) and pseudodata sets (blue box). The error bar of pseudo data sets result represents
the 1 σ uncertainty from statistical fluctuations and systematic errors.
The pseudo data have been generated with the best fit value in NH expanded FV θ13 constrained fit shown in Table 7.13. . . . . . . . . . . . . . 189

xxiv

List of Figures

D.4 Zenith angle distributions for of data (open circle) and best fit MC for
both NH (blue line) and IH (orange line). Multi-GeV single ring νe -likeevents are shown here. Left figures are for data taken before Apr. 2016,
while right figures are for data taken between Apr. 2016 to Apr. 2018.
The momentum range are 1330 MeV ∼ 2500 MeV, 2500 MeV ∼ 5000
MeV, 5000 MeV ∼ 10000 MeV, > 10000 MeV from most top figures,
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
D.5 Zenith angle distributions for of data (open circle) and best fit MC for
both NH (blue line) and IH (orange line). Multi-GeV multi ring ν̄e -likeevents are shown here. Left figures are for data taken before Apr. 2016,
while right figures are for data taken between Apr. 2016 to Apr. 2018.
The momentum range are 1330 MeV ∼ 2500 MeV, 2500 MeV ∼ 5000
MeV, > 5000 MeV from most top figures, respectively. . . . . . . . . . . . 191
D.6 Distributions of the difference in best fit χ2 values between fits withsin2 θ23 fixed to 0.4 and 0.6 to pseudo data sets. In the cyan (orange)
histogram the pseudo data have been generated assuming sin2 θ23 = 0.4(0.6) hierarchy at normal hierarchy. Shaded portions of the histograms
denote the fraction of pseudo data sets with more extreme values thanthat observed in the data, ∆χ2data = −0.73. . . . . . . . . . . . . . . . . . 192
D.7 Distributions of the best fit χ2 of observed data (red triangle) and pseudodata sets (blue box) generated with true sin2 θ23 = 0.6. The error bar ofpseudo data sets result represents the one σ uncertainty from statisticalfluctuations and systematic errors. The pseudo data have been generated
with the best fit value in NH expanded FV θ13 constrained fit shown in
Table 7.13. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
D.8 Zenith angle distributions for of data (open circle) and best fit MC forsin2 θ23 = 0.4 (magenta line) and 0.6 (green dashed line). Sub-GeV single
ring µ-like 1-decay electron events are shown here. Left figures are for datataken before Apr. 2016, while right figures are for data taken between
Apr. 2016 to Apr. 2018. The momentum range are 100 MeV ∼ 200 MeV,
200 MeV ∼ 300 MeV, 300 MeV ∼ 600 MeV, 600 MeV ∼ 1000 MeV, 1000
MeV ∼ 1330 MeV from most top figures, respectively. . . . . . . . . . . . 193
D.9 Zenith angle distributions for of data (open circle) and best fit MC for
both sin2 θ23 = 0.4 (magenta line) and 0.6 (green dashed line). Multi-GeV
single ring νe -like events are shown here. Left figures are for data taken
before Apr. 2016, while right figures are for data taken between Apr.
2016 to Apr. 2018. The momentum range are 1000 MeV ∼ 2500 MeV,
2500 MeV ∼ 5000 MeV, 5000 MeV ∼ 10000 MeV, >10000 MeV frommost top figures, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . 194
E.1 Distribution of the decay time of tagged Michel electron (or position)from cosmic ray muon. The left figure shows the sub-GeV cosmic ray
muon while the right figure is for multi-GeV muon. Open circles are theelectrons tagged by APFit, and full triangles are for fiTQun. The solidline is the expectation based on the number of cosmic ray muons. . . . . . 196
E.2 Tagging efficiency for the Michel electron from cosmic ray muon. The left
figure shows the sub-GeV cosmic ray muon while the right figure is for
multi-GeV muon. Open circles are the electrons tagged by APFit, andfull triangles are for fiTQun. The error bars indicate the statistical error. 196

List of Figuresxxv

E.3 Goodness as a function of hit time for an event with a mutli-GeV muon
and a Michel electron. The black line shows the goodness as a functionof hit time with the vertex fixed to the pre-fix vertex (muon vertex). The
red line is for the goodness with the vertex at the decay electron position.
Green curves denot threshold for identifying candidate peaks. . . . . . . 197
E.4 Tagging efficiency for the Michel electron from cosmic ray muon after
improvement. The left figure shows the sub-GeV cosmic ray muon while
the right figure is for multi-GeV muon. Open circles are the electrons
tagged by APFit, and full triangles are for fiTQun. . . . . . . . . . . . . . 197
E.5 Tagging efficiency for the Michel electron from cosmic ray muon. The
horizontal axis shows the decay time, while the vertical axis represents
the momentum of parent muon. The efficiency is denoted by the color. . . 198

List of Tables
1.1

Summary of global best fit to the parameters of neutrino oscillation. The
number in the parentheses represents the result for inverted hierarchy.
Taken from [27] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1
2.2

Momentum thresholds of Cherenkov radiation in pure water. . . . . . . . 18
Specification of the 20-inch ID PMT for Super-Kamiokande experiment.
Taken from [28]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.1

Summary of absolute energy scale error. The final absolute energy scalevalue is obtained from the most discrepant sample, i.e. decay electron inthis study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Summary of energy scale check. The energy scale error is the quadraticsum of absolute energy scale error and the time variation error. The
absolute energy scale error value is obtained from the most discrepant
sample, i.e. decay electron in this study. . . . . . . . . . . . . . . . . . . 58

4.2

5.1

5.2

6.1

6.2

6.3

Selection efficiencies and number of events for FC sample with livetimeof 3118.45 days for SK-IV. Selection efficiencies are for events whose realvertex is within the FV and number of OD hits < 16 and visible energy >30 MeV. The FV cut is 200 cm and the variable reconstructed by APFit
is used in this Table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Estimated contamination for each background source. The term “subGeV” represent events with Evis < 1.3 GeV, while “multi-GeV” represent
events with Evis > 1.3 GeV. . . . . . . . . . . . . . . . . . . . . . . . . . 70
Basic performance of reconstruction algorithms on the FC CCQE singlering event sample. Both result of APFit and fiTQun are shown here. The
visible energy of the sample used here is 1 GeV. . . . . . . . . . . . . . . 82
Ring counting performance on FC atmospheric neutrino events. Bothresult of APFit and fiTQun are shown here. The number of reconstructedrings are denoted by columns and the number of true rings are denoted byrows. The true rings are defined as only final state particles with energy
30 MeV higher than the Cherenkov threshold. . . . . . . . . . . . . . . . . 83
Performance of CC1π 0 events reconstruction and selection in atmospheric
neutrino MC events with three-ring final state. The reconstructed PID of
the three rings in order of momentum are denoted by eee or µee, which
represent candidates of νe CC1π 0 and νµ CC1π 0 events, respectively. The
number of events passing the π 0 invariant mass cut is shown in the event
rate column. The purity of target events in the π 0 invariant mass regionis shown in the purity column. . . . . . . . . . . . . . . . . . . . . . . . . 85
xxvii

xxviii
7.1

List of Tables

Interaction mode composition of multi-GeV single-ring νe -like and ν e -likesamples. MC is normalized to data livetime 3118.5 days. The fiducial
volume is Dwall > 200 cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
7.2 Performance of multi-ring e-like events selection. The definition of efficiency is the fraction of the selected events among all true νe or ν̄e events.
The fraction of the true νe or ν̄e events over all selected events is definedas the purity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7.3 Purity of different neutrino flavor of each samples with a FV cut of 200 cm.
The neutrino oscillations with ∆m232 = 2.52×10−3 eV2 and sin2 θ23 = 0.51
has been considered. The result within new region (50 cm ¡ Dwall ¡ 200 cm)
is shown by the number in the parentheses. . . . . . . . . . . . . . . . . . 105
7.4 Time variation of atmospheric neutrino event rate in different detector
region. number of OD hits in cluster with the largest charge is required
to be less than 16, and visible energy is required to be larger than 30
MeV. For the last row, Dwall < 50 cm , there are many cosmic ray muon
background events contaminated therefore the event rate is much higher
than the expected value of MC. . . . . . . . . . . . . . . . . . . . . . . . 110
7.5 Time variation of atmospheric neutrino event rate in different detector
region. number of OD hits with the largest charge in cluster is required
to be less than 16, and visible energy is required to be larger than 30 MeV. 110
7.6 Summary of the oscillation parameters fixed in the analysis. Both ofthe value and associated uncertainty are shown. It should be noted thatsin2 θ13 is not fixed in the “θ13 -free”analyses. . . . . . . . . . . . . . . . . . 116
7.7 Oscillation parameter points scanned in this analysis. . . . . . . . . . . . . 116
7.8 Summary table of subsample categorization for different MC likelihood
distribution. The true number of rings is counted using only particles
with energy more than 30 MeV above Cherenkov threshold in the final
states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7.9 Systematic errors related to the ring counting, particle identification andmulti-GeV multi-ring e-like separation. The related distribution can befound in Appendix C. Errors from a particular uncertainty are assumed
to be fully correlated or fully anti-correlated among all of the analysis
samples. The sign of numbers in the table indicates the correlation. The
systematic error from different sources is assumed to be independent fromeach other. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.10 Uncertainty from two-ring π 0 event selection. . . . . . . . . . . . . . . . . 122
7.11 Uncertainty from decay electron tagging. . . . . . . . . . . . . . . . . . . . 122
7.12 CC backgrounds, including CCνµ and CCν̄µ , in multi-GeV e-like sample. 124
7.13 Best-fit result of oscillation parameter for each analysis. The term “NH”(“IH”) refers to the fit assuming normal (inverted) hierarchy. The fits
without and with a constraint on sin2 θ13 are termed as “Free” and “Constrained”, respectively. The best-fit result of sin2 θ23 in both octants areshown, with the best-fit octant enclosed in a box. . . . . . . . . . . . . . . 132
7.14 Summary of the systematic error used in this analysis. The best-fit valueof the systematic error parameter in percent is shown in second column.
The estimated 1σ error size in percent is shown in third column. The
result of the θ13 − constrained fit within expanded FV is shown here. . . 135

List of Tablesxxix

7.15 Normal hierarchy significance summarized in terms of the probability ofobserving a χ2 preference for the NH more extreme than that observedin the data assuming an IH, p0 (IH) , and CLH
s values for a range ofassumed parameters. The first row shows the true θ23 used to generate
MC ensembles used in the calculations. Other oscillation parameters aretaken from the analysis’ best-fit. . . . . . . . . . . . . . . . . . . . . . . . 143
B.1 Sample purity broken down by neutrino flavor assuming neutrino oscillations with ∆m232 = 2.52 × 10−3 eV2 and sin2 θ23 = 0.51. The righttwo columns show the sample size of the data and MC for the entire SK
data set, respectively. Only fully contained events are reconstructed byfiTQun. Partially contained events and upward-going muon are reconstructed by APFit. All of the samples shown here have a FV cut of 200
cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
B.2 Sample purity broken down by neutrino flavor assuming neutrino oscillations with ∆m232 = 2.52 × 10−3 eV2 and sin2 θ23 = 0.51. The right twocolumns show the sample size of the data and MC for the entire SK dataset, respectively. Only fully contained events in the new region, within
the range of the distance to the wall from 50cm to 200cm are shown. . . 178
B.3 Sample purity broken down by neutrino flavor assuming neutrino oscillations with ∆m232 = 2.52 × 10−3 eV2 and sin2 θ23 = 0.51. The right twocolumns show the sample size of the data and MC for the entire SK dataset, respectively. Only fully contained events are reconstructed by fiTQun
and have a FV cut of 50 cm. PC and Up-µ samples are reconstructed by
APFit and no FV cut change for these two samples. . . . . . . . . . . . . 179
D.1 Summary of parameter estimates for analysis with data until Apr. 2016
and data until Apr. 2018. . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
E.1 Decay e tagging efficiency for cosmic ray muon. . . . . . . . . . . . . . . . 198
E.2 Decay e tagging efficiency for cosmic ray muon in new region 50 cm<
Dwall < 200 cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

Chapter 1

Introduction
The neutrino is the second most abundant fundamental particle in the Standard Model,
after only photons. Tens of billions of neutrinos pass through every square centimeter ofthe Earth every second. The neutrino was thought to be massless until the discovery ofneutrino oscillation in late 1990s. Neutrino physics has achieved significant progress sincethen, however, many of the neutrino’s characteristics are still unknown since neutrinos
only interact with other matter via weak interactions. Nonetheless neutrinos are thought
to have deep connections to the physics of the early universe and further study of these
elusive particles may give insight into those processes.

1.1

Neutrino

The neutrino was first proposed by W. Pauli in 1930 to explain the continuum energy
spectrum from the β decay (n → p + e− + ν̄e ). The first observation of neutrino wasin 1956 by F. Reines and C. L. Cowan [29] by detecting anti-electron-neutrinos froma nuclear reactor via inverse β decay interaction: ν̄e + p → n + e+ . The second typeof neutrinos νµ was discovered by L. M. Lederman, M. Schwartz and J. Steinberger in
1962 in the pion decay process at the Brookhaven’s Alternating Gradient Synchrotron(AGS) [30]. In 2000, DONUT (Direct Observation of the NU Tau) experiment announced the discovery of the third species of neutrinos: ντ [31]. The result from thedecay of Z 0 produced by e+ e− collisions at Stanford Linear Collider (SLC) and Large
Electron Positron collider (LEP) indicates that there are three (2.984±0.008) type oflight neutrinos 1 , which are affected by weak interaction [32].
One of the most frontier questions to today’s astrophysicist and particle physicist is why
our universe is matter-dominated. One of the conditions for such a matter-dominated1

Light means the mass of neutrino is less than half of Z 0 mass (∼91GeV).

1

2

Chapter 1. Introduction

universe, called Sakharov conditions, is the existence of CP violation: the violation of
the combined charge and parity symmetries. The CP violation in quark sector has been
discovered but its size is not large enough to explain the matter-antimatter imbalance
observed today. Recent neutrino oscillation result from the T2K experiment indicatesthat the CP violation also exists for neutrinos with more than 95% probability [34], whichmeans that the neutrino might play an important role in the creation of the matterantimatter imbalance in the universe. The determination of δCP in the T2K experimentis based on the measurement of the oscillation probability difference between neutrino(e.g. νµ → νe ) and antineutrino (e.g. ν̄µ → ν̄e ). However, because of its fixed baseline,
degeneracies between CP-violation and mass hierarchy in the oscillation probabilities
limit the overall sensitivity. The sensitivity to the CP-violation can be improved byconstraining the mass ordering (discussed below), for instance, from the measurementof atmospheric neutrinos sample, which is one of the goals of this thesis, to resolve sucha degeneracy issue. Furthermore, atmospheric neutrinos themselves also offer additional
sensitivity to the CP violation as well as other topics in neutrino oscillation physics.

1.2

Neutrino Oscillation

In the Standard Model of Particle Physics, neutrinos are assumed to be massless. However, neutrino oscillations observed in atmospheric [1], solar [35], accelerator [36] andreactor experiments [37, 38] indicate that neutrino flavor eigenstates actually consist of
superposition of neutrino mass eigenstates, suggesting that the Standard Model needed
revision.
In the neutrino oscillation diagram proposed by Maki, Nakagawa, Sakata [39] and Pontecorvo [40], the flavor eigenstates of neutrino can be written as mixing of mass eigenstates:|να i =

3
X

Uαi |νi i.

(1.1)

i=1

Here, |να i represents the flavor eigenstate νe , νµ or ντ , |νi i represents the mass eigenstateν1 , ν2 or ν3 . U is the Pontecorvo-Maki-Nakagawa-Sakata matrix (PMNS matrix), which
is a 3 × 3 matrix describing the mixing of the flavor eigenstates and the mass eigenstates:


1


U =
0

0
cosθ23

0





 

sinθ23 ×

0 −sinθ23 cosθ23

cosθ1300 sinθ13 e−iδ



1


 
 × −sinθ12 cosθ12 0
 

0
0
1 ,
(1.2)

−sinθ13 eiδ 00cosθ13cosθ12sinθ120Chapter 1. Introduction

3

where θij (i, j = 1, 2, 3) represents the mixing angles between the mass eigenstates |νi i
and |νj i and δ denotes the CP violation phase. The three matrices control the oscillations observed by atmospheric neutrinos experiments, reactor experiments 2 and solar
neutrino experiments, respectively, which will be discussed later in this Chapter. The
time evolution of the flavor eigenstates is|να (t)i =X

Uαi e−iEi t |νi (t = 0)i

i

=

XX
i

Uαi Uα∗0 i e−iEi t |να0 (t = 0)i.

(1.3)

α0

Here, Ei represents the eigenvalue of the mass eigenstate |νi i. The probability for flavor
eigenstate να at t = 0 to oscillate to state νβ at time t is:
P (να → νβ ) = |hνβ (t)|να (0)i|2
X∗ −iEi t 2=|
Uβi Uαi
e
|

(1.4)

i

=

X

|Uαi Uβi |2 +

X

i

∗ ∗
Uαj Uβj e−i(Ei −Ej )t ,Uαi Uβii6=j

The above equation shows that the transition phenomenon between different flavors,
also known as neutrino oscillation, is due to the mass eigenstate mixing of neutrinos
with finite mass.
The PMNS matrix U can be simplified for the case of the two-flavor mixing:

U=

cosθ

sinθ

!
(1.5)

−sinθ cosθ .

Transition probability shown in Equation 1.4 can be simplified into:22

P (να → νβ ) = sin 2θsin
2(Ei − Ej )t
2


.

(1.6)

Strictly speaking, here refers to the reactor experiments with a short baseline such as Daya Bay(∼ 50 km) and RENO (∼ 1 km). The reactor experiments with a long baseline such as KamLAND(∼ 180 km) is more sensitive to θ12 , like solar neutrino experiments.

4

Chapter 1. Introduction

For neutrinos with small finite mass, the following approximation can be made:Ei − Ej =qq(m2i + p2i ) − (m2j + p2j )

!1
1
m2i 2
m2i 2
− pj 1 + 2= pi 1 + 2
pi
pj
!


m2j
m2
≈ p+ i − p+
2p
2p


≈

(1.7)

∆m2ij
.
2E

Here, ∆m2ij ≡ m2i − m2j is defined as the mass-squared difference of the neutrino mass
eigenstates. The momentum pi and pj are approximated to p and further approximatedto E due to the small mass of neutrino. Substituting Equation 1.7 into Equation 1.6,
with neutrino flight length Lν and neutrino energy Eν , in two-flavor mixing case, thesurvival probability of να can be expressed as:
P (να → να ) = 1 − P (να → νβ )= 1 − sin2 2θsin2= 1 − sin2 2θsin2∆m2ij Lν
4Eν!

1.27∆m2ij (eV2 )Lν (km)Eν (GeV)

(1.8)
!
.

The oscillation probability therefore depends on the mixing angle θ, the mass squaredifference ∆m2ij , flight length of neutrinos Lν and energy of neutrinos Eν . The oscillation
amplitude is at maximum, i.e. the survival probability is at minimum whenLν (km)π∼.
Eν (GeV)
2.54 × ∆m2ij (eV2 )(1.9)

Therefore, neutrino oscillation experiments can measure the value of neutrino squared
mass difference |∆m2ij | and mixing angle sin2 2θ with 2-flavor oscillation paradigm bymaking suitable choices of Lν and Eν . To extract the sign of the mass difference and theprecise mixing angle θ, oscillation between three flavor neutrinos needs to be considered,as will be discussed in Section 1.5.

1.3

Neutrino Oscillation Experiments

Neutrino oscillations have been confirmed by many experiments with various sources.
Currently, the data are described well by the PMNS formalism involving all of thethree active neutrino flavors, as introduced in previous section. Most of the oscillation

Chapter 1. Introduction

5

parameters have been experimentally measured [27]. In this section, some of thoseexperiments which contributed to the discovery of neutrino oscillation or the oscillation
parameter measurement will be introduced.

1.3.1

Atmospheric Neutrino and Accelerator Neutrino

As cosmic rays propagate in the atmosphere, they generate many pions and kaons ininteractions with air nuclei as:
p + N → N 0 + π ± /K 0 /K ± ,(1.10)

where N and N’ represent the nucleons in atmosphere. Atmospheric neutrinos are produced by decays of those mesons:
π + → µ+ +νµë

e+ + νe + ν̄µ ,(1.11)

π − → µ− +ν̄µ
ëe− + ν̄e + νµ .

(1.12)

Therefore, both neutrinos and antineutrinos of both electron and muon type are contained in the atmospheric neutrino flux. The flavor ratio is expected to be (νµ + ν̄µ ) :(νe + ν̄e ) ∼ 2 : 1. This ratio increases in high energy region due to the increase of theprobability for µ± to be observed before decaying. The flight length of atmospheric neutrinos ranges between 10 ∼ 104 km determined by the zenith angle of neutrino’s path,
which corresponds to the thickness of atmosphere and the diameter of the Earth, respectively. Together with energy range from 10−1 to 103 GeV, which can be detected bySuper-Kamiokande detector (Super-K, SK), atmospheric neutrinos observed by Super-K
cover a wide range of L/E, which means the sensitivity to ∆m2 is 10−1 ∼ 10−4 eV2
based on Equation 1.9.
In the early 1990’s, a large deficit in the flavor ratio (νµ + ν̄µ ) : (νe + ν̄e ) was observed by experiments using large underground water Cherenkov detectors such as
Kamiokande [45] and IMB-3 [46]. The measured ratio was only a half of the expectation.
This discrepancy is known as atmospheric neutrino anomaly, suggesting neutrino oscillations of a different kind from that affecting solar neutrinos, which will be introducedin next section, due to the very different L/E involved.
In 1998, a large deficit in the upward-going νµ events compared to the expectation
assuming no oscillation (Figure 1.1) was reported by the Super-K collaboration [1].
This was the first evidence for the neutrino oscillation. Moreover, the zenith angle

6

Chapter 1. Introduction

sub-GeV
250

250

e-likep < 0.4 GeV/c

200

200

multi-GeV
50

e-likep > 0.4 GeV/c40

75

e-likep < 2.5 GeV/c

60

150

150

30

45

100

100

20

30

50

50

10

15

0
200

0
300

0
100

0
125

µ-likep < 0.4 GeV/c

160

240

µ-likep > 0.4 GeV/cµ-like

80

100

120

180

60

75

80

120

40

50

40

60

20

25

0
-1

-0.6 -0.2 0.2
cosΘ

0.6

1

0
-1

-0.6 -0.2 0.2
cosΘ

0.6

1

0
-1

-0.6 -0.2 0.2
cosΘ

0.6

1

0
-1

e-likep > 2.5 GeV/c

Partially Contained-0.6 -0.2 0.2
cosΘ

0.6

1

Figure 1.1: Zenith angle distributions for electron-like (top) and muon- like(bottom) events in sub-GeV (left) and multi-GeV (right) energy region, observed bySuper-Kamiokande. Upward- and downward-going events are represented by negativeand positive value of cos θ, respectively. The expectations without oscillations are
represented by the hatched bands, while the black lines denote the best-fit to datawith the oscillation between νµ and ντ . Taken from [1].

distributions of upward-going muons, which are created by νµ interactions in the rockaround the detector such as MACRO [47] and Super-K [48], also show a disagreement
to the non-oscillation expectation but consistency to the νµ ↔ ντ oscillation hypothesis.
A dip in the L/E distribution predicted from the neutrino oscillation was observed andreported by Super-K[49].
The atmospheric neutrino oscillation was then confirmed by experiments using neutrinos
from accelerators. In such experiments, an intense νµ beam is generated by the hadronic
showers from the collision between a high energy proton beam and a fixed target, andthen the neutrinos are observed at a far detector, whose distance from the neutrino sourceis O(100 km). The accelerator neutrino beams generated in this way are almost pure
νµ or ν̄µ beams, which is controlled by changing the polarity of charge of the hadrons
to be focused. In the accelerator neutrino beam, the contamination from νe or ν̄e issmall. The neutrino energy is also tuned ∼ 1 GeV to maximize the disappearance effectνµ → νx at the far detector based on the ∆m2 obtained from the atmospheric neutrino.
K2K [50] and MINOS [51] are the first of accelerator neutrino experiments with a long
baseline. They reported consistent significant deficits in the number of observed eventsand distortions in the νµ energy spectrum. Their results are in a good agreement withthe atmospheric neutrino measurement by Super-Kamiokande.
Many ongoing experiments are currently measuring the atmospheric neutrino mixing
parameters (θ23 and ∆m232,31 ) with increasing precision. It should be noticed that theChapter 1. Introduction

7

sensitivity of current atmospheric and accelerator neutrinos experiment to the sign of∆m232 is limited, since the matter effect, which will be discussed in Section 1.5, is subdominant. 3
The latest results for these two oscillation parameters from a combination of various
experiments are as follows [27].
For normal mass hierarchy , ∆m232 > 0:
sin2 θ23 = 0.51 ± 0.04|∆m232 | = (2.45 ± 0.05) × 10−3 eV2 .

(1.13)For inverted mass hierarchy, ∆m232 < 0:sin2 θ23 = 0.50 ± 0.04|∆m232 | = (2.52 ± 0.05) × 10−3 eV2 .

1.3.2(1.14)

Solar Neutrino and Reactor Neutrino

Electron neutrinos are generated by nuclear fusion reactions in Sun. The combined effectof the interaction in Sun can be written as:
4p → 4 He + 2e+ + 2νe + γ.

(1.15)The energy of solar neutrinos varies from 0.1 MeV ∼ 10 MeV. The first experiment thatobserved solar neutrinos is the Homestake experiment [35]. A detector utilized 520 tonsof chlorine to detect the electron neutrinos by counting the number of Ar produced by thereaction νe + 37 Cl → e− + 37 Ar. Only about 1/3 of the prediction of the Standard Solar
Model (SSM) were observed. Later experiments such as SAGE [52] and GALLEX [53],which utilize νe +71 Ga → e− +71 Ge, and Kamiokande experiment [54], a water Cherenkov
detector, also observed such deficits. This discrepancy between the measurement andprediction of solar neutrino flux, which was known as the solar neutrino problem, hadbeen unresolved for more than 30 years until the discovery of the neutrino oscillation by
Sudbury Neutrino Observatory (SNO) experiment [55].
The SNO [56], which is a heavy water (D2 O) Cherenkov detector, measures both νe flux
and the total neutrino flux (νe + νµ + ντ ) from the Sun. Neutrinos in SNO are detectedby the charged-current (CC) and neutral-current (NC) interactions on deuterium and3

It should be noted that this is not the case for ∆m221 , since the electron densities in the Sun is verylarge which makes the matter effect is strong. Such effect observed in solar neutrinos (i.e. νe ) are only
possible if ∆m221 > 0 which is similar to the atmospheric case discussed in Section 1.5.

Chapter 1. Introductionφµτ (× 10 6 cm -2 s-1)

8

BS05

φSSM 68% C.L.

6

NC

φµ τ 68%, 95%, 99% C.L.

5
4
3
SNO

2

φCC 68% C.L.
SNO

φNC 68% C.L.
SNO

1

φES 68% C.L.
SK

φES 68% C.L.
0
0

0.5

1

1.5

2

2.5

3

3.5

φe (× 106 cm-2 s-1)

Figure 1.2: 8 B solar neutrino flux measured by SNO with different channels. The
horizontal axis shows the νe flux and the vertical axis shows the sum of the νµ and ντ
flux. The allowed regions at 68% from the measurements of CC, NC and ES channels
are represented as the red, blue and green bands respectively. The gray band denotes
the ES measurement result from Super-Kamiokande [2]. The dashed lines represents
the prediction of the total neutrino flux based on Standard Solar Model. The point
and the contours show the best estimate and the allowed regions at 68, 95 and 99%
confidence levels by combining the results from CC and NC. Taken from [3].

also by elastic-scattering (ES) on electron as the following:
CC : νe + d → p + p + e−

(1.16)

NC : νx + d → p + n + νx

(1.17)

ES : νx + e− → νx + e− ,(1.18)

where νx denotes any of the three neutrino flavors. The total number of neutrinos
observed is consistent with the prediction from the SSM. As shown in Figure 1.2, about
one third of the observed neutrinos are electron neutrinos, which is also consistent with
the SSM prediction. The deficit of νe and presence of non-νe component in the solar
neutrino flux was also observed by Super-Kamiokande experiment, an H2 O Cherenkov
detector [2]. Combination results from SNO and Super-K showed that the non-electron
neutrino flux is due to the flavor transition of the solar neutrinos via the mechanism
called the large mixing angle (LMA) MSW solution4 [57, 58].
Similar deficit of ν̄e flux was also confirmed by the KamLAND experiment [59], which
detects ν̄e from nuclear reactors with energy from 0.9 MeV to 8.5 MeV using liquid
4

The large mixing angle means a large value of θ in Equation 1.8. The value of tan2 (θ) is around
0.4 for LMA, while it is only 4 × 10−4 for the small mixing angle solution, which was another possible
solution at that time.

Chapter 1. Introduction

9

scintillator. In KamLAND, e+ and neutron will be generated by inverse β-decay interaction ν̄e + p → e+ + n after ν̄e enter the detector. By combining the prompt signal
from e+ and the delayed signal from the gamma generated by the neutron capture, thebackground level can be suppressed significantly. Considering the energy of the ν̄e fromthe nuclear reactors is about 3 MeV and the average distance to the rectors is 180 km,
the sensitivity to ∆m212 is at the order of 10−5 eV, based on Equation 1.9.
The current known values for θ12 and ∆m221 obtained from the combination of oscillation
results of various solar and reactor neutrino experiments are [27]:1.3.3∆m221 = (7.53 ± 0.18) × 10−5 eV 2 ,(1.19)

sin2 θ12 = 0.307 ± 0.013.

(1.20)

Mixing Angle θ13

The third mixing angle θ13 was thought to be small since both solar and atmospheric
neutrino results could be explained assuming it is zero, i.e. two flavor mixing, withtwo different ∆m2 . The first attempt to measure θ13 was made by the CHOOZ experiment [60], which detected reactor ν̄e with a much shorter base line ∼ 1 km than thatof KamLAND. The dominant contribution at this distance is θ13 and ∆m231 rather thanthe “solar” oscillation parameters like KamLAND. The survival probability of reactorν̄e can be expressed as:
P (ν̄e → ν̄e ) = 1 − sin2 2θ13 sin2 (1.27∆m231 L).
Eν

(1.21)CHOOZ finally found no evidence for ν̄e disappearance. The upper limit of sin2 2θ13 isgiven as less than 0.15 at 90% confidence level [60].
In 2012, the reactor experiments with a short baseline including Daya Bay[37], Double
Chooz[61] and RENO[38] reported that sin2 2θ13 ∼ 0.1 separately. A clear deficit in thereactor ν̄e flux was observed. The hypothesis of a zero-value of θ13 was excluded byDaya Bay and RENO at 5.2σ and 4.9σ, respectively.
Besides the reactor experiments, the value of θ13 can also be obtained by observing theappearance of νe in a νµ beam generated by accelerator. The appearance probability ofνe is proportional to sin2 2θ13 , as Equation 1.25. The first decisive evidence for the νeappearance was shown by T2K in 2013 at 7.3 σ [62].
The current best constrains on θ13 are provided by the reactor experiments. The global
averaged value is [27]:sin2 θ13 = 0.0210 ± 0.0011.

(1.22)

10

Chapter 1. Introduction

Figure 1.3: Two possible mass hierarchies for the three neutrino mass eigenstates.
The varying width of the shadings shows the variation with respect to cosine of theCP violating phase δ. Taken from [4].

1.4Unresolved Issues

The global best fit results to the parameters of neutrino oscillation are summarized inTable 1.1. Though most of these parameters have been experimentally measured [27],the ordering of the mass states with the largest splitting,5 the octant of the atmosphericmixing angle θ23 (sin2 θ23 < 0.5 or > 0.5), and the value of the CP-violating phase areunknown.
Parameter∆m221
sin2 θ12
sin2 θ13
sin2 θ23|∆m231 |Best fit ± 1σ C.L.
(7.53 ±0.18) × 10−5 eV20.307 ±0.013
0.0210 ±0.0011
0.51(0.50) ±0.04(2.45(2.52) ± 0.05) × 10−3 eV2Table 1.1: Summary of global best fit to the parameters of neutrino oscillation. The
number in the parentheses represents the result for inverted hierarchy. Taken from [27]

While the value of the neutrino mass splitting have been measured with a certain precision, the sign of ∆m232,31 has not been determined yet. As shown in Figure 1.3, thecase ∆m232,31 > 0 is termed as normal hierarchy (NH) which means that the mass eigenstate ν3 has the largest mass, whereas ∆m232,31 < 0 is called inverted hierarchy (IH) in
which case the mass eigenstate ν3 has the smallest mass. For the problem related to theneutrino mass, besides the hierarchy, the absolute mass and its origin are still unknown.
5

Also known as the mass hierarchy, and indicated by the sign of ∆m232 .

Chapter 1. Introduction

11

Ca
Se

Zr Nd
Te
Cd
Te

1

Ge Mo

10−1

KamLAND-Zen (136Xe)
IH

m

(eV)

Xe

10−2
NH

10−3
10−4

10−3 10−2 10−1mlightest (eV)50 100 150
A

Figure 1.4: Effective Majorana neutrino mass hmββ i as a function of the lightest
neutrino mass mlightest . The dark shaded region shows the expectations based on thebest-fit values of neutrino oscillation parameters for both normal hierarchy (NH) andinverted hierarchy (IH), respectively. The light shaded regions considers the 3σ
uncertainties of those parameters. The cyan horizontal bands indicate 90% C.L. upper
limits on hmββ i with 1 36Xe from KamLAND-Zen. Right panel shows thecorresponding limits for each nucleus as a function of the mass number. Taken
from [5].

Some theories assume that neutrinos are Majorana fermions, which means that a neutrino is its own antiparticle. The Majorana fermion is a new type of elementary particles
and has never been observed until now. The next generation of neutrino-less double beta
decay experiments might give us some insight on this issue if the mass hierarchy is determined to be inverted as shown in Figure 1.4 [5]. Therefore, the determination of
the mass hierarchy could also provide valuable input to study the massive neutrinos’fundamental properties.
Furthermore, it is also not known whether θ23 mixing is exactly maximal (sin2 θ23 = 0.5),
and if not, whether the angle is smaller (sin2 θ23 < 0.5, first octant) or larger (sin2 θ23 >
0.5, second octant). This issue is known as the octant of θ23 . A new underlying fundamental symmetry of the neutrino mixing matrix and of the lepton sector of particle
physics might be revealed if θ23 mixing is found to be maximal. Furthermore, the sizeof matter effect, which will be discussed in Section 1.5, is crucial to the determinationof the mass hierarchy and depends on the value of sin2 θ23 . The high precision measurement of sin2 θ23 will also provide a test of many theories of neutrino masses and mixing.
Current primary approach to measure θ23 is to observe the disappearance of νµ , which

12

Chapter 1. Introduction

depends on sin2 2θ23 dominantly. The measurement on subdominant oscillation effect,
such as the oscillation between νµ and νe with a leading term depending on sin2 θ23 , isnecessary to determine the octant of θ23 .
The final, and probably most important question is whether CP is violated for leptons.
In some leading theories, the asymmetry between matter and antimatter in current
matter-dominated universe can be contributed by the CP violation in the lepton sector [63]. As T2K and reactor experiments have shown a non-zero θ13 , the question ofCP-violation in the lepton sector might be addressed by neutrino oscillation measurement, as the CP-violation phase δCP appears as a combination with θ13 in Equation
1.2. The determination of δCP is mainly based on the measurement of the oscillation
probability difference between neutrino (e.g. νµ → νe ) and antineutrino (e.g. ν̄µ → ν̄e ).
Since the matter effects could also cause such a difference on the oscillation probability, there is a degeneracy between CP-violation and mass hierarchy, especially for theaccelerator experiment with a fixed beamline. The sensitivity to δCP of several ongoingand proposed neutrino oscillation experiments can be improved significantly if the mass
hierarchy is known [64].
Some latest results, e.g. T2K [22], NOvA [41] and Super-Kamiokande [6] experiments
have given us some insight to those unresolved issues. Next-generation experiments
planned in both the U.S. [42], China [43], and Japan [44] also focus on these issues andexhibit the possibility to resolve them. Since the atmospheric neutrino cover a wide
range of L/E, so it can contribute to the measurement of those parameters, as we will
see in next Section.

1.5

Sub-Dominant Effects of Atmospheric Neutrino Oscillation

Beside the dominant oscillation channel between νµ and ντ , the oscillation νµ → νe induced by θ13 and θ12 also affects the survival possibility for νµ and appearance possibility
for νe .
As has been stated in previous section, the atmospheric neutrinos cover a wide rangeof L/E, which results in the sensitivity to ∆m2 from 10−1 to 10−4 eV2 . Therefore,both ∆m232,31 (∼ 10−3 eV2 ) and ∆m221 (∼ 10−5 eV2 ) affect the transition possibility ofνµ → νe . Their contributions vary as the neutrino energy.
For the multi-GeV region, which means the energy is larger than 1.33 GeV for the SuperK experiment, the contribution from ∆m221 is enough small to be ignored, the leadingChapter 1. Introduction

13

term of the probability for the oscillation between νe and νµ in vacuum therefore can be
expressed as:
P (νe → νe ) = 1 − sin2 2θ13 sin2

1.27∆m232 LE

P (νµ → νµ ) ∼= 1 − 4 cos2 θ13 sin2 θ23 (1 − cos2 θ13 sin2 θ23 )2
2 1.27∆m32 L
× sin
EP (νµ → νe ) = P (νe → νµ )


2
2
2
2 1.27∆m32 L= sin θ23 sin 2θ13 sin.
E(1.23)(1.24)(1.25)

When the atmospheric neutrinos traverse the Earth, the potential from the Earth’s
matter effect due to the amplitude of forward scattering via charged current interactions
between electrons and νe /ν̄e has to be taken into account. Since νe , and only νe caninteracts with the electrons in the Earth via W± bosons, the amplitude of forward
scattering for νe differs from those of the other flavors (i.e. νµ and ντ ). The effective
potential for the interaction via W± bosons is therefore applied to νe , which is as thematter effect or the MSW effect of neutrino oscillations [57, 58]. For the scenario above
the oscillation parameters sin2 θ13 and ∆m232 in Equation 1.25 are replaced by theirmatter-equivalents [65],sin2 2θ13,M =sin2 2θ13,(cos 2θ13 − ACC /∆m232 )2 + sin2 2θ13

(1.26)q(cos 2θ13 − ACC /∆m232 )2 + sin2 2θ13 .

(1.27)∆m232,M = ∆m232√
Here ACC = ±2 2GF Ne Eν is the effective potential and it is positive (> 0) for neutrinoor negative (< 0) for antineutrino. The electron density Ne is assumed to be constantand GF denotes the Fermi constant.
When ACC /∆m232 = cos 2θ13 , these matter variables are resonantly enhanced. Sincecos 2θ13 is positive, the enhancement only happens for neutrinos if the hierarchy isnormal (both of ACC and ∆m232 are positive) while it only occurs for antineutrinos forthe case of the inverted hierarchy (both of ACC and ∆m232 are negative). Actually,
electron density Ne changes as neutrinos travel through the Earth due to the varying
matter profile. Neutrinos experience a variety of matter effects and similar enhancements
to the oscillation probability. The most strong matter effect occurs for the neutrinos
with a few to ten GeV of energy through the Earth’s core, as shown in the top rightfigure in Figure 1.5.

14

Chapter 1. Introduction(a) P (νµ → νµ )(b) P (νµ → νe )(c) P (ν̄µ → ν̄µ )(d) P (ν̄µ → ν̄e )Figure 1.5: Oscillation probabilities for neutrinos (upper panels) and antineutrinos(lower panels) as a function of energy and zenith angle assuming a normal mass
hierarchy. Matter effects in the Earth distorted the neutrino oscillation probability
between 2 and 10 GeV, while there is no such distortion in the antineutrino figures.
For an inverted hierarchy, the distortion caused by matter effects appear in theantineutrino figures. The discontinuities near cosθzenith = −0.5 and -0.8 are due tothe matter density change between crust, mantle and core. Here the oscillation
parameters are taken to be ∆m232 = 2.5 × 10−3 eV2 , sin2 θ23 = 0.5, sin2 θ13 = 0.0219,
and δCP = 0. Taken from [6]Chapter 1. Introduction

15

On the other hand, the patterns on the oscillation probability below 1 GeV are driven
by the solar neutrino mixing parameters, θ12 and ∆m221 . The effect of θ13 can be ignored
since it is subdominant, then the oscillation probabilities can be written as [67]:P (νe → νµ ) = P (νµ → νe )= Pex cos2 θ23 ,(1.28)P (νe → νe ) = 1 − Pex ,
where
2

2

Pex = sin 2θ12,M sin

1.27∆m221,M L(1.29)!

E.

(1.30)

Here the matter effect has been considered and the corrected parameter ∆m221,M andsin2 2θ12,M can be written easily by replacing the mass splitting and mixing angle andin Equation 1.26 and 1.27 by θ12 and ∆m221 . Considering contribution from both νe andνµ produced in the atmosphere, the observed νe flux after oscillation Φe is
Φe = Φ0e (1 − Pex ) + Φ0µ (Pex cos2 θ23 ),(1.31)

where Φ0e and Φ0µ represent the flux of νe and νµ before oscillation. For simplicity, letr := Φ0e /Φ0µ , which means the ratio of νµ and νe before oscillation, and then the Equation
1.31 can be rewritten as:Φe− 1 = Pex (r cos2 θ23 − 1).
Φ0e(1.32)

The flux flavor ratio r is about 2 for low energy atmospheric neutrinos. Therefore, an
excess of sub-GeV νe flux would be observed for θ23 < π/4 case while a deficit would beseen instead when θ23 > π/4.
For the full three flavor oscillation analysis, besides the θ13 , θ12 and their interference,
the oscillation probability is also affected by the CP violation phase δCP , which results
in a change about ∼ 2% on the total νe flux observed in sub-GeV at maximum. The
atmospheric neutrinos therefore have some sensitivity to CP violation. In Section 7.2,
the oscillation effects discussed here will be revisited.

1.6

Thesis Overview

This thesis shows an analysis of 3118.5 days of SK-IV atmospheric data with a 30% larger
fiducial volume (FV) than previous Super-K analyses for a 253.9 kiloton·year exposure.
The detector is introduced in Chapter 2. In Chapter 3 the simulation of atmospheric

16

Chapter 1. Introduction

neutrino flux, neutrino interaction and data are discussed. The calibration method
and its result, which are essential to the atmospheric neutrino oscillation analysis, aredetailed in Chapter 4. The data reduction of atmospheric neutrino sample and event
categorization are discussed in Chapter 5. The new reconstruction algorithm and its
performance in comparison to the conventional reconstruction are detailed in Chapter 6.
Based on this performance improvement, an analysis of the atmospheric neutrino datawith the expanded fiducial volume and corresponding systematic errors is presented in
Chapter 7, before concluding in Chapter 8.

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. The
schematic view and the location of the detector are shown in Figure 2.1. The detector
is located in a obsolete zinc mine in Mountain Ikenoyama. The mean rock overburden
over the detector is ∼ 1000 m, which is ∼ 2700 m water equivalent. The flux of cosmic
ray muon observed by Super-Kamiokande is therefore reduced by 5 orders comparing tothat on the surface of the Earth.

Figure 2.1: Super-Kamiokande detector and its location in Mountain Ikenoyama.
Take from [7]

17

18

Chapter 2. The Super-Kamiokande Detector

The main physics goal of the Super-Kamiokande experiment is the detection of nucleon
decays and the studies of neutrinos from difference sources, including atmospheric neutrinos, solar neutrinos, supernova neutrinos and other astrophysical sources. Furthermore,
the Super-Kamiokande detector is also utilized as a far detector for the T2K experiment,a long-baseline neutrino oscillation experiment using accelerator neutrinos [26].
The first phase of Super-Kamiokande experiment (SK-I) started observing in April, 1996
and continued to take the data for five years until the maintenance in July, 2001. An
accident occurred during water refilling after the maintenance in November 2001. Morethan half of the PMTs were destroyed in that accident. The Suepr-Kamiokande detector
resumed observation with half PMT density in the inner detector (ID) from October,
2002, which is referred to as SK-II period. SK-II kept measuring for three years andfinished in October 2005 for PMT density resumption work. The observation of SK-III
period is restarted in June, 2006 with the same density of PMT as SK-I. This phase of theexperiment ended in August 2008 to upgrade electronics. The new phase, referred to as
SK-IV, collected data from September 2008 till May 2018. In June 2018, refurbishment
work starts as a preparation for Super-Kamiokande Gadolinium Project [69], which is
aimed to increase the detection efficiency to low energy anti-neutrinos with neutron
tagging, via the addition of water-soluble gadolinium salt.
In this thesis, only the data observed in SK-IV (2008-2018) is used. The configurationof the Super-K detector in SK-IV will be introduced in this Chapter.

2.1

Cherenkov Radiation

The charged particles are observed by the Super-Kamiokande detector by Cherenkov
light detection. When a charged particle travel through the medium with a speed fasterthan the light speed in the medium, an electromagnetic radiation named as Cherenkov
radiation will be emitted. The momentum thresholds of the Cherenkov radiation in purewater for charged particles frequently observed in Super-K detector are summarized inTable 2.1.
Particle type
Momentum threshold (MeV/c)electron
0.57

muon
118

charged pion
156

proton1051

Table 2.1: Momentum thresholds of Cherenkov radiation in pure water.

The Cherenkov radiation is emitted in a cone shape with a half opening angle θC withrespected to the direction of the particle’s momentum. The half opening angle (alsoreferred to as Cherenkov angle) is determined by cos θC = 1/nβ, where β = v/c and nChapter 2. The Super-Kamiokande Detector

19

represents the refractive index of the medium. The Cherenkov angle in water is about
42◦ for a relativistic particle with β ∼ 1 and n = 1.33.
The number of photons produced by the Cherenkov radiation is a function of wavelength
λ as [27]:
d2 N
2πα
1
= 2 (1 − 2 2 ).
dxdλ
λ
n β

(2.1)

Here, x represents the path length of the particle and α represents the fine structure
constant. In the sensitive region of the Super-K PMTs, whose wavelength ranges from
300 nm to 600 nm, about 340 photons are emitted per centimeter.
The projection of the Cherenkov light on the wall of the inner detector is a ring. The
information of the particles, including its type, position and momentum are extracted
from such a ring by the event reconstruction algorithm, which will be introduced in
Chapter 6. A example of neutrino event observed by Super-K detector is shown in
Figure 2.2.
Super-Kamiokande IV
Run 77472 Sub 279 Event 285409251
18-01-20:05:35:14
I nner : 1164 hi t s, 3559 pe
Out er : 1 hi t s, 1 pe
Tr i gger : 0x10000007
D_wal l : 436. 3 cm
Evi s: 394. 7 MeV
mu- l i ke, p = 556. 2 MeV/ cChar ge(pe)>26.7
23.3-26.7
20.2-23.3
17.3-20.2
14.7-17.3
12.2-14.7
10.0-12.2
8.0-10.0
6.2- 8.0
4.7- 6.2
3.3- 4.7
2.2- 3.3
1.3- 2.2
0.7- 1.3
0.2- 0.7< 0.2

380
0 mu- edecays

304
228
152
76
0
0

500

1000

1500

2000

Times (ns)Figure 2.2: A example of neutrino event observed by Super-K detector. Each point
denotes a PMT hit, whose charge is represented by the color. The Cherenkov ringpattern can be seen clearly.

20

Chapter 2. The Super-Kamiokande Detector

2.2

Detector

2.2.1

Water Tank

The water tank of Super-Kamiokande, in which 50 kt ultra pure water is filled, is in acylindrical shape with a height of 41.4 m and a diameter of 39.3 m. The water tank isseparated into two concentric cylindrical regions by the PMT support structure, whichis covered by opaque sheets on both surface as shown in Figure 2.3. The two regions
are referred to as the inner detector (ID) and outer detector (OD), respectively. Thereis an 55 cm insensitive region between ID and OD to contain the supporting structure
and cables for the PMTs.
Tyvek
8" PMT

Top

Black sheet

Barrel20" PMT

BottomFigure 2.3: Supporting structure of PMTs. Taken from [7].

The inner detector (ID) is a cylinder with a height of 36.2 m height and a diameter of
33.8 m. There are 11,146 inward facing 20-inch PMTs attached to the supporting frame
to detect the Cherenkov light generated from the 32 kton pure water contained. The
photocathode coverage is around 40% in the ID, and the rest of the ID wall is covered
with black polyethylene terephthalate sheet to reduce reflection and to separate the ID

Chapter 2. The Super-Kamiokande Detector

21

and the OD optically. The photocathode coverage was reduced to about 20% in SK-IIdue to the accident and was resumed to 40% from SK-III.
The outer detector (OD), which surrounds the ID, is monitored by 1,885 outward facing
PMTs with a size of 8 inches. The thickness of OD is 2.2 m along the barrel and 2.05 m
on the top and bottom. To improve the light collection efficiency, a wavelength shifting
plate with a size of 60 cm × 60 cm is attached to each OD PMT and the surface of
supporting frame is covered by tyvek sheets as reflective material. The OD is used toreject cosmic ray muons and to distinguish the charged particles which penetrate the
ID wall. The water layer in the OD also prevents the ID from the neutrons and gamma
rays from surrounding rocks.

2.2.2

Photomultiplier Tube

The 20-inch PMTs in the ID (Hamamatsu R3600) were originally developed by Hamamatsu Photonics K.K. with the cooperation of Kamiokande collaborators [71]. The
bleeder circuit and the dynode structure were upgraded later to improve timing response and photon-collection efficiency for Super-Kamiokande [72]. The schematic viewand the specification of the PMT are shown in Figure 2.4 and Table 2.2.

Figure 2.4: Schematic view of the 20-inch ID PMT for Super-Kamiokande
experiment. Taken from [7].

The photocathode of the PMT is coated with bi-alkali (Sb-K-Cs) for its high quantum
efficiency in the Cherenkov wavelength region (∼ 22% for wavelength from 360 to 400nm) and low thermionic emission. The collection efficiency at the first dynode is around
70% with a uniformity of 7% for different incident position on the photocathode. ID
PMT’s gain is 107 with a high voltage around 2000 V. The single photoelectron (p.e.)

22

Chapter 2. The Super-Kamiokande Detector
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 diameter
Bialkali (Sb-K-Cs)
22% at λ = 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 σ) for 1 p.e. equivalent signals
13 kg
6 kg/cm2 water proof

Table 2.2: Specification of the 20-inch ID PMT for Super-Kamiokande experiment.
Taken from [28].

peak can be seen clearly in the charge distribution for single p.e. signal as shown in
Figure 2.5. The transit time spread for the single p.e. signal is about 2.2 ns, while the

distribution (arbitrary unit)

average rate of dark hits at 0.25 photoelectrons is about 3 kHz.
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

0

1

2
3
photo-electron

4

5

Figure 2.5: Single photoelectron distribution of the 20-inch PMT.

On November 12th, 2001, one PMT at the bottom wall of the ID tank imploded during
the SK tank water refilling after the maintenance work. This implosion triggered acascade implosions of PMTs and resulted in about 60% ID and OD PMTs destroyed.
To prevent such an accident in the future, from SK-II all the inner PMTs are equippedwith covers, which consists of a clear UV-transparent acrylic dome with a thicknessof 12 mm and Fiber-Reinforced Plastic (FRP) shield for photocathode and side area
protection, respectively. As shown in Figure 2.6, several holes are on the FRP to enable
the water flow in and out of the acrylic cover freely. Transparency of the acrylic coveris more than 96% for photons with wavelength larger than 350 nm in water.

Chapter 2. The Super-Kamiokande Detector

23

Figure 2.6: PMT case for 20-inch ID PMT since SK-II period.

For the OD PMT (Hamamatsu R1408), as mentioned before, their 8-inch photocathodeare attached to 60 cm × 60 cm × 1.3 cm wavelength shift plate to increases light collection efficiency by 60% [73]. Although the timing resolution at single p.e. is degradedfrom 13 ns to 15 ns due to the wavelength shift plate, considering that OD is used asa calorimeter and veto counter rather than a particle tracker, a higher light collection
efficiency is more important comparing with the sacrifice on the timing resolution.

2.2.3

Outer Detector (OD)The OD aims to reject cosmic ray muons for the neutrino event samples. To reject thecorner clipping muons more efficiently, tyvek sheet was installed between the end cap
and barrel region in the OD before SK-III launch, as shown in Figure 2.7.
OD

ID
OD segmentation

Figure 2.7: A schematic view of outer detector (OD). Taken from [8]As an example, the distribution of OD hits for a simulated partially contained event isshown in Figure 2.8. The partially contained event means that a particle produced in
the ID tank penetrated the ID wall and entered the OD. The exiting point on the wall is

24

Chapter 2. The Super-Kamiokande Detector

around the edge of the tank’s bottom wall. Both cases with and without tyvek sheet are
shown in the figure. The OD signal for a detector with tyvek sheet can be seen clearlyat the barrel region of the wall, while the OD signal is spread to the bottom wall if notyvek sheet is installed, which makes it hard to distinguish from a corner clipping event.

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

NUM
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

1990/00/00:NoYet:NoYet
RunMODE:MonteCarlo
0:NoYetTRG ID :00000011 1990/00/00:;R=R :Z : PHI : GOOD
T diff.: 0.00
us0.00: 0.00: 0.00:0.000
FEVSK :80000003 CANG : RTOT : AMOM : MSnOD 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;

Comnt;

Figure 2.8: OD event display for a simulated partially contained event, without(with) tyvek sheet for left (right) figure. The distribution of OD hit is shown in thecenter of each figure, while the distribution of ID hits is shown on the top left corner.

2.3

Water Purification and Air Purification System

The Super-Kamiokande detector utilizes spring water in the mine as the medium in thetank. The water used in the tank is purified with a flow rate of 60 ton/hour by thewater purification system continuously for a high purity. Furthermore, the radioactive
impurities in the water, especially radon, which is one of the background sources forthe observation of neutrino in the MeV energy region such as solar neutrino, are also
removed by the water purification system.
To prevent the radon in air dissolving into the purified water, radon-free air producedby air purification system is pumped into the space above the water surface in thetank continuously. Radon contamination in the radon-free air is suppressed to lessthan 3 mBq/m3 . As a comparison, a typical radon concentration in the mine tunnel
air without any purification in summer is about 1200 Bq/m3 , which is five orders ofmagnitude higher than the value after purification [74].

2.4Electronics and Data Acquisition System for SK-IV

The front end electronics of Super-K were upgraded from Analog Timing Module [75]in September 2008 to a dead-time free data acquisition system based on new electronics

Chapter 2. The Super-Kamiokande Detector

25

called QBEE (QTC-Based Electronics with Ethernet) [9].
The QTC (charge-to-time converter) for PMT signal readout used in QBEE and its
surrounding are shown in Figure 2.9. For each QTC chip, there are three input channels
with a charge dynamic range from 0.2 to 2500 pC. Each channel has three gain ranges,
whose relative gain ratio are 1/49: 1/7 : 1 and corresponding to small, medium andlarge gain, respectively.

Figure 2.9: Block diagram and surroundings of QTC. Taken from [9].

Each QTC channel has a built-in discriminator to trigger itself and convert the integrated
charge of the input signal to a timing signal. The leading edge of the time signal
represents the signal timing and the width denotes the charge of input signal. The block
diagram of one channel is shown in Figure 2.10. Input signals from PMTs are first
amplified by a low-noise-amplifier (LNA), and then delayed by a low-pass filter (LPF).
After processed by a voltage-to-current converter (V/I), the signals are finally integrated
by a capacitor. All of the charge accumulated on the capacitors are monitored by an
output signal called PMTSUM.
Besides the time information, which is obtained by measuring the leading edge of input
signal from PMT, the QTC is also sensitive to charge. Integration of the input charge
starts when the amplified input signal goes over the threshold given by discriminator.
At the same time, a signal is generated to register a hit for the trigger. After the end of
the charge integration, which is controlled by the timer block, the charge integrated on
the capacitor starts to be discharged with a constant current. The discharging time, also
the width of the output signal, is then proportional to the integrated charge of input
signal. Hence, both timing and charge information are encoded into the timing output
signal.
The timer block has three timers for charging, discharging and VETO, respectively.
The time lengths of the gates are controlled by digital-to-analog converters (DACs) via
adjusting the discharge current and threshold of comparator.

26

Chapter 2. The Super-Kamiokande Detector
Pedestal
DSC (Large)
DSC (Medium)
Discriminator (DSC)CALDSC Qtrg
Logic
VETO

HIT
Timer Block

Charging Timer
LNA

protection
diode
impedanceadjusting

+

Input

DAC
LPF
(delay)LPF

Attenuator

Small

LNA

Termination
Resistor

MediumDischarging
Timer

VETO
Timer
Charging andDischarging Block
D

Q
Q
CLR

V/I
Converter
PMTSUMDAC

DSC

DSC Logic

Charging Timer

LPF

V/I

DAC

LVDS

Comparator

Discharging Timer

Charging and Discharging BlockLVDS

PMTSUM

LNA
Large

DSC

DSC Logic

LPF

V/I

Charging Timer

Discharging Timer

Charging and Discharging BlockLVDS

PMTSUM
PMTSUM

Figure 2.10: Block diagram of one QTC channel. There are three gain ranges foreach channel. Taken from [9].

The timing chart for QTC operation is shown in Figure 2.11. The timer for charging
after triggered by the output signal from discriminator, opens a charge gate for 400 ns.
The switch between charging capacitor and the V/I converter closes in the charge gate,
the input signal therefore accumulates in the capacitor. A discharge gate is opened by
the discharging timer for ∼ 350 ns after the charge gate. The switch between capacitor
and the discharging current source closes, then the input signals are ignored by opening
the switch near the V/I converter. The time when the voltage of the integrated signal
decreases to the threshold level of comparator is represented by the trailing edge of
output signal from QTC. The output signal from QTC is then proportional to the chargeof input signal. Reset and VETO signals are issued at the end of the discharge gate toreinitialize other QTC circuits. The time consumption for one input signal processingis ∼ 900 ns in total.
A hardware trigger is used in the triggering system for SK-I, SK-II and SK-III [75], while
the software trigger is used in the new DAQ readout system used in SK-IV instead byrecording and analyzing all hits. Seventeen microseconds worth of PMT data is readout at a frequency of 60 kHz and particle interactions are identified using the software
trigger therein, which results in a dead-time free system in SK-IV and the decay electron
tagging efficiency is therefore increased. In addition, the upgraded electronics also hasa better linearity and a wider dynamic range, which make the precision of high energy
event reconstruction higher.

Chapter 2. The Super-Kamiokande Detector

Figure 2.11: Timing chart for QTC operation. Taken from [9].

27Chapter 3

Simulation of Atmospheric
Neutrino
The simulation of atmospheric neutrino events in Super-Kamiokande is based on MonteCarlo method and consists of three parts: flux of the atmospheric neutrino, interactions
between neutrino and nucleons, and response of the detector. The events expected tobe observed in the detector are simulated based on the atmospheric neutrino flux modeland interaction model. The basic information of the particles in the final state, including
particle type, vertex, direction, momentum and so on, are generated in this step. Then,the particle’s track in the detector, emission of Cherenkov radiation, light propagation
and response of detector hardware are simulated based on those information. Finally,
the simulated events, also called Monte Carlo events, are saved in the same structure
and reconstructed as the real observed data for the oscillation analysis.
This chapter describes the atmospheric neutrino flux model, neutrino interaction modeland detector simulation, which are important to the systematic error evaluation andoscillation parameter fitting.

3.1

Atmospheric Neutrino Flux

In this analysis, atmospheric neutrino flux model calculated by M. Honda et al. [10,76](Honda flux) is used as the default model, while other models, such as G. Battistoni
et al. [77] (Fluka flux) and G. Barr at al. [78] (Bartol flux) are compared to the Honda
flux for the systematic uncertainties estimation.
The primary cosmic ray flux model, as the input of the neutrino flux calculation, isdetermined by experimental measurements, including the BESS [79, 80] and AMS [81]2930

Chapter 3. Simulation of Atmospheric Neutrino

experiments. The cosmic ray flux is affected by solar activities, which changes periodically. The flux is higher for a intense solar activity (solar maximum) than that for amild solar activity (solar minimum). The fluctuation of the cosmic ray flux in a solar
cycle is more than a factor of two in 1 GeV region, while the difference is around 10 %when energy is around 10 GeV.
When primary cosmic ray particles, mostly protons, penetrate atmosphere, they interact
with air molecules and produce secondary particles such as pions and kaons. The US
Standard Atmosphere ‘76 model [82] is used in the Honda flux to describe the profileof the atmosphere’s density and calculate the dependence of atmospheric neutrinos on
zenith angle. Effects from geomagnetic field are estimated based on the IGRF2005 model[83].
Two theoretical models are used for the hadronic interactions between the air molecules
and the cosmic rays: JAM [84] and DPMJET-III [85], which are interaction models forenergies less and greater than 32 GeV, respectively. Mesons generated from those hadroninteraction further decay into muons and neutrinos. Therefore, the DPMJET-III model,
which is used in the Honda flux, is tuned based on the measurements of the cosmic raymuons flux by BESS [80, 86] and L3+C [87].
The track of the cosmic ray particles in the atmosphere are fully simulated in threedimensions during the flux calculation. The zenith angle distribution of the atmospheric
neutrino at Super-K calculated by the Honda model is shown in Figure 3.1. The flux
peak at the horizon is mainly due to the longer track length in the atmosphere comparing
with vertical direction, which means more time for decay. The large up-down asymmetry
in the most left plot, in which neutrino’s energy is around 0.32 GeV, is due to thegeomagnetic field, while the cosmic rays with higher energy are less affected therefore
no obvious up-down asymmetry in the most right plot.
The energy spectrum of atmospheric neutrinos at the Super-K site after averaging overall direction is shown in Figure 3.2. Besides the Honda flux (solid line), the Bartol flux
(dashed line) and Fluka flux (dotted line), which have different hadronic interaction
model and cosmic ray data set used for tuning, are also shown. The difference between
different flux models is taken as systematic uncertainties.
The atmospheric neutrino MC events are generated with the predicted flux at Super-K
without any oscillations. The effects of oscillation are applied by reweighting to eachMC event based on its oscillation probability calculated using Equation 1.4.

Chapter 3. Simulation of Atmospheric Neutrino

31

Figure 3.1: Dependence of atmospheric neutrino flux on zenith angle for different
energy regions. The atmospheric neutrino is vertically downward going when cosθ = 1
and it is upward going when cosθ = -1. Taken from [10].

Figure 3.2: Atmospheric neutrino fluxes (left) and flavor ratio (right) after
averaging over all directions. The prediction based on Honda 11 flux is shown by redsolid line, while the result of the Bartol flux and the FLUKA flux are represented bydashed line and dotted line. Taken from [10].

3.2

Neutrino Interaction

Simulation of the interaction between atmospheric neutrinos and nuclei in water in thetank and the rock surrounding the Super-K detector is done by NEUT (version 5.4.0)[88]. The interactions between atmospheric neutrino and electrons are neglected insimulations due to the three orders of magnitude smaller cross sections comparing withthe one between neutrino and nuclei.

32

Chapter 3. Simulation of Atmospheric Neutrino

Neutrino interactions can be classified into charged-current (CC) and neutral-current(NC) interactions determined by the type of bosons exchanged. The CC interaction,
when a neutrino interacts with the target by exchanging a W ± boson, produces a charged
lepton (e.g. electron or muon) whose flavor corresponds to that of neutrino (e.g. νe orνµ ). Therefore, the incoming neutrino in a CC event can be identified by distinguish theflavor of charged lepton, which is important in the observation of the effect of neutrino
flavor change in an oscillation analysis. As for the NC interactions via Z boson exchange,
no signal about the neutrino flavor left in the detector since the outgoing lepton is alsoa neutrino.
Besides electron and muon neutrinos, which exist in the original neutrino flux, tau
neutrino (ντ ) can also be detected by Super-K due to oscillation. The CC interactionfor ντ can only occur in the multi-GeV energy region since tau lepton has a large mass
value of 1.78 GeV/c2 . Interaction with ντ are simulated in the same manner as νe andνµ by NEUT, while the produced tau leptons, which decays with a short life time of
2.9×10−13 s, are simulated by TAUOLA [89].
In NEUT, the following interactions are considered:
1. CC/NC (quasi-)elastic scattering: ν + N → l + N 0
2. CC meson exchange interaction: ν + N N 0 → l + N 00 N 000
3. CC/NC single meson production: ν + N → l + N 0 +meson
4. CC/NC coherent pion production: ν +16 O → l +16 O + π
5. CC/NC deep inelastic scattering: ν + N → l + N 0 + hadrons.
Here, N , N 0 , N 00 and N 000 represent nucleons (e.g. proton or neutron) and l represents a
lepton.

3.2.1(Quasi-)Elastic Scattering

NC elastic scattering is a process in which a neutrino simply scatters of a nucleon
target by transferring momentum without producing any new particles. In CC Quasielastic scattering, the neutrino changes into the corresponding charged lepton and thetarget nucleon also changes to preserve the total electric charge, without generation ofother particles. In NEUT, such interactions on free nucleon target are simulated by the
Llewellyn-Smith model [90]. The differential cross section for the interaction with freeChapter 3. Simulation of Atmospheric Neutrino

33

nucleon is:2
M 2 G2F cos2 θC
dσ ν̄
2
2 s−u
2 (s − u)
A(q
)
+
B(q
)
.
=
+
C(q
)
dq 2
8πEν2
M2
M4

(3.1)

Here, M represents the nucleon mass, which equals 0.938 MeV, GF represents the Fermi
coupling constant, θC denotes the Cabibbo angle, Eν denotes the neutrino energy, q =pν − pl represents the transferred four-momentum, s and u are Mandelstam variables[90]. The factors A, B and C are expressed as:




q2
q2
m2 − q 22
4 − 2 |FA | − 4 + 2 |FV1 |2A(q ) =
4M 2
M
M


2
1
4q FV ξFV2
q2
q2−
− 2 |ξFV2 |2 1 +
M
4M 2
M2
i
m
−
((FV1 + ξFV2 )2 + |FA |2 )M
q2
B(q 2 ) =(FA (FV1 + ξFV2 ))
M2

q2
1
2
1 2
2 2
2
|FA | + |FV | −|ξFV | .
C(q ) =
4
4M 2
2

(3.2)
(3.3)
(3.4)

Here, the mass of the outgoing lepton is represented as m, ξ is defined as the anomalous
magnetic moment 3.71µN 1 . Axial vector form factor FA (q 2 ), vector form factors FV1 (q 2 )and FV2 (q 2 ) are expressed as:−2q2
FA (q 2 ) = −1.232 1 − 2
MA




−1
q2
q2
2
2
1 2
GE (q ) −GM (q )FV (q ) =
1−
4M 2
4M 2

−1
q2
2 2
ξFV (q ) =1−[GE (q 2 ) − GM (q 2 )]
4M 2

−2
q2
−1
2
GE = (1 + ξ) GM (q ) = 1 − 2
,
MV

(3.5)
(3.6)
(3.7)
(3.8)

where GE and GM represent the electric form vector and the magnetic form vector,
respectively. The vector mass MV is determined as 0.84 GeV by the electron scatteringexperiments, and the MiniBooNE [91] experiment shows that the axial vector masses
MA is 1.05 GeV. The value of MA is also consistent with former determinations of thenucleon axial mass as shown in Figure 3.3 [11].
For a nucleon bound in an oxygen nucleus, the local Fermi-Gas model made by Nieves[92, 93], is used for taking the nuclear effect such as Fermi motion and Pauli blocking
1e~
The nuclear magneton µN is defined as 2m
.
pc

34

Chapter 3. Simulation of Atmospheric Neutrinoof the nucleons into account. The effect of random phase approximation correction andcontribution from multi-nucleon effect are considered.
Argonne (1969)
Frascati (1970)
Frascati (1970) GEn=0
Frascati (1972)
DESY (1973)
Daresbury (1975) SP
Daresbury (1975) DR
Daresbury (1975) FPV
Daresbury (1975) BNR
Daresbury (1976) SP
Daresbury (1976) DR
Daresbury (1976) BNR
DESY (1976)
Kharkov (1978)
Olsson (1978)
Saclay (1993)
MAMI (1999)
AverageArgonne (1973)
CERN (1977)
Argonne (1977)
CERN (1979)
BNL (1980)
BNL (1981)
Argonne (1982)
Fermilab (1983)
BNL (1986)
BNL (1987)
BNL (1990)
Average
0.85

0.95

1.05

1.15

1.25

0.85

MA [GeV]

0.95

1.05

1.15

1.25

MA [GeV]Figure 3.3: Axial mass MA extractions. Left panel: From (quasi)elastic neutrino
and antineutrino scattering experiments. The weighted average is
MA = (1.026 ± 0.021) GeV. Right panel: From charged pion electroproduction
experiments. The weighted average is MA = (1.069 ± 0.016) GeV. Taken from [11].

Figure 3.4 shows the interaction cross section of quasi-elastic scattering calculated by
NEUT and experimental data. It should be noticed that the cross section of antineutrino
is smaller than that of neutrino since the weak interaction only includes left-handed
particle and right-handed antiparticle under relativistic limit and the angular momentum
conservation constrains the scattering angle distribution of the output leptons for the
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

-

(a) νµn→µ p

ANL
BNL
SKAT

Serpukhov
GGM (1977)
GGM (1979)

σ (10-38 cm2)

σ (10

-38

2

cm )

antineutrino interaction.

0

2

4

6
8
10
Eν (GeV)

12

14

2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

–

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

0

2

4

6
8
10
Eν (GeV)12

Figure 3.4: Cross section of Charged current quasi-elastic interaction of νµ (left)and ν̄µ (right), with experimental data from ANL [12], Gargamelle [13] [14], BNL [15],Serpukhov [16] and SKAT [17]. Solid line represents the result for scattering off a free
proton, the dashed line represents the result for scattering off bound nucleons in 16 O.
Taken from [18]14Chapter 3. Simulation of Atmospheric Neutrino

3.2.2

35

Single Meson Production

The resonance production of single meson, e.g. π, η and K is modeled in NEUT mainly
as an intermediate resonance decay:
ν + N → l + N ∗ , N ∗ → N 0 + meson.

(3.9)

Here N and N 0 represent nucleons in the initial and final state, respectively. The intermediate resonance state is denoted by N ∗ . Only the events with an invariant mass
W of the intermediate baryon resonances less than 2 GeV/c2 is considered here. Theinteraction with W greater than 2 GeV/c2 , are simulated as a part of deep inelastic
scattering, which is described in Section 3.2.4.
This interaction is simulated based on the model from Rein and Sehgal [94] with therevised form factor by Graczyk and Sobczyk [95], and the parameters characterizing theform factor were determined by bubble chamber data [96].
The angular distribution of the final state pion with respect to the ∆(1232) resonanceis determined using Rein’s method [97], while this distribution for other resonances
are assumed to be isotropic in the rest frame of the resonance. For the decay from abaryon resonance, nucleon momentum is required to be greater than the Fermi surface
momentum to simulate the Pauli blocking effect. Moreover, the absorption of mesons ina nucleus results in the absence of mesons in the resonance events [98]. The fraction ofsuch phenomena is estimated to be 20% in NEUT.
Figure 3.5 shows the cross section of charged current single pion productions for νµ [18].

3.2.3

Coherent Pion Production

Coherent pion production, in which the incident neutrino interacts with the entire oxygen
nucleus producing a pion, is simulated based on the Berger and Sehgal’s model [99]. The
pions and outgoing leptons are peaked at the forward direction since the momentum
transferred to the oxygen nucleus is small.

3.2.4

Deep Inelastic Scattering

Multiple hadrons are often generated in deep inelastic scattering interactions by theinteraction between incident neutrino interacts and the constituent quarks in the target
nucleon. This process is dominant in multi-GeV region and considered in NEUT when
the hadronic invariant mass W larger than 1.3 GeV/c2 . The DIS interaction is modeledChapter 3. Simulation of Atmospheric Neutrinocm2)

νµ p → µ− p π+σ(ν µ )(10

−38

1.2

σ(ν µ )(10

−38

cm2)

36

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

νµ n → µ p π−

0Eν (GeV)1

10Eν (GeV)0
ANL Radecky, Phys.Rev.D 25, 1161 (1982)

0.8

ANL Campbell, Phys.Rev.Lett. 30, 225 (1973)

0.7
ANL Barish, Phys.Rev.D 19, 2521 (1979)

0.6
BEBC Allen Nucl.Phys.B 264, 221 (1986)

0.5
0.4

BEBC Allen Nucl.Phys.B 176, 269 (1980)

0.3

BEBC Allasia Nucl.Phys.B 343, 285 (1990)

0.2
BNL Kitagaki Phys.Rev.D 34, 2554 (1986)

0.1
0FNAL Bell Phys.Rev.Lett. 41, 1008 (1978)

1

10

Eν (GeV)Figure 3.5: Cross section for νµ charged current single pion productions. Solid lines
represents the calculation result of NEUT, points represent data from experiments
shown in bottom right figure. Taken from [18].

using the GRV98 parton distribution function [100] and utilizing CKM matrix elements
when calculating structure functions therefrom. Corrections for low q 2 scattering havebeen updated to those of Bodek and Yang [101].
For the interaction with W in the range from 1.3 GeV/c to 2.0 GeV/c, only pions are
considered in the simulation as the outgoing mesons. The average multiplicity of pionis estimated from the results of bubble chamber experiment [102, 103]. Only the eventwith two or more pions are considered in DIS channel in this energy region since there
is an overlap with the resonance pion production, as introduced before.
Cross sections of charged current DIS interactions for νµ and ν̄µ are shown in Figure
3.6.

3.2.5

Nuclear Effects

The interactions of secondary particles produced in the neutrino-nucleon interactions
inside the 16 O nuclei are also considered in NEUT. The meson, such as π, K and η,
produced within 16 O nuclei are tracked from their creation to either their absorption
inside the nuclei or exiting. This simulation is based on the cascade model and externaldata.

σ/Eν (10-38cm2 GeV-1)

Chapter 3. Simulation of Atmospheric Neutrino

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

37

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

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

CCFRR 84
SKAT

100

200

30 50

150

250

Eν (GeV)Figure 3.6: Cross section of charged current DIS for νµ (upper lines and points) andν¯µ (lower lines and points) interactions. Dashed lines show the prediction of NEUT,
which uses GRV98 with Bodek-Yang correction. The experimental data is represented
by the points. Taken from [18].

The most frequently seen hadrons at the Super-K detector is pions since the large production cross section from neutrino interaction and large pion-nucleon interaction crosssection for Eν > 1 GeV. NEUT considers four pion interactions in 16 O nuclei: quasielastic scattering in which a single pion of the same charge remains after interaction,
charge exchange in which a charged pion is converted into a π 0 or π 0 to charged pion,
absorption in which no pion remain after interaction, and pion production.
The nuclear density distribution, which determines the interaction probabilities within
a nucleus is described by the Wood-Saxon density profile [104]. The probabilities forfour interactions are calculated at each simulation step.
The cross sections of π + −12 C scattering as a function of the momentum of π + after
tuning based on π + −12 C data are shown in Figure 3.7 [19].
Besides pion, the quasi-elastic scattering and charge exchange interaction of kaon are
also considered in NEUT, whose model is tuned based on the cross sections measured by
K ± − N scattering experiments [105–107]. The absorption of η (ηN → N ∗ → π(π)N )is also considered [108].

3.3

Detector Simulation

Secondary particles produced by neutrino interactions are then propagated in the detector, which is simulated by a GEANT3[109]-based detector simulator called SKDETSIM.
The interaction of the particles with water is simulated by GEANT while GCALOR[110]Chapter 3. Simulation of Atmospheric Neutrino

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

450

π- 12
6 C

400
350

σ (mb)σ (mb)

38

250

ABS Geant4

ABS DUET (stat+syst)
CX DUET (stat+syst)
ABS Previous Exp.
CX Previous Exp.

CX Geant4
ABS FLUKA

200

CX FLUKA300

ABS NEUTCX NEUT

250

150

200

100

150
100

50
50
0
0

200

400

600

800

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

0

100

150

200

250

300

350

400

450

pπ (MeV/c)

Figure 3.7: Cross section of π + −12 C scattering as a function of the momentum of
π + . Solid lines represent the result of NEUT calculation before tuning, while theresult after tuning are represented by dashed lines. Experimental data are shown as
the points with error bars. Taken from [19]. Right figure is the update for theabsorption (ABS) and charge exchange (CX) based on new scattering data [20]is used to simulate hadronic interactions except the interaction with pions below 500
MeV/c, for which a cascade model is used.
The Cherenkov photons produced by the particles are then propagated in the detector as
described in Section 2.1. Reflectivity of the PMT and the black sheet are also modeledusing the result obtained in Section 2.2.
The charge and time responses of the PMT and the electronics are simulated based on
the calibration measurements. Then the detector simulation provides the same data
structure as the observed data so that these two can be analyzed in the same manner.

Chapter 4

Calibration
Calibrations are performed in the SK detector in order to understand the detector andkeep its performance high. The calibration results are utilized for the detector simulation
described in Section 3.3 as well as for the data analysis. Further details of the Super-K
calibration can be found in [21].
The signal observed and recorded by Super-K is the charge and time information from
PMT through the electronics. Besides the response of PMT itself, which is affectedby the individual difference on quantum efficiency and the gain of PMTs, the water
quality and other detector’s characteristics also need to be measured precisely in orderto model the propagation of Cherenkov photon for the detector simulation and event
reconstruction. A good detector calibration and a precise evaluation for the calibration
result relate to the event reconstruction performance and therefore play important rolesin the neutrino oscillation analysis.

4.1

Detector Calibration

4.1.1

Relative Gain Calibration

The high voltage of each PMT is set individually to get a uniform charge response for all
the PMTs in the detector. The relative gain calibration is performed by flashing a laser
with a similar apparatus as the timing calibration described in Section 4.1.4 to measure
and correct the remaining individual variations of the PMT gain.
A pulsed laser near the ID center is flashed isotropically and repeatedly with two different
intensities: a high intensity and a low intensity. For the high intensity case, every PMT
detects a proper number of photons, creating an average charge Q(i) for each ID PMT i.
39

40

Chapter 4. Calibration

For the other case, the laser is flashed at a low intensity, only a few PMTs get a hit at atime and each hit is a single p.e.. The number of hits observed by i-th PMT is counted
and recorded as N (i). The complicating factors in estimating those two intensities Q(i)and N (i) are similar since the light source is at the same position:
Q(i) ∝ IH × a(i) × qe (i) × G(i)(4.1)

N (i) ∝ IL × a(i) × qe (i),(4.2)

where IH (IL ) represents the average intensities of the high (low) intensity flashes. The
value of a(i), qe and G(i) represent the acceptance, quantum efficiency and gain of i-th
ID PMT, respectively. The relative gain of each PMT G(i) can then be obtained bytaking the ratio of Q(i) to N (i) as:
G(i) ∝ Q(i)/N (i).

(4.3)

The standard deviation of the gain for all PMTs is found to be 5.9%. The individual
relative gain factor obtained above is used to correct the coefficient to converse theoutput charge to the number of photoelectrons observed for each PMT.

4.1.2

Absolute Gain Calibration

The absolute gain is used to convert the charge recorded by a PMT in pico Coulomb(pC) into the number of incident photoelectroncs. The absolute gain was measured
by observing the charge distribution of single photoelectron signals with a spherical
gamma-emitting nickel source, which emits 9 MeV gamma rays isotropically by capturing
neutrons emitted from a 252 Cf source in it. The nickel source was placed near the centerof the ID tank and delivered 0.004 p.e/event for each PMT, which ensures more than
99% hits are single p.e. hits.
The observed charge distribution for the single p.e. signals is shown in Figure 4.1. The
hits from all PMTs are corrected for the relative gain variation and accumulated. The
conversion factor from the observed charge to the number of p.e.s is determined to be
2.658 pC/p.e, which is the peak of the single p.e.. This distribution is also used for thedetector simulation described in Section 3.3.
Furthermore, as shown in Figure 4.2, the peak of charge distribution for the dark hits,
which is proportional to the absolute gain of PMT, increased as time passes and theincreasing rate depends on the production year of PMT. The reason of such an increaseof the gain is not clear yet, but a correction is applied on the gain to reduce the effect.

Chapter 4. Calibration

10

41

6

10

5

10

4

10

3

10

2

10
1
0

5

10

15

20

25

30

35

40

45

50
pC

Figure 4.1: Distribution of the observed charge for single p.e. signals obtained fromthe calibration data with nickel source. Taken from [21].

Figure 4.2: Time variation of the peak of the charge distribution for the PMT darkhits. Different color denote the different production year of PMT. A correction on
PMT gain is applied based on this measurement.

4.1.3

Quantum Efficiency Calibration

Besides the gain, the relative difference of quantum efficiency (Q.E.) is also measured foreach PMT since it affects the charge response for small number of incident photons. The
measurement utilizes the same nickel source as the absolute gain calibration describedin Section 4.1.2.

42

Chapter 4. Calibration

As Equation 4.2, the average number of hits on a PMT with a low intensity of incident
light is proportional to the quantum efficiency qe (i). The relative Q.E. for each PMT
qe (i) is obtained by comparing the simulated events, which account for the acceptancefactor a(i) but does not include the variations in qe (i) and the real nickel source data.
The resulting Q.E. for each individual PMT is also used in the detector simulation.

4.1.4

Relative Timing Calibration
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. Takefrom [21].

The time response is important for the precise reconstruction of event vertices and track
directions. It varies between PMTs due to many factors such as the difference on lengthof cable, the process time of electronics etc. Moreover, the time response also depends on
the observed charge, since hits with larger charge exceed threshold of TDC discriminator
earlier than those with less charge. This phenomenon is known as time-walk effect. The
goal for the timing calibration is to make a correction table for the time-walk effect foreach PMT in the detector with also consideration of the overall process time.
The schematic view of the system for the timing calibration is shown in Figure 4.3. A
fast pulse of 0.4 ns FWHM at a wavelength of 337 nm is generated by a nitrogen laser
and monitored by a 2-inch PMT with a fast response for trigger. The laser light pulse
is then shifted to 398 nm by a dye, where the overall response including Cherenkov

Chapter 4. Calibration

43

emission spectrum, light absorption in water and quantum efficiency of the PMTs isalmost maximum. The light intensity can be changed by adjusting the optical filter,
which enable us to measure the time responses at various pulse height, i.e. the charge.
The TQ distribution is obtained for each PMT individually, as shown in Figure 4.4, andthen fitted by a polynomial as a function of charge to extract the calibration constants,
named the TQ map. The uniformity of the time response of the PMTs in the entire
detector is ensured.
T (ns)

tqmap of cable # 00010
3102

0 1 2 3 4 5 6 7 8 9 10
1230

10

Q (p.C.)
1.0
0.9

1220

0.8
0.7

1210
0.6
0.5

1200

0.4
1190
0.3

Entries 561679
1180

0.2
0.1

1170
0

20

40

60

80

100

120

140

160

180

0.0

QBin

Figure 4.4: A typical TQ distribution for ID PMT. The horizontal axis and thevertical axis show the charge and TOF-corrected hit time, respectively. The scale ofhorizontal axis is linear up to 10 pC and then goes to log scale. A larger T value
corresponds to earlier hit in this plot. Taken from [21].

4.1.5Water Property Calibration

The photon propagation in water is modeled empirically with considering the lightscattering and absorption.

The attenuation of light in water can be expressed asexp(−l/L(λ)), where l is the distance the light traveled and L(λ) represents the attenuation length as a function of wavelength λ.
In Super-K simulation, L(λ) is defined as
L(λ) =

1αabs (λ) + αasym (λ) + αsym (λ)(4.4)

where αabs , αasym and αsym are coefficients for absorption, asymmetric scattering andsymmetric scattering, respectively. The asymmetric term αasym is used to account forforward Mie scattering, while the effect of symmetric scattering αsym takes the Rayleighscattering and symmetric Mie scattering into account and can be described by 1 + cos2 θ,
where θ represent the direction of the scattered photon.

44

Chapter 4. Calibration

LASER
337, 375 Optical fiber!
405, 445
473 nm!
B1!
B2!
B3!

Scattering!

B4!
B5!

Figure 4.5: The laser injector system for the water property measurement. Taken
from [21].

Figure 4.5 shows the laser injector system to measure the water constants. An collimated laser beam with adjustable wavelength is injected at the top of Super-K tank andvertically down. The scattered and reflected light is detected by the PMT belonging
to five divisions of the barrel region labeled from B1 to B5, and the top wall of thedetector. The hit time distribution after subtracting the time of flight for the PMTs ineach detector region is shown in Figure 4.6. The sharp peaks on the right between 1830
to 1900 ns represent photons reflected by the PMTs and black sheets on the bottom
wall, while those earlier hits are due to the photon scattered in water.
The water calibration constants, αabs , αasym and αsym are tuned to achieve the best
agreement between simulation and data. Figure 4.7 shows the result based on thedata taken in Apr 2009. The water parameters are always monitored by this system
during the Super-K operation. The time variation of the measured coefficients at various
wavelengths is shown in Figure 4.8.

4.2

Energy Scale

The momentum reconstruction of neutrino event is mainly based on the charge observedby the PMTs in the tank. The factors discussed above, including water property or
PMT gain will affect the performance of momentum reconstruction. On the other hand,the precise momentum determination of the neutrino event is necessary for the neutrinoChapter 4. Calibration

45

0.004
0.003
0.002
0.001
1500
0.004
0.003
0.002
0.001

1600

1700

1800

1900

1500
0.004
0.003
0.002
0.001

1600

1700

1800

1900

1500
0.004
0.003
0.002
0.001

1600

1700

1800

1900

1500
0.004
0.003
0.002
0.001

1600

1700

1800

1900

1500
0.004
0.003
0.002
0.001

1600

1700

1800

1900

1500

1600

1700

1800

1900

Figure 4.6: PMT hit time distribution in different detector regions for data (opencircle) and Monte Carlo (red line) after tuning by scattering and absorption
parameters. Taken from [21].

oscillation analysis since the oscillation probability is highly related to the energy ofneutrinos. Four well-known independent control samples are used for the energy scalecalibration [6] and systematic evaluation in different momentum range:• Track length of high energy stopping muons (1∼ 10 GeV/c)• Cherenkov angle of low energy stopping muons (200 ∼ 500 MeV/c)• Invariant mass of π 0 produced by neutrino interactions (∼ 130 MeV/c)• Momentum distribution of decay electron (∼ 40MeV/c)
The absolute energy scale error, the time variation and the detector uniformity are discussed in this section. The data except the π 0 samples for the absolute error estimation
was taken in April 2009, which is also used to calibrate the simulation software, while
π 0 sample taken in full SK-IV are used. Time dependent attenuation length correction
and PMT-by-PMT gain correction, which is introduced in Section 4.1.2, are applied.
A new reconstruction algorithm fiTQun, which will be introduced in Section 6.2, is used
to reconstruct calibration samples here for the consistency with the neutrino oscillation
analysis. As will be discussed in Section 7.3, recent Super-Kamiokande results have

46

Chapter 4. Calibration

Inverse water transparency (1/m)

10-1
Total

Absorption

Symmetric

Asymmetric

10-2

10-3

10-4
300

325

350

375

400

425

450

475

500

Wavelength (nm)Figure 4.7: An example of water parameter fitting. The data used here is taken in
Apr 2009. Points represent the average value of coefficients. The errors on the averagevalue are represented as error bars. The vertical axis represent the coefficient α inEquation 4.4. Taken from [21]been limited by a lack of statistics, so a larger fiducial volume (where the distancefrom the vertex to the nearest wall, Dwall > 50 cm) of the Super-K detector than its
predecessor (Dwall > 200 cm) is used for the oscillation analysis. The energy scale
for both the conventional fiducial volume (Dwall > 200 cm) and the expanded fiducial
volume (Dwall > 50 cm) are evaluated to validate such a fiducial volume expansion.

4.2.1

High Energy Stopping Muons

The high energy stopping muon events are used to check the energy scale around 110 GeV/c by taking ratios of the reconstructed momentum of muons to their track
lengths (p/range), since the energy loss dE/dx is approximately constant. The range isestimated from the muon entering position and the Michel electron vertex, both of which
are assumed to be independent on the momentum reconstruction. Selection criteria for
stopping muons are as follows:Chapter 4. Calibration

47

Figure 4.8: Time variation of the light absorption and scattering coefficientsmeasured at various wavelengths. The coefficients for absorption αabs , asymmetricscattering αasym and symmetric scattering αsym for various wavelength are shown inthe blue (top), black (middle) and purple (bottom), respectively. In 2008, thewavelength was changed from 365 nm to 375 nm, from 400 nm to 405 nm, and from
420 nm to 445 nm. The time of this change is indicated by the black vertical bars.
Taken from [21]1. Total number of p.e. in ID is larger than 1000 p.e.
This cut is to ensure the energy of muon is high enough to have a long track length.
2. Event is reconstructed as single-ring µ-like event.
3. Entering position of the cosmic ray muon is on the top wall of the detector: z >1760 cm and r < 1640 cm.
4. Direction of the stopping muon is downward (cosθzenith > 0.94).
Cuts 2 ∼ 4 are used to select the muon events which have a clear Cherenkov ring
therefore can be reconstructed well.
5. Only one decay electron is detected.
6. Muon track length is longer than 500 cm.
7. Number of PMT hits within a 50 ns sliding time window is larger than 60.
This cut is used to remove accidental coincidence gamma signal from radioactive
isotopes.
8. Decay time is longer than 2000 ns.
This cut is to ensure that the reconstruction of the decay electron is not affectedby the light from the parent muon.
9. Vertex position of the decay electron is reconstructed within the fiducial volume.

48

Chapter 4. Calibration

Figure 4.9 shows p/range distributions of data and MC in the range from 5 m to 35 mwith a binning width of 5 m and a fiducial volume cut of 50 cm on the decay electron.
The errors are evaluated at each bin by comparing the peak value of both of data andMC after fitting to Gaussian function. The largest error, which comes from the longest
range bin, i.e. the highest energy bin, is −1.68%. For the conventional fiducial volume
case (Dwall > 200 cm), similar distribution can be observed and the largest error is

Data:2.233± 0.002
Error:1.15 ± 0.15 %

250

100

100

50

50

1.8

2

2.2

2.4

250

Data:2.185± 0.002
MC:2.205± 0.002

200

0

2.6
2.8
3
3.2
3.4
Momentum/range [MeV/c/cm]

Error:0.94 ± 0.17 %150

MC:2.207± 0.002
Error:1.01± 0.14 %

150

1.6

Data:2.185± 0.002

200

150

0

Number of Events

MC:2.259± 0.002

200

Number of Events

250

Number of Events

Number of Events−1.54%, which also comes from the highest energy bin.

1.6

1.8

2

2.2

2.4

2.6
2.8
3
3.2
3.4
Momentum/range [MeV/c/cm]

220
200

Data:2.194± 0.002

180

MC:2.213± 0.002

160

Error:0.85 ± 0.17 %

140
120
100

100

80
60

50

40
20
1.6

1.8

2

2.2

2.4

0

2.6
2.8
3
3.2
3.4
Momentum/range [MeV/c/cm]

180
160

Data:2.179± 0.002

140

MC:2.180± 0.002

120

Error:0.05 ± 0.20 %100

Number of Events

Number of Events

0

1.6

1.8

2

2.2

2.4

2.6
2.8
3
3.2
3.4
Momentum/range [MeV/c/cm]

Data:2.199± 0.004

100

MC:2.162± 0.00380Error:-1.68± 0.33 %

60

80
40

60
40

20

20
0

1.6

1.8

2

2.2

2.4

2.6
2.8
3
3.2
3.4
Momentum/range [MeV/c/cm]

0

1.6

1.8

2

2.2

2.4

2.6
2.8
3
3.2
3.4
Momentum/range [MeV/c/cm]

Figure 4.9: Momentum over the range of the stopping muon events with various
track length for the data (black dots) and MC (blue line). From the top left plot to thebottom right plot, the track length is 5-10 m, 10-15 m, 15-20 m, 20-25 m, 25-30 m and30-35 m, respectively. The fiducial volume for the decay electron used to determine
the track length is defined as Dwall > 50 cm. MC are normalized to data by area.

Chapter 4. Calibration

4.2.2

49

Low Energy Stopping Muons

The momentum of stopping muons with low energy (< 500 MeV/c) can be estimatedby Cherenkov angle as following:mµp(θ) = p2
n cos2 (θ) − 1(4.5)

where θ, n, β, mµ and p(θ) represent the Cherenkov angle, refraction index, ratio between
velocity and the light of speed, muon mass and momentum respectively.
Selection criteria for the low energy stopping muons are:
1. Entering point is at the top wall of the detector: z > 1720 cm.
2. Direction of the stopping muon is downward: cosθzenith > 0.87.
3. Momentum estimated based on the Cherenkov angle is between 200 MeV and 440
MeV.
4. Goodness of reconstruction based on the Cherenkov angle is larger than 0.6.
5. Total number of p.e. in ID is between 500 p.e. and 5000 p.e.
6. Only one decay electron is detected.
7. Decay time is longer than 1200 ns.

Cuts 1 ∼ 5 are used to select the muon events which have a clear Cherenkov ring andcan be reconstructed well. The event selection is optimized for the Cherenkov angle
reconstruction tool. For cuts 6 ∼ 7, although the information of the decay electron is
not used like the high energy cosmic ray muon, the requirement of one decay electron
is to increase purity of muon in the sample. The requirement on the decay time is toensure the reconstruction of the parent muon is not affected by the decay electron.
The ratios of the reconstructed momentum p(p.e.) to p(θ) are evaluated at three p(θ)
bins. Figure 4.10 shows the distribution of p(p.e.)/p(θ) for different p(θ) range. The
largest error, which comes from the lowest energy bin, is -0.85%. This value is common
for both of the conventional fiducial volume (Dwall > 200cm) and the expanded fiducial
volume (Dwall > 50 cm) since the event selection doesn’t rely on the fiducial volume.

50

Chapter 4. Calibration

Number of Events

450
400

Data:1.269±0.003

350

MC:1.258±0.003

300

Error:-0.85±0.26 %

250
200
150
100
50
0.5

1

Number of Events

0
0

1.5

2

2.5
P(pe)/P(θ)

350

Data:1.164±0.003

300

MC:1.164±0.003
Error:-0.02±0.30 %

250
200
150
100
50
0.5

1

Number of Events

0
0

1.5

2

2.5
P(pe)/P(θ)

350

Data:1.085±0.003

300

MC:1.089±0.003

250

Error:0.32±0.33 %

200
150
100
50
0
0

0.5

1

1.5

2

2.5
P(pe)/P(θ)

Figure 4.10: Distribution of p(p.e.)/p(θ) at each p(θ) bin for data (black dots) andMC (blue line). From the top, the p(θ) range is 200-280 MeV/c, 280-360 MeV/c and360-440 MeV/c, respectively. MC are normalized to data by area.

Chapter 4. Calibration

4.2.3

51

Neutrino Induced π 0 Events

Single π 0 events are produced by neutral current interactions of atmospheric neutrinosin the detector. Since the produced π 0 decays into two photons almost immediately,
the invariant mass of π 0 can be calculated with the reconstructed momentum of the two
photons (Pγ1 and Pγ2 ):qMπ0 = 2Pγ1 Pγ2 (1 − cosθ)(4.6)

where θ represents the opening angle between the two photons. The selection criteria
for the neutral current π 0 events are:
1. Pass the FC reduction, which is introduced in Section 5.1.
2. Two e-like rings are detected.
3. No decay electron is detected
4. Vertex position is reconstructed within the fiducial volume.
The distribution of the invariant π 0 mass for data and MC within different fiducial
volume cuts are shown in Figure 4.11. The error is calculated by comparing the Gaussian
fitted peak positions for both data and MC. The error in the new region (200cm >Dwall > 50cm), 1.44 ±0.80%, is at the same level of the conventional FV (Dwall >200cm), -0.63 ±0.34%. The final error for Dwall > 50 cm is -0.23± 0.31%.

4.2.4

Decay Electrons

Stopping cosmic ray muons produce large number of decay electron events. The selection
criteria for the decay electrons are listed as follows:
1. Total number of p.e. in ID is larger than 1000 p.e. for parent event.
2. Parent event is single-ring µ-like event.
3. Only one decay electron is detected.
4. Number of PMT hits within a 50 ns sliding time window is larger than 60.
5. Decay time is longer than 1200 ns.
6. Vertex position of the decay electron is reconstructed within the fiducial volume.

Chapter 4. Calibration

Number of Events

52

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

Data: 139.41 ± 0.47
MC : 138.54 ± 0.06
Error : -0.63 ± 0.34 %

50

100

150

200

250

300

Number of Events

Reconstructed mγ γ (MeV/c2)
50
Data: 138.83 ± 1.08
40

MC : 140.83 ± 0.15
Error : 1.44 ± 0.80 %

30
20
10
0
0

50

100

150

200

250

300

Number of Events

Reconstructed mγ γ (MeV/c2)

250
Data: 139.34 ± 0.43
MC : 139.02 ± 0.05200

Error : -0.23 ± 0.31 %
150
100
50
0
0

50

100

150

200

250

300

Reconstructed mγ γ (MeV/c2)
Figure 4.11: Invariant mass of neutral pions for the data (black dots) and MC (bluelines). From the top, the Dwall range is > 200 cm, 50 ∼ 200 cm and > 50 cm. MC arenormalized to data by livetime.

Chapter 4. Calibration

53

Cuts 1∼3 are used to ensure the decay electron comes from a muon. Cut 4 rejects
γ rays with energy of 6 MeV from µ− capture on nucleons. Cuts 5 and 6 are used
to select good electrons, which are not affected by the light from the parent muon andwithin the target region in the detector. Figure 4.12 shows the reconstructed momentum
distribution with different fiducial volume cuts. The error is calculated by comparing
the mean value of the spectrum for both data and MC. The error in the new region(50cm < Dwall < 200cm), -1.07 ±0.29%, is at the same level of the conventional FV(Dwall > 200cm), -2.09 ±0.14%. The final error in Dwall > 50 cm is -1.92±0.13 %.

4.2.5

Time Variation of Energy Scale

Stability of the detector’s energy scale is confirmed with the high energy stopping muonsand decay electrons as Figure 4.13. The p/range value of the stopping muons is stable(-0.01%±0.02%/year) while the average momentum of the decay electrons is increasing
as 0.19%±0.02%/year. As introduced in Section 4.1.2, the PMT gain increases as time
passes. The dark hit rate also increases due to the unchanged trigger threshold. The
increase of the PMT dark hit rate, which cannot be cancelled by the gain correction, is
considered to be the reason of the increase in the decay electron momentum. The time
variation uncertainties are defined as the larger value of the ratios between the RMS
to the mean in the two samples. The time variation error of the decay electrons in the
new region (50 cm < Dwall < 200 cm) is 0.88%, which is at the same level as the one inconventional FV, 0.59%. The final time variation uncertainty is 0.62% from the decay
electron in the region of Dwall > 50 cm.

4.2.6

Detector Uniformity of Energy Scale

Uniformity of the detector is evaluated with the decay electrons from the stopping cosmic
ray muons. Vertexes and the direction of the decay electrons distribute almost uniformly
in the fiducial volume, which makes them a good calibration sample for the detector
uniformity check. Besides the selection criteria shown in Section 4.2.4, decay electrons
are also required to be perpendicular to their parent muon direction in order to avoid
muon polarization effect. This condition is -0.25 < cos θeµ < 0.25, where cos θeµ is the
opening angle between the decay electron and the parent muon directions. The error isdefined as the central value most deviated from one, which comes from the horizontal
direction as shown in Figure 4.14, for all the three regions. The value for 50 cm < Dwall <200 cm, 1.08%, is at the same level as the value for Dwall > 200 cm, 0.58%. The final
energy uniformity error for Dwall > 50 cm is 0.67%.

Chapter 4. Calibration

Number of Events

54

2200
2000

Data:37.80±0.03 MeV

1800

MC:37.01±0.03 MeV

1600

Error:-2.09±0.14 %

1400
1200
1000
800
600
400
200

Number of Events

0
0

10

20

30

40

50

60

70
80
90
100
Momentum [MeV]

Data:38.38±0.06 MeV

500

MC:37.97±0.06 MeV
400

Error:-1.07±0.29 %

300

200

100

Number of Events

0
0

10

20

30

40

50

2500

60

70
80
90
100
Momentum [MeV]

Data:37.91±0.02 MeV
MC:37.18±0.02 MeV

2000

Error:-1.92±0.13 %

1500

1000

500

0
0

10

20

30

40

50

60

70
80
90
100
Momentum [MeV]

Figure 4.12: Momentum of decay electrons for the data (black dots) and MC (blueline). From the top, the Dwall range is > 200 cm, 50 ∼ 200 cm and > 50 cm. MC arenormalized to data by area.

Mean Mom/Range[MeV/cm]

2.45

RMS/Mean=0.008/2.202=0.38%

2.4
2.35
2.3
2.25

2.45

2.35
2.3
2.25
2.2

2.15

2.15

2009

2010

2011

2012

2013

2014

2015

2.1

2016
Year

42

RMS/Mean=0.225/38.148=0.59%

41

RMS/Mean=0.009/2.192=0.39%

2.4

2.2

2.1

Mean momentum[MeV]55

Mean momentum[MeV]Mean Mom/Range[MeV/cm]

Chapter 4. Calibration

40
39
38
37

2011

2012

2013

2014

2015

2016
Year

RMS/Mean=0.236/38.250=0.62%

41
40
39
38
37
36

35

35

34

34
33
2009

2010

2011

2012

2013

Mean momentum[MeV]

32

2010

42

36

33

2009

2014

2015

32

2016
Year

2009

2010

2011

2012

2013

2014

2015

2016
Year

42

RMS/Mean=0.343/38.941=0.88%

41
40
39
38
37
36
35
34
33
32

2009

2010

2011

2012

2013

2014

2015

2016

2017
Year

Figure 4.13: Time variation plots of the p/range of muon (top figures) and themomentum of decay electron (bottom three figures) for fiTQun. The left two figures
show the result for Dwall > 200 cm, while the right two figures are for Dwall > 50 cm.
The bottom figure shows the time variation of the momentum of decay electron with200cm > Dwall > 50 cm. The dashed line shows the range of average value with ±1%
error.

4.2.7

Summary of the Energy Scale Error

The absolute energy scale error is evaluated by various samples for different momentum
ranges and summarized in Figure 4.15 and Table 4.1. The final absolute energy scaleerror is obtained from the most discrepant sample, which is 1.92% (2.09%) from thedecay electron for Dwall > 50(200) cm. The statistical uncertainty shown here is justfor reference and not considered in the final uncertainties determination. By adding theabsolute energy scale uncertainties to the time variation uncertainties in quadrature,
the final energy scale uncertainties is 2.02% (2.17%) for Dwall > 50(200) cm. Theenergy scale related uncertainties in the new region ( 50 cm < Dwall < 200 cm) are also

56

Chapter 4. Calibration

Pe(MC)/Pe(Data)

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
cosθzenith

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8
1
cosθzenith

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8
1
cosθzenith

Pe(MC)/Pe(Data)

1.05
1.04
1.03
1.02
1.01
1
0.99
0.98
0.97
0.96
0.95
-1

Pe(MC)/Pe(Data)

1.05
1.04
1.03
1.02
1.01
1
0.99
0.98
0.97
0.96
0.95
-1

Figure 4.14: Uniformity of the detector as a function of zenith angle of decay
electrons. Y axis is the ratio of MC to data of the average momentum of decay
electrons. X axis is direction of the electron on z axis, a positive value represents a
downward direction. From the top, the Dwall range is > 200 cm, 50 ∼ 200 cm and> 50 cm. The dashed line shows the ±1% error range.

Chapter 4. Calibration

57

summarized in Table 4.1. The uncertainty evaluated based on the low energy cosmic ray
muon sample is common for all three detector regions since the event selection doesn’t
rely on the FV cut. The energy uniformity error is assumed to be independent from theenergy scale error since it is shown that the energy scale related systematic errors keep

(MC-Data)/Data(%)

stable even if we expand the fiducial volume to Dwall > 50 cm, as shown in Table 4.2.

4Decay Electron

3

Neutral current π0
Sub-GeV stopping µ2

Multi-GeV stopping µ

1
0
-1
-2
-3
-4

(MC-Data)/Data(%)

10

4

102

3

10

Momentum [MeV]

Decay Electron

3

Neutral current π0
Sub-GeV stopping µ2

Multi-GeV stopping µ

1
0
-1
-2
-3
-4
10

102

3

10

Momentum [MeV]

Figure 4.15: Absolute energy scale error with Dwall > 200 cm (top) and Dwall > 50
cm (bottom). The statistical uncertainty is denoted by the vertical error bars. The
momentum range of each sample used to evaluate energy scale error is shown by thehorizontal error bar. The final absolute energy scale error 2.09% (1.92%) for
Dwall > 200(50) cm.

58

Chapter 4. Calibration

Dwall range> 50 cmHigh energy stopping muon
Track length
5 m ∼ 10 m
1.15% ± 0.15%
10 m ∼ 15 m
1.01% ± 0.14%
15 m ∼ 20 m
0.94% ± 0.17%
20 m ∼ 25 m
0.85% ± 0.17%
25 m ∼ 30 m
0.05% ± 0.20%
30 m ∼ 35 m
−1.68% ± 0.33%
Low energy stopping muon
Momentum range
200 MeV ∼ 280 MeV
280 MeV ∼ 360 MeV
360 MeV ∼ 440 MeV

50 cm∼200 cm> 200 cm

1.60% ± 0.50%−0.11% ± 0.64%−0.80% ± 1.03%−0.54% ± 0.94%
0.33% ± 1.26%
1.08% ± 2.03%

1.13% ± 0.16%
0.92% ± 0.14%
1.00% ± 0.17%
0.96% ± 0.17%
0.09% ± 0.21%−1.54% ± 0.35%-0.85%±0.26%-0.02%±0.30%
0.32%±0.33%

Neutrino induced π 0-0.23%±0.31%

1.44%±0.78%-0.62%±0.34%-1.92%±0.13%-1.07%±0.29%-2.09%±0.14%

Decay electron

Table 4.1: Summary of absolute energy scale error. The final absolute energy scalevalue is obtained from the most discrepant sample, i.e. decay electron in this study.

Dwall range
Energy scale error> 50 cm> 200 cm

2.02%

2.17%

1.92%

2.09%

0.62%
Energy uniformity error
0.67%

0.59%

Absolute energy scale
Time variation

0.58%

Table 4.2: Summary of energy scale check. The energy scale error is thequadratic-sum of absolute energy scale error and the time variation error. The
absolute energy scale error value is obtained from the most discrepant sample, i.e.
decay electron in this study.

Chapter 5

Neutrino Event Selection
At trigger level, the Super-Kamiokande event sample consists mainly of downward-goingcosmic ray muons and low energy radioactivities from contaminants in the water suchas radon. The neutrino data selection method directly determines the quality of thesample used for analysis. An efficient selection method is essential to achieve a higher
sensitivity for the atmospheric neutrino oscillation analysis.
Atmospheric neutrino events observed in SK are categorized into three types:• Fully contained (FC)• Partially contained (PC)• Upward-going muon (Up-µ)For events classified as FC and PC, the neutrino interacts within the fiducial volume,
defined as the region located more than 200 cm from the ID wall (Dwall > 200 cm) in
previous analyses. The fiducial volume has been expanded to (Dwall > 50 cm) for ahigher statistics in this analysis, which will be described in Section 7.3. Events withno activity in the outer detector are classified as FC. If energy deposition in the OD isobserved, generally from a high energy muon exiting the ID, the event is classified as
PC.
Muons created by neutrino interactions in the rock around SK or in the OD waterand traveling upward through the detector form the Up-µ sample. The reason why
only upward muons are used is that those events are hard to be distinguished fromthe cosmic rays muons, which travels in downward direction. The FC data are subdivided into several categories according to the visible energy 1 , the number of observed
1Visible energy is defined as the energy of an electromagnetic shower producing the same amount ofCherenkov light as observed in the event.

59

60

Chapter 5. Data Selection

Cherenkov rings, the particle ID (PID) of the most energetic ring, and the numberof observed electrons from muon decays. The detailed category for FC events will be
discussed in Section 7.1.1. All FC events have been reconstructed and categorized by
the new reconstruction algorithm, which will be introduced in Section 6.2.
The events categorized as PC and Up-µ samples after reduction are further divided into
“through-going” and “stopping” subsamples according to the muon stopping point. The
Up-µ events categorized as “through-going” are further divided into “showering” if anassociated electromagnetic shower is found in the ID or “non-showering” if not. These
two samples are reconstructed using the conventional reconstruction algorithm, which
will be introduced briefly in Section 6.1. Only the FC reduction will be focused in thissection since the new reconstruction algorithm is currently only used for FC events as
has been stated above. More detail about the data reduction and event reconstruction
for PC and Up-µ events can be found in [111].
The schematic view of each event type are summarized in Figure 5.1. The neutrinomean energies are ∼1 GeV for FC events, ∼10 GeV for PC events and stopping Up-µevents, ∼100 GeV through-going Up-µ events, as shown in Figure 5.2.

Fully contained

Upward muon

Partially contained

Figure 5.1: Schematic view of observed atmospheric neutrino in SK. Dashed lines
represent the track of atmospheric neutrinos, while the solid lines show the lepton’s
track.

250
2500

Events/5000 Days!

1400
1200

FC Sub-GeV (νe+νe)

1000

FC Multi-GeV (νe+νe)

Up-µ Stop

FC Sub-GeV (νµ+νµ)

Up-µ Through200

FC Multi-GeV (νµ+νµ)

2000

Up-µ Showering

PC Stop

150

800

1500

600

1000

100

500

50

PC Through

400
200
0
10−1

1

10

102

103

104

Neutrino Energy (GeV)!

105

0
10−1

1

10

102

103

104

Neutrino Energy (GeV)!

105

0
10−1

1

10

102

103

104

Neutrino Energy (GeV)!

Figure 5.2: Expected neutrino energy spectra of different event categories. Taken
from [6].

105Chapter 5. Data Selection

61

The event reduction is aimed to remove background events including cosmic ray muonand flasher events, which are caused by internal coronal discharge of a PMT, to purify
the final sample in neutrino events. Each sample, including FC, PC and Up-µ, have theirown reduction processes, which are optimized for the topology and the OD activity ofneutrino events.
It should be noticed that the criteria in event reduction are different for different SuperK phases since the condition of the detector changed. Only the criteria for Super-K
IV is discussed here, the detail of the event reduction for other Super-K phases can befound in [8].

5.1Reduction for Fully Contained Events

FC events are distinguished from PC events based on the number of OD hits in thehighest charge cluster. The process of the FC data reduction consists of five steps.

5.1.1

First Reduction

The event trigger is fired at a rate of 106 times a day, which consists mainly of downwardgoing cosmic ray muons and radioactivities with low energy due to contaminants in thewater. Two criteria are applied to remove those backgrounds in the first step of the FC
reduction.

1. PE300 > 200 p.e.s
where PE300 is the maximum number of total photoelectrons (p.e.s) observed by
the ID PMTs within a sliding 300 ns time window.
2. NHITA800 ≤ 55
where NHITA800 denotes the number of OD hits in a fixed time window from 500
ns to 1300 ns, which is around the event trigger at 900 ns.

Most of the low energy background events can be rejected by the first cut. The second
cut is aimed to reject the cosmic ray muon events, which have many OD hits. After the
first reduction, the event rate is decreased to the order of 103 events/day.

5.1.2

Second Reduction

Selection criteria for the second reduction include:

62

Chapter 5. Data Selection
1. NHITA800 ≤ 30 or PEtotal > 100,000 p.e.s
where PEtotal is the sum of the observed p.e.s of the ID PMT.
2. PEmax /PE300 < 0.5
where PEmax is the maximum number of p.e.s observed by a single ID PMT.

Stronger rejection to events with OD activity is done by the first cut while very highenergy FC events with light leakage into the OD side are accepted. The second cut
mainly rejects the electrical noise events caused by a single PMT with a high charge.
Event rate becomes ∼ 102 events/day after the FC second reduction.

5.1.3

Third Reduction

Remaining background events are mainly noise events and cosmic ray muons with asmall number of OD hits or high momentum. They are rejected by the following cutsin the third reduction.

Hard Muon Cut
Cosmic ray muons with a very high energy (e.g. > 1 TeV), also called “hard muon”,
would be identified and rejected by the following cut:
1. NHITA500 ≥ 40
where NHITA500 is the number of OD hits in a sliding 500 ns time window before
1300 ns, which is just after the main trigger (∼ 900 ns).

Through-going Muon Cut
Through-going muons have very high energy and deposit large energy in the ID. A
through-going muon fitter is applied when:
1. PEmax > 231
2. Number of hits for ID PMTs ≥ 1000
This fitter finds the entering and exiting points of the through-going muon in the ID andcalculated the goodness of fit based on the observed and expected hit time informationof PMTs. The maximum value of the goodness is 1 when the expected and observed

Chapter 5. Data Selection

63

distribution of PMT hit time are identical. A minus value would be returned if the fitter
failed finding the entrance or exiting points.
The criteria to reject the through-going muon is:
1. Goodness of the through-going muon fit > 0.75
2. NHITAin ≥ 10 or NHITAout ≥ 10
where NHITAin (NHITAout ) denotes the number of OD hits within 8 m from theentrance (exit) in a fixed 800 ns time window.
Events that passed those criteria are rejected as through-going muon.

Stopping Muon Cut
A stopping muon fitter, which is similar to the through-going muon fitter, is applied tofind the entrance point of muon for the stopping muon rejection. Events fulfilling thefollowing criteria are rejected as stopping muons:
1. Goodness of the stopping muon fit ≥ 0
which means that the fitter finds the entrance point successfully
2. NHITAin ≥ 10

Cable Hole Muons
There are twelve cable holes on top of the SK detector tank. No OD PMT was installedon the cable holes to enable the high voltage supply cables and signal cables to passthrough. Therefore, cosmic ray muons are able to enter the detector without OD activity
if they cross those holes.
Four 2 m × 2.5 m plastic scintillation counters were installed in April, 1997 on four
cable holes near the ID wall among all twelve holes to reject those cable hole muons.
The rejection criteria with the veto counters are:

1. One hit on plastic scintillation counter.
2. Both Lveto,x and Lveto,y < 4 mwhere Lveto,x and Lveto,y are the distances between the cable hole and the reconstructed vertex in the direction of x and y, respectively.

64

Chapter 5. Data Selection

Flasher Event Cut
Flasher events caused by the internal coronal discharges of PMT usually have a border
hit timing distribution than that of neutrino events. A TOF fitter aimed to find thelight source assuming all photons are emitted from a point-like source simultaneously,
will also be used to fit the event when the total number of hits in ID is less than 250.
The goodness of fit is expected to be small for the flasher events since the process ofinternal discharge is slow which makes the emitting of photons is not able to be regarded
as at the same time.
The criteria for the flasher events rejection are:
1. NMIN100 ≥ 20
where NMIN100 represents the minimum number of ID hits in a sliding 100 ns time
window from +300 ns to +800 ns after the trigger.
2. Goodness of the TOF fitter ≤ 0.4

Accidental Coincidence Events Cut
When a cosmic ray muon event follows a low energy event within the trigger gate, thecosmic ray muon might survive from the cuts on OD hits since the OD activities causedby the muon are much later than the trigger timing. Those events, which is termed as
accidental coincidence events, are removed by applying cut on the hit happened later
than the trigger time:
1. NHITAoff ≥ 20
where NHITAoff is the number of OD hits in a fixed 500 ns time window which isfrom +400 ns to +900 ns after the trigger timing, also called off-time window.
2. PEoff > 5000 p.e.s
where PEoff is the number of p.e.s in the ID within the off-time window.

Low Energy Events Cut
The radioactive background and electrical noise events which survive from the previous
reduction step are suppressed by:

1. NHIT50 < 50
where NHIT50 is the number of ID hits in a sliding 50 ns time window.

Chapter 5. Data Selection

65

Figure 5.3: Two flasher event displays in SK-IV. Each circle represents a hit on
PMT. A larger circle denotes a higher charge signal for the PMT. The flash light for
both events come from the same PMT on the left part of plots, therefore similar hit
pattern observed.

The time windows defined here considers the time of flight for each photon, which isassumed to be from a common light source. NHIT50 = 50 is a typical value for anelectron with an energy of 9 MeV. This cut is optimized for a high efficiency of theneutrino events selection in the oscillation analysis, which requires the visible energy
larger than 30 MeV.

5.1.4

Fourth Reduction

The FC fourth reduction is mainly aimed to reject the flasher events. Failing tubes
often produce multiple events with similar topologies as these discharges repeat. Typical
flasher events have spatially similar hit distributions, as shown in Figure 5.3.
Two methods, including a pattern matching algorithm and the Kolmogorov-Smirno (KS)test, are used to evaluate the similarity of the past events which passed the third stepof the FC reduction and the event for test. Only events within latest two years will be
used as the past events since the possibility that two neutrino events have similar hit
pattern increases.
The PMTs in the ID is divided into 1450 part for 2 m × 2 m square, called patches.
The pattern matching algorithm calculate a correlation parameter r defined as
r=

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

(5.1)

66

Chapter 5. Data Selection

1000

800

600

400

200

0
-6

-4

-2

0

2

4

6
8
10
Flasher Likelihood

Figure 5.4: Final likelihood for flasher event rejection. Points show the distributionof data, while the black (red) solid line shows the distribution of nominal MC (fittedMC).

where N represents the number of patches, hQA i and σA represent the average charge
and its standard deviation for event A, respectively. The meaning of hQB i and σB are
defined similarly.
On the other hand, the KS test will also be done based on the patches. The final selection
will consider the result from both two methods and reject the event with high similarity
to past events. Figure 5.4 shows the final likelihood distribution to reject the flasher
events. A larger value represents a higher possibility for a flasher event. The threshold
is shown as the dashed line. Points show the distribution of data, while the black (red)solid line shows the distribution of nominal MC (fitted MC). The difference between thenominal MC and the fitted MC is 1.3%, which is regarded as the uncertainty from theflasher rejection.

5.1.5

Fifth Reduction

The final reduction consists of several criteria optimized for different background sources.
Event that passed the third reduction will be reconstructed by APFit, which is a conventional reconstruction for neutrino events. The reconstruction information can beutilized for the background rejection with higher precision. The detail about APFit will
be introduced in Section 6.1.

Chapter 5. Data Selection

67

Invisible Muon Cut
Cosmic ray muons with energy lower than the Cherenkov threshold after entering ID
generate no ID PMT hits. The Michel electron from such invisible muons can emit lightinside the ID, like a neutrino event. A typical invisible muon has an OD signal earlier
than the trigger timing and a low energy ID signal from the Michel electron. Events
passing the following criteria are rejected as invisible muons:
1. PEtot <1000 p.e.s
2. NHITACearly ≥ 5
where NHITACearly is the maximum number of OD hits in the cluster in a sliding
200 ns time window from −8800 ns to −100 ns.
3. NHITACearly + NHITAC500 ≥ 10 if DISTclust < 500 cm
NHITACearly ≥ 10 otherwise
where NHITAC500 is the number of OD hits in the cluster in a fixed 500 ns time
window from −100 ns to +400 ns. DISTclust represents the distance between the
OD hit clusters used for the calculation of NHITACearly and NHITAC500 .

Coincidence Muon Cut
The accident coincidence muon events which survive from the third reduction are rejectedby the following criteria:
1. PE500 < 300 p.e.s
where PE500 is the total number of p.e.s observed in the ID in a fixed 500 ns time
window from −100 ns to +400 ns.
2. PElate ≥ 20 p.e.s
where PElate is the maximum number of OD hits in a 200 ns sliding time window
from +400 ns to +1600 ns.

Long-tail Flasher Cut
A tighter flasher cut than the one in the third reduction step is used to remove remaining
flasher events:
1. NMIN100 ≥ 6 if goodness of TOF fitter < 0.4
where NMIN100 is the minimum number of ID hits in a sliding 100 ns time window

68

Chapter 5. Data Selection
from +300 ns to +800 ns. The definition is the same as the one in the third
reduction step.

Stopping Muon Cut
A tighter cut is applied when the stopping muon event is reconstructed well, as
1. NHITAin ≥ 5 when goodness of stopping muon fit ≥ 0.5
Furthermore, the reconstructed variable from APFit is also utilized to reject the stoppingmuon.
1. NHITAAPFit ≥ 4
where NHITAAPFit is the number of OD hits within 8 m from the muon’s entrance
point, reconstructed by APFit, in a sliding 200 ns time window from -400 ns to+400 ns.
Events that pass either cut will be rejected as stopping muon events.

Cable Hole Muon Cut
Similarly, the cut for cable hole muon is tighter when the stopping muon event is reconstructed well, as
1. Goodness of the stopping muon fit ≥ 0.4
2. PEtot > 4000
3. cos value of the muon’s direction on z axis < -0.6
4. the distance from the muon entering point to the nearest cable hole < 250 cm

Backgrounds due to hardware issue
Some non-physical background events might be contaminated due to some hardware
issues. Events are rejected if any of the following conditions are satisfied.

• N0 ≥ 250 and N0 - N1 ≥ 100– N0 (N1 ) is the number of hits for ID PMTs with less (larger) than single p.e.

Chapter 5. Data Selection

69– This cut is used to reject electronic noise from high voltage system or electronic boards, which create lots of hits with small charge.
• NHITA < 1 for any electronic hut– NHITA is the number of OD hits.
– This cut is used to reject events with electronic hut doesn’t work.
• Reference PMT for laser or xenon lamp are hit.
– This cut is used to reject the event for calibration, which utilized laser orxenon lamp.

5.1.6

FC Reduction Summary

The final FC neutrino samples are selected by:
1. Event vertex is within the fiducial volume (FV)The definition of FV is 200 cm away from the ID wall in the previous analysis. In
this analysis, FV will be extended to 50 cm, which will be discussed in Section 7.3.
2. Number of OD hits in the highest charge OD cluster (NHITAC) < 16.
3. Visible energy (Evis ) > 30 MeV.
Selection efficiencies in each step of the FC reduction are evaluated based on atmospheric
neutrino MC and summarized in Table 5.1. The final event rate is 8.09 events / day,within the conventional FV (Dwall > 200 cm). Although the reconstruction tool used inTable 5.1 is APFit rather than fiTQun, the result for fiTQun is expected to be similar
due to the similarity of the vertex reconstruction performance of these two reconstruction
tools, as we will see in next Chapter.
The systematic error from the FC reduction is estimated by comparing the cut variables’
distribution between data and MC. The largest discrepancy between data and MC is
found to be from the flasher on the fourth reduction and the discrepancy is evaluated tobe 1.3% as presented in Section 5.1.4. This value is regarded as the final FC reduction
uncertainty.
Most of the background contamination in the final FC samples are flasher events andcosmic ray muons. All events passing the reduction are scanned by eye to determine
the level of background contamination in the final analysis sample and to estimate theuncertainty inherent to the reduction process. Table 5.2 summarizes the contaminationfor each sample.

70

Chapter 5. Data Selection
Reduction step
1st
2nd
3rd
4th
5th
Final

Number of events for data
6295064111596213835313044612065925231

Event rate for data
2018.65 / day
357.86 / day
44.36 / day
41.83 / day
38.69 / day
8.09 / day

Efficiency
100.0%
99.99%
99.83%
99.00%
98.95%
98.02%

Table 5.1: Selection efficiencies and number of events for FC sample with livetime of
3118.45 days for SK-IV. Selection efficiencies are for events whose real vertex is within
the FV and number of OD hits < 16 and visible energy > 30 MeV. The FV cut is 200
cm and the variable reconstructed by APFit is used in this Table.

Background contamination
Sub-GeV
Multi-GeV

e-like (flasher)
0.03%
0.07%

µ-like (cosmic ray muons)
0.02%
0.14%

Table 5.2: Estimated contamination for each background source. The term“sub-GeV” represent events with Evis < 1.3 GeV, while “multi-GeV” represent
events with Evis > 1.3 GeV.

5.2

Reduction for Partially Contained Events and Up-µEvent

The PC and Up-µ samples also have their own reduction processes, which are optimized
for the topology of OD activity of neutrino events. These two samples are reconstructedusing the pre-existing reconstruction algorithm (APFit) and are divided into “stopping”
and “through-going” subsamples based on the estimated muon stopping point for PC
and Up-µ events. The “through-going” events in the Up-µ sample are further divided
into “showering” and “non-showering” based on whether the event induces an electromagnetic shower while traversing the ID. More detail about the data reduction and event
reconstruction for PC and Up-µ events can be found in [111]. The final event rate for
PC event is 0.66 events/day, while it is 1.41 events/day for Up-µ events.

Chapter 6

Event Reconstruction
The Super-K event reconstruction algorithms determine the physical properties of theevent based on the PMT hit information. The properties include the interaction vertex,
number of particles, particle types, momenta and directions. There are two different
reconstruction tools, APFit and fiTQun. Since the beginning of SK, APFit has been theoriginal fitter and serves for more than 20 years, and fiTQun is a newly developed fitter
based on the maximum likelihood method. Before moving to describe the algorithm usedin the thesis, fiTQun, the reconstruction scheme of APFit will be introduced briefly. The
performance of fiTQun and its comparison with that of APFit will also be discussed. As
we will see, fiTQun exhibits better performance than APFit, which allows us to expand
the fiducial volume (FV) without reducing the sample purity during categorization of
atmospheric neutrino event, resulting in a higher sensitivity.

6.1

Conventional Event Reconstruction

The conventional event reconstruction algorithm, APFit, was introduced at the beginning of Super-K. APFit contributed to the discovery of atmospheric neutrino oscillationin 1998 [1] and has been also used in both the K2K [50] experiment and the T2K experiment [62].
APFit is a fitter with single iteration based on the time and charge information observed by PMTs [112]. The interaction vertex is reconstructed based on the hit timing
information after the time of flight correction of photon. Then the direction of the firstfound ring, usually with the highest energy, is determined based on the observed charge
distribution with respect to the interaction vertex. Additional ring candidates are foundusing a Hough-transform based method and selected by a optimized likelihood function71

72

Chapter 6. Event Reconstruction

for spurious ring candidates rejection. After determining the number of Cherenkov ringsin the event, the particle type of each ring is determined based on the Cherenkov ringpattern and opening angle. Rings from electrons often have rough edges from the light
generated by their electromagnetic showers, while rings from muons or charged pions
mainly produce crisp edges, as shown in Figure 6.1.

Figure 6.1: Event display of single-ring electron (right) and single-ring muon (left)
neutrino MC event. Each circle denotes a PMT hit, whose observed charge isrepresented by the size of the circle.

In the final step of the fit, the momentum of each ring is evaluated based on the charge
information observed by the PMTs whose angular position with respect to the line fromthe interaction vertex to each ring’s center is less than 70◦ . Corrections and adjustments
would be applied if two rings were overlapped.
It is worth noting that in APFit the hit time information is only used in the first step
to find the initial vertex candidate and not used in the following steps such as particle
identification or ring counting.

6.2

New FC Event Reconstruction

The new algorithm called fiTQun uses the maximum likelihood method to reconstructparticle types and, at the same time, determine their kinematics in the detector. The
algorithm is based on methods developed for the MiniBooNE experiment [113]. FiTQun
was developed from scratch for Super-K with additional functions such as reconstruction
for multi-ring events. In comparison to the fitting procedure of APFit, more information,
including the PMT hits outside the Cherenkov cone and the hit time, is taken into account by fiTQun. As we will see later, the fit process in fiTQun will run multiple times toChapter 6. Event Reconstruction73

find the best kinematics parameters (e.g. vertex, momentum, direction) for each possible
particle configuration hypothesis (e.g. particle types) and choose the best one finally.
On the other hand, APFit only fits once and determines the kinematic parameters or
particle configuration one by one. Since the launch of Super-Kamiokande in 1996, theremarkable evolution of computing power has allowed fiTQun to achieve a greater accuracy of reconstruction at a similar time consumption as APFit used to be. Currently,
fiTQun has already been used in the T2K analyses for fiducial volume expansion due toits improved resolution of reconstructed quantities and particle identification [34].

6.2.1

Likelihood Function

An event topology hypothesis Γ (e.g. single-ring µ-like) is taken into account in thefitting along with its associated kinematic parameters θ, including vertex positions,particle generation times, particle directions and their momenta. The likelihood function
for a given hypothesis to estimate the kinematic variables in fiTQun is based on theobserved charge and hit time of each PMT and defined as:L(Γ, θ) =unhit
YPj (unhit|Γ, θ)

hit
Y

j

{1 − Pi (unhit|Γ, θ)}fq (qi |Γ, θ)ft (ti |Γ, θ). (6.1)

i

In this equation, index j runs over all PMTs with no hits, also called unhit PMTs.
The probability of not registering a hit with a fitting hypothesis (Γ, θ) is calculated
as Pj (unhit|Γ, θ) for each unhit PMT. The index i labels PMTs that registered a hit.
The charge likelihood, fq (qi |Γ, θ), represents the likelihood density for observing charge
qi for these PMTs under hypothesis (Γ, θ). In the same way, the likelihood density of
producing a hit at the observed time ti is defined as ft (ti |Γ, θ).
Indeed, the propagation processes of particle and optical photon in the detector are
independent from the response of the PMT and the electronics. The charge likelihood
therefore can be rewritten in terms of the expected number of photoelectrons generated(the predicted charge) at the i-th PMT given the hypothesis, µi (Γ, θ). The likelihood
function then becomeseL(Γ, θ) =

unhit
Y
j

Pj (unhit|µj )hitY{1 − Pi (unhit|µi )}fq (qi |µi )ft (ti |Γ, θ).
i

(6.2)

74

Chapter 6. Event Reconstruction

Here, the unhit probability Pj (unhit|µj ) and the charge likelihood density fq (qi |µi ) areonly characteristics of the PMT. The processes of Cherenkov photon emission and propagation in water are considered in the calculation of the predicted charge µi (Γ, θ) andseparated from the electronics of detector.
During the calculation process of the predicted charge µi , the observed photons are
divided into two categories: one is generated by the Cherenkov process and hit the
PMT directly through water in the detector, also called unscattered light or direct light;
the other is the photons which have been scattered or reflected before observed by PMT,
called indirect light. These two kinds of photons are considered separately and summed
to form the final predicted charge.
The predicted charge from direct light arriving at a PMT is calculated by integrating
the Cherenkov radiation profile along the track with considering the correction from thelight transmission in water, the distance between the light source and the PMT, and
the PMT angular acceptance. The charge generated by indirect light arriving at the
PMT is predicted by integrating the product of the direct light emission profile and ascattering function. This function is generated in advance with considering the effects
from the position of the PMT, light source and the wall. In the multiple ring case,the predicted charge of each ring is first calculated separately and then summed up to
calculate a total expected charge. The final charge likelihood fq (qi |µi ) is calculated bycomparing the prediction assuming that the photoelectrons generation obeys the Poisson
distribution to the observed charge in the PMT.
The time likelihood term is written as ft (ti |texp
i , Γ, p, µi ), where p is the particle momentum under the topology hypothesis Γ, and texp
represents the expected hit time, whichiis defined as the arrival time of unscattered photons emitted from the midpoint of track
and arriving at the PMT directly as
texp= t + smid /c + |RiPMT − x − smid d|/cn .
i

(6.3)

Here t and x are the creation time and vertex of the particle, respectively, d is theparticle direction, RiPMT is the position of i-th PMT and smid represents half of the track
length. The parameters c and cn are the group velocity of Cherenkov light in vacuum
and water, respectively. The reason why the time likelihood depends on the predictedcharge is that the first photon reaching a PMT is recorded as the hit time, which
results in a narrower time likelihood density distribution for more incident photons, i.e.
more predicted charge. The track length of a particle s determined by the topology
Γ and momentum p also affects the shape of likelihood density since not all photons
are generated in the middle of the track. The shape of time likelihoods from direct andindirect photon hits are determined based on particle gun simulations. During the fitting

Chapter 6. Event Reconstruction

75

process, contributions from those two likelihoods are calculated separately in the same
way as the charge likelihood. The final time likelihood for a single particle is obtained bymerging those two likelihoods according to their relative intensities. For multi-particle
topology hypothesis, the time likelihood is calculated ring-by-ring and then merged toa final likelihood function assuming the photons from a particle with earlier texp
alwaysiarrive earlier than the photons from any other particles whose texp
values are later.
i
The best set of the kinematic parameters θ̂ for a given event topology hypothesis Γ is
obtained by maximizing likelihood value L(θ|Γ). The best estimation of a given event
is determined by comparison of L(Γ, θ̂) among all hypotheses, Γ. It worth noting thatthe best estimation is not always the hypothesis with the largest likelihood value. A
hypothesis with more rings, i.e. more parameters, always fits the observation betterand therefore has a higher likelihood value due to the increased number of degrees of
freedom. Therefore care is needed when determining the best hypothesis. The detailof the topology selection, including ring counting and particle identification, will be
introduced in next section.

6.2.2

Fitting Procedure

The fiTQun reconstruction process can be divided into four steps.
1. Vertex pre-fitting
The interaction vertex is estimated based on the PMT timing information roughly.
2. Hit clustering
PMT hits are clustered in time to find candidates for particle activities.
3. Single-ring reconstruction
Events are fitted assuming event topologies with only a single light-producting
particle,
4. Multi-ring reconstruction
Events are fitted using hypotheses with multiple particles. The results from singlering reconstruction are used as seeds for fitting.
MINUIT[114] package is used to minimize the negative log likelihood -ln L(Γ, θ) withrespect to θ during the fits.

76

Chapter 6. Event Reconstruction

Vertex pre-fitting
The vertex pre-fitter is a fast algorithm which estimates an initial vertex position with
only the hit time information from PMTs around the primary event trigger. It is assumed
that all observed light is emitted from a single position in this step. The estimation isdone by searching for time t and the vertex position x which maximize the goodness
function:G(x, t) =hit
Xres /σ)2 /2)e(−(Ti,(6.4)

Tires = ti − t − |RiPMT − x|/cn

(6.5)

i

where

is the residual hit time calculated assuming a point-like light source at the interaction
vertex and accounting for the photon’s time-of-flight. The position of the i-th PMT isdenoted by RiPMT . As the time and vertex approach their true values during a gridsearch process, the Tires distribute near zero, resulting in a large goodness value. The
vertex returned during this step is just a rough estimation, called the pre-fit vertex.
This vertex will be fitted again with higher precision during the minimization of L(Γ, θ)
in Step 3 and Step 4.

Hit clustering
Events in Super-K are defined by detector activity in an O(10 µs) time window aroundan event trigger. Multiple subevents representing clusters of PMT hits separated intime from the primary trigger might be contained in one event. Muon decay is one
example: the primary event trigger is generated by muon while a delayed subevent is
produced by the additional hits from its decay electron. In order to separate the hits
from the particles created in different time for further fitting, the activity around theprimary trigger and additional subevents, if any, are searched for and selected by the
hit clustering algorithm.
The algorithm starts by searching for the subevent activities around the time of theevent trigger with a peak-finding algorithm. The vertex goodness G(x, t) from Equation
6.4 is scanned in t while fixing the vertex x to the pre-fit vertex for the additional
delayed peaks’ search. Figure 6.2 shows the goodness distribution from a muon decay

Chapter 6. Event Reconstruction

77

event as an example. The parent muon and its Michel electron are the two dominantpeaks. Scattered or reflected light can cause fake peaks in or near a true cluster of hit
activity. Therefore, peaks are required to be above a minimum threshold F (t), which isshown as the blue curve in Figure 6.2. The definition of F (t) is
F (t) := 0.25 arg max{i∈MG(x, ti )} + η,(1 + ((t − ti )/γ)2(6.6)

where M represents all local maxima of the goodness function G(x, t). The time constantγ is assigned as 25 ns (70 ns) when t < ti (t > ti ). In order to suppress hits from darknoise of the PMTs, the threshold function is offset by η = 9. The minimum goodness
between any two peaks is required to be below 0.6 × F (t), which is shown as green
dashed curve. The figure shows only two peaks that pass these criteria (marked by red

G(x,t)

triangles) which will be subject to further fitting.

Goodness

103

Threshold for peak, F(t)Threshold for minimum value between peaks, 0.6×F(t)

102

10

1000

1500

2000

2500

3000
t [ns]

Figure 6.2: An example of the goodness function with respect to hit time. A muon
event with its decay electron is shown here. The goodness function with respect to hit
time is shown as the black line, while the vertex is fixed to the pre-fit vertex.
Thresholds for identifying candidate peaks: F (x) and 0.6 × F (x) are denoted by blue
(dashed) and green (dotted) curves. Candidates found by the hit clustering algorithm
are marked by red triangles and their associated time windows are indicated by grayvertical lines. Overlapped time windows are merged into one.

All vertex positions are assumed to be close to the pre-fit vertex in the first step of
hit clustering. However, this assumption is broken when the primary particle (e.g.
muon with high momentum) travels a long distance from the interaction vertex. In thiscase, the goodness distribution will be smeared and might be lower than the thresholdresulting in a subevent missing and a lower decay electron tagging efficiency. To improve
the tagging efficiency, the vertex pre-fitting is rerun after masking the hits caused by the
primary particle for a vertex close to the vertex of the secondary particle. After that the

78

Chapter 6. Event Reconstruction

peak-finding algorithm uses the new vertex as the position in the goodness function x to
search for possible missing goodness peaks. More details can be found in Appendix E.
For each found peak, its associated hits are contained by a time window defined as -180
ns < Tires < 800 ns. Those hits will be input to the vertex pre-fitter and peak finder
once again to get the final subevent candidates with a higher precision for the full event
reconstruction. The interaction time and vertex fitted in each window will be used as
the seeds for fitting in next steps.

Single-ring reconstruction
The single-ring fitter, whose topology hypothesis Γ in Equation 6.1 is single-particle,
is applied to each time window determined by the previous step firstly. The likelihood
function is maximized against the charge and time distribution observed by PMTs bychanging the kinematic parameters of the event, including the created vertex and time,
the particle momentum, and direction. This process will run iteratively for each of
the hypothesis considered by the single ring fitter: electron, muon, and charged-pion.
Particle identification (PID) is then determined by comparing the best-fit likelihood
values of them. For example, the distinguishment between electrons and muons of aevent is based on the value of the logarithm of the ratio between the likelihood value
with best-fit electron and muon hypotheses: ln(Le /Lµ ). Figure 6.3 demonstrates the
distribution of this variable for the FC atmospheric neutrino samples. The distributionsof data and MC are in good agreement. A clear separation of the likelihood ratio between
electron-like (e-like) and muon-like (µ-like) events can be seen for both sub-GeV andmulti-GeV energy regions.
The distribution of the distance from the reconstructed to true vertex of the FC singlering charged-current quasi-elastic (CCQE) event sample in the atmospheric neutrino MC
whose true interaction vertex lies within the conventional fiducial volume (Dwall > 200cm) is shown in Figure 6.4, which is used to evaluate the vertex resolution by calculation
the 68 percentile. Figure 6.5 shows vertex resolution as the function of visible energy.
The vertex resolution of fiTQun for CCQE νe events is stable at 20.6 cm with the visibleenergy in the range from 100 MeV to 1330 MeV, while the resolution of APFit varies
from 34.6 cm to 25.3 cm in the same range. The vertex resolution of fiTQun for CCQE
νµ events, which varies from 29.2 cm to 15.9 cm, is better than that of APFit, whichchanges from 34.2 cm to 18.2 cm. Similar plots for direction resolution are shown in
Figure 6.6 and Figure 6.7. The performance of fiTQun for direction reconstruction is
the same as that of APFit. The distribution of the ratio between the reconstructed
momentum and true momentum are shown in Figure 6.8. In the same energy range, theChapter 6. Event Reconstruction

79

2500
400
µ-like350
Number of events

Number of eventsµ-likee-like

2000

1500

1000

e-like

300
250
200
150
100

500

50
0
-3000

-2000

-1000
0
1000
PID Likelihood

2000

0-10000

3000(a) Sub-GeV events

-5000

0
PID Likelihood

5000

10000

(b) Multi-GeV events

a.u

a.u

Figure 6.3: PID likelihood distribution of FC single-ring events. Left figure is for
sub-GeV events and right figure is for multi-GeV events. Distribution for neutrino
data is denoted by points and that for atmospheric neutrino MC after oscillation isshown as histograms. The true hierarchy is assumed to be normal hierarchy. The
oscillation parameters is taken to be sin2 θ13 = 0.0210, sin2 θ23 = 0.5,∆m223 = 2.4 × 10−3 eV2 , and δCP = 0. Same parameters are applied below.
Charged-current νµ interaction component is shown as the shaded histograms. The
statistical error are denoted by error bars. The reconstructed event vertex to thenearest ID wall (Dwall ) is required to be larger than 200 cm.

0.05

fiTQun

0.04

APFit

0.03

APFit

0.04

0.03

0.02

0.02

0.01

0
0

fiTQun

0.05

0.01

10

20

30

40
50
60
70
80
90 100Distance between true/rec vertex[cm]

0
0

10

20

30

40
50
60
70
80
90 100Distance between true/rec vertex[cm]Figure 6.4: Single-ring electron(left) and muon(right) vertex resolution for FC
true-fiducial CCQE events in atmospheric neutrino MC, compared between
APFit(dashed line) and fiTQun(solid line). The resolution is defined as the 68
percentile of the respective distributions, which is shown by corresponding verticalline.

momentum resolution of fiTQun for CCQE νe events improves from 5.39% to 2.58% as
the visible energy increases, while the resolution of APFit varies from 7.04% to 3.32%, asshown in the top left plot of Figure 6.9. The momentum resolution for CCQE νµ eventsis stable across the energy range, being lower than 2.5% for fiTQun and slightly worse for
APFit. FiTQun also shows a higher ability to discriminate between electrons and muons
with less than a 1% mis-identification rate in the same energy range (Figure 6.10). The
typical values of the metrics used to measure fit quality are summarized in Table 6.1,
which show that fiTQun performs as well or better than APFit in general.

Vertex Resolution [cm]Chapter 6. Event Reconstruction

Vertex Resolution [cm]

80

35
30
25
20
15

30
25
20
15

10

10

fiTQun
APFit

5
0

35

0

200

400

fiTQun
APFit

5
600

800

1000

0

1200
1400
E vis [MeV]

0

200

400

600

800

1000

1200
1400
E vis [MeV]

a.u

a.u

Figure 6.5: Vertex resolution of FC single-ring charged current quasi-elastic(CCQE) event as a function of visible energy. Left figure is for CC νe events and rightfigure is for CC νµ events. The performance of fiTQun is indicated by the full
triangles, while that for APFit is indicated by the open circles.

0.08

0.12

fiTQun0.07

fiTQun
0.1

0.06

APFit

APFit

0.08

0.05
0.06

0.04
0.03

0.04

0.02
0.02

0.01
0
0

2

4

6

8
10
12
14
16
18
20
Angle between tru/rec direction [deg]

0
0

2

4

6

8
10
12
14
16
18
20
Angle between tru/rec direction [deg]Direction Resolution [°]Direction Resolution [°]Figure 6.6: Single-ring electron (left) and muon (right) direction resolution for FC
CCQE events in atmospheric neutrino MC, compared between APFit (dashed line)
and fiTQun (solid line). The resolution is defined as the 68 percentile of the respective
distributions which is shown by corresponding vertical line.

5

4

3

2.4
2.2
2
1.8
1.6
1.4
1.2
1

2

0.8

0

0

0.6

fiTQun
APFit

1

200

400

fiTQun
APFit

0.4
0.2
600

800

1000

1200
1400
E vis [MeV]

0

0

200

400

600

800

1000

1200
1400
E vis [MeV]

Figure 6.7: Direction resolution of single-ring electron (left) and muon (right) eventsin the FC CCQE event sample in the atmospheric neutrino MC, plotted as a functionof visible energy. The full triangles indicate the performance of fiTQun and the open
circles are for APFit.

81

a.u

a.u

Chapter 6. Event Reconstruction

0.012

fiTQun

0.018
0.016

fiTQun

0.01
0.014

APFit

APFit

0.012

0.008

0.01
0.006
0.008
0.006

0.004

0.004
0.002
0.002
0
0.8

0.85

0.9

0.95

1

1.05
1.1
1.15
1.2
Ratio of rec/true Momentum

0
0.8

0.85

0.9

0.95

1

1.05
1.1
1.15
1.2
Ratio of rec/true Momentum

Momentum Resolution [%]Momentum Resolution [%]Figure 6.8: Single-ring electron (left) and muon (right) momentum resolution for FC
CCQE events in atmospheric neutrino MC, compared between APFit (dashed line)
and fiTQun (solid line). The bias (resolution) is defined as the mean (RMS) value ofthe ratio distribution between the reconstructed momentum and true momentum.

7
6
5
4
3

fiTQun
APFit

1

1.5

0

200

400

fiTQun
APFit

0.5

600

800

1000

0

1200
1400
E vis [MeV]5

Momentum Bias [%]Momentum Bias [%]

2

1

2

0

2.5

4
3
2
1
0

0

400

600

800

1000

1200
1400
E vis [MeV]

600

800

1000

1200
1400
E vis [MeV]

5
4
3
2
1
0

-1

-1

-2

-2

fiTQun
APFit

-3
-4
-5

200

0

200

400

fiTQun
APFit

-3
-4
600

800

1000

1200
1400
E vis [MeV]

-5

0

200

400

Figure 6.9: Momentum resolution and bias as a function of visible energy. Top two
figures show the momentum resolution and bottom ones show the momentum bias.
Left two figures are for FC CCQE νe events in the atmospheric neutrino MC whileright ones are for νµ events. The performance of fiTQun is indicated by the full
triangles, while that for APFit is indicated by the open circles.

Chapter 6. Event Reconstruction

3.5

Mis-PID rate [%]Mis-PID rate [%]82

fiTQun

3

APFit

2.5

3.5

2

1.5

1.5

1

1

0.5

0.5

200

400

600

800

1000

1200
1400
Evis [MeV]

APFit

2.5

2

0
0

fiTQun

3

0
0

200

400

600

800

1000

1200
1400
Evis [MeV]

Figure 6.10: Mis-identification rate as a function of visible energy. Left figure shows
the result of FC CCQE νe events in the atmospheric neutrino MC, while right figure
shows the result of CCQE νµ events. The performance of fiTQun is indicated by thefull triangles, while that for APFit is indicated by the open circles. The statisticalerror are denoted by error bars.

Reconstruction
fiTQunTrue CCQE νe sample
Vertex Resolution
20.6 cm
Direction Resolution
1.48◦
Momentum Bias
0.43%
Momentum Resolution 2.90%
Mis-PID rate
0.02%
True CCQE νµ sample
Vertex Resolution
15.8 cm
Direction Resolution
1.00◦
Momentum Bias
-0.18%
Momentum Resolution 2.26%
Mis-PID rate
0.05%

APFit
24.9 cm
1.68◦
0.63%
3.56%
0.50%
17.3 cm
1.28◦
0.54%
2.60%
0.91%

Table 6.1: Basic performance of reconstruction algorithms on the FC CCQE
single-ring event sample. Both result of APFit and fiTQun are shown here. The
visible energy of the sample used here is 1 GeV.

Multi-ring reconstruction
A large fraction of atmospheric neutrino events with multi-GeV energies, which havehigh sensitivity to the mass hierarchy, have multi-particle final states, therefore thereconstruction performance for multiple light-producing particles is essential to the atmospheric neutrino oscillation analysis. In fiTQun, the multi-ring fitter is only applied to
the first time window, which is around the primary event trigger and not to any delayedones, to reduce the computing time consumption. The multi-ring hypothesis for fitting
is constructed by adding a new ring to the previous fit result iteratively. After adding
a new ring, the kinematic parameters in the likelihood function (c.f. Equation 6.2) arevaried to find their best fit values. All of the three hypotheses used in the single-ringChapter 6. Event Reconstruction
True Numberof Rings
True 1 ring
True 2 rings
True ≥3 rings

fiTQun Reconstruction
1 ring 2 rings ≥ 3 rings
95.0% 4.64%
0.41%
27.8% 66.7%
5.56%
7.04% 25.5%
67.5%

83
APFit Reconstruction
1 ring 2 rings ≥ 3 rings
95.9% 3.85%
0.29%
42.5% 52.8%
4.63%
20.2% 33.0%
46.8%

Table 6.2: Ring counting performance on FC atmospheric neutrino events. Bothresult of APFit and fiTQun are shown here. The number of reconstructed rings aredenoted by columns and the number of true rings are denoted by rows. The true rings
are defined as only final state particles with energy 30 MeV higher than the
Cherenkov threshold.

reconstruction, e-like, µ-like and π + -like rings are tested. The new ring is validated
based on the likelihood ratio between the result of the updated fit and the original one
for each of the hypotheses and accepted only if the likelihood ratio pass the criterion
for ring counting. The processes to search for new rings are repeated until either the
hypothesis with new ring is rejected by the criterion or six rings exist in the hypothesis.
The likelihood ratio between the 2-ring and 1-ring hypotheses for both sub-GeV andmulti-GeV events are shown in Figure 6.11. A cut on this likelihood ratio, which is determined at 9.35 (11.83) based on MC study, is used for single- and multi-ring separation
for the hypothesis with e-like (µ-like) first ring. Furthermore, MC studies indicate thatring candidates that have an angular separation of less than 20◦ with other rings are
typically due to the particle scattering rather than a new particle, so those candidates
are discarded as spurious. When this occurs, a new topology hypothesis with one ring at
the position of the original two close rings is generated to merge them. The merged ringis refitted for all particle hypotheses while keeping all other rings fixed. This procedure,ring merging and refitting, is run over for all ring candidates of the event in descendingorder of their energies.
The performance of the ring counting in both APFit and fiTQun using atmospheric
neutrino MC events is summarized in Table 6.2. In this table, only charged particles withenergy more than 30 MeV above the Cherenkov threshold in the final states, which makes
them observable, are considered as true ring candidates and their angular separation
with other particles are not required. FiTQun shows a greater ability to reconstructmulti-ring events than APFit and the ability to correctly identify single-ring topologies
is at the same level for both algorithms. Although the fake ring ratio of fiTQun isslightly larger than that of APFit for true single-ring or two-ring event reconstruction,
the increase of fake ring ratio is not large enough to neutralize the benefit from thehigher fraction of the events reconstructed correctly.
The PID likelihood distribution of the ring with the highest energy in fully contained
multi-ring events is shown in Figure 6.12. The separation between e-like and µ-like is not

84

Chapter 6. Event Reconstruction

as good as that for single-ring events due to the overlap of Cherenkov rings. However,the peak for both of e-like events and µ-like events still can be seen clearly.

1600
500
1R like

MR like

1R like

1200

Number of events

Number of events

1400

1000
800
600

MR like

400

300

200

400
100
200
0
-10

0

10
20
30
Ring Counting Likelihood

40

0

50

(a) Sub-GeV events

0

20
40
60
Ring Counting Likelihood

80

100

(b) Multi-GeV events

Figure 6.11: The distribution of the likelihood ratio between the best-fit result of
single-ring hypothesis and multi-ring hypothesis of FC atmospheric neutrino events.
The most energetic ring is required as e-like ring. Left figure shows the result of
sub-GeV events, while right figure shows multi-GeV events’ result. Distribution foratmospheric neutrino data is denoted by the points and the MC prediction including
neutrino oscillations is denoted by the histogram. The events with single-ring final
state is shown by the shaded histogram. The statistical error are denoted by error
bars. The terms “1R” and “MR” in the figures represent single-ring and multi-ring,
respectively. The reconstructed event vertex to the nearest ID wall (Dwall ) is required
to be larger than 200 cm.

1600µ-likeµ-like500

e-like1200

Number of events

Number of events

1400

1000
800
600

e-like

400
300
200

400
100
200
0
-3000

-2000

-1000
0
1000
PID Likelihood(a) Sub-GeV events

2000

3000

0
-10000-5000

0PID Likelihood

5000

10000

(b) Multi-GeV events

Figure 6.12: PID likelihood distributions of the most energetic ring for fully
contained multi-ring events. Left figure shows the results of sub-GeV events whileright figure are for multi-GeV events. Distribution for atmospheric neutrino data isdenoted by the points and the MC prediction including neutrino oscillations is denotedby the histogram. The component of νµ charged-current interactions is shown by theshaded histogram. The statistical error are denoted by error bars. The reconstructed
event vertex to the nearest ID wall (Dwall ) is required to be larger than 200 cm.

The distribution of reconstructed invariant mass of π 0 s in charged-current single pion
(CC1π 0 ) events is studied to check the multi-ring fitter’s kinematic reconstruction performance. The invariant mass based the second and the third rings in order of energy forevents with three e-like rings (eee events) is shown in Figure 6.13. The true CCνe 1π 0

85

70

70

60

60

50

50

Number of events

Number of events

Chapter 6. Event Reconstruction

40
30
20
10
0

40
30
20
10

50

100
150
200
250
300
350
Reconstructed Invariant Mass [MeV/c](a) Event selected by fiTQun

400

0

50

100
150
200
250
300
350
Reconstructed Invariant Mass [MeV/c]

400(b) Event selected by APFit

Figure 6.13: Reconstructed invariant mass calculated based the second and thethird rings in order of energy for events with three e-like rings in the sub-GeV
atmospheric neutrino MC sample. Left figure shows the events reconstructed byfiTQun, and right figure is for the events reconstructed by APFit. True CCνe 1π 0events are denoted by shaded histograms. The invariant mass range used for the event
rate and purity calculation is shown by arrows. The reconstructed event vertex to thenearest ID wall (Dwall ) is required to be larger than 200 cm.

fiTQun selection
Event rate Purity
eee events (target: νe CC1π 0 )Sub-GeV
278.5
52.1%
Multi-GeV
112.1
47.0%
µee events (target: νµ CC1π 0 )Sub-GeV
384.5
54.6%
Multi-GeV
143.5
64.6%

Topology

APFit selection
Event rate Purity
175.9
32.8

42.9%
36.2%

201.2
51.6

38.0%
32.0%

Table 6.3: Performance of CC1π 0 events reconstruction and selection in atmospheric
neutrino MC events with three-ring final state. The reconstructed PID of the threerings in order of momentum are denoted by eee or µee, which represent candidates ofνe CC1π 0 and νµ CC1π 0 events, respectively. The number of events passing the π 0invariant mass cut is shown in the event rate column. The purity of target events inthe π 0 invariant mass region is shown in the purity column.

components are shown by shaded histograms. Comparing with the peak near the π 0mass in the invariant mass distribution of APFit, the one of fiTQun is larger, sharperand has fewer backgrounds, which indicates a higher selection efficiency, better energy
resolution and lower background contamination. Table 6.3 shows the event rate and thepurity of the CCνe 1π 0 events with 85 < m < 185 MeV (indicated by the arrows in the
figure). The study on CCνµ 1π 0 events (µee) is also done in the same way and the result
is summarized in Table 6.3. As a conclusion, the fiTQun algorithm shows an improved
performance for the multi-ring event reconstruction over APFit.
The performance of multi-ring event reconstruction determines the separation ability ofneutrino and antineutrino components for multi-GeV multi-ring events directly, which

86

Chapter 6. Event Reconstruction

affects the sensitivity to the mass hierarchy significantly. The separation between theneutrino and antineutrino components for multi-ring events is done by a two-stage likelihood method. The detail of the method and its performance will be discussed in
Section 7.1.2. As a conclusion, fiTQun shows higher purity and efficiency than that of
APFit in multi-GeV multi-ring e-like events selection.
From these distributions and numbers, we can see that fiTQun has better performance
than that of APFit for both of single-ring and multi-ring events. Better sensitivities
to sin2 θ13 and mass hierarchy are expected due to higher reconstruction precision andlower background.

6.2.3

FiTQun in SK-I to SK-III

As mentioned in Chapter 2, there are four distinct detector phases in Super-K, and only
the data from the latest phase SK-IV is used in this thesis. The data collected from SK-I
to III phases is not used at the this moment due to the large data-MC discrepancies
observed in the reconstructed variable distributions from fiTQun. Further improvementsin the detector simulation is necessary to use those data. In fact, comparing with APFit,
fiTQun utilizes more information such as hit information outside the Cherenkov cone andtime information during fitting and does the fit with higher precision, so discrepancies
between data and MC that were unseen in previous, less-precise reconstruction before
might cause a significant effect for fiTQun.
As an example of such a data-MC discrepancy, Figure 6.14 shows the data-MC comparison of the distribution of PID likelihood the for the FC sub-GeV single-ring atmospheric
neutrino events for SK-I. Large discrepancies between data and MC can be found in the
e-like sample, namely, the single electron events in data are shifted to the left. Somework has been done to resolve this issue (see Appendix A), but the systematic error
evaluation is not finished yet, therefore the data taken in SK-I is not available for the
analysis in this thesis.
In general, SK-IV is the most stable phase among all four detector phases, due to various hardware improvements including the electronics and the water system. A moresophisticated calibration method described in Chapter 4 also helps the detector simulation tuned more precisely. The data-MC agreement is therefore overall better in SK-IV
compared to the other phases.
One of the potential improvements one can make in the detector simulation for SK-I
to III is the simulation on detector response with small p.e. for atmospheric neutrino
events. Due to the computation power limitation, the simulation for high energy events

Chapter 6. Event Reconstruction

87

SK4 Data
νe CC
νe CC
νµ CC
νµ CC
NC

450
400
350
300
250
200
150
100
50
0
-3000

-2000

-1000

0

1000

2000

3000

Figure 6.14: PID Likelihood distribution of SK-I for sub-GeV single-ring events.
Error bars represent the statistical uncertainty of the data. Events with negative(positive) likelihood values are designated µ-like (e-like). Color denotes the true flavorinteraction component.

(e.g. atmospheric neutrino events) are different from those for low energy events (e.g.
solar neutrino events) before SK-IV. Therefore, the detector response with small p.e.
for atmospheric neutrino events for SK-I to SK-III is not calibrated as well as SKIV. Although Super-K does have several important results with low energy events (e.g.
solar neutrinos, [115–117]) in early SK phase, the simulation and calibration method are
different for analysis using low energy sample and atmospheric neutrino. Unlike APFit,
fiTQun considers the unhit probability therefore the those hits with small charge cancause a large discrepancy on fiTQun while their effects on APFit are limited.
Another possible improvement is to introduce the effect of time variation to MC simulation. The time variation of water quality for SK-I to III is larger than that for SK-IV, as
shown in Figure 6.15. The reconstruction would be affected since the charge pattern of
the Cherenkov ring changes due to the variations on the water quality. Such a variation
is not implemented yet in the current atmospheric neutrino MC. The simulation is done
based on the calibration constants measured at a single point in time in each detector
phase. For instance, the water in SK-IV is simulated based on the water calibration datataken in April 2009. When the property of the water deviates in data significantly fromthe calibration point for simulation, the difference manifests as data-MC discrepancies
in the fiTQun likelihoods.
It is thus expected that the detector simulation in SK-I to III can be improved in
the future based on the available calibration data so that one can benefit from theadvanced reconstruction performance of fiTQun while analyzing the data from the otherSK phases.

Chapter 6. Event Reconstruction

Attenuation length [m]

88

130
120
110
100
90
80
70
60
2000

2005

2010

2015
Year

Figure 6.15: Time variation of the light attenuation length in water. Different color
denote different Super-K phases, SK-I, II, III and IV are shown in black, green, blue
and red, respectively.

Chapter 7

Neutrino Oscillation Analysis
7.1

Overview

In this chapter, a neutrino oscillation analysis using the SK atmospheric neutrino data is
described in detail. As mentioned in Chapter 1, in atmospheric neutrino data, oscillation
effects manifest in the energy and the zenith angle distributions of the observed neutrino
events. In this analysis, the atmospheric neutrino data is first selected into several
subsamples in order to separate the events in terms of neutrino type, and for each sample
the events are binned in the 2D distribution of the observed momentum and the zenith
angle of the direction. The oscillation parameters are then extracted by comparing thebinned data to the expectation calculated from MC simulation at different oscillation
parameters.
All the data samples including FC, PC and Up-µ in SK-IV are used for this analysis.
The livetime is 3118.45 days. Five hundreds years equivalent MC atmospheric neutrino
events are used. The FC events are reconstructed by fiTQun while PC and Up-µ samples
are reconstructed by APFit since the fiTQun tuning processing for those two samples
was not ready at the time of this thesis. However, since the selection of these samples
is distinct from the FC sample and those samples have sensitivity to the mass hierarchy
and the atmospheric mixing parameters, they are included here. For the oscillation
parameter estimation, a χ2 method is employed to make predictions on the oscillation
parameters based on the observed data. The following sections describe each of theanalysis elements in detail.

89

90

Chapter 7. Neutrino Oscillation Analysis

7.1.1

FC Event Selection and BinningThis section describes the event selection and the binning which are used as an input
for the oscillation analysis. FC data is categorized into subsamples based on the reconstructed particle information from fiTQun, and then binned in reconstructed momentum
and zenith angle. Since the oscillation effects manifest differently between the flavoursof atmospheric neutrinos as shown in Figure 1.5, it is important to implement an event
selection scheme which is capable of categorizing the observed neutrino events by theneutrino flavour (electron or muon). In addition, statistical separations of neutrinos
and antineutrinos are done in order to improve the sensitivity to the oscillation effects
such as the ones from the mass hierarchy, which happen differently for neutrinos andantineutrinos, as discussed in Section 1.5.
As introduced in Chapter 5, the FC events used in the analysis are required to have avisible energy greater than 30 MeV. The visible energy, Evis , is defined as the sum of
the reconstructed kinetic energy above the Cherenkov threshold for all rings, taking into
account the assumed particle type (electron, muon or π + ) of each ring. The fiducial
volume cut is also applied, which requires that the distance from the reconstructedvertex of the most energetic ring to the nearest point on the wall, also called Dwall tobe greater than the threshold. Both of the conventional FV cut used in the previous
Super-K atmospheric analysis (Dwall > 200 cm) and the expanded FV cut ( Dwall > 50cm) are studied. The FC events which pass these basic selections are further divided
into 13 event categories based on the reconstructed information from fiTQun as follows:

1. Sub-GeV: Evis < 1.33 GeV(a) single-ring e-like 0-decay electron sample
Single e-like ring with p > 100 MeV/c, ndcy = 0.
(b) single-ring e-like 1-decay electron sample
Single e-like ring with p > 100 MeV/c, ndcy ≥ 1.
(c) single-ring µ-like 0-decay electron sample
Single µ-like ring with p > 200 MeV/c, ndcy = 0.
(d) single-ring µ-like 1-decay electron sample
Single µ-like ring with p > 200 MeV/c, ndcy = 1.
(e) single-ring µ-like 2-decay electron sample
Single µ-like ring with p > 200 MeV/c, ndcy ≥ 2.
(f) FC sub-GeV 2-ring π 0 -like sample
Two e-like rings with invariant mass between 85 and 215 MeV/c2 , ndcy = 0.

Chapter 7. Neutrino Oscillation Analysis

91

2. Multi-GeV: Evis > 1.33 GeV(a) single-ring νe -like sample
Single e-like ring , ndcy ≥ 1.
(b) single-ring ν e -like sample
Single e-like ring , ndcy = 0.
(c) single-ring µ-like sample
Single µ-like ring.
(d) multi-ring νe -like sample
2 or more rings and the most energetic ring is e-like
Passed through the MME likelihood cut defined below and multi-ring νe -likecut(e) multi-ring ν e -like sample
2 or more rings and the most energetic ring is e-like
Passed through the MME likelihood cut and but failed the multi-ring νe -likecut(f) multi-ring µ-like sample
2 or more rings and the most energetic ring is µ-like
Evis > 600 MeV and pµ > 600 MeV/c(g) multi-ring e-like other2 or more rings and the most energetic ring is e-like
Failed the MME likelihood cut

Here, e-like and µ-like are the reconstructed PID information of the output charged
lepton generated by neutrino interaction as presented in Section 6.2.2, ndcy represents
the number of detected decay electrons.
Figure 7.1 shows the schematic diagram which summarizes the steps for selection of FC
samples.
The reason to do categorization based on the number of detected decay electrons is todistinguish the interaction mode. Comparing with the quasi-elastic interactions, more
decay electrons would be generated from the neutrino interaction with charged pion
production.
PC and Up-µ samples after the reduction process are reconstructed by APFit and divided
into “stopping” and “through-going” subsamples based on the energy deposition within
the OD. The “through-going” events in Up-µ sample are further divided into “showering”
and “non-showering”. The detail about the data reduction and event reconstruction for

92

Chapter 7. Neutrino Oscillation Analysis

Number of Rings
Evis

PID
e-like0 decay e
1 or more decay e

SubGeV elike_0dcy
SubGeV_elike_1dcy

0 decay e
1 decay e
2 or more decay e

SubGeV_mulike_0dcy
SubGeV_mulike_1dcy
SubGeV_mulike_2dcy

Single Ring

SubGeV

mu-like
Two Rings

Both rings are e-like and Minv in π0 range

SubGeV_pi0like

FC events
Number of Rings PID
e-likeSingle Ring

1 or more decay e
0 decay e

MultiGeV_mulike

mu-like
MultiGeVνe-νebar separation likelihood
MME likelihood
PID of M.E.R
passed νe-likee-likefailedMulti Ring

MultiGeV_elike_nue
MultiGeV_elike_nuebar

νe bar-likemu-like

MultiRing_elike_nue
MultiRing_elike_nuebar
MultiRingOther
MultiRing_mulike

Figure 7.1: Summary of event categories for FC samples. The texts in the dashedline boxes shows the result classified by the variables which is on the boxes.

PC and Up-µ events can be found in [111]. After all selections, there are a total of 13
FC analysis samples, 2 PC samples and 3 Up-µ samples.
The binning is defined by momentum and zenith angle as shown in Figure 7.2. All
samples in the sub-GeV region are divided into 5 momentum bins. Among them, e-likesample without decay electron and µ-like sample with 0 or 1 decay electron are divided
into 10 zenith angle bins. All multi-GeV samples, including FC, PC and Up-µ samples,are divided into 10 zenith angle bins. There are 515 bins in total in current analysis.
The expected contribution to the atmospheric neutrino oscillation sensitivity, including
mass hierarchy and CP violation, of each sample will be discussed in Section 7.2. Briefly
speaking, the largest contribution comes from the FC sample, while the PC and Up-µsamples also contribute.

7.1.2

Selection for Multi-GeV e-like Sample

The key to distinguish the normal from the inverted mass hierarchy is to determine
whether the excess of upward-going e-like events at multi-GeV region is from νe or ν̄e ,as has been stated in Section 1.2. It is important to separate neutrino and antineutrino
components.

Chapter 7. Neutrino Oscillation Analysis

93

Figure 7.2: Definition of the binning used in the neutrino oscillation analysis. FC:
405 bins, PC: 60 bins, Up-µ: 50 bins. The samples which is divided into 10 zenith
angle bins between -1 < cos θ< 1 for FC and PC, -1 < cos θ< 0 for UPMU are denotedby white boxes, while no zenith angle bins are divided for the samples represented byshaded boxes.

Multi-GeV Single-Ring e-like Sample
The multi-GeV single-ring e-like sample is divided into νe -like and ν e -like samples byconsidering the difference between the interactions of νe and ν e in water:
For CC νe interaction:
νe + N → e− + N 0 + π +ëµ+ + νµë e+ + νe + ν µ

An decay electron (actually a positron) is produced finally.
For CCν e :ν e + N → e+ + N 0 + π −

(7.1)

94

Chapter 7. Neutrino Oscillation Analysis

1000

SK-IV Data
Total MC
Charged Current ν eCharged Current ν e

800

600

400

200

0
0

1

2

3

4

5

6
7
8
9
10
Number of Decay Electron

Figure 7.3: Distribution of number of decay electrons in both data and MC for
SK-IV multi-GeV single-ring e-like sample. Cyan (magenta) shaded histogram
represent the distribution of CCνe (CCν̄e ) events, red solid line represents total
atmospheric neutrino MC and open circle indicates the distribution of data. Error
bars show the statistical uncertainty of data.

where π − is more easy to be absorbed by oxygen nuclei and no decay electron is observed.
Therefore multi-GeV single-ring e-like sample can be separated based the number ofdecay electrons as follows:
Number of decay electrons > 0 → νe -like,
Number of decay electrons = 0 → ν e -like.
The distribution of number of decay electrons for multi-GeV single-ring e-like sample
is shown in Figure 7.3. The fractions for different interaction modes after separating
multi-GeV single-ring νe -like sample and ν e -like sample are summarized in Table 7.1.
Interaction modeCC νeCC ν e
CCνx
NC
Total

νe -like
72.6%
7.7%
8.4%
11.3%
208.1

ν e -like
55.3%
37.9%
1.2%
5.6%
975.2

Table 7.1: Interaction mode composition of multi-GeV single-ring νe -like and ν e -likesamples. MC is normalized to data livetime 3118.5 days. The fiducial volume is
Dwall > 200 cm.

Chapter 7. Neutrino Oscillation Analysis

95

Multi-GeV Multi-Ring e-like Sample
As the fraction of multi-ring events increases at higher energy, the multi-ring event
samples are also included in the analysis in multi-GeV region. Due to the complexityof multi-ring events, it is more difficult to achieve high purity of the targeted event
types compared to the single-ring case generally. The multi-ring e-like sample, which is
selected by requiring the most energetic ring to be e-like, has rather high contaminationof CCνµ and NC events.
A two-stage method based on likelihood functions was developed to reject the νµ /µ̄µbackgrounds and purify the νe and ν̄e components for multi-ring e-like events. In thefirst stage of the separation (Multi-GeV Multi-ring separation, MME separation), CC
νe + ν¯e interactions is extracted with the method introduced in [118]. However, here theinput variables for the likelihood function are reconstructed by fiTQun.
The key to select the CCνe events from CCνµ and NC backgrounds is to confirm that themost energetic ring, which is also e-like ring, is from the electron generated by neutrinoCCνe interaction rather than the γ ring from π 0 decay in the hadronic system. Fourvariables are used in this likelihood: visible energy fraction of the most energetic ring,visible energy fraction of the most energetic charged pion like ring, number of decay
electrons and the distance between the event vertex and the farthest decay electrons.
The energy fraction of the most energetic ring tends to be small for the background
events since larger energy is carried by the hadrons for CCνµ or NC. The reason tochoose the visible energy fraction of the most energetic charged pion like ring is also
based on the similar reason. For the last two variables about the decay electron, it isdue to the fact that the hadronic system carrying large energy produces more pions,whose decays produce electrons, and the travel length of muon from CCνµ is longer
than the charged pion from CCνe .
Figure 7.4 shows the distributions of these variables. The likelihood function is definedas the logarithm value of the ratio between the possibility distribution function (p.d.f.)
for the target (CCνe and CCν̄e ), ΓSi (xi ), and the p.d.f. for the background (CCνµ andNC), ΓBi (xi ):

LMME =

X

log(ΓSi (xi )/ΓBi (xi )).

(7.2)

xi

Here, the four variables used in the likelihood are represented by xi . The likelihood
distribution of the first stage separation is shown in Figure 7.5. Events that fail this

96

Chapter 7. Neutrino Oscillation Analysis

SK-IV Data
Total MCMulti Ring e-likeCCν e+ν e

350
300

1400
1200

250

1000

200

800

150

600

100

400

50

200

0
0

0.1

0.2

0.3

0.4

0.5

0.6
0.7
0.8
0.9
1
Momentum fraction of M.E.R

0
0

0.1

0.2

0.3

0.4

0.5
0.6
0.7
0.8
0.9
1
Momentum fraction of π-like M.E.R(a) Visible energy fraction of most energetic ring (b) Visible energy fraction of charged pion-like(M.E.R)
M.E.R

900

120

800
100

700
600

80

500
60

400
300

40

200
20
100
0
0

1

2

3

4

5

6
7
8
9
10
Number of decay electrons(c) Number of decay electrons

0
0

0.05

0.1

0.15

0.2
0.25
0.3
Distance to farest decay e(d) Distance to the farthest decay electron divided by total visible energy

Figure 7.4: Distributions of the variables used in the multi-GeV multi-ring e-like(MME) likelihood for SK-IV. Shaded histogram represents CCνe + ν e , red solid line
represents total atmospheric neutrino MC and open circle indicates the distribution ofdata. Error bars represent the statistical uncertainty of the data.

selection are classified as “multi-ring other” and the remains are termed as “multi-ringe-like”.
The second stage of the separation process focuses on separating neutrino and antineutrino interactions from the multi-ring e-like data. Unlike the multi-GeV single-ring e-likesample, which is separated with a simple cut on the number of decay electrons, a likelihood method presented in [6] is used, while the variables reconstructed by fiTQun areused.
It is known that νe scattering tends to be less forward peaked and have larger energy
transferred to the hadronic system compared to ν̄e since CCνe interaction has a wider
inelasticity y := (E 0 − E)/E distribution due to the nature of the weak interaction. HereE and E 0 represent the initial energy of neutrino and the energy of outgoing electron,
respectively. This implies, comparing with CCν̄e , :• Transverse momentum tends to be larger for CCνe .

Number of Events

Chapter 7. Neutrino Oscillation Analysis

97

SK-IV Data
Total MCCCνe+νe

700
600

other

e-like

500
400
300
200
100
0

-10

-8

-6

-4
-2
0
2
4
6
8
Multi-Ring e-like - other Separation Parameter

Figure 7.5: Likelihood distribution for the first stage of SK-IV multi-GeV multi-ringe-like events separation. The statistical error are denoted by error bars. The cut isdetermined at -0.25 according to the result of MC study.

• Momentum fraction for the most energetic ring tends to be smaller for CCνe .
• More rings are expected to be observed for CCνe .
• More decay electrons are expected to be observed for CCνe .
A likelihood with parameters including the number of decay electrons, number of rings
and transverse momentum of the most energetic ring are used to separate νe -like and ν e like samples. The distributions of those variables are shown in Figure 7.6. The likelihood
function is similar to Equation 7.2, which is made by just replacing the variable used
and the definition of signal and background to CCνe and CCν̄e . Table 7.2 shows theefficiency and purity for selecting νe /ν̄e events in the multi-ring e-like samples, andidentifying true CC νe (ν̄e ) events.

7.2

Oscillation Effect

As discussed in Section 1.5, the electron neutrino oscillation is observable by subdominant oscillation. For the electron neutrino, the oscillation effect due to θ13 isdominant from a few GeV to 10 GeV, as shown in Figure 1.5.
The amplitude of neutrino oscillation is resonantly enhanced when ACC /∆m232 = cos 2θ13 .
Considering that cos 2θ13 is positive, the enhancement only happens when the sign of
ACC and ∆m232 are the same: neutrinos for the normal hierarchy (both of ACC and∆m232 larger than 0) or antineutrinos for the inverted hierarchy (both of ACC and ∆m232

98

Chapter 7. Neutrino Oscillation Analysis

900
SK-IV Data
Total MC
Charged Current ν eCharged Current ν e

800
700

500

400

600
500

300

400
200

300
200

100
100
0
0

1

2

3

4

5

0
0

6
7
8
9
10
Number of decay electrons(a) Number of decay electron

1

2

3

4

5

6
7
Number of rings(b) Number of rings

140
120
100
80
60
40
20
0
0

100

200

300

400

500

600 700 800 900 1000
Transverse momentum [MeV](c) Transverse momentum (MeV)

Number of Events

Figure 7.6: Distributions of the variables used in the MME νe and ν e separation
likelihood for SK-IV passed the first selection. Cyan (magenta) shaded histogram
represent the distribution of CCνe (CCν̄e ) events, red solid line represents total
atmospheric neutrino MC and open circle indicates the distribution of data. Error
bars show the statistical uncertainty of data.

SK-IV Data
Total MC
Charged Current ν eCharged Current ν e

400
350
300
250
200
150
100
50
0
-1.5

-1

-0.5

0

0.5
1
1.5
2
2.5
3
Multi-Ring ν e - ν e Separation Parameter

Figure 7.7: Likelihood distribution for the second stage of SK-IV multi-GeV
multi-ring e-like events separation. The statistical error are denoted by error bars.
Events with likelihood values larger (less) than 0 are selected as ν̄e -like (νe -like).

Chapter 7. Neutrino Oscillation Analysis

First stage
Multi-ring e-like events
Efficiency
PuritySecond stage
Multi-ring True CCνe events
Efficiency
PurityMulti-ring True CCν̄e events
Efficiency
Purity99
fiTQun

APFit

75.7%
77.8%

69.7%
69.5%

56.8%
58.8%

53.6%
52.6%

68.4%
30.0%

70.9%
25.9%

Table 7.2: Performance of multi-ring e-like events selection. The definition ofefficiency is the fraction of the selected events among all true νe or ν̄e events. The
fraction of the true νe or ν̄e events over all selected events is defined as the purity.

are less than 0). Therefore, the mass hierarchy can be determined by observing theincrease in number of upward events in νe -like and ν̄e -like samples. Figure 7.8 shows thesensitivity to the mass hierarchy of each sample. The largest contribution comes fromthe multi-GeV sample, including multi-GeV FC sample, PC sample and Up-µ sample.
Figure 7.9 shows the zenith angle distribution of ratio of number of events with oscillation at (∆m212 , ∆m232 , sin2 θ12 , sin2 θ13 , δCP ) = (7.49 × 10−5 eV2 , 2.5× 10−3 eV2 , 0.309,
0.021, 200◦ ), to that without oscillation. The enhancement in the upward direction(cos θ < 0) for νe sample can be seen while a deficit is seen in the νµ sample, whose size
depends on the value of sin2 θ23 . A sizable difference can be seen between the normal
hierarchy, which is denoted by dashed line, and the inverted hierarchy, which is denotedby solid line. The reason of larger electron neutrino appearance observed in upward
direction in the normal hierarchy than that in the inverted hierarchy is due to moreneutrinos from the primary flux and larger absolute cross section of neutrinos than thatof antineutrinos as discussed in Chapter 3.
The CP phase also has effect on the oscillation possibility via the interference between
θ13 -dependent oscillation and the oscillation discussed in Equation 1.25 [68]. Figure 7.10
shows the expected sensitivity to different δCP of each sample. The largest contribution
comes from the sub-GeV FC sample. The ratio between number of events with andthat without oscillations as a function of zenith angle at three different CP phase values
is shown in Figure 7.11. Besides the excess in upward direction for multi-GeV e-likesamples, the interference gives an overall ∼ 2% effect on sub-GeV e-like samples.

100

Chapter 7. Neutrino Oscillation Analysis

Showering µsin2 θ23=0.4
sin2 θ23=0.5
sin2 θ23=0.6

Non-showering µ
Up Stop µ
PC Thru
PC Stop
Multi-Ring Unclassified
Multi-Ring µ-like
Multi-Ring e-like ν eMulti-Ring e-like ν eMulti-GeV µ-like
Multi-GeV e-like ν eMulti-GeV e-like ν e
Sub-GeV π 0 -like M-R
Sub-GeV µ-like 2-dcy e
Sub-GeV µ-like 1-dcy e
Sub-GeV µ-like 0-dcy e
Sub-GeV e-like 1-dcy e
Sub-GeV e-like 0-dcy e

00.2

0.4 0.6 0.81
1.2 1.4
Sensitivity to reject wrong MH ∆ χ2

Figure 7.8: Sensitivity to reject wrong mass hierarchy (MH) of each sample foratmospheric neutrino. The vertical axis shows the ∆χ2 value between the true
oscillation point at normal hierarchy and the best fit point at inverted hierarchy. The
true oscillation paramters are (∆m212 , ∆m232 , sin2 θ12 , sin2 θ13 , δCP ) = (7.49 × 10−5
eV2 , 2.5× 10−3 eV2 , 0.309, 0.021, 200◦ ). True MH is assumed to be normal hierarchy.
Oscillations at different values of sin2 θ23 = 0.4 (red), 0.5 (blue), 0.6 (magenta) are
shown.

7.3

Fiducial Volume Expansion

The fiducial volume (FV) in previous analyses with APFit was defined as the region
where the distance to the nearest wall in the inner detector is larger than 200 cm(Dwall > 200 cm), which results that approximately 30% loss of the Super-K innerdetector mass. Nevertheless, statistics of the observation still constrains the physicalsensitivity. Therefore, by making use of the improved reconstruction performance offiTQun, the fiducial volume is able to be expanded while keeping a high signal purity
and low background to increase the sensitivity.

7.3.1

MC Study for Fiducial Volume Expansion

A larger fiducial volume doesn’t only provide more statistics, but also risk for lower
purity of signal interactions in neutrino sample. The reconstruction performance might
deteriorate for events close to the ID wall since some particles that travel toward theChapter 7. Neutrino Oscillation Analysis

101

Multi-GeV e-like νe
N/NNo osc.

N/NNo osc.

Multi-GeV e-like νe
1.3
1.2

1.2
1.15
1.1

1.1

1.05

1

1
0.95

0.9
0.9
0.8
0.7
-1

0.85
0.8
-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

cos (θ)1cos (θ)Multi-Ring e-like νeMulti-Ring e-like νe
N/NNo osc.

N/NNo osc.

0.8

1.3
1.2

1.2

1.1

1.1
1

1

0.9

0.9

0.8
0.8
0.7
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

cos (θ)Multi-Ring µ-like
N/NNo osc.

N/NNo osc.

1.4

1.4

1.2

1.2

1

1

0.8

0.8

0.6

0.6

-0.8

-0.6

-0.4

-0.2

0

0.2

1

cos (θ)Multi-GeV µ-like

-1

0.8

0.4

0.6

0.8

1

cos (θ)

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

cos (θ)Figure 7.9: Oscillation effect on zenith angle distribution for multi-GeV samples.
The vertical axis shows the ratio between the number of events with oscillations andthe ones without oscillations. Oscillation parameters are (∆m212 , ∆m232 , sin2 θ12 ,sin2 θ13 , δCP ) = (7.49 × 10−5 eV2 , 2.5× 10−3 eV2 , 0.309, 0.021, 200◦ ). Oscillations at
different values of sin2 θ23 = 0.4 (red), 0.5 (blue), 0.6 (green) for normal hierarchy are
shown.

102

Chapter 7. Neutrino Oscillation Analysis

Showering µδCP=0°
δCP=80°δCP=300°

Non-showering µ
Up Stop µ
PC Thru
PC Stop
Multi-Ring Unclassified
Multi-Ring µ-like
Multi-Ring e-like ν eMulti-Ring e-like ν eMulti-GeV µ-like
Multi-GeV e-like ν eMulti-GeV e-like ν e
Sub-GeV π 0 -like M-R
Sub-GeV µ-like 2-dcy e
Sub-GeV µ-like 1-dcy e
Sub-GeV µ-like 0-dcy e
Sub-GeV e-like 1-dcy e
Sub-GeV e-like 0-dcy e

00.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8
Sensitivity to reject wrong δCP ∆ χ2

Figure 7.10: Sensitivity to δCP of each sample for atmospheric neutrino. The y-axis
shows the ∆χ2 value between the true oscillation point and the best fit point. The
true oscillation parameters are (∆m212 , ∆m232 , sin2 θ12 , sin2 θ13 , sin2 θ23 , δCP ) = (7.49
× 10−5 eV2 , 2.5× 10−3 eV2 , 0.309, 0.021,0.425, 200◦ ). True MH is assumed to benormal hierarchy. Oscillations at different values of δCP = 0◦ (red), 80◦ (blue), 300◦
(magenta) are shown.

wall in those events will only hits a few PMTs and results in poorly imaged Cherenkov
rings. Furthermore, the possibility of that events with interaction vertices outside of the
ID wall being reconstructed within the FV increases as the FV cut is moved closer tothe wall, due to the vertex resolution. Although most of those events are rejected by the
OD cut, some events may generate too few photons to be rejected and contaminate theanalysis sample if they originate in the 55 cm insensitive region between the wall of ID
and OD or very near the OD wall. Due to the low precision of the detector modeling in
those region, large systematic errors might be introduced by those “entering events”.
The fraction of signal and backgrounds of the FC sub-GeV µ-like 0 decay-e eventschanges as a function of the distance from the reconstructed vertex to the nearest ID wall
(Dwall ) as shown in Figure 7.12. Until 50 cm, the purity of signal events, including νµ CC
and ν̄µ CC interactions (red) for this sample, keeps stable. The component fractions forother samples can be found in Appendix B. Similar behavior is observed in all of 13
FC event categories, which shows that the fiducial volume cannot be expanded to theregion with Dwall less than 50 cm due to the entering ν events. Table 7.3 shows theChapter 7. Neutrino Oscillation Analysis

103

Multi-GeV e-like νe
N/NNo osc.

N/NNo osc.

Sub-GeV e-like 0-dcy e
1.06

1.2

1.04
1.1

1.02
1

1

0.98

0.9

0.96
0.8
0.94
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

cos (θ)1cos (θ)Multi-Ring e-like νe
N/NNo osc.

Multi-GeV e-like νe
1.15

1.2

1.1
1.1

1.05
1

1

0.95

0.9

0.9
0.8
0.85
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1

-0.8

-0.6

-0.4

cos (θ)

-0.2

0

0.2

0.4

0.6

0.8

1

cos (θ)Multi-Ring e-like νe
N/NNo osc.

N/NNo osc.

0.8

1.2
1.15
1.1
1.05
1
0.95
0.9
0.85
0.8
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

cos (θ)Figure 7.11: Zenith angle distributions of the ratio of number of events withoscillation to that without oscillation, assuming oscillation parameters (∆m212 , ∆m232 ,sin2 θ12 , sin2 θ23 ) = (7.49 × 10−5 , 2.5 × 10−3 , 0.309, 0.5) in normal hierarchy case for
SK-IV. Cases with δCP = 80◦ (red), 220◦ (blue) and 300◦ (green) are shown. The
error bars represents statistical uncertainty.

Chapter 7. Neutrino Oscillation Analysis

Fraction [%]

104

100
90

Expanded FVConventional FV

80
70
60

νµ + νµ CC

50

νx CC

40

NC

30

Entering ν Events

20
10
0
0

50

100

150

200

250

300

350

400

Distance to nearest wall [cm]Figure 7.12: The composition of the FC sub-GeV µ-like sample without decay
electron. Horizontal axis shows the distance between the reconstructed vertex and thenearest ID wall.

purity of the 13 FC samples within the new region (50 cm < Dwall < 200 cm) and theconventional FV definition (Dwall > 200 cm).
To investigate the effect of the FV expansion, including both of the purity decrease andstatistics increase, a study based on MC sample was performed. The sensitivity to reject
the inverted mass hierarchy assuming the true mass hierarchy is normal with different
FV cut values is shown in Figure 7.13. Systematic errors used in this figure are assumed
to be unchanged from the values obtained in the conventional FV region. From thisfigure it can be expected for a FV cut at 50 cm to bring an improved mass hierarchy
sensitivity if the systematic errors keep stable even in the new region (50 cm< Dwall <200 cm). The effect of the FV expansion on the systematic errors and a more precise
sensitivity study is presented below.
In fact, besides the Dwall value, a cut on “towall” value, which means the distance fromthe vertex to the wall along the direction of the particle as shown in Fig 7.14, is also
considered. Events that are close to the ID wall, but point away from the wall (small
Dwall but large towall), generally hit enough PMTs to have a well-imaged Cherenkov
ring. On the other hand, events that are close to the ID wall but point towards thewall (small Dwall and small towall), generally hit too few PMTs, and as a result havepoorly imaged Cherenkov rings. Figure 7.15 shows the purity of each component for
sub-GeV single ring e-like 1 decay electron as function of the towall value. The fractionof νe CC events is lower than 20% since too many background events from νµ CC arecontaminated.

Chapter 7. Neutrino Oscillation Analysis

SampleCC νeCC ν̄eFully Contained (FC) Sub-GeV
e-like, Single-ring
0 decay-e
0.728
0.242
(0.702) (0.227)
1 decay-e
0.907
0.020
(0.712) (0.015)µ-like, Single-ring
0 decay-e
0.010
0.004
(0.034) (0.011)
1 decay-e
0.000
0.000
(0.001) (0.000)
2 decay-e
0.000
0.000
(0.000) (0.000)
π 0 -likeTwo-ring
0.051
0.016
(0.109) (0.036)
Fully Contained (FC) Multi-GeV
Single-ringνe -like
0.726
0.077
(0.748) (0.066)
ν̄e -like
0.553
0.379
(0.566) (0.371)µ-like0.000
0.000(0.000) (0.000)
Multi-ringνe -like
0.588
0.117
(0.609) (0.112)
ν̄e -like
0.526
0.300
(0.541) (0.301)µ-like0.010
0.001(0.016) (0.002)
Other
0.283
0.026
(0.302) (0.032)105

CC νµ + ν̄µ

CC ντ

NC

0.001
(0.025)0.033(0.208)0.000(0.001)0.001(0.001)0.028(0.045)0.040(0.063)0.795(0.805)0.974(0.968)0.984(0.980)0.001(0.001)0.000(0.000)0.000(0.000)0.191(0.150)0.026(0.031)0.016(0.019)0.011(0.018)0.000(0.000)0.922(0.837)0.058(0.064)0.003(0.003)0.996(0.995)0.027(0.016)0.008(0.007)0.003(0.004)0.113(0.105)0.056(0.053)0.001(0.001)0.054(0.059)0.021(0.023)0.959(0.946)0.342(0.342)0.036(0.032)0.020(0.016)0.004(0.005)0.053(0.051)0.204(0.188)0.134(0.118)0.026(0.031)
0.295(0.274)

Table 7.3: Purity of different neutrino flavor of each samples with a FV cut of
200 cm. The neutrino oscillations with ∆m232 = 2.52 × 10−3 eV2 and sin2 θ23 = 0.51
has been considered. The result within new region (50 cm ¡ Dwall ¡ 200 cm) is shownby the number in the parentheses.

Chapter 7. Neutrino Oscillation Analysis

Incorrect Hierarchy Rejection ∆ χ2

106

1.2
1
0.8
0.6
Conventional FV

0.4
0.2
0
50

100

150

200

250
300
FV Cut [cm]Figure 7.13: Sensitivity for incorrect mass hierarchy rejection. The exposure time is3118.5 days. Horizontal axis shows different FV cut position. The true hierarchy isassumed to be normal hierarchy. The true oscillation parameters are taken as:
sin2 θ13 = 0.0210, sin2 θ23 = 0.5, ∆m223 = 2.4 × 10−3 eV2 , and δCP = 0. The systematic
error for the analysis with a FV cut at 200 cm is used for all sensitivity calculation
with different FV cut.

Figure 7.14: Defination of “Dwall ” and “Towall”.

Fraction [%]Chapter 7. Neutrino Oscillation Analysis

107

100
90
80
70

νe CC νe CC
νµ CC νµ CC
ντ + ντ NC

60
50
40
30
20
10
0
0

50

100

150

200

250

300

350

400

Towall [cm]Figure 7.15: Purity of each component for sub-GeV single ring e-like 1 decay
electron as function of towall value. A Dwall cut at 50 cm has been applied.

To investigate the necessity of a cut on towall, the MC sensitivities are also studied foreach towall cut value. Figure 7.16 shows the sensitivity as the function of both Dwall
cut and towall cut. Left figure shows the contribution from the sub-GeV single-ringe-like 1 decay sample. Obviously, the best point for this sample is not near (50, 50) due
to too many νµ CC background events. However, the contribution from this sample issmall. On the other hand, those samples with higher sensitivity such as multi-GeV e-likeevents, the statistics of events with small towall is low, as shown in Figure 7.17. A highenergy event with small towall will easily penetrate the ID wall, enter the OD and thenbe categorized as a PC event. Therefore, the total sensitivity from all samples indicatesthat the towall cut is not critical to change the sensitivity. The cut on the towall value
is not used for simplicity.

7.3.2

OD Hits

OD hits in the atmospheric neutrino analysis are used to reject cosmic ray muon backgrounds and separate PC from FC events. The selection efficiency of FC events might beaffected if we extend the FV to Dwall > 50 cm due to the poor modeling of the responseof the OD. This can thereby potentially introduce biases and relative inefficiencies in
the FC sample. The distributions of the number of OD hits in both region have beenstudied (Figure 7.18). Data and MC are in good agreements near the FC cut, which
requires that the number of OD hits in cluster with the largest charge (nhitac) less than

0.0074250

0.0072
0.007

200

∆χ2

300

Dwall cut [cm]∆χ2

Chapter 7. Neutrino Oscillation Analysis

Dwall cut [cm]

108

300
1.15
250
1.1
200

1.05

0.0068150

1500.00660.0064

100

1
0.95

100

0.0062
50
50

100

150

0.9
50
50

200
250
300
Towall cut [cm]

100

150

200
250
300
Towall cut [cm]Single-ring e-likeSingle-ring ν e-like

Fraction [%]Fraction [%]Figure 7.16: Sensitivity to reject wrong mass hierarchy. True mass hierarchy is NH.
The left figure shows the contribution from sub-GeV single-ring e-like 1 decay sample,
while the right figure shows the total sensitivity from all samples. The open circle at(200,200) in right figure represents the sensitivity at the FV cut used in previous
APFit analysis.

6
5
4
3

14
12
10
8
6

2
4
1
0
0

2
50

100

150

200

250

300

350

0
0

400

50

100

150

200

250

Multi-ring e-like

Multi-ring ν e-like

Fraction [%]

16
14
12
10

350

400

Towall [cm]Fraction [%]Towall [cm]

300

10

8

6

ν e CCν e CCν µ CCν µ CCντ + ντNC

SK4 Data

8
4

6
4

2

2
0
0

50

100

150

200

250

300

350

400

Towall [cm]

0
0

50

100

150

200

250

300

350

400

Towall [cm]Figure 7.17: Towall distribution for multi-Gev e-like events. The statistics withsmall towall value is low.

Chapter 7. Neutrino Oscillation Analysis

109

Number of events

Number of events

104

104

103

103

102

102

10

10

1

1
10-1

10-1

10-2

10-2

0

5

10

15

20

25
30
35
40
Number of OD hits in cluster

0

5

10

15

20

25
30
35
40
Number of OD hits in cluster

Figure 7.18: Distribution of the number of OD hits in cluster. Left figure shows the
distribution for data with in conventional FV (Dwall > 200 cm). Right figure shows
the result in new region (50 cm < Dwall < 200 cm). The peak around 20 in rightfigure is due to cosmic ray muon events, which is confirmed by eye-scanning. The
threshold line to select FC events is denoted by the red dashed line.

Figure 7.19: Number of OD bad channels.

16. After removing the non-neutrino backgrounds confirmed by eye-scanning, the number of FC events with nhitac larger than 16 for data is 8, which is in consistent with thenumber from MC, 11.5.
Another thing that needs to be noted is that the FC event rate in the region near wallmight depend on OD hits or status of OD PMTs highly, since more entering background
events might be contaminated due to inactive channels.
As shown in Figure 7.19, the number of bad OD channels increased significantly since
SK-IV start. Fortunately, no large change is observed in the event rate of atmospheric
neutrino samples as in Table 7.4 and Table 7.5, which show the time variation of theatmospheric neutrino event rate within different detector regions. Therefore, the effectof OD bad channel on rejection efficiency is limited.

7.3.3

Data/MC Comparison and Background Estimation

In addition to the entering neutrino and event mis-categorization, there may be otherbackgrounds from non-neutrino (non-ν) sources such as cosmic ray muons and flashing
PMTs due to the FV expansion.

110

Chapter 7. Neutrino Oscillation Analysis
Livetime [days]
Dwall > 200 cm
50 cm < Dwall < 200 cm
Dwall < 50 cm

0∼10008.06±0.09
2.21±0.05
1.22±0.03

1000∼2000
8.07±0.09
2.17±0.05
1.16±0.03

2000∼3118.5
8.17±0.10
2.25±0.05
1.22±0.04

MC
8.29
2.15
0.96

Table 7.4: Time variation of atmospheric neutrino event rate in different detector
region. number of OD hits in cluster with the largest charge is required to be lessthan 16, and visible energy is required to be larger than 30 MeV. For the last row,
Dwall < 50 cm , there are many cosmic ray muon background events contaminatedtherefore the event rate is much higher than the expected value of MC.

Livetime [days]
1810 cm > Z > 900 cm
900 cm > Z > 0 cm
0 cm > Z > -900 cm-900 cm > Z > -1810 cm

0∼10002.95±0.05
2.93±0.05
2.86±0.05
2.75±0.05

1000∼2000
2.85±0.05
2.93±0.05
2.88±0.05
2.73±0.05

2000∼3118.5
3.03±0.06
2.92±0.06
2.85±0.06
2.83±0.06

MC
2.83
2.88
2.88
2.76

Table 7.5: Time variation of atmospheric neutrino event rate in different detector
region. number of OD hits with the largest charge in cluster is required to be lessthan 16, and visible energy is required to be larger than 30 MeV.

Before being observed in the ID, cosmic ray muons must pass through the OD, therefore
most of them are rejected by cuts on OD activity. The true vertex of cosmic ray muons
events observed in ID is the entrance point on the wall, meaning that the FV cut can help
to reject any events remaining after OD cuts are applied. However, due to the vertex
resolution of the reconstruction algorithm, which is around 15 cm ∼ 20 cm, cosmic ray
background events might contaminate the FC sample when the FV cut is near the wall.
Another kind of non-neutrino background is flasher events, which are caused by theinternal electrical discharge of a PMT. The reconstructed vertex for flasher event is also
expected to be on the wall, more specifically, the position of discharged PMT. Due to thesame reason as the cosmic ray muon case, more flasher backgrounds might be introduced
into the FC sample when the FV cut is near the wall.
The distribution of the distance from the reconstructed vertex to the nearest wall of FC
events is shown in Figure 7.20. Several distributions, including the events’ momentum,
direction, particle ID, ring counting likelihood of the FC sample have also been checked
for both the conventional FV region (Dwall > 200 cm) and new region (50 cm < Dwall <200 cm), respectively. The observations are in good agreement with the expectation.
In fact, all FC events within the fiducial volume (Dwall >50 cm in this analysis) werechecked by eye with a graphical event display program to evaluate the background rate
more precisely. The eye-scanning result shows that one flasher event and 24 cosmic ray
muon events were contaminated in the FC sample within the expanded FV (Dwall >Number of Events

Chapter 7. Neutrino Oscillation Analysis

111

450
400
350
300
250
200
150
100

Expanded FVConventional FV

MC / Data

50
0

50

100

150

50

100

150

1.4

200

250

300

350

400
Dwall[cm]

1.2
1
0.8
0.6
0

200 250 300 350 400
Distance to nearest wall [cm]Figure 7.20: Number of events as the function of the distance between thereconstructed event vertex and the nearest ID wall for FC samples. Points represent
the result of data and solid line is for MC. The ratio between MC and data is shownin bottom figure.

50 cm). Most of the cosmic ray muon backgrounds were categorized as multi-ring µ-likeevents with downward direction and momentum which is larger than 5 GeV/c. Due tothe presence of the fake ring found by fiTQun, their vertices were reconstructed within
the expanded fiducial volume. Independent cosmic ray muon samples also show consistent result for both the rate and type of such mis-reconstructions, which enable us
to remove the expected cosmic ray background in the final analysis sample statistically
based on the Dwall distribution of the cosmic ray muon events. By comparison of theevent classification of this control sample with those observed in the eye-scanned background, it is shown that the FC event reduction process does not depend strongly on
the muon’s direction and momentum. Figure 7.21 and Figure 7.22 show the distributionof Dwall , the momentum and zenith angle distribution respectively. The shape of thecosmic ray muon sample (solid line) and the eye-scanned background events (points) in
the FC sample are in good agreement.
As has been stated above, only one flasher event is observed within the expanded FV(Dwall > 50 cm) by eye-scanning. This contamination is expected to have a limited
impact on the final FC sample. Therefore, the effect of the flasher event is considered
by a systematic error on the event rate in the oscillation analysis presented below.

112

Chapter 7. Neutrino Oscillation Analysis

MultiGeV Multi Ring µ
Number of Background Events

Number of Background Events

MultiGeV Single Ring µ

4.5
4
3.5
3
2.5
2
1.5
1

10

8

6

4

2

0.5
0
0

50

100

150

200

250

300

350

0
0

400

50

Distance to nearest wall [cm]

100

150

200

250

300

350

400

Distance to nearest wall [cm]Figure 7.21: The distribution of the distance from the reconstructed vertex to thenearest ID wall of µ-like samples. Left figure shows the distribution of single-ring
events while right figure shows the one of multi-ring events. Both of the independent
cosmic ray muon sample (solid line) and eye-scanned background events in FC sample(points) are shown. Histograms are normalized by the total number of backgrounds
events seen in the FC sample.

Zenith Angle

MultiGeV 1R µ

2-D
0.8

Zenith Angle

2.5
1

0.6
2

0.4
0.2

10 -1

1

1.5

0
-0.2

10 -2

1

-0.4
-0.6

0.5
10

-0.8

Zenith Angle-1
3

MultiGeV MR µLog(Momentum)

1

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6
4.8
5
Log(Mom) [MeV]

0

3

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6
4.8
5
Log(mom) [MeV]

-3

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6
0.8
1
Zenith Angle

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6
0.8
1
Zenith Angle

1
0.8

8

0.6

7

0.4

6

0.2

10

10

1

5

1

0
4

10 -1

-0.2
3
-0.4
2

-0.6

10 -2

1

-0.8
-1
2

10

-1

2.5

3

3.5

4

4.5
5
Log(Mom) [MeV]

0

2

2.5

3

3.5

4

4.5
5
Log(mom) [MeV]-1

Figure 7.22: The distribution of the momentum and zenith angle of single-ring(top) and multi-ring (bottom) µ-like samples. The left two figures show the 2-d
distribution, where points represent the background events identified by eye scanningin FC and color box indicate the distribution of cosmic ray muon sample. In rightfour figures, solid line represents the result of cosmic ray muon sample and the crossshows the distribution of the eye-scanned background events in FC. Histograms arenormalized by the total number of events.

1R e-like, > 200cm

1R µ -like, > 200cm
Event

113

Event

Chapter 7. Neutrino Oscillation Analysis

1000

1400
1200
1000

800

800
600
600
400
400
200
0
10-2

200

10-1

1

10

102

0
10-2

3

10

10-1

1

10

3

10

1R e-like, 50~200cm

1R µ -like, 50~200cm
Event

Mom [GeV]

Event

Mom [GeV]

102

350

400
350

300

300

250

250
200
200
150
150
100

100

50
0
10-2

50
10-1

1

10

102

3

10

0
10-2

10-1

Mom [GeV]

1

10

102

3

10

Mom [GeV]

Figure 7.23: Momentum distribution of single ring event in different detector
region. Samples are divided based on the reconstructed particle ID. Upper two plotsshow the events within conventional FV, Dwall > 200 cm, while the lower two plotsshow the events within new region, 50 cm < Dwall < 200 cm. Good agreement
between data and MC can be observed.

Another possible issue is momentum reconstruction. The influence of reflection andscattering would be stronger when the particle vertex is near to the wall. The decreaseof number of hit PMTs might enhance this influence further. Furthermore, part of the
track might penetrate the ID wall and enter the 55 cm insensitive region between ID
and OD, which might cause bias in momentum reconstruction. Figure 7.23 shows thatthe momentum distribution of single-ring events of MC is in good agreement with data,
which means that those effects have been simulated in the MC and will not result in
additional systematic uncertainty. The other detailed basic distribution of each event
category in both FV regions can be found in Appendix B.
Figure 7.24 shows the distributions of reconstructed direction for each analysis sample

114

Chapter 7. Neutrino Oscillation Analysis

within the expanded FV. As indicated in Figure 7.25, the event rate of FC sample isstable at 8.3(2.2) per day for the conventional FV (new region) since the start of SK-IV,
with 0.73 events per day for PC sample and 1.49 events per day for Up-µ sample.

Sub-GeV e-like 1-dcy e

Sub-GeV µ -like 0-dcy e

Sub-GeV e-like 0-dcy e

Sub-GeV µ -like 1-dcy e

1000

1000

200

500

100

Up Stop µ

200
200
500

100

0

0

0

0

0

1.2
1
0.8

1.2
1
0.8

1.2
1
0.8

1.2
1
0.8

1.2
1
0.8

Multi-GeV e-like ν e

Sub-GeV µ -like 2-dcy e
200

200

Multi-GeV e-like ν e

Non-showering µ

PC Stop

50
50
500
100

100

0

0

2

1.2
1
0.8

1
0
200

Sub-GeV π 0 -like M-R

0

0

0

1.2
1
0.8

1.2
1
0.8

1.2
1
0.8

Multi-Ring e-like ν eMulti-Ring e-like ν e

400Showering µ

PC Thru
100

100

500

100

200
50

0

1.2
1
0.8

500

500

1000

1000

0

0

0

0

1.2
1
0.8

1.2
1
0.8

1.2
1
0.8

1.2
1
0.8

200

Multi-GeV µ -like

Multi-Ring µ -like

100

0
-1

0

1

1.2
1
0.8
-1

lepton momentum (MeV)

Multi-Ring Unclassified

400

200

200

100

0
-1

0

1

1.2
1
0.8
0

cos zenith

1

-1

0
-1

0

1

0

1

1.2
1
0.8
0

1

-1

cos zenith

cos zenith

cos zenith

Figure 7.24: Comparisons between SK-IV data and MC within expanded FV(Dwall > 50 cm). The sample has been divided into 18 categories. The reconstructed
momentum distributions for the samples with only one zenith angle bin are shown in
the first column. The other samples are shown as zenith angle distributions fromsecond to fifth column. The best-fit MC assuming the normal hierarchy is denoted bysolid lines. The ratio between MC to data is shown in the narrow panels below each
distribution. The error bars in each panels represent the statistical error.

7.4

Analysis Method

Atmospheric neutrino data are fit to the MC prediction under different oscillation parameter sets with a binned χ2 method assuming Poisson statistics to find the most
preferred one. Different error sources are assumed to be uncorrelated and scaling factors
on the MC in each bin are assigned for each error to reflect their effects [119]. The χ2
is defined as: X  2
X
On
i
χ =2
En − On + On ln
+
,
E
σ
n
i
n
2

i

(7.3)

Chapter 7. Neutrino Oscillation Analysis

115

Rate [ events / day ]101

FC (Conventional FV, dwall > 200 cm)
FC (New region, 200 cm > dwall > 50 cm)
PC
Up- µ

10-1
0

500

1000

1500
2000
2500
Days since SK4 started

3000

3500

Figure 7.25: Final event rates variation with respect to running time since SK-IV
launch. The error bars represent the statistical error. The fully contained event rate isdenoted by circle and the partially contained (upward-going muon) event rates aredenoted by upward-facing (downward-facing) triangles. The fiducial volume cut value
for FC events are shown in the parentheses.

where
En = En,0 (1 +X

fni i ).

(7.4)

i

In this equation the observed number of events in the n-th analysis bin is represented by
On , the nominal MC expectation and the one after considering systematic errors in thatbin are denoted by En,0 and En , respectively. The fractional change of the event rate
under a 1σ variation of the systematic error source is described as fni , where index n
and i represent the number of bin and systematic error, respectively. The penalty termfrom the systematic errors adjustment for a better agreement between MC and data isdefined as the sum of (i /σi )2 . MC is fit to the data by finding the error parameters set
i which has minimum χ2 for each oscillation parameter set.
Two fits, including a θ13 -free fit, in which θ13 is a free parameter, and a θ13 -constrained
fit, in which it is constrained to its global best fit value, are performed over 515 analysis
bins in SK-IV. Besides the expanded FV (Dwall > 50 cm), the fits are also done within
the conventional FV (Dwall > 200 cm) for comparison. The fit with θ13 -constrained is
performed over three parameters, |∆m232,31 |, sin2 θ23 and δCP with a total of 72 systematic errors and sin2 θ13 is constrained to 0.0210 ± 0.0011 [27]. The number of oscillation
parameters for fit with θ13 -free is increased to four to include sin2 θ13 and the numberof the systematic errors is reduced to 71: the error from θ13 uncertainty is removed inthis fit. Although the sin2 θ13 has been measured very precisely by other experiments
already, measurements with atmospheric neutrinos can test the validity of the analysis’116

Chapter 7. Neutrino Oscillation Analysis
Parameter∆m221
sin2 θ12
sin2 θ13Value(7.53 ± 0.18) × 10−5 eV20.307 ± 0.013
0.0210 ± 0.0011

Table 7.6: Summary of the oscillation parameters fixed in the analysis. Both of thevalue and associated uncertainty are shown. It should be noted that sin2 θ13 is not
fixed in the “θ13 -free”analyses.

Parameter|∆m232,31 |sin2 θ13
sin2 θ23
δCP

Range
1.5 × 10−3 ∼ 3.5 × 10−3 eV20.0 ∼ 0.06
0.3 ∼ 0.725
0.0 ∼ 2π

Scan points
40 points
25 points
35 points
19 points

Table 7.7: Oscillation parameter points scanned in this analysis.

hierarchy preference since θ13 drives the appearance of upward-going electron events
expected for a mass hierarchy signal. During the fitting, the solar mixing parameters,∆m221 and sin2 θ12 , are fixed to the values in Table 7.6. The uncertainty in the fixed
oscillation parameters are considered with the corresponding systematic errors in theanalysis. Each combination of the oscillation parameter shown in Table 7.7 is used tofind the best-fit parameter set which minimizes the χ2 as defined in Equation 7.3. The
normal and inverted hierarchy are fitted separately. The value of ∆m232 is used during
the fits to the normal hierarchy, while ∆m231 value is used for the inverted hierarchy
fitting. The one with smaller minimum χ2 value among the two hierarchy fits is taken
as the global best-fit. The difference between the two hierarchies is used to evaluate themass hierarchy preference.

7.5

Systematic Errors

The same systematic errors as those introduced in [6] are used for those related to theflux of atmospheric neutrino and cross section of interaction. The errors for PC or
Up-µ events are also the same as in the reference, but the systematic errors related toFC event reconstruction are evaluated with the fiTQun reconstructed samples for bothconventional fiducial volume and expanded fiducial volume. The detailed description forother error sources and evaluation method can be found in [28, 120].

Chapter 7. Neutrino Oscillation Analysis

7.5.1

117

Energy Calibration

The systematic from the energy calibration is obtained by comparing of the energy
scales between data and MC. To cover the whole energy range of the neutrino sample,
three calibration samples: decay electrons, π 0 from atmospheric neutrino neutral current
interaction and stopping cosmic ray muons are used to evaluate the absolute energy
scale error. The value of the most discrepant sample is taken as the uncertainty on theabsolute energy scale in the oscillation analysis. The total systematic error from theenergy calibration is the quadrature-sum of absolute energy scale error and the time
variation of the absolute energy scale. The total energy scale uncertainty for fiTQun
within conventional FV (expanded FV) is 2.17% (2.02%), which is at the same level ofthat for APFit, 2.1% [6].
In addition, the uncertainty from the up/down asymmetry of the detector is also considered. It is evaluated by studying the difference on momentum distribution between data
and MC of Michel electrons from cosmic ray muons with different zenith angle. The
difference at the most deviated direction is taken as the uncertainty from the detector
asymmetry. The uncertainty from up/down asymmetry for fiTQun within conventional
FV (expanded FV) is 0.67% (0.58%), which is at the same level of that for APFit, 0.4%
[6].
The detailed information can be found in Section 4.2.

7.5.2

Ring Counting, Particle Identification and Multi-GeV Multi-ringe-like Event Separation

Systematic errors on the event selection including the particle identification (PID), number of rings (ring counting), and the two-stage separation of the multi-GeV multi-ringe-like event samples are evaluated using the atmospheric neutrino data itself since noother control samples span the same energies and event topologies as the atmospheric
neutrinos.
The likelihood distribution of MC is fitted to the data by shifting and smearing as:
L0m = β 1 Lm + β 0 ,(7.5)

where β 0 is a shift parameter that shifts the Lm distribution, while β 1 is a scale parameterthat adjusts the Lm distribution to be wider (for β 1 > 1) or narrower (for β 1 < 1).
The original distribution of MC is further divided into several subsamples based on MC
truth information. The categorization of the subsamples is summarized in Table 7.8. An

118

Chapter 7. Neutrino Oscillation Analysis

Likelihood distribution
Ring counting
Particle ID
Multi-GeV multi-ring separation

Subsample
True single-ring events, true multi-ring events
True νe /ν̄e events, true νµ /ν̄µ events
True νe events, true ν̄e events, other

Table 7.8: Summary table of subsample categorization for different MC likelihood
distribution. The true number of rings is counted using only particles with energy
more than 30 MeV above Cherenkov threshold in the final states.

example of fitting the ring counting likelihood within the new region (50 cm < Dwall <200 cm) for the associated systematic error using true single- and multi-ring events isshown in Figure 7.26. The dashed lines represent the distribution of the subsample
while the solid lines represent the overall distribution. The blue and red colors denotethe distribution before and after fitting, respectively. The final fractional change in the
total sample content across the threshold between before and after fitting is used as the

Events / ( 2 )systematic error, as shown in Figure 7.27.

70
1R like MR like
60
50
40
30
20
10
0

0

50

100
RC likelihood

Figure 7.26: Ring counting likelihood distribution for multi-GeV e-like events in the
new region (Dwall from 50 cm to 200 cm). The original p.d.f constructed from MC(blue line), the fitted p.d.f. (red line) and the data (points) are shown. The dashedlines show the contribution from different components: true single-ring subsample andtrue multi-ring sample. The true number of rings is counted using only particles withenergy more than 30 MeV above Cherenkov threshold in the final states. Error bars
represent the statistical error.

A Markov chain Monte Carlo method based on Metropolis-Hastings algorithm [121] isused to estimate the effect of the fitting error. The fit result in the previous step is chosen
as the start point, and the ∆χ2 from a random move is used to determine whether we
accept this move. The 68.7% percentile points with larger absolute ∆χ2 value is usedas the final systematic error value (Figure 7.28).

Number of Events

Chapter 7. Neutrino Oscillation Analysis

119

70
60

1R like

MR like

0

20

50
40
30
20
10
0

40

60
80
100
Ring Counting likelihood

Number of steps

Figure 7.27: Ring counting likelihood distribution for multi-GeV e-like events in the
new region. (dwall from 50 cm to 200 cm.) Point shows the data, and solid (dashed)line shows fitted (nominal) distribution. The dotted line is the cut line forsingle-ring/multi-ring event separation. The shaded histograms show the after fitting
distribution for true single-ring events in final state. The term “1R” and “MR”
represent “single-ring” and “multi-ring” respectively.

2500

2000

1500

1000

500

0

0

2

4

6

8
10
Fraction change [%]Figure 7.28: Fractional change of the single ring component distribution ofmulti-GeV e-like ring counting likelihood systematic error evaluation for conventional
FV (Dwall > 200 cm, black) and new region (50 cm < Dwall < 200 cm, blue). The
dashed lines show the final systematic error value, which is defined as the 68.7%
percentile points with larger absolute value. The systematic error in new region islarger than the one in conventional FV.

120

Chapter 7. Neutrino Oscillation Analysis

The result of systematic error evaluation of each of the FC analysis samples for both theconventional fiducial volume (Dwall > 200 cm) and the new region (50 cm < Dwall < 200cm) are summarized in Table 7.9. The related distributions can be found in Appendix
C. Errors from a particular uncertainty are assumed to be fully correlated or fully anticorrelated among all of the analysis samples. The sign of numbers in the table indicatesthe correlation. The systematic error from different sources is assumed to be independent
from each other. In the conventional fiducial volume, the sizes of the systematic errors
evaluated by fiTQun is similar or better than that for APFit. FiTQun reconstruction
also shows consistent errors between the conventional fiducial volume and the new region(50 cm < Dwall < 200 cm) for many error categories, although larger systematic errors
are observed for the new region in some cases: such as the ring counting uncertainty
for multi-GeV single-ring events. However, the benefit to the sensitivity of neutrino
oscillation by including events in the new region into the analyses outweighs this slightdownside. The final coefficient (fni ) used in the analysis within the expanded FV isobtained by merging the numbers for the different FV regions. The sensitivity with the
expanded FV and updated systematic errors is presented in Section 7.6.

7.5.3

Other Systematic Error related to Event Reconstruction

Two-ring π 0 selection
The uncertainty from two-ring π 0 events selection is estimated by considering the uncertainty from ring counting and PID for both first ring and second ring. The evaluation
method for each ring is similar to the one introduced in previous section. The ringcounting and PID likelihood distributions for data and MC for both rings within the
expanded FV are shown in Figure 7.29. The errors for each item are summarized inTable 7.10. The final uncertainty from two-ring π 0 selection is 1.03%, which is used
for both analyses within the conventional FV and the expanded FV due to the similar
ring counting and PID performance within the conventional FV and the new region for
sub-GeV events (Table 7.9).

Decay electron tagging
Decay electron tagging is important for the atmospheric neutrino oscillation since it is
crucial for the classification between νe -like events and ν̄e -like events, as discussed in
Section 7.1.2: an event with no decay electrons would be easier to be reconstructed asν̄e -like events. The systematic error in decay electron detection efficiency is estimatedChapter 7. Neutrino Oscillation Analysis

Reconstruction
Dwall range
Ring counting
Sub-GeV , Single-ringe-like, p < 400 MeV
e-like, p > 400 MeVµ-like, p < 400 MeVµ-like, p > 400 MeV
Sub-GeV, Multi-ringe-likeµ-like, Evis > 600MeVMulti-GeV Single-ringe-likeµ-likeMulti-GeV Multi-ringe-likeµ-like121

fiTQun50 cm∼200 cm

fiTQun> 200 cm

APFit[6]> 200 cm

1.94%
0.59%
1.08%
1.25%

1.20%
0.48%
0.42%
1.21%

1.6%
1.0%
-3.0%
0.6%

-3.58%-2.32%-2.39%-1.66%

-1.9%
2.3%

8.61%
2.65%

1.21%
-2.33%

1.0%-1.2%-4.21%-3.07%-0.62%
0.72%

-0.9%
2.4%

0.36%
-0.37%-0.28%
0.22%

0.06%-0.06%-0.35%
0.35%

Particle ID of single-ring events
Sub-GeV
e-like0.99%µ-like-0.89%
Multi-GeV
e-like0.23%µ-like-0.21%

Particle ID of the brightest ring in multi-ring events
Sub-GeV
e-like-3.19%-0.72%
4.19%µ-like1.31%
0.31%-1.56%
Multi-GeV
e-like
1.94%
1.10%
3.33%
µ-like-1.06%-0.66%-1.56%
Multi-GeV Multi-ring e-like - other separatione-like-0.88%-0.67%
other
0.50%
0.53%

3.0%
-3.4%

Multi-GeV Multi-ring νe -like - ν̄e -like separation
νe -like-3.64%-2.33%
ν¯e -like
4.51%
2.10%

6.82%
-6.04%

Table 7.9: Systematic errors related to the ring counting, particle identification andmulti-GeV multi-ring e-like separation. The related distribution can be found in
Appendix C. Errors from a particular uncertainty are assumed to be fully correlated
or fully anti-correlated among all of the analysis samples. The sign of numbers in thetable indicates the correlation. The systematic error from different sources is assumed
to be independent from each other.

Chapter 7. Neutrino Oscillation Analysis

Number of Events

Number of Events

122

700
1R like

2R like

600
500
400

250

2R like

3R like

200

150

300
100
200
50

100
0

0
0

10

220

20

µ like

200

30
40
50
Ring Counting likelihood-10

Number of Events

Number of Events-10

e like

180
160
140

0

400

10

20

µ like30
40
50
Ring Counting likelihoode like

350
300
250

120
200

100
80

150

60

100

40
50

20
0
-1000

-500

0

500

0
-1000

1000
1500
PID likelihood

-500

0

500

1000
1500
PID likelihood

Figure 7.29: Likelihood distribution of ring counting (top) and PID (bottom) for
both first ring (left two plots) and second ring (right two plots). Fiducial volume cut
is Dwall > 50 cm. The shaded histogram show the distribution of CC 1π 0 events. The
solid red (dashed blue) line denote the fitted MC (nominal) while the points are fordata. Error bars show the statistical uncertainty of data.

Ring counting
PID

First ring
0.43%
0.27%

Second ring
0.12%
0.89%

Table 7.10: Uncertainty from two-ring π 0 event selection.

Conventional FV (Dwall > 200 cm)
New region (50 cm < Dwall < 200 cm)

Sub-GeV
0.7%
0.7%

Multi-GeV
0.7%
2.1%

Table 7.11: Uncertainty from decay electron tagging.

by comparing the number of tagged Michel electrons from cosmic ray muons data and
MC. The error is summarized in Table 7.11.
Furthermore, the uncertainty from decay probability of π + is also considered, since italso affects the number of detected decay electrons. The decay probability of π + depends
on whether it is scattered or not and how it is scattered, which varies as a function ofits momentum. The uncertainty from π + interaction is estimated from the difference
between experimental data and the Monte Carlo prediction, as shown in Figure 3.7.

Chapter 7. Neutrino Oscillation Analysis

123

The uncertainty of π + decay probability is then evaluated by generating π + MC withincreased or decreased mean free path based on the interaction uncertainties [122]. The
final uncertainty for π + decay electron tagging varies from 0.3% to 0.6% for FC sub-GeV
single-ring e-like 0-decay sample.

7.6

Results and Discussion

7.6.1

Fit Results and Discussion with Free θ13

The measurements with atmospheric neutrinos on θ13 represent an important test of
the analysis’ hierarchy preference since the appearance of upward-going electron events
expected for the mass hierarchy signal is driven by θ13 . Due to the limited impact of thesolar mixing parameters, ∆m221 and sin2 θ12 on the oscillations of neutrinos in the energy
range considered in the analysis, they are fixed in this analysis, while their associatedSensitivity to reject sin2θ13 = 0 ∆ χ2

uncertainties are considered as systematic errors.

9

fiTQun Expanded FV

8
fiTQun Conventional FV
7
APFit Conventional FV
6
5
4
3
2
1
0

0.4

0.45

0.5

0.55

0.6
True sin2θ23

Figure 7.30: Expected median sensitivity of the atmospheric neutrino sample toreject a zero value of sin2 θ13 with a livetime of 3118.5 days. Horizontal axis shows theassumed true value of sin2 θ23 . The result of the events reconstructed by fiTQun
within the conventional FV and expanded FV are denoted by grey and blue lines,
respectively. The sensitivity with events reconstructed by APFit and conventional FV
is denoted by orange line.

The expected sensitivity to reject a zero value of sin2 θ13 as a function of the true valueof sin2 θ23 is shown in Figure 7.30. The fiTQun-based analysis samples with both theconventional (grey dashed line) and expanded FV (blue solid line) as well as an analysis
with APFit-reconstructed samples with the conventional FV (orange dotted line) are
shown. Similar sensitivities for the two algorithms within the same FV are due to124

Chapter 7. Neutrino Oscillation Analysis
Reconstruction
Dwall range
Single-ringνe -likeν̄e -like
Multi-ringνe -likeν̄e -likefiTQun50 ∼ 200 cm

fiTQun> 200 cm

APFit
> 200 cm

0.064
0.003

0.058
0.003

0.109
0.009

0.059
0.023

0.054
0.021

0.121
0.040

Table 7.12: CC backgrounds, including CCνµ and CCν̄µ , in multi-GeV e-likesample.

the large fraction of common events between them: > 97% and consistency of theirsystematic errors. The remaining 3% difference comes from the vertex resolution near
the boundary of FV. The slight boost in the sensitivity for the fiTQun-selected sample
within the conventional FV, is due to the lower contamination of νµ events in the e-likesamples, which is the most sensitive to θ13 , as shown in Table 7.12. More signal events
are incorporated naturally by expanding the FV in the analysis and the improvementof sensitivity is therefore more significant.
Figure 7.31 and Figure 7.32 show one dimensional allowed regions for θ13 , |∆m232,31 |,sin2 θ23 and δCP within the conventional FV and the expanded FV, respectively. A
smaller χ2 can be obtained for the normal hierarchy hypothesis than that for the inverted hierarchy hypothesis with χ2NH,min − χ2IH,min = −1.81(−1.41) for the expanded(conventional) FV. The value of sin2 θ13 is preferred from 0.003 to 0.033 (from 0.001
to 0.023) for the normal (inverted) hierarchy fit within 1σ. This result also disfavors azero-value of sin2 θ13 at ∼ 1.8σ (1.2 σ) for the normal (inverted) hierarchy fit, which isconsistent with the value preferred globally. Table 7.13 summarizes the best-fit information and constraints on the oscillation parameter.
As discussed in Section 1.2, upward-going electron neutrino appearance at multi-GeV
energies determines both sin2 θ13 and the mass hierarchy. The up-down asymmetry of
the multi-GeV single- and multi-ring electron-like samples is shown in Figure 7.33. Here,the asymmetry is defined as the ratio of the difference of the number of upward-going(cosθzenith < −0.4) and downward-going (cosθzenith > 0.4) events relative to theirsum. The normal hierarchy preference is mainly contributed by the excesses between
a few and ten GeV in the multi-GeV single-ring e-like νe and the multi-ring e-like νesamples, as shown in Figure 7.34.
The best-fit atmospheric mixing parameters from the normal hierarchy fit are sin2 θ23 =+0.051+0.10
20.588+0.030−0.062 for second octant (best-fit), 0.425−0.034 for first octant and ∆m32 = 2.63−0.21 ×

10−3 eV2 . The result disfavors the maximal mixing (sin2 θ23 = 0.5) weakly at ∼ 1σ. This

Chapter 7. Neutrino Oscillation Analysis

125

10

10

8

8
99%

99%

∆ χ2

6

∆ χ2

6

95%

4

4

90%

90%

2

2
68%

0

95%

0

0.02

68%

0.04

0
0.0015

0.06

0.002

0.0025sin2θ130.003

0.0035|∆ M223| eV 2

10

10

8

8
99%

99%

∆ χ2

6

∆ χ2

6

4

95%

4

90%

95%
90%

2

2
68%

0
0.3

68%

0.4

0.5

0.6

sin θ23
2

0.7

0

0

2

4

6

δ CP

Figure 7.31: Difference of χ2 value between global best fit result in this analysis andthe result with the oscillation parameter shown in the horizontal axis. In these plots,
no external constraints are applied and the events within conventional FV (Dwall >200 cm) are used. The normal hierarchy result is denoted by blue solid lines. The
result of inverted hierarchy is denoted by orange dashed lines and has been offset fromthe normal hierarchy result by the difference of minimum χ2 values between them.

preference is driven by the data deficits of the multi-GeV µ-like sample as well as the data
excesses in the upward-going regions of the single-ring e-like ν̄e sample and multi-ringe-like νe sample. Since both sin2 θ23 and sin2 θ13 measurements feature in the νµ → νeoscillation probability, as shown in Equation 1.25, the preference for these two parameters are coupled. As we will see in the next section, this coupling will affect the resultof the θ13 -constrained analysis.
The normal (inverted) hierarchy fits show the best-fit value of δCP at 3.84 (4.19) radians. It is expected to observe more electron neutrino appearance in the sub-GeV e-likesamples due to νµ → νe oscillations for δCP = 3.84 comparing with the least preferredδCP value of 0.8 radians. It should be noted that δCP similarly influences the amount of126

Chapter 7. Neutrino Oscillation Analysis

10

10

8

8
99%

99%

∆ χ2

6

∆ χ2

6

95%

4

4

90%

90%

2

2
68%

0

95%

0

0.02

68%

0.04

0
0.0015

0.06

0.002

0.0025sin2θ130.003

0.0035|∆ M223| eV 2

10

10

8

8
99%

99%

∆ χ2

6

∆ χ2

6

4

95%

4

90%

95%
90%

2

2
68%

0
0.3

68%

0.4

0.5

0.6

0.7

sin θ23
2

0

0

2

4

6

δ CP

Figure 7.32: Difference of χ2 value between global best fit result in this analysis andthe result with the oscillation parameter shown in the horizontal axis. In these plots,
no external constraints are applied and the events within expanded FV (Dwall > 50cm) are used. The normal hierarchy result is denoted by blue solid lines. The result of
inverted hierarchy is denoted by orange dashed lines and has been offset from thenormal hierarchy result by the difference of minimum χ2 values between them.

electron neutrino appearance in the multi-GeV region, but its effect is subdominant tothat of θ13 . Considering the fact that neutrinos have higher flux and interaction crosssection than those of antineutrinos in the atmospheric sample, the predicted excess of
upward-going e-like events is smaller for the inverted hierarchy since matter effects only
influence antineutrino events in that case. The value of θ13 can be therefore adjusted
for a better agreement between data and MC more easily, which results in a weakerconstraint on δCP for the inverted hierarchy fit.

Up - Down / Up + Down

Chapter 7. Neutrino Oscillation Analysis

0.4

0.4

0.2

0.2

0

0

-0.2

-0.2

-0.4 Multi-GeV e-like νe-0.4 Multi-GeV e-like νe

127

Data
Normal Hierarchy
Inverted Hierarchy

0.4

104 0.4

0.2

0.2

0.2

0

0

0

-0.2

-0.2

-0.2

-0.4 Multi-Ring e-like νe-0.4 Multi-Ring e-like νe-0.4 Multi-Ring Other

104

104

104

0.4

104

Momentum [MeV/c]

Prefer IH ∆ χ2

Figure 7.33: Upward- (cosθ < −0.4) to downward-going (cosθ > 0.4) event ratio fordifferent momentum within expanded FV. The momentum for multi-ring event isdefined as the sum of the momentum of each ring. Statistical uncertainties are denotedby the error bars. The cyan line denotes the best fit from the normal hierarchy
hypothesis, and the orange line the best fit from the inverted hierarchy hypothesis.

4
3
2
1

Prefer NH

0
-1
-2
-3
-4

SK4

S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
_SuK4_Su K4_SinK4_Su K4_SuK4_Su K4_Tw K4_MuK4_MuK4_MuK4_MuK4_MuK4_MuK4_MuK4_PCK4_PCK4_UpK4_UpK4_Up
bGe bGe gle bGe bGe bGe oRin ltiG ltiG ltiG ltiR ltiR ltiR ltiR Sto Thr Sto Thr Thr
pMu uMu uMuu
VEli VEli Ring VM
VM
VM
gPi0 eV_e eV_e eV_m ing_ ing_ ing_ ingO puuu
Non Shoeemke0 ke1Ptlike like like
like like
Dec Dec i0
Sho wer
0De 1De 2De_nu _ba ulike like_nulike_ba ulike herayay
wer ing
cay cay cay
rnu
e
rnu
e
e
ing
e

Figure 7.34: The preference for MH for each event types within expanded FV. The
y-axis shows the difference of χ2 between normal hierarchy (NH) and inverted
hierarchy (IH) contributed by each event sample at the best-fit oscillation parameters.
The definition of the vertical axis is ∆χ2 = χ2NH − χ2IH , which means that the negativevalue represent the preference for the normal hierarchy.

128

7.6.2

Chapter 7. Neutrino Oscillation Analysis

Fit Results and Discussion with Constrained θ13

In general, it is hard for Super-K to reconstruct the neutrino kinematics fully since atmospheric neutrinos come from all directions with a wide range of energies, and often
produce particles without Cherenkov radiations. As a result, the neutrino mixing parameter sensitivity from atmospheric neutrino experiments is not at the same level asthat from experiments which utilize artificial neutrino sources, such as in accelerator orreactor. As shown in Equation 1.26, the size of MSW effect directly relies on the valueof sin2 θ13 , which is constrained by reactor neutrino experiments with high precision:
sin2 θ13 = 0.0210 ± 0.0011 [27]. The sensitivity to the mass hierarchy and δCP can be
expected to be increased by introducing such a constraint. During the θ13 -constrained
fit, the value of sin2 θ13 is fixed at its central value 0.0210. A systematic error to reflect the impact of the uncertainty of sin2 θ13 is added to the analysis. The same data
samples, binning, and parameter grid for all other oscillation parameters as the analysis
presented above are used in this fit.
Besides the constraint on sin2 θ13 from reactor experiments, the constraint on sin2 θ23from accelerator neutrinos could also help increase the sensitivity. More details will be
discussed in Chapter 8.
As shown in the Figure 7.35, the expected sensitivity to the mass hierarchy with eventsreconstructed by fiTQun is higher than the one with events reconstructed by APFit for
all assumed values of sin2 θ23 . The sensitivity strengthens further after expanding FV.
The χ2 value as a function of the atmospheric neutrino mixing parameters and δCP in
the θ13 -constrained fit to the data is illustrated in Figure 7.36. Similar to the result of
the unconstrained fit, the normal mass hierarchy is preferred with ∆χ2 = −2.45(−2.18)
by the Super-K IV data for the expanded (conventional) FV. The fit result with the
expanded FV is in good agreement with the one with the conventional FV, but showshigher significance to reject inverted hierarchy, which is consistent with the expectationfrom the sensitivity.
The preferred region at 1σ for |∆m232 | is [2.41,2.75] ×10−3 eV2 ( [2.36, 2.67] ×10−3 eV2 )when the true mass hierarchy is assumed to be normal (inverted). This result is consistent with the one from the unconstrained fit. The jagged nature of the |∆m232 | comesfrom the limited data and MC statistics in the low energy region. As shown in Equation 1.25, the oscillation probability is proportional to sin2 (1.27∆m232 L/E), which meansthat it will change drastically for a large L/E value and is therefore very sensitive to∆m232 . Therefore, without sufficient data and MC to average over these oscillations,
small changes in ∆m232 result in the rapid change of the χ2 , as seen in the figure.

Chapter 7. Neutrino Oscillation Analysis

fiTQun Expanded FV

3Incorrect Hierarchy Rejection ∆χ2

129

fiTQun Conventional FV
APFit Conventional FV

2.5
2
1.5
1
0.5
0

0.4

0.45

0.5

0.55

0.6

0.55

0.6

sin2θ23

Incorrect Hierarchy Rejection ∆χ2

3

fiTQun Expanded FV
fiTQun Conventional FV
APFit Conventional FV

2.5
2
1.5
1
0.5
0

0.4

0.45

0.5
sin θ232

Figure 7.35: Expected median sensitivity with a livetime of 3118.5 days to mass
hierarchy. Top figure shows the sensitivity to normal mass hierarchy and bottom
figure shows the sensitivity to inverted hierarchy. Horizontal axis shows the assumed
true value of sin2 θ23 . Here the value of sin2 θ13 is fixed to 0.0210. The result of theevents reconstructed by fiTQun within the conventional FV and expanded FV are
denoted by the gray and blue bands. The sensitivity with events reconstructed by
APFit algorithm and the conventional FV is denoted by the orange band. The bands
represents the effect of the uncertainty from δCP .

Figure 7.37 shows the two-dimensional atmospheric mixing contour for the preferredregions at 90% C.L.. While the best fit value of sin2 θ23 is changed to the first octant
compared to both the previous Super-K analysis with APFit [6] and the unconstrained
fit with fiTQun, it is within errors of the unconstrained fit and in good agreement withother measurements, as shown in Figure 7.38. Comparing with the fitting result with
θ13 -free, the preferred value of θ23 shifts from the second to first octant after adding theconstraint on sin2 θ13 , though both octants are allowed at 1σ. It is due to the degeneracy
on the appearance probability for certain combinations of sin2 2θ13 and sin2 θ23 , as shown

130

Chapter 7. Neutrino Oscillation Analysis

in Equation 1.25. Figure 7.39 shows the preference for the sin2 θ23 octant of each sample
for the constrained sin θ13 fit (0.02, red triangles) and the unconstrained sin θ13 fit (whose
best fit value is 0.0075, blue boxes). Although the preference for the second octant
from multi-ring νe -like events is stronger for a larger sin θ13 value due to the upwardgoing event excesses that drive the mass hierarchy preference (c.f. Figure 7.9), themulti-ring ν̄e -like and µ-like samples prefer the first octant more significantly. This isbecause the θ13 and octant preferences in the unconstrained analysis cannot be supported
simultaneously by the excess of νe and deficits of νµ , as shown in Figure 7.40. The datais accommodated for a larger sin θ13 value by reducing the amplitude for both the excessof e-like events and the deficit of µ-like events by moving θ23 to the first octant. In fact,
the difference in χ2 between the minimum value of each octant is ∆χ21st −2nd = -0.73(0.13) for the constrained (unconstrained) fit. The preference for both octant is weak,
as we will discuss below.
The fit result of δCP is consistent with the result of the unconstrained fit but a tighter
constraint is obtained: the 1σ allowed region is [1.79, 5.81] ([0, 0.12] ∪ [1.43, 6.28])
for the normal (inverted) hierarchy. Table 7.13 shows the best-fit oscillation parametervalues and their 1σ errors.
10

10

8

10

8
99%

8
99%

99%

4

95%

∆ χ2

6

∆ χ2

6

∆ χ2

6

4

90%

95%

4

90%

2

90%

2
68%

0
0.0015

2
68%

0.002

0.0025

0.003

0
0.3

0.0035

68%

0.4

0.5

0.6

0

0.7

10

8

∆ χ2
95%

4

90%

2
68%

0.00252|∆ M23| eV 20.003

0.0035

95%
90%

2
68%

6

6

∆ χ2

∆ χ2

4

90%

2

4

99%

6

95%

6

8
99%

6

4

10

8
99%

0.002

2
δ CP

10

0
0.00150

sin2θ23|∆ M223| eV 2

4

95%

0
0.3

68%

0.4

0.5
sin2θ23

0.6

0.7

0

0

2
δ CP

Figure 7.36: Difference of χ2 value between global best fit result in this analysis andthe result with the oscillation parameter shown in the horizontal axis. In these plots,
no external constraints are applied. Top three figures show the result withinconventional FV (Dwall > 200 cm) and bottom three figures show the result within
expanded FV (Dwall > 50 cm). The normal hierarchy result is denoted by blue solidlines. The result of inverted hierarchy is denoted by orange dashed lines and has been
offset from the normal hierarchy result by the difference of minimum χ2 values
between them.

131

0.0035

0.003

0.003
|∆ m223| eV 2

0.00350.00250.0025

0.002

0.002

0.0015
0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7
sin2θ230.0015
0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7
sin2θ23

Figure 7.37: Two-dimensional atmospheric mixing contour for the preferred regions
at 90% C.L.. The left (right) figure shows 90% C.L. for the normal (inverted)hierarchy. The solid line denotes the fit result within expanded FV while the dashedline is for the result within conventional FV. The full (open) star denotes the best-fit
value for expanded FV (conventional FV). In each contour sin2 θ13 is fixed to 0.0210.
Both contours have been drawn with respect to the global best-fit for each FV.

0.0035∆ m232 [eV2]|∆ m223| eV 2

Chapter 7. Neutrino Oscillation Analysis

0.003

Normal Hierarchy, 90% C.L.
SK IV with fiTQun (This work, 3118.5 days)
SK I-IV with APFit (5326 days)
T2K
NOvA
IceCube
MINOS

0.0025

0.002
0.2

0.4

0.6

0.8

sin2 θ23

Figure 7.38: Constraints on neutrino oscillation contours at the 90% C.L. fromanalyses assuming the normal mass hierarchy. The contour from Super-K IV with
fiTQun (red) is taken from the analysis with sin2 θ13 constrained to be
0.0210 ± 0.0011. Contours from the Super-K I-IV with APFit (dashed cyan) [6], T2K(dotted yellow) [22], NOvA (dashed green) [23], IceCube (dashed black) [24] andMINOS+ (dashed blue) [25] experiments are also shown.

Chapter 7. Neutrino Oscillation Analysis

Prefer 2nd∆ χ2

132

4
3
2
1

Prefer 1st

0
-1
-2
-3
-4

SK4

S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
_SuK4_Su K4_SinK4_Su K4_SuK4_Su K4_Tw K4_MuK4_MuK4_MuK4_MuK4_MuK4_MuK4_MuK4_PCK4_PCK4_UpK4_UpK4_Up
bGe bGe gle bGe bGe bGe oRin ltiG ltiG ltiG ltiR ltiR ltiR ltiR Sto Thr Sto Thr Thr
pMu uMu uMuu
VEli VEli Ring VM
VM
VM
gPi0 eV_e eV_e eV_m ing_ ing_ ing_ ingO puuu
Non Shoeemke0 ke1Ptlike like like
like like
Dec Dec i0
Sho wer
0De 1De 2De_nu _ba ulike like_nulike_ba ulike herayay
wer ing
cay cay cay
rnu
e
rnu
e
e
ing
e

Figure 7.39: Distributions of the ∆χ2 for octant preference(χ2sin2 θ23 =0.4 − χ2sin2 θ23 =0.6 ) for each sample. Blue boxes represents the fit result withsin θ13 free, whose the best-fit value is 0.0075, while the red triangles show the fit
result with sin θ13 fixed to 0.02.

Fit within conventional FV
Hierarchy
χ2
sin2 θ13
sin2 θ23 (1st oct.)

NH
540.9
0.013+0.032−0.0110.425+0.075−0.036IH
542.3
0.005+0.018−0.0050.438+0.062−0.040θ13 Constrained
NH
IH
541.0
543.2
–
–
+0.0460.425−0.038
0.425+0.057−0.037sin2 θ23 (2nd oct.)|∆m232,31 | [×10−3 eV2 ]δCP
Fit within expanded FV
Hierarchy
χ2
sin2 θ13
sin2 θ23 (1st oct.)0.575+0.031
0.563+0.046−0.075−0.063+0.07
2.63
2.37+0.33
−0.44
−0.16
3.49+1.80
3.84+2.44
−3.84
−1.79
θ13 Free
NH
IH
576.3
578.1
+0.0150.008+0.025
0.008−0.005−0.007+0.0750.425+0.051
0.425−0.034−0.027

Excluded in 1σ 0.588+0.016−0.034+0.12
2.58
2.63+0.08
−0.39
−0.38
3.14+2.25
3.14+3.05
−1.71
−1.21
θ13 Constrained
NH
IH
576.5
579.0
–
–
+0.0460.425−0.037
0.425+0.055−0.036sin2 θ23 (2nd oct.)|∆m232,31 | [×10−3 eV2 ]δCP

0.588+0.030−0.062
2.63+0.10−0.21
3.84+2.00−2.14

θ13 Free0.575+0.034−0.0752.53+0.14−0.08
4.19+2.09−4.19

0.600+0.013−0.0302.53+0.22−0.12
3.14+2.67−1.35

0.588+0.022−0.0372.53+0.14−0.31
4.89+1.51−3.46

Table 7.13: Best-fit result of oscillation parameter for each analysis. The term“NH” (“IH”) refers to the fit assuming normal (inverted) hierarchy. The fits without
and with a constraint on sin2 θ13 are termed as “Free” and “Constrained”,
respectively. The best-fit result of sin2 θ23 in both octants are shown, with the best-fit
octant enclosed in a box.

Up - Down / Up + Down

Chapter 7. Neutrino Oscillation Analysis

0.4

0.4

0.2

0.2

0

0

-0.2

-0.2

-0.4 Multi-GeV e-like νe-0.4 Multi-GeV e-like νe

133

Data
sin2θ23=0.4
sin2θ23=0.6

0.4

104 0.4

0.2

0.2

0.2

0

0

0

-0.2

-0.2

-0.2

-0.4 Multi-Ring e-like νe-0.4 Multi-Ring e-like νe-0.4 Multi-Ring �-like

104

104

0.4

103

104

Up - Down / Up + Down

Momentum [MeV/c]

0.4

0.4

0.2

0.2

0

0

-0.2

-0.2

-0.4 Multi-GeV e-like νe-0.4 Multi-GeV e-like νe

Datasin2θ23=0.4
sin2θ23=0.6

0.4

104 0.4

0.2

0.2

0.2

0

0

0

-0.2

-0.2

-0.2

-0.4 Multi-Ring e-like νe-0.4 Multi-Ring e-like νe-0.4 Multi-Ring �-like

104

104

104

0.4

103

Momentum [MeV/c]

Figure 7.40: Upward- (cosθ < −0.4) to downward-going (cosθ > 0.4) event ratio asa function of energy for expanded FV. Statistical error are denoted by the error bars.
For the multi-ring samples the momentum is defined as the sum of each ring’s
momentum. The cyan line denotes the best fit when sin2 θ23 = 0.4, and the orange
line the best fit when sin2 θ23 = 0.6. Top five plots show the result whensin2 θ13 = 0.0075, which is the best-fit value in the sin2 θ13 -free analysis, while thebottom ones show the result when sin2 θ13 = 0.02, which is the global best-fit value.

134

Chapter 7. Neutrino Oscillation Analysis

Note that the absolute χ2 value increased from around 540 to 576 after expanding thefiducial volume, which means worse agreement between data and best-fit MC since thenumber of degrees of freedom in the fit have not changed. To validate the fit result,
MC ensembles were generated at the best-fit point with statistical and systematic error
variations applied. Statistical variations is estimated based on the current detector
exposure, and systematic errors have been varied according to Gaussian distribution

Fraction of Pseudo Data Sets

with the deviation parameter specified by their uncertainties (σ in Equation 7.3).

10-1

Best fit, SK θ13 Constrained
SK True NH
p-value: 0.049

10-2

10-3

10-4
100

200

300

400

500

600

700

800

900 1000

χ2

Figure 7.41: Distributions of the best-fit χ2 values fits to pseudo data sets. The
pseudo data have been generated with the best-fit value in NH expanded FV θ13
constrained fit shown in Table 7.13. Shaded portions of the histograms denote thefraction of pseudo data sets with more extreme values than that observed in the data,
χ2N H = 576.5.

Figure 7.41 shows the distribution of the best-fit χ2 values fit to MC ensembles. During
the pseudo data generation, only the events in the new region (50 cm < Dwall < 200cm) were generated with fluctuations and then merged with the observed data within
the conventional FV (Dwall > 200 cm). The p-value for observing a χ2 greater than thatseen in the data, 576.5, is 0.049. Figure 7.42 shows the absolute χ2 distribution for each
event category. The error bar of pseudo data (blue box) represents one σ uncertainty
from statistical fluctuations and systematic error. The observed data in this analysis isshown by red triangle. The top two categories which have the largest deviation frompseudo data center value are sub-GeV e-like 0 decay sample and multi-GeV single-ringνe -like sample. The zenith angle distributions for both two samples are shown in Figure7.43 and Figure 7.44. Those events were also checked by eye-scanning and no abnormal
was observed. As a conclusion, there is no issue observed in the oscillation analysis withthe expand FV. More statistical validation can be found in Appendix D.
The systematic errors used and their best-fit values in the sin2 θ13 -constrained analysis
have been discussed in Section 7.5 and summarized in Table 7.14. The best-fit values ofChapter 7. Neutrino Oscillation Analysis

135

the systematic parameters are within 1.5 standard deviations of their priors.

Table 7.14: Summary of the systematic error used in this analysis. The best-fit
value of the systematic error parameter in percent is shown in second column. The
estimated 1σ error size in percent is shown in third column. The result of theθ13 − constrained fit within expanded FV is shown here.

Systematic Error
Flux normalization

Fit Value (%)σ (%)

Eν < 1 GeV1

7.3

25

> 1 GeV2

9.6

15Eν < 1 GeV

1.3

2

1 < Eν < 10 GeV

2.7

3> 10 GeV3

6.6

5Eν < 1 GeV

−1.1

5

1 < Eν < 10 GeV

−2.7

5

Eν > 10 GeV4

−0.34

8

Eν < 1 GeV

0.36

2

1 < Eν < 10 GeV

3.24

6

> 10 GeV5

9.2

15

e-like0.079

0.1µ-like0.24

0.3

0-decay µ-like

0.87

1.1

e-like0.63

0.8µ-like0.40

0.5

0-decay µ-like

1.34

1.7

e-like0.55

0.7µ-like0.16

0.2

Multi-ring Sub-GeV e-like

0.32

0.4

µ-like0.16

0.2

Multi-ring Multi-GeVe-like

0.24

0.3

µ-like

0.16

0.2

0.16

0.2

e-like−0.0230.1µ-like−0.0230.1

0-decay µ-like−0.0690.3

e-like−0.32

1.4µ-like−0.44

1.9

0-decay µ-like

−0.32

1.4

e-like

−0.74

3.2

Eν
(νµ + ν̄µ )/(νe + ν̄e )Eνν̄e /νeν̄µ /νµEν
Up/down ratio< 400 MeV

> 400 MeV

Multi-GeVPC
Horizontal/vertical ratio< 400 MeV

> 400 MeV

Multi-GeV

Continued on next page

136

Chapter 7. Neutrino Oscillation Analysis
Table 7.14 – Continued from previous page

Systematic Error

Fit Value (%)σ (%)µ-like−0.53

2.3

Multi-ring Sub-GeV e-like

−0.32

1.4

µ-like−0.30

1.3

Multi-ring Multi-GeVe-like

−0.64

2.8

µ-like

−0.35

1.5

−0.39

1.7

K/π ratio in flux calculation6

−5.3

10

Neutrino path length

−2.3

10

−0.22

5

2.46

5

Matter effects

0.06

6.8

Solar Activity

0.31

7

MA in QE

−4.79

10

Single π Production, Axial Coupling

−7.03

10

Single π Production, CA5

9.33

10

Single π Production, BKG

PC

Sample-by-sample

FC Multi-GeV
PC + Stopping UP-µ

−3.03

10

CCQE cross section7

7.37

10

CCQE ν̄/ν ratio7

4.81

10

CCQE µ/e ratio7

−2.74

10

DIS cross section

5.72

10

DIS model comparisons8

4.84

10

DIS Q2 distribution (high W)9

−5.70

10

DIS Q2 distribution (low W)9

−9.21

10

Coherent π production

−9.41

100

NC/CC

−0.10

20

9.08

25

ντ cross section
Single π production, π 0 /π ±

1.75

40

(i=e, µ)10

2.30

10

NC fraction from hadron simulation

−5.07

10

e-like 0-decay−0.0500.6µ-like 0-decay−0.0660.8

e-like 1-decay

−0.34

4.1

µ-like 1-decay−0.0250.3µ-like 2-decay

−0.47

5.7

Single π production, ν̄i /νi

π + decay uncertainty Sub-GeV 1-ringFinal state and secondary interactions11
Continued on next page

Chapter 7. Neutrino Oscillation Analysis

137

Table 7.14 – Continued from previous page
Systematic Error
1-ringMulti-ring

Fit Value (%)

σ (%)

−6.83

10

6.93

10

Meson exchange current12

1.08

10

∆m221 [27]0.00.0002

2

0.05

1.3

2

sin (θ13 ) [27]

0.00

0.11

FC reduction

0.27

1.3

PC reduction

−0.26

1

0.000

0.02

PC stopping/through-going separation (bottom)

−1.75

6.8

PC stopping/through-going separation (barrel)

1.13

8.5

sin (θ12 ) [27]FC/PC separation

PC stopping/through-going separation (top)

−29.0

40

Non-ν background

Sub-GeV µ-like−0.001

0.02

Multi-GeV µ-like−0.0020.07

Sub-GeV 1-ring 0-decay µ-like−0.0010.02

PC−0.0150.49

Sub-GeV e-like (flasher event)−0.000

0.03

Multi-GeV e-like (flasher event)−0.001

0.07

−0.40

2.00

e-like

−0.44

−0.86

µ-like0.43

0.84

e-like0.19

0.37µ-like1.03

2e-like1.32

2.57µ-like

−0.74

−1.44

Multi-ring Sub-GeV e-like

−0.42

−0.82

µ-like0.42

0.82

Multi-ring Multi-GeVe-like

−0.66

−1.28

µ-like0.59

1.14

e-like

−0.11

0.52

µ-like

0.10

−0.49

e-like

−0.02

0.09

µ-like

0.02

−0.09

e-like

0.29

−1.33

Fiducial Volume
Ring separation< 400 MeV
> 400 MeV
Multi-GeV

Particle identification (1 ring)
Sub-GeV
Multi-GeV
Particle identification (multi-ring)
Sub-GeV

Continued on next page

138

Chapter 7. Neutrino Oscillation Analysis
Table 7.14 – Continued from previous page

Systematic Error

Fit Value (%)σ (%)µ-like

−0.12

0.53

e-like

−0.27

1.25

µ-like

0.16

−0.73

νe ,ν̄e

−0.42

−0.71

0.32

0.52

Energy calibration

−0.24

2.02

Up/down asymmetry energy calibration

−0.53

0.67

Stopping

0.048

0.5

Through-going

0.029

0.3

UP-µ stopping/through-going separation

0.018

0.6

Energy cut for stopping UP-µ

0.30

1.7

Path length cut for through-going UP-µ

−0.33

1.5

Through-going UP-µ showering separation

3.26

3

−4.9

11

2.3

17

−6.55

24

Multi-GeV
Multi-ring likelihood selectionMulti-ring e-like
Multi-ring Other

UP-µ reduction

Background subtraction for UP-µStopping13
Non-showering13
Showering13
νe /ν̄e Separation

1.89

2.57

Sub-GeV 2-ring π 0−0.0391.03

Decay-e tagging

0.29

0.7

FC stopping muon subtraction

89.2

67

The significance of the preferences for the θ23 octant and the mass hierarchy in thisanalysis is addressed with the CL method [128]. The definition of the CLO for the θ
s

1

s

23

Uncertainty decreases linearly with log Eν from 25 %(0.1 GeV) to 7 %(1 GeV).
Uncertainty is 7 % up to 10 GeV, linearly increases with log Eν from 7 %(10 GeV) to 12 %(100 GeV)
and then to 20 %(1 TeV)
3
Uncertainty linearly increases with log Eν from 5 %(30 GeV) to 30 %(1 TeV).
4
Uncertainty linearly increases with log Eν from 8 %(100 GeV) to 20 %(1 TeV).
5
Uncertainty linearly increases with log Eν from 6 %(50 GeV) to 40 %(1 TeV).
6
Uncertainty increases linearly from 5% to 20% between 100GeV and 1TeV.
7
Difference from the Nieves [123] model is set to 1.0
8
Difference from CKMT [124] parametrization is set to 1.0
9Difference from GRV98 [125] is set to 1.0
10
Difference from the Hernandez[126] model is set to 1.0
11
Ref. [127]12
Difference from NEUT without model from [123] is set to 1.0
13
The uncertainties in BG subtraction for upward-going muons are only for the most horizontal bin,−0.1 < cos θ < 0.
2

Chapter 7. Neutrino Oscillation Analysis

139

60
50
40
30
20
10
0

Sub Sub Sub Sub Sub Su Mult Mult Mult Mult Mult Mult Mult PC PC Up Non Sho-Ge -Ge -Ge -Ge -Ge b-Ge i-Ge i-G i-G i-Rin i-R i-R i-R Stop Thr Stop -sh weru
V e V e V µ V µ V µ V π 0 V e eV e eV µ g e ing ing ingµ ower ing µ-like -like -lik -lik -lik
-like e-lik µ -lik Unc-li -liing
lass
0-d 1-d e 0-d e 1-d e 2-dlike M ke ν ke ν likeµνe e ν ecy e cy e cy e cy e cy e -R
ified
e
e
e

Figure 7.42: Distributions of the best-fit χ2 of observed data (red triangle) andpseudo data sets (blue box). The error bar of pseudo data sets result represents the1σ uncertainty from statistical fluctuations and systematic errors. The pseudo data
have been generated with the best-fit value in NH expanded FV θ13 constrained fit
shown in Table 7.13. The variation shows the effect of the statistical fluctuations of
the new region (50 cm < Dwall < 200 cm).

octant study is as:CLOs =p0 (O2 ),1 − p0 (O1 )(7.6)

Here, O1 and O2 represent the hypotheses with the first octant (sin2 θ23 = 0.4) and thesecond octant (sin2 θ23 = 0.6), respectively. The χ2 values of the best-fit result with thefirst octant hypothesis (O1 ) and the second octant hypothesis (O2 ) are defined as χ2O1
and χ2O2 , respectively. The value of p0 (O1 ) represents p-value with respect to obtaining
a value of ∆χ2O1 −O2 := χ2O1 − χ2O2 larger than the value of data under the hypothesis
with the first octant. The definition of p0 (O2 ) is similar, but the p-value is calculatedwith respected to obtaining a smaller value than the observation. Figure 7.45 shows thepseudo MC distributions of both octant and both hierarchies hypotheses. The shaded
histograms show the definition of p0 (Oi ). The second octant is disfavored at 90.2% and69.5% for the normal and inverted hierarchies, respectively, which means that no strong
preference for the octant is observed.
The parameter CLH
s for the mass hierarchy is defined in the same way, asCLHs =p0 (IH),1 − p0 (NH)(7.7)

which just replaces the hypothesis of the first octant (O1 ) and the second octant (O2 )used in Equation 7.6 with the normal hierarchy (NH) and the inverted hierarchy (IH).
The distribution for the mass hierarchy CLH
s calculation is shown in Figure 7.46. As

Chapter 7. Neutrino Oscillation Analysis

Dwall > 200cm

50 cm < Dwall < 200 cm

EventsEvents

140

400
Data
SK4 Normal
SK4 Inverted

Data
SK4 Normal
SK4 Inverted

100

200

0
-1

1
Events0cos (θ)Events0-10cos (θ)10cos (θ)10cos (θ)10cos (θ)10cos (θ)

1

100

200
50

0
-1

1
Events0cos (θ)Events

0
-1

100

200
50
100

0
cos (θ)0-1

1
EventsEvents

0
-1

200

50
100

0
cos (θ)

100

40

20

50

0
-1

0
-1

1
EventsEvents0-10cos (θ)1

0-1

Figure 7.43: Zenith angle distributions for of data (open circle) and best-fit MC for
both NH (blue solid line) and IH (orange dashed line). Sub-GeV single ring e-like
0-decay electron events are shown here. Left figures are for conventional FV (Dwall >200) cm, while right figures are for new region (50 cm < Dwall < 200 cm). The
momentum range are 100 MeV ∼ 200 MeV, 200 MeV ∼ 300 MeV, 300 MeV ∼ 600
MeV, 600 MeV ∼ 1000 MeV, 1000 MeV ∼ 1330 MeV from most top figures,
respectively.

Chapter 7. Neutrino Oscillation Analysis

141

50 cm < Dwall < 200 cm
EventsEventsDwall > 200cm
Data
SK4 Normal
SK4 Inverted

Data
SK4 Normal
SK4 Inverted

10

20
5

0
cos (θ)0-1

1

0cos (θ)10cos (θ)10cos (θ)10cos (θ)1

EventsEvents

0
-1

20

5
10

0
-1

1
Events0cos (θ)Events

0
-1

4

10
2

0
cos (θ)

10

5

0
-1

0
-1

1
EventsEvents

0
-1

2

1

0
cos (θ)1

0-1

Figure 7.44: Zenith angle distributions for of data (open circle) and best-fit MC for
both NH (blue solid line) and IH (orange dashed line). Multi-GeV single ring νe -likeevents are shown here. Left figures are for conventional FV (Dwall > 200) cm, whileright figures are for new region (50 cm < Dwall < 200 cm). The momentum range are1000 MeV ∼ 2500 MeV, 2500 MeV ∼ 5000 MeV, 5000 MeV ∼ 10000 MeV, >10000
MeV from most top figures, respectively.

Fraction of Pseudo Data Sets

Chapter 7. Neutrino Oscillation Analysis

Fraction of Pseudo Data Sets

142

Best fit, SK θ13 Constrained
sin2θ23=0.6
sin2θ23=0.4
p-value: 0.073
p-value: 0.258

10-2

10-3

Best fit, SK θ13 Constrained
sin2θ23=0.6
sin2θ23=0.4
p-value: 0.236
p-value: 0.224

10-2

10-3

10-4
-30

-20

-10

0

10

χ2 2

10-4
-30

20

-χ2 2

sin θ23=0.4

-20

-10

0

10

χ2 2

sin θ23=0.6

20

-χ2 2

sin θ23=0.4

sin θ23=0.6

10-1

Fraction of Pseudo Data Sets

Fraction of Pseudo Data Sets

Figure 7.45: Distributions of the difference in best-fit χ2 values assuming the firstoctant (sin2 θ23 = 0.4) and second octant (sin2 θ23 = 0.6). The value of sin2 θ13 is fixed
to 0.0210 and expanded FV are used. The result of pseudo data generated assuming
sin2 θ23 = 0.4 (sin2 θ23 = 0.6) is shown as the cyan (orange) histogram. The left figure
shows the true NH case, while the right figure show the true IH case. The calculationof p-value to obtain a value of ∆χ2 more extreme than the observed value,∆χ2data = −0.73, is according to the shaded histogram.

Best fit, SK θ13 Constrained
SK True IH
SK True NH
p-value: 0.025
p-value: 0.919

10-2

10-3

10-1

Best fit, SK θ13 Constrained
SK True IH
SK True NH
p-value: 0.033
p-value: 0.873

10-2

10-3

10-4
-15

-10

-5

0

10-4

5

-15

-10

-5

0

5

χ2 - χ2NH

χ2 - χ2IH

NH2(b) True sin θ23 = 0.425 (best fit)

Fraction of Pseudo Data Sets

Fraction of Pseudo Data Sets(a) True sin θ23 = 0.4

Best fit, SK θ13 Constrained
SK True IH
SK True NH
p-value: 0.065
p-value: 0.717

10-2

10-3

-15

-10

-5

0

Best fit, SK θ13 Constrained

10-2

10-3

10-4

5

SK True IH
SK True NH
p-value: 0.072
p-value: 0.497

-15

-10

-5

0

χ2 - χ2NH
IH(c) True sin2 θ23 = 0.5

IH

2

5

χ2 - χ2NH

IH(d) True sin2 θ23 = 0.6

Figure 7.46: Distributions of the difference in best-fit χ2 values assuming thenormal hierarchy and inverted hierarchy. The value of sin2 θ13 is fixed to 0.0210 and
expanded FV are used. The result of pseudo data generated assuming normal(inverted) hierarchy is shown as the cyan (orange) histogram. The results withdifferent assumed true sin2 θ23 are shown. The calculation of p-value to obtain a valueof ∆χ2 more extreme than the observed value, ∆χ2data = −2.45, is according to theshaded histogram.

Chapter 7. Neutrino Oscillation Analysis

True sin2 θ
p0 (IH)
CLHs23

SK θ13 Constrained Fit
0.4
0.425
0.5
0.025
0.033
0.065
0.308
0.260
0.229

143

0.6
0.072
0.143

Table 7.15: Normal hierarchy significance summarized in terms of the probability ofobserving a χ2 preference for the NH more extreme than that observed in the dataassuming an IH, p0 (IH) , and CLH
s values for a range of assumed parameters. The
first row shows the true θ23 used to generate MC ensembles used in the calculations.
Other oscillation parameters are taken from the analysis’ best-fit.

shown in Figure 7.35, the sensitivity to the mass hierarchy changes significantly for
different value of θ23 . The result of this analysis doesn’t show strong constraint on thisparameter. MC ensembles therefore have been generated with different θ23 values. The
other oscillation parameters are fixed to their best-fit values during the MC generation
process. The p-values and CLH values for the hierarchy test are summarized in Tables

7.15. The observed CLH
s values vary from a 30.8% C.L. preference for the inverted
mass hierarchy to 14.3% as sin2 θ23 increases from 0.4 to 0.6. Less electron neutrino
appearance is expected for a smaller value of sin2 θ23 , which results in a smaller p0 (IH)
values and larger CLH . The preference at the analysis best-fit value is 26.0%, whichsmeans that the inverted hierarchy is disfavored at 74.0% by SK-IV data with fiTQun
reconstruction and the expanded FV. Comparing with the result of NH from the fullSK I-IV analysis based on APFit: 93.0% [6], the preference from this analysis is smaller
since only data from SK-IV is used. Full analysis with fiTQun will be done in the near
future.

Chapter 8

Conclusions and Outlook
We have developed a new algorithm based on a maximum likelihood method for reconstruction of atmospheric neutrino events observed by Super-K. The new algorithm
has improved performance with various metrics including resolution of event vertex andmomentum, ring counting and particle identification compared with those of the conventional reconstruction algorithm. Reliable performance is also demonstrated by the newalgorithm over a larger fiducial volume of the detector. The fiducial volume definition
has been therefore expanded to include all events whose distance to the nearest ID wallis larger than 50 cm for the first time in this analysis, which means a 32% increase in
the number of events used in analysis comparing with the previous SK analyses, whose
cut is 200 cm. An analysis of a 253.9 kton-year exposure of the SK-IV atmospheric data
reconstructed by the new algorithm within the expanded fiducial volume definition has
been performed. The result is consistent with both previous Super-K measurements andthe ones from other experiments. The best-fit value and 1σ error on the atmospheric+0.013+0.0462
mixing parameters are ∆m232 = 2.53+0.22−0.12 , sin θ23 = 0.425−0.037 (0.600−0.030 ) for first(second) octant and δCP = 3.14+2.67−1.35 for the normal mass hierarchy assumption. A weak
preference for the normal mass hierarchy is indicated by the data. The inverted mass
hierarchy is disfavored at 74.0% assuming the oscillation parameters at the best-fit pointin the analysis. Such a preference varies from 69.2% to 85.7% as the value of sin2 θ23changes from 0.4 to 0.6. For the θ23 octant, the second octant is disfavored at 90.2%
and 69.5% assuming the normal and inverted hierarchy, respectively, which means thatcurrent data doesn’t indicate strong preference.
The analysis presented here has demonstrated the potential of the new reconstruction
algorithm to improve our understanding of the neutrino oscillations with only a subsetof the Super-Kamiokande data. The improvement on the expected median sensitivity
with the new algorithm becomes more significant when more data is used, as shown in
145

146

Chapter 8. Conclusions and Outlook

Figure 8.1. Currently the new reconstruction has only been applied to the 3118.5-day
SK-IV fully contained data set, but further studies are ongoing to include other Super-K
samples and run periods, which brings an additional 2,800 days of livetime. Some work

Sensitivity to reject wrong hierarchy ∆χ2

to achieve this goal has already started such as the work shown in Appendix A.

3.5
fiTQun Expanded FV

3fiTQun Conventional FV

2.5

APFit Conventional FV

2
1.5
1
0.5
0

3000

4000

5000

6000

7000

8000
9000
Livetime [days]

Figure 8.1: Median expected sensitivity to mass hierarchy for different livetime.
Normal hierarchy is assumed to be true and the true value of sin2 θ23 is assumed to be
0.5. Here the value of sin2 θ13 is fixed to 0.0210. The result of the eventsreconstructed by fiTQun within the conventional FV and expanded FV are denotedby the gray and blue lines. The sensitivity with events reconstructed by APFit
algorithm and the conventional FV is denoted by the orange line.

Furthermore, the size of the mass hierarchy signal depends on oscillation parameters
including θ13 , θ23 and |∆m232 | as shown in Equation 1.25, 1.26 and 1.27. As a result thelarge uncertainties on them translate directly into reduced hierarchy sensitivity. Besides
the value of θ13 , which is constrained by reactor experiments as discussed in Section 7.6.2,
the value of θ23 and |∆m232 | can also be constrained by accelerator experiments, such as
the T2K experiment [6]. For this reason, the sensitivity to the mass hierarchy, sin2 θ23
octant and δCP of the joint analysis of atmospheric neutrinos and accelerator neutrinos
for both present and future experiments will be discussed below.

Combination with accelerator neutrinos of T2K experiment
In the T2K experiment, the neutrino beam produced at the Japan Proton Accelerator
Research Complex (J-PARC) is directed 2.5 degrees away from the Super-K detector fora narrow band beam with a spectrum peaked at 600 MeV, the first oscillation maximum
for the 295 km, which is the baseline length from neutrino production point to Super-K
detector. A near detector located 280 m downstream of the neutrino production point

Chapter 8. Conclusions and Outlook

147

and at the same off-axis angle is used to measure the unoscillated beam spectrum andto thereby constrain the expected spectrum at Super-K, the far detector (Figure 8.2).

Figure 8.2: A schematic of a neutrino’s journey from the neutrino beamline at
J-PARC, through the near detectors (green dot) which are used to determine theproperties of the neutrino beam, and then 295 km underneath the main island ofJapan to Super-Kamiokande. Taken from [26]Currently the constraint on θ23 and |∆m232 | from T2K [34] is tighter than that fromSuper-K and is statistically independent from the Super-K result. Therefore the sensitivity of the analysis in this thesis can be expected to be increased by taking the T2K
measurements and its systematic error sources into consideration during oscillation analysis.
Indeed, the observable of the atmospheric neutrinos from Super-K and the accelerator
neutrinos from T2K are the same since Super-K serves as the far detector for the latter
experiment. Therefore, the simulated T2K νe appearance and νµ disappearance samples
and associated systematic errors, rather than the effective model used in [6], can be
integrated into the atmospheric analysis directly to incorporate T2K’s measurement.
The sensitivity analysis specifies a set of oscillation parameters for the MC, which isoscillated accordingly to calculate the expected number of events in each bins. Similar
to the analysis described in Section 7.4, a binned χ2 method assuming Poisson statistics
is used to judge the agreement between the oscillated MC with true parameter and test
parameter. Since the systematic errors associated to the accelerator neutrino sample
provided by the T2K experiments are correlated, the definition of χ2 function modified
from Equation 7.3 to :
2

χ =2

X
n

On
En − On + On ln
En


+

XX
i

2 −1
i [σij
] j

(8.1)

j

2 ]−1 is the covariance matrix representing the correlation between i-th and j-th
where [σijsystematic errors. The definition of the penalty term for adjusting the systematic errors
2 ]−1 becomes a diagonal
is consistent with the previous one defined in Equation 7.3 as [σijmatrix if there is no uncertainty correlation.
Another difference between accelerator neutrino sample and atmospheric neutrino sample is that the effect of systematic errors, which are parameterized by different methods.

148

Chapter 8. Conclusions and Outlook“Sub-Binning” based ontrue informaon

Binning based onreconstructed informaon…

Interac on mode

En,0

50 100 150

Sub-Binning for bin(150,30)

CCQE CCMEC CCDIS

10 20 30 …

Reconstructed direc on [° ]e-like sample in -mode

En,m,0En,0,0 En,1,0 En,2,0 …
50 100 150…

…

True neutrino energy [MeV]Reconstructed momentum [MeV]

50 100 150…

Interac on mode

Sub-Binning for bin(150,20)

CCQE CCMEC CCDIS

10 20 30 …

Reconstructed direc on [° ]-like sample in -mode

50 100 150…

…

True neutrino energy [MeV]Reconstructed momentum [MeV]… Binning for other sample… Sub-Binning for other bin

Figure 8.3: Sub-binning structure for the T2K systematic uncertainty
parameterization. The binnings shown on the left are based on reconstructed
information, while the ones shown on the right are based on true information. En,0
represents the number of events of nominal MC prediction in n-th bin based onreconstructed information, while En,m,0 denotes the number of events in the m-th
sub-bin for n-th bin.

In the Super-K analysis, the original definition of the expectation of the number of eventsin n-th bin based on reconstructed information after incorporating the systematic error
En in Equation 7.4:
En = En,0 (1 +X

fni i )iassumes that the systematic uncertainty affects the number of events in each bin ofreconstructed energy directly. However, in the T2K experiment, a more detailed parameterization method is used: the systematic uncertainties of the neutrino flux orinteraction cross-section also depend on the true neutrino energy or interaction mode
(i.e. charged-current or neutral-current) besides the reconstructed energy and direction.
To emulate the parameterization of T2K in Super-K structure, each reconstruction energy bin En is further divided into several “sub-bin”, which considers the true neutrinoenergy and interaction mode for MC (c.f Figure 8.3).

Chapter 8. Conclusions and Outlook

149

The definition of En,0 , which represents the nominal MC expectation in n-th bin and
En is also updated to:
En,0 =

X

En,m,0

m

En =

X
m

En,m,0 (1 +

X

i
fn,m
i )

i

i
where m is the sub-binning number in each bin n based on reconstructed bin, fn,mrepresents the coefficient for the i-th systematic uncertainty on the m-th sub-bin of n-thbin. Such a “sub-bin” definition of En is only used for accelerator neutrinos since theeffect of the dependency on true neutrino energy and interaction mode is not consideredduring the systematic error evaluation process for atmospheric neutrinos. The error
values are the same as the one in [129].
Currently, the correlation between the systematics of the two kinds of neutrino sources
is not considered except the energy scale since the evaluation method is different forthese two experiments. Further efforts are needed to unify the systematic error related
to the cross-section and detector.
The expected sensitivities as the function of true sin2 θ23 with different running time
are shown in Figure 8.4. The atmospheric neutrino data collected by all four Super-K
phases within the convention FV are considered here.
The sensitivity to neutrino mass hierarchy with joint analysis (dashed line) of the atmop
spheric neutrino sample and accelerator neutrino sample is expected be ∆χ2 > 2.5(3.0)
for sin2 θ23 > 0.55 and true δCP = 0◦ (270◦ ) at the end of 2026, while the result withp
atmospheric neutrino only (solid line) is ∆χ2 > 2 at the same condition. The atmospheric neutrinos show higher sensitivity comparing with accelerator neutrinos sincematter effects are strong in the resonance-enhanced oscillation region of the atmospheric
neutrino energy spectrum (2-10 GeV). On the other hand, for accelerator neutrinos, theirenergy is low (∼ 600 MeV) and the baseline for the neutrino oscillation is short (295km). Furthermore, as shown in Figure 8.4, the sensitivity of accelerator neutrinos depends on the true value of δCP largely due to the degeneracy between the CP violationand the mass hierarchy caused by its fixed baseline. It is therefore difficult to determine
the mass hierarchy by accelerator neutrino measurement only. Anyway, accelerator neutrino data provides a clean measurement of the atmospheric mixing parameters which
determines the size of the matter effects, and the sensitivity to the mass hierarchy istherefore improved.
Figure 8.5 shows the sensitivity to the CP violation (sin δCP 6= 0) when the true mass
hierarchy is assumed to be the normal hierarchy. Although the sensitivity with theChapter 8. Conclusions and Outlook∆χ2 Wrong Hierarchy Rejection∆χ2 Wrong Hierarchy Rejection

150

3
2.5
2
1.5
1

3
2.5
2
1.5
1

0.5
0

3.5

0.5

0.4

0.45

0.5

0.55

0

0.6
sin2θ23

0.4

0.45

0.5

0.55

0.6
sin2θ23

∆ χ2

∆ χ2

Figure 8.4: Sensitivity to mass hierarchy as function of true sin2 θ23 . The solid lines(dotted lines) are the sensitivity with atmospheric neutrino only (accelerator neutrinoonly), while the dashed lines represent the result of joint analysis. The result shown inleft figure assumes true δCP = 0◦ , while the right figure is for true δCP = 270◦ . Theorange, purple, magenta and green lines represent the result at the end of year 2026,
2023, 2020 and 2017, respectively.

SK+T2K

3.5

SK+T2K

4

T2K
3

T2K
3.5

SK

SK

3

2.5

2.5
2
2
1.5
1.5
1

1

0.5
0

0.5
-150

-100

-50

0

50

100

150
δCP [degree]

0

-150

-100

-50

0

50

100

150
δCP [degree]

Figure 8.5: Sensitivity to CP violation (sin δCP 6= 0) as function of true sin δCP .
The true mass hierarchy of left figure is assumed to be normal hierarchy, while forright figure it is assumed to be inverted hierarchy. The green, blue and red lines
represent the sensitivity with atmospheric neutrino only, accelerator neutrino only
and combined analysis at the end of year 2026, respectively.

accelerator neutrinos from the T2K experiment (blue line) is higher than that withatmospheric neutrinos (green line) overall, the degeneracy between the mass hierarchy
and the CP phase constrains the sensitivity. By combining these two samples, thedegeneracy is resolved and the sensitivity is increased more than 1σ in the region where
the degeneracy was observed before.
For the sin2 θ23 octant, as shown in Figure 8.6, although the sensitivity from accelerator
neutrino samples is larger than that from atmospheric neutrino samples, the sensitivity
can be increased further by combining these two samples.

∆χ2 Wrong Octant Rejection

Chapter 8. Conclusions and Outlook

151

SK+T2K

6

T2K
SK

5
4
3
2
1
0

0.4

0.45

0.5

0.55

0.6
sin2θ23

Figure 8.6: Sensitivity to sin2 θ23 octant ( > 0.5 or < 0.5) as function of truesin2 θ23 . The green, blue and red lines represent the sensitivity with atmospheric
neutrino only, accelerator neutrino only and combined analysis at the end of year
2026, respectively.

Future experiment: Hyper-Kamiokande
It has been shown that current sensitivity to the mass hierarchy has been limited by thelack of statistics. A next generation gigantic underground water Cherenkov ring imaging
detector, Hyper-Kamiokande (Hyper-K) is proposed to obtain more statistics. It consistsof at least one and possibly two water tanks with the dimensions of the 74 m (D) × 78
m (H) for each. The total fiducial water mass will be 0.37 million tons, approximately
17 times as large as that of Super-K detector [44]. The first tank is planed to be placedin Japan at 295 km from the J-PARC neutrino beam with 2.5◦ off-axis angles as theSuper-K detector, while the second detector if built, is being considered in Korea, with
an off-axis angle of 1.3◦ and a baseline of 1,088 km[130]. The Hyper-K experiment is
expected to start its operation around 2026 [44].
A combined analysis of accelerator beam and atmospheric neutrino data assuming aHyper-K detector in Japan (JD) and at the Mt. Bisul in Korean (KD) and comparewith sensitivities assuming two detectors in Kamioka, Japan (JD × 2) are shown. The
treatment of the atmospheric neutrino samples and their systematic errors follows thatof Super-Kamiokande, with no assumed improvements, while the systematic errors foraccelerator neutrino samples is estimated based on the T2K experiment taking into
account improvements expected in the future T2K data and analysis [44, 130].
The physics sensitivity shown here assumes that one or two 187 kton detectors will be
operated over 10 years (10 ×107 sec) at 1.3 MW beam power, corresponding to 27 × 1021

Chapter 8. Conclusions and Outlook∆χ2 Wrong Hierarchy Rejection

152

14

Atm+Beam (JD+KD)
Beam (JD+KD)

12

Atm+Beam (JD × 2)
Atm+Beam (JD × 1)
Atm (JD+KD)

10
8
6
4
2
0

0.4

0.45

0.5

0.55

0.6
sin2 θ23

Figure 8.7: Sensitivity to the normal mass hierarchy for components of a combined
measurement of beam and atmospheric neutrinos for a 10 year exposure. Here JD
refers to a single Hyper-K detector in Kamioka, Japan, and JD ×2 refers two to suchdetectors operating simultaneously. The horizontal axis shows the assumed value ofsin2 θ23 and the width of the bands shows the variation in sensitivity with δCP .

protons on target. The total exposure for atmospheric neutrinos is expected to be 1.87
Mton·year for one tank (3.74 Mton·year for two tanks).
Figure 8.7 shows the sensitivity to the mass hierarchy for the combined analysis. After10 years of running the expected ability to determine the mass hierarchy is better thanp∆χ2 = 7. Though the combined JD and Mt. Bisul beam measurement has better
sensitivity than the atmospheric neutrino measurement alone, the sensitivity improves
further when all of the samples are combined. The power of this improvement results
in an earlier realization of the hierarchy as shown in the left plot of Figure 8.10. Thepsensitivity is expected to exceed ∆χ2 = 4 within two years of operations in thisexperimental configuration.
Atmospheric neutrinos provide additional constraints on δCP as shown in the Figure 8.8
although it has less sensitivity than the measurement with accelerator neutrinos. Similarly, the constraint on the CP violation from atmospheric neutrino is weaker thanthat from the accelerator and provides only a slight improvement in sensitivity shown
Figure 8.9. However, as with the other oscillation measurements, the power of the combined accelerator and atmospheric measurement comes in the early realization of thissensitivity as shown in the right plot of Figure 8.10.
As discussed above, the sensitivity to the mass hierarchy for Super-K can be increasedby analyzing the accelerator beam neutrinos and atmospheric neutrinos simultaneously.
p
The sensitivity expected be ∆χ2 > 2.5(3.0) for sin2 θ23 > 0.55 at the end of 2026 fortrue δCP = 0◦ (270◦ ).

∆χ2Chapter 8. Conclusions and Outlook

153

14
Beam (JD+KD)
Atm+Beam (JD+KD)

12
JD Beam Only
Mt. Bisul Beam Only10

Atm (JD+KD)

8
6
4
2
0

-150

-100

-50

0

50

100

150

δCP

∆χ2

∆χ2

Figure 8.8: Sensitivity to δCP = 0 for components of a combined measurement ofbeam and atmospheric neutrinos for a 10 year exposure. Here JD refers to a single
Hyper-K detector in Kamioka, Japan.

Atm+Beam (JD+KD)
Beam (JD+KD)
Atm+Beam (JD × 2)
Atm+Beam (JD × 1)
Atm (JD+KD)

10

8

8

6

6

4

4

2

2

0

-150

-100

-50

0

50

Atm+Beam (JD+KD)
Beam (JD+KD)
Atm+Beam (JD × 2)
Atm+Beam (JD × 1)
Atm (JD+KD)

10

100

150

0
δCP

-150

-100

-50

0

50

100

150

δCP

Figure 8.9: Sensitivity to CP violation (sin δCP 6= 0) for components of a combined
measurement of beam and atmospheric neutrinos for a 10 year exposure. Here JD
refers to a single Hyper-K detector in Kamioka, Japan, and JD × 2 refers two to suchdetectors operating simultaneously. The horizontal axis shows the assumed true valueof δCP .

On the other hand, Hyper-K has strong potential to determine neutrino mass hierarchyand discover leptonic CP violation by the observation of accelerator and atmospheric
neutrino oscillation. Joint analysis of these two kinds of neutrinos at Hyper-K can
improve its sensitivity further. After 10 years running, the significance to determine themass hierarchy is expected to reach 5σ, when sin2 θ23 = 0.5 and a CP violation discovery
at 3σ or more can be made for 70% of the δCP phase space.

Chapter 8. Conclusions and Outlook10

sin2θ23 = 0.4

Atm + Beam ( JD + KD)

sin2θ23 = 0.5

Atm + Beam ( JD × 2 )sin2θ23 = 0.6

Atm + Beam ( JD × 1 )Fraction of δcp for 3σ CPV∆χ2 Wrong Hierarchy Rejection

154

8

6

1
0.9
0.8
0.7
0.6
0.5

sin2θ23 = 0.4

Atm + Beam ( JD + KD)

0.3

sin θ23 = 0.5

Atm + Beam ( JD × 2 )0.2

sin2θ23 = 0.6

Atm + Beam ( JD × 1 )

0.4
4

2

2

0.1
0

1

2

3

4

5

6

7

8
9
10
Running Time (Years)

0

1

2

3

4

5

6

7

8
9
10
Running Time (Years)

Figure 8.10: Sensitivity as a function of year to the mass hierarchy (left) and thefraction of δCP phase space for which CP violation (sin δCP 6= 0) can be determinedat 3σ or better. Red lines show a combined beam and atmospheric neutrino
measurement with one Hyper-K detector in Kamioka, Japan (JD) and one at the Mt.
Bisul site in Korea. Cyan lines show the same analysis assuming two detectors in
Kamioka (JD × 2) and grey lines assume only one detector in Kamioka (JD × 1).
Different symbols show the assumed value of sin2 θ23 .

Appendix A

SK-I retuning
As has been introduced in Section 6.2.3, the new reconstruction tool fiTQun, is expected
to be used for ATMPD analysis, but large data/MC discrepancy were found for the PID
likelihood of SK-I FC samples, while such discrepancy is not observed for SK-I APFit,
as shown in Figure A.1. The distribution of such sample for SK-IV data and MC are
also in good agreement, as shown in Figure 6.3.
The key to distinguish e-like events and µ-like events is the difference on charge pattern:rings from electrons tend to have rough edges produced by the light from their electromagnetic showers, while rings from muons or charged pions predominately produce
crisp edges. Figure A.2 and Figure A.3 show the charge profile and hit profile withrespect to the vertex and direction of particles for sub-GeV single-ring events of SK-I
and SK-IV for. The ratio between data and MC is shown in Figure A.4, which indicatesa good agreement on the charge profile for both SK-I and SK-IV (left), while a large
discrepancy on the hit profile for SK-I between data and MC can be seen clearly. Morehits observed for SK-I MC than that of data.

SK4 Data
νe CC
νe CC
νµ CC
νµ CC
NC

450
400
350
300
250

SK4 Data
νe CC
νe CC
νµ CC
νµ CC
NC

600
500
400
300

200
200

150
100

100
50
0
-3000

-2000

-1000

0

1000

2000

0
-10

3000

-8

-6

-4

-2

0

2

4

6

8

10

Figure A.1: PID likelihood distribution of sub-GeV single-ring events for SK-I. Left(right) figure shows the distribution reconstructed by fiTQun (APFit).

155

156

Appendix A. SK-I retuning

Figure A.2: Charge profile with respect to the reconstructed vertex and direction of
sub-GeV single-ring events for SK-I (left) and SK-IV (right). Points denote the dataand solid line is for MC.

Figure A.3: Hit profile with respect to the reconstructed vertex and direction for
sub-GeV single-ring events for SK-I (left) and SK-IV (right). Points denote the dataand solid line is for MC.

Figure A.4: Ratio between data and MC of charge profile (left) and hit profile(right) for sub-GeV single-ring events for SK-I (black) and SK-IV (red).

Appendix A. SK-I retuning

157

Figure A.5: Charge distribution around the Cherenkov angle (42◦ ) for sub-GeV
single-ring events of SK-I (left) and SK-IV (right). Points denote the distribution ofdata, while the solid line is for MC.

Figure A.6: Charge distribution around the Cherenkov angle (42◦ ) for Michelelectron from cosmic ray muon of SK-I (left) and SK-IV (right). Points denote the
distribution of data, while the solid line is for MC.

Figure A.5 and Figure A.6 show the charge distribution around the Cherenkov angle(42◦ ) for sub-GeV single-ring events and Michel electron from cosmic ray muon, respectively. Large discrepancy is observed in low charge region for SK-I, which results in the
disagreement on hit pattern. Such discrepancy has limited effect on the charge patternsince the charge for those hits are small.
The ideal method to resolve issue is to retune the single p.e. distribution for SK-I with
the nickel data as introduced in Section 4.1.2. The reason to double the gain of PMT
is for the measurement of the shape below the trigger threshold. However, those data is
not found yet since they were taken 20 years ago. The retuning of single p.e. distribution
is done by adding the observed discrepancy between MC and data with considering theeffect from trigger threshold. The distribution under the trigger threshold is tuned tokeep the mean value of single p.e. unchanged since the charge pattern agrees well.
The event selection for the nickel data is:
1. N50 > 30

158

Appendix A. SK-I retuning

Figure A.7: Hit time distribution of each PMT for Nickel source after event
selection. The on-time region and off-time region are marked with blue line.

2. Distance between the vertex reconstructed by low energy fitter and the positionof nickel source is less than 4 m.
3. Time difference between two events is longer than 10µs.
The data from Nickel source taken on 10, August, 1996 for calibration. The hit timeand charge distribution of each PMT are shown in Figure A.7 and Figure A.8. To
remove the effect of dark hits caused by electronic noise, the charge distribution wereshown for on-time region and off-time region respectively. The final observed single p.e.
distribution is obtained from the difference from these two charge distribution.
The tuned single p.e. distribution with and without is shown in Figure A.9. Tuned MC
has good agreement with the nickel data.
The charge profile and hit profile after tuning is shown in Figure A.10. The agreement
between data and MC on hit profile increased a lot while the charge profile is almost
not affected as our expectation.
The momentum and PID of Michel decay electron from cosmic ray muon before andafter tuning is shown in Figure A.11 and Figure A.12. The discrepancy between data
and MC for the momentum is at the same level (∼ 1.2%, while the agreement on the
PID distribution increased.
Similarly, the PID distribution for sub-GeV single-ring event after tuning is shown in
Figure A.13. The agreement improved large although some discrepancy remained dueto the remaining discrepancy shown in Figure A.14, which need further investigation.

Appendix A. SK-I retuning

Figure A.8: Charge distribution of each PMT for Nickel source. Black (red) line
denote the distribution from on-time (off-time) region.

Figure A.9: Charge distribution for single p.e. with (left) and without (right) the
trigger effect. The points on the right shows the data from Nickel source taken on 10,
August, 1996 for calibration.

Figure A.10: Charge profile (left) and hit profile (right) for sub-GeV single-ring
events for SK-I (black) and SK-IV (red).

159

160

Appendix A. SK-I retuning

Figure A.11: PID likelihood distribution for SK-I Michel electron from cosmic ray
muon before (left) and after (right) retuning.

Figure A.12: Momentum distribution for SK-I Michel electron from cosmic ray
muon before (left) and after (right) retuning.

SK4 Data
νe CC
νe CC
νµ CC
νµ CC
NC

500

400

300

200

100

0
-3000

-2000

-1000

0

1000

2000

3000

Figure A.13: PID likelihood distribution of sub-GeV single-ring events for SK-I
after retuning.

Appendix A. SK-I retuning

Figure A.14: Charge distribution around the Cherenkov angle (42◦ ) for sub-GeV
single-ring events of. Points denote the distribution of data, while the solid line is for
MC.

161

Appendix B

Performance check for fiducial
volume expansion
Table B.1, B.2, and B.3 show the event rate and component fraction for each analysis
sample in conventional FV (Dwall > 200 cm), new region (50 cm < Dwall < 200 cm) andexpanded FV (Dwall > 50 cm) respectively.
Basic distribution, including the event rate, interaction fraction, direction resolution,
momentum bias, energy distribution and zenith angle distribution, for all 13 FC event
samples are shown here to validate fiducial volume expansion. Different figures share thesame legends, which indicate the different interaction mode. The event reconstruction
and selection are done by fiTQun.
There are no serious issues observed when expanding the FV to Dwall > 50 cm.

163

164

Appendix B. Performance check for FV expansion

SampleCC νe CC ν¯e CC νµ + ν¯µ CC ντ NC Data MC
Fully Contained (FC) Sub-GeV
e-like, Single-ring
0 decay-e
0.728 0.242
0.001
0.000 0.028 57965422.3
1 decay-e
0.907 0.020
0.033
0.001 0.040 596 559.3µ-like, Single-ring
0 decay-e
0.010 0.004
0.795
0.001 0.191 10371011.9
1 decay-e
0.000 0.000
0.974
0.000 0.026 53955086.3
2 decay-e
0.000 0.000
0.984
0.000 0.016 493 493.5
π 0 -likeTwo-ring
0.051 0.016
0.011
0.000 0.922 13461304.5
Fully Contained (FC) Multi-GeV
Single-ringνe -like
0.726 0.077
0.058
ν̄e -like
0.553 0.379
0.003µ-like0.000 0.000
0.996
Multi-ringνe -like
0.588 0.117
0.054
ν̄e -like
0.526 0.300
0.021µ-like0.010 0.001
0.959
Other
0.283 0.026
0.342
Partially Contained (PC)
Stopping
0.084 0.033
Through-going 0.007 0.002

0.840
0.985

Upward-going Muons (Up-µ)Stopping
0.008 0.003
0.987
Through-goingNon-showering 0.002 0.001
0.996
Showering
0.001 0.000
0.998

0.027 0.113
0.008 0.056
0.003 0.001

217 208.1
963 975.2
882 907.0

0.036 0.204
0.020 0.134
0.004 0.026
0.053 0.295

759 773.4
644 654.6
19821943.1
10381024.5

0.000 0.043
0.000 0.006

374 374.4
16851711.2

0.000 0.003

843 776.6

0.000 0.001
0.000 0.001

30132956.3
547 550.8

Table B.1: Sample purity broken down by neutrino flavor assuming neutrino
oscillations with ∆m232 = 2.52 × 10−3 eV2 and sin2 θ23 = 0.51. The right two columns
show the sample size of the data and MC for the entire SK data set, respectively.
Only fully contained events are reconstructed by fiTQun. Partially contained eventsand upward-going muon are reconstructed by APFit. All of the samples shown here
have a FV cut of 200 cm.

Appendix B. Performance check for FV expansion

165

Component Fraction (fQ)
2

Fraction

Number of Events

Event Rate Distribution (fQ)
400
350

1.5

300

ν e CCν e CCν µ CCν µ CCν τ// ν τ CC

NC

Entering ν Events

250
1

200
150

0.5

100
50
0
0

50

100

150

200

250

300

0
0

350
400
Dwall [cm]

50

150

200

250

300

lepton

40

1.2

1.15

p rec/p

35
30
25

1.1

1.05

20

1

15

0.95

10

0.9

5

0.85

0
0

50

100

150

200

250

300

350

0.8
0

400

50

100

150

200

250

300

Dwall [cm]Dwall > 200 cm
SK4 Data
ν e CCν e CCν µ CCν µ CCντ
NC

Momentum

1400
1200
1000
800
600

350

400

Dwall [cm]200 cm > Dwall > 50 cm
450

SK4 Data
ν e CCν e CCν µ CCν µ CC
ντ
NC

400
350
300
250

Dwall > 50 cm
SK4 Data
ν e CCν e CCν µ CCν µ CC
ντ
NC

1800
1600
1400
1200
1000

200

800

150

600

100

400

400
200
0
0

50
200

400

600

800

1000
1200
1400
Momentum [MeV]

0
0

200
200

400

600

800

1000
1200
1400
Momentum [MeV]

600
500

220

900

200

800

180

700

160

200

400

600

800

1000
1200
1400
Momentum [MeV]

600

140

400
300
200

120

500

100

400

80

300

60

200

40

100

100

20
0
-1

0
0

240

700

Zenith Angle

350
400
Dwall [cm]fQ Momentum Bias (Rec/True)

Direction Resolution fQ

Dir Res. to lepton [Deg]

100

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6
0.8
1
Zenith angle

0
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6
0.8
1
Zenith angle

0
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6
0.8
1
Zenith angle

Figure B.1: Basic distribution for SubGeV single-ring e-like 0 decay-e sample
reconstructed and selected by fiTQun. Top four figures shows the event rate,component fraction, direction resolution and momentum bias as the function of the
distance to nearest wall (dwall), respectively. The error bar shows the statisticaluncertainty. Bottom six figures shows the energy distribution and zenith angle
distribution in different detector region. Same plots are shown for next 12 figures.

166

Appendix B. Performance check for FV expansion

Component Fraction (fQ)
2

Fraction

Number of Events

Event Rate Distribution (fQ)
45
40

1.5

35

ν e CCν e CCν µ CCν µ CCν τ// ν τ CC

NC

Entering ν Events

30
25

1

20
15
0.5

10
5
0
0

50

100

150

200

250

300

0
0

350
400
Dwall [cm]

50

150

200

250

300

lepton

40

1.2

1.15

p rec/p

35
30
25

1.1

1.05

20

1

15

0.95

10

0.9

5

0.85

0
0

50

100

150

200

250

300

350

0.8
0

400

50

100

150

200

250

300

Dwall [cm]Dwall > 200 cm
SK4 Data
ν e CCν e CCν µ CCν µ CCντNC

120

Momentum

350
400
Dwall [cm]fQ Momentum Bias (Rec/True)

Direction Resolution fQ

Dir Res. to lepton [Deg]

100

100

80

350

400

Dwall [cm]200 cm > Dwall > 50 cm
60

SK4 Data
ν e CCν e CCν µ CCν µ CC
ντ
NC

50

40

30

60

Dwall > 50 cm
SK4 Data
ν e CCν e CCν µ CCν µ CC
ντ
NC

180
160
140
120
100
80

20

40

60
40

10

20

20
0
0

200

400

600

800

1000
1200
1400
Momentum [MeV]

0
0

200

400

600

800

1000
1200
1400
Momentum [MeV]

0
0

200

400

600

800

1000
1200
1400
Momentum [MeV]Zenith Angle

80
30
100

70
25
60

80

50

20

40

15

60

30

40
10

20

0
-1

20

5

10
-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6
0.8
1
Zenith angle

0
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6
0.8
1
Zenith angle

0
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6
0.8
1
Zenith angle

Figure B.2: Basic distribution for SubGeV single-ring e-like with 1 or more decay-esample reconstructed and selected by fiTQun.

Appendix B. Performance check for FV expansion

167

Component Fraction (fQ)
2

Fraction

Number of Events

Event Rate Distribution (fQ)

120
100

1.5

ν e CCν e CCν µ CCν µ CCν τ// ν τ CC

NC

Entering ν Events

80
1

60
40

0.5
20
0
0

50

100

150

200

250

300

0
0

350
400
Dwall [cm]

50

150

200

250

300

lepton

40

1.2

1.15

p rec/p

35
30
25

1.1

1.05

20

1

15

0.95

10

0.9

5

0.85

0
0

50

100

150

200

250

300

350

0.8
0

400

50

100

150

200

250

300

Dwall [cm]Dwall > 200 cm
SK4 Data
ν e CCν e CCν µ CCν µ CCντNC

300

Momentum

350
400
Dwall [cm]fQ Momentum Bias (Rec/True)

Direction Resolution fQ

Dir Res. to lepton [Deg]

100

250

200

350

400

Dwall [cm]200 cm > Dwall > 50 cm
SK4 Data
ν e CCν e CCν µ CCν µ CC
ντ
NC

100

80

60

150

Dwall > 50 cm
SK4 Data
ν e CCν e CCν µ CCν µ CC
ντ
NC

400
350
300
250
200

40

100

150
100

20

50

50
0
0

200

400

600

800

1000
1200
1400
Momentum [MeV]

0
0

200

400

600

800

1000
1200
1400
Momentum [MeV]Zenith Angle

140

0
0

200

400

600

800

1000
1200
1400
Momentum [MeV]

200
60
180

120

160

50
100

140
40

120

80

100

30

60

80

40

20

20

10

0
-1

0
-1

60
40
20

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6
0.8
1
Zenith angle

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6
0.8
1
Zenith angle

0
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

Figure B.3: Basic distribution for SubGeV single-ring µ-like 0 decay-e sample
reconstructed and selected by fiTQun.

0.6
0.8
1
Zenith angle

168

Appendix B. Performance check for FV expansion

Component Fraction (fQ)
2

Fraction

Number of Events

Event Rate Distribution (fQ)

300

1.5

250

ν e CCν e CCν µ CCν µ CCν τ// ν τ CC

NC

Entering ν Events

200
1
150
100

0.5

50
0
0

50

100

150

200

250

300

0
0

350
400
Dwall [cm]

50

150

200

250

300

lepton

40

1.2

1.15

p rec/p

35
30
25

1.1

1.05

20

1

15

0.95

10

0.9

5

0.85

0
0

50

100

150

200

250

300

350

0.8
0

400

50

100

150

200

250

300

Dwall [cm]Dwall > 200 cm
SK4 Data
ν e CCν µ CCν µ CCντ

Momentum

1200
1000
800

NC

600

350

400

Dwall [cm]200 cm > Dwall > 50 cm
SK4 Data
ν e CCν µ CCν µ CC
ντ

400
350
300
250

NC

Dwall > 50 cm
1800

SK4 Data
ν e CCν µ CCν µ CC
ντ

1600
1400
1200
1000

200

800

150

600

100

400

NC

400
200
0
0

50
200

400

600

800

1000
1200
1400
Momentum [MeV]700

Zenith Angle

350
400
Dwall [cm]fQ Momentum Bias (Rec/True)

Direction Resolution fQ

Dir Res. to lepton [Deg]

100

0
0

200
200

400

600

800

1000
1200
1400
Momentum [MeV]

500

180

800

160

700

140

600

120

400

400

600

800

1000
1200
1400
Momentum [MeV]

500

100
300

200

900

200

600

0
0

400

80
300
60

200

200

40
100

100

20
0
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6
0.8
1
Zenith angle

0
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6
0.8
1
Zenith angle

0
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

Figure B.4: Basic distribution for SubGeV single-ring µ-like 1 decay-e sample
reconstructed and selected by fiTQun.

0.6
0.8
1
Zenith angle

Appendix B. Performance check for FV expansion

169

Component Fraction (fQ)
2

Fraction

Number of Events

Event Rate Distribution (fQ)
30
25

1.5

ν e CCν e CCν µ CCν µ CCν τ// ν τ CC

NC

Entering ν Events

20
1

15
10

0.5
5
0
0

50

100

150

200

250

300

0
0

350
400
Dwall [cm]

50

150

200

250

300

lepton

40

1.2

1.15

p rec/p

35
30
25

1.1

1.05

20

1

15

0.95

10

0.9

5

0.85

0
0

50

100

150

200

250

300

350

0.8
0

400

50

100

150

200

250

300

Dwall [cm]MomentumDwall > 200 cm
100

SK4 Data
ν µ CC80

ν µ CC
ντ

60

NC

350

400

Dwall [cm]200 cm > Dwall > 50 cm
30
25

20

Dwall > 50 cm

SK4 Data
ν e CCν µ CCν µ CC
ντ

140

NC

80

15

SK4 Data
ν e CCν µ CCν µ CC
ντ

120
100

NC

60

40
10
20

0
0

40

5

200

400

600

800

1000
1200
1400
Momentum [MeV]

0
0

20

200

400

600

800

1000
1200
1400
Momentum [MeV]24

Zenith Angle

350
400
Dwall [cm]fQ Momentum Bias (Rec/True)

Direction Resolution fQ

Dir Res. to lepton [Deg]

100

0
0

200

400

600

800

1000
1200
1400
Momentum [MeV]

100

22

70

20
60

80

18
16

50

14
40

60

12
10

30

40

8
20

6
20

4

10

2
0
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6
0.8
1
Zenith angle

0
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6
0.8
1
Zenith angle

0
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6
0.8
1
Zenith angle

Figure B.5: Basic distribution for SubGeV single-ring µ-like with 2 or more decay-esample reconstructed and selected by fiTQun.

170

Appendix B. Performance check for FV expansion

Component Fraction (fQ)
2

Fraction

Number of Events

Event Rate Distribution (fQ)
80
70

1.5
60

ν e CCν e CCν µ CCν µ CCν τ// ν τ CC

NC

Entering ν Events

50
1

40
30

0.5

20
10
0
0

50

100

150

200

250

300

0
0

350
400
Dwall [cm]

50

150

200

250

300

350
400
Dwall [cm]fQ Momentum Bias (Rec/True)

Direction Resolution fQ

100
90

p rec/pν

Dir Res. to ν [Deg]

100

80

1.4
1.2

70

1

60
0.8

50
40

0.6

30

0.4

20
0.2

10
0
0

50

100

150

200

250

300

350

0
0

400

50

100

150

200

250

300

Dwall [cm]Dwall > 200 cm
SK4 Data
ν e CCν e CCν µ CCν µ CCντ
NC

Momentum

700
600
500
400

400

200 cm > Dwall > 50 cm
SK4 Data
ν e CCν e CCν µ CCν µ CC
ντ
NC

200
180
160
140
120

Dwall > 50 cm
900

SK4 Data
ν e CCν e CCν µ CCν µ CC
ντ
NC

800
700
600
500

100
400

300

80
300
60

200

200

40
100
0
0

100

20
200

400

600

800

1000
1200
1400
Momentum [MeV]

0
0

200

400

600

800

1000
1200
1400
Momentum [MeV]

0
0

200

400

600

800

1000
1200
1400
Momentum [MeV]

220

180

Zenith Angle

350

Dwall [cm]

200

160

50
180

140

160

40

120

140

100

120

30

100

80

80

20

60

60
40

40

10
20
0
-1

20
-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6
0.8
1
Zenith angle

0
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6
0.8
1
Zenith angle

0
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6
0.8
1
Zenith angle

Figure B.6: Basic distribution for SubGeV two-ring π 0 sample reconstructed andselected by fiTQun.

Appendix B. Performance check for FV expansion

171

Component Fraction (fQ)
2

Fraction

Number of Events

Event Rate Distribution (fQ)
14
12

1.5

10

ν e CCν e CCν µ CCν µ CCν τ// ν τ CC

NC

Entering ν Events

8

1

6
4

0.5

2
0
0

50

100

150

200

250

300

0
0

350
400
Dwall [cm]

50

150

200

250

300

lepton

40

1.2

1.15

p rec/p

35
30
25

1.1

1.05

20

1

15

0.95

10

0.9

5

0.85

0
0

50

100

150

200

250

300

350

0.8
0

400

50

100

150

200

250

300

Dwall [cm]MomentumDwall > 200 cm

40

30

350

400

Dwall [cm]200 cm > Dwall > 50 cm

SK4 Data
ν e CCν e CCν µ CCν µ CCντNC

50

SK4 Data
ν e CCν e CCν µ CCν µ CC
ντ
NC

12

10

8

Dwall > 50 cm
SK4 Data
ν e CCν e CCν µ CCν µ CC
ντ
NC

70
60
50
40

6

30

20
4
10

20

2

0
10

Zenith Angle

350
400
Dwall [cm]fQ Momentum Bias (Rec/True)

Direction Resolution fQ

Dir Res. to lepton [Deg]

100

4

10
Momentum [MeV]

5

10

0
10

4

10
Momentum [MeV]

5

0

12

50

30

10

40

25

8

10 4

10
Momentum [MeV]

35

30
20

6

15

20
4

10

0
-1

10

2

5

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6
0.8
1
Zenith angle

0
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6
0.8
1
Zenith angle

0
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6
0.8
1
Zenith angle

Figure B.7: Basic distribution for MultiGeV single-ring νe -like sample reconstructed
and selected by fiTQun.

5

172

Appendix B. Performance check for FV expansion

Component Fraction (fQ)
2

Fraction

Number of Events

Event Rate Distribution (fQ)

50

1.5

40

ν e CCν e CCν µ CCν µ CCν τ// ν τ CC

NC

Entering ν Events
30
1
20
0.5
10

0
0

50

100

150

200

250

300

0
0

350
400
Dwall [cm]

50

150

200

250

300

lepton

40

1.2

1.15

p rec/p

35
30
25

1.1

1.05

20

1

15

0.95

10

0.9

5

0.85

0
0

50

100

150

200

250

300

350

0.8
0

400

50

100

150

200

250

300

Dwall [cm]Dwall > 200 cm
Momentum

300
250
200

350

400

Dwall [cm]200 cm > Dwall > 50 cm

SK4 Data
ν e CCν e CCν µ CCν µ CCντNC

350

SK4 Data
ν e CCν e CCν µ CCν µ CC
ντ
NC

90
80
70
60
50

Dwall > 50 cm
SK4 Data
ν e CCν e CCν µ CCν µ CC
ντ
NC

450
400
350
300
250

40

200

30

150

20

100

150
100
50

10

0

10 4

10
Momentum [MeV]

5

50

0

10 4

10
Momentum [MeV]

30

160

Zenith Angle

350
400
Dwall [cm]fQ Momentum Bias (Rec/True)

Direction Resolution fQ

Dir Res. to lepton [Deg]

100

140

5

0

10 4

10
Momentum [MeV]

180
160

25

140

120
20

120

100
80

100

15

80
60

10

60

40

40
5

20
0
-1

20
-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6
0.8
1
Zenith angle

0
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6
0.8
1
Zenith angle

0
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6
0.8
1
Zenith angle

Figure B.8: Basic distribution for MultiGeV single-ring ν̄e -like sample reconstructed
and selected by fiTQun.

5

Appendix B. Performance check for FV expansion

173

Component Fraction (fQ)
2

Fraction

Number of Events

Event Rate Distribution (fQ)

60

1.5

50

ν e CCν e CCν µ CCν µ CCν τ// ν τ CC

NC

Entering ν Events
40
1
30
20

0.5

10
0
0

50

100

150

200

250

300

0
0

350
400
Dwall [cm]

50

150

200

250

300

lepton

40

1.2

1.15

p rec/p

35
30
25

1.1

1.05

20

1

15

0.95

10

0.9

5

0.85

0
0

50

100

150

200

250

300

350

0.8
0

400

50

100

150

200

250

300

Dwall [cm]Dwall > 200 cm
300
250
200

200 cm > Dwall > 50 cm
SK4 Data
ν e CCν e CCν µ CCν µ CC
ντ
NC

70
60
50

20

50

10

0

10 4

10
Momentum [MeV]

5

300
250
200
150
100
50

0

140

40

120

35

100

30

SK4 Data
ν e CCν e CCν µ CCν µ CC
ντ
NC

350

30

100

400

Dwall > 50 cm
400

40
150

350

Dwall [cm]80

SK4 Data
ν e CCν e CCν µ CCν µ CCντNC

350

Momentum

350
400
Dwall [cm]fQ Momentum Bias (Rec/True)

Direction Resolution fQ

Dir Res. to lepton [Deg]

100

10 4

10
Momentum [MeV]

5

0

10 4

10
Momentum [MeV]Zenith Angle

160
140
120

25

100

80
20

80

15

60

10

40

60
40
20
0
-1

5
-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6
0.8
1
Zenith angle

0
-1

20
-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6
0.8
1
Zenith angle

0
-1

-0.8

-0.6

-0.4

Figure B.9: Basic distribution for MultiGeV single-ring νµ -like sample
reconstructed and selected by fiTQun.

-0.2

0

0.2

0.4

0.6
0.8
1
Zenith angle

5

174

Appendix B. Performance check for FV expansion

Component Fraction (fQ)
2

Fraction

Number of Events

Event Rate Distribution (fQ)
45
40

1.5

35

ν e CCν e CCν µ CCν µ CCν τ// ν τ CC

NC

Entering ν Events

30
25

1

20
15
0.5

10
5
0
0

50

100

150

200

250

300

0
0

350
400
Dwall [cm]

50

150

200

250

300

p rec/pν

40
35
30

1.4
1.2
1

25

0.8

20
15

0.6

10

0.4

5

0.2

0
0

50

100

150

200

250

300

350

0
0

400

50

100

150

200

250

300

Dwall [cm]Visible Energy

Dwall > 200 cm
120
100
80

350

400

Dwall [cm]200 cm > Dwall > 50 cm

SK4 Data
ν e CCν e CCν µ CCν µ CCντNC

140

SK4 Data
ν e CCν e CCν µ CCν µ CC
ντ
NC

35
30
25
20

Dwall > 50 cm
SK4 Data
ν e CCν e CCν µ CCν µ CC
ντ
NC

180
160
140
120
100
80

60

15

40

10

20

5

60
40

0

Zenith Angle

350
400
Dwall [cm]fQ Momentum Bias (Rec/True)

Direction Resolution fQ

Dir Res. to ν [Deg]

100

10 4

10
Visible Energy [MeV]

5

20

0

120

30

100

25

80

20

60

15

40

10

20

5

10 4

10
Visible Energy [MeV]

5

0

10 4

10
Visible Energy [MeV]

140
120
100
80
60
40

0
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6
0.8
1
Zenith angle

0
-1

20

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6
0.8
1
Zenith angle

0
-1

-0.8

-0.6

-0.4

-0.2

Figure B.10: Basic distribution for MultiGeV multi-ring νe -like sample
reconstructed and selected by fiTQun.

0

0.2

0.4

0.6
0.8
1
Zenith angle

5Appendix B. Performance check for FV expansion

175

Component Fraction (fQ)
2

Fraction

Number of Events

Event Rate Distribution (fQ)
40
35

1.5

30

ν e CCν e CCν µ CCν µ CCν τ// ν τ CC

NC

Entering ν Events

25
1

20
15

0.5

10
5
0
0

50

100

150

200

250

300

0
0

350
400
Dwall [cm]

50

150

200

250

300

p rec/pν

40
35
30

1.4
1.2
1

25

0.8

20
15

0.6

10

0.4

5

0.2

0
0

50

100

150

200

250

300

350

0
0

400

50

100

150

200

250

300

Dwall [cm]Dwall > 200 cm
160

Visible Energy

350
400
Dwall [cm]fQ Momentum Bias (Rec/True)

Direction Resolution fQ

Dir Res. to ν [Deg]100

200 cm > Dwall > 50 cm

SK4 Data
ν e CCν e CCν µ CCν µ CC
ντ
NC

140
120
100

350

400

Dwall [cm]SK4 Data
ν e CCν e CCν µ CCν µ CC
ντ
NC

50

40

30

Dwall > 50 cm
SK4 Data
ν e CCν e CCν µ CCν µ CC
ντ
NC

200
180
160
140
120

80

100

60

80

20

60

40

40

10
20

20

0

10 4

10
Visible Energy [MeV]

5

0

10 4

10
Visible Energy [MeV]

5

0

35

140

30

120

25

100

20

80

15

60

10

40

10 4

10
Visible Energy [MeV]Zenith Angle

100

80

60

40

20
5
0
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6
0.8
1
Zenith angle

0
-1

20

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6
0.8
1
Zenith angle

0
-1

-0.8

-0.6

-0.4

-0.2

Figure B.11: Basic distribution for MultiGeV multi-ring ν̄e -like sample
reconstructed and selected by fiTQun.

0

0.2

0.4

0.6
0.8
1
Zenith angle

5

176

Appendix B. Performance check for FV expansion

Component Fraction (fQ)
2

Fraction

Number of Events

Event Rate Distribution (fQ)

100

1.5
80

ν e CCν e CCν µ CCν µ CCν τ// ν τ CC

NC

Entering ν Events

60

1

40
0.5
20
0
0

50

100

150

200

250

300

0
0

350
400
Dwall [cm]

50

150

200

250

300

p rec/pν

40
35
30

1.4
1.2
1

25

0.8

20
15

0.6

10

0.4

5

0.2

0
0

50

100

150

200

250

300

350

0
0

400

50

100

150

200

250

300

Dwall [cm]Visible Energy

Dwall > 200 cm

300
250
200

SK4 Data
ν e CCν e CCν µ CCν µ CC
ντ
NC

90
80
70
60
50

150

350

400

Dwall [cm]200 cm > Dwall > 50 cm

SK4 Data
ν e CCν e CCν µ CCν µ CCντNC

350

Dwall > 50 cm
SK4 Data
ν e CCν e CCν µ CCν µ CC
ντ
NC

450
400
350
300
250

40

200

30

150

20

100

100
50

10

0

10 4

10
Visible Energy [MeV]300

Zenith Angle

350
400
Dwall [cm]fQ Momentum Bias (Rec/True)

Direction Resolution fQ

Dir Res. to ν [Deg]

100

5

50

0

10 4

10
Visible Energy [MeV]

80

5

0

10 4

10
Visible Energy [MeV]

350

70

250

300
60
250

200
50

200
150

40
150

30

100

100

20
50

50

10
0
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6
0.8
1
Zenith angle

0
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6
0.8
1
Zenith angle

0
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6
0.8
1
Zenith angle

Figure B.12: Basic distribution for MultiGeV multi-ring µ-like sample reconstructed
and selected by fiTQun.

5

Appendix B. Performance check for FV expansion

177

Component Fraction (fQ)
2

Fraction

Number of Events

Event Rate Distribution (fQ)

50

1.5
40

ν e CCν e CCν µ CCν µ CCν τ// ν τ CC

NC

Entering ν Events

30

1

20
0.5
10
0
0

50

100

150

200

250

300

0
0

350
400
Dwall [cm]

50

150

200

250

300

p rec/pν

40
35
30

1.4
1.2
1

25

0.8

20
15

0.6

10

0.4

5

0.2

0
0

50

100

150

200

250

300

350

0
0

400

50

100

150

200

250

300

Dwall [cm]Visible Energy

Dwall > 200 cm
160
140
120
100

350

400

Dwall [cm]200 cm > Dwall > 50 cm

SK4 Data
ν e CCν e CCν µ CCν µ CCντNC

180

SK4 Data
ν e CCν e CCν µ CCν µ CC
ντ
NC

50

40

30

Dwall > 50 cm
240

SK4 Data
ν e CCν e CCν µ CCν µ CC
ντ
NC

220
200
180
160
140
120

80

100
20

80

60

60

40

10

40

20

20

0
10

Zenith Angle

350
400
Dwall [cm]fQ Momentum Bias (Rec/True)

Direction Resolution fQ

Dir Res. to ν [Deg]

100

4

10
Visible Energy [MeV]

5

0
10

4

10
Visible Energy [MeV]

5

0

160

40

180

140

35

160

120

30

140

100

25

80

20

60

15

40

10

20

5

10 4

10
Visible Energy [MeV]

120
100
80
60

0
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6
0.8
1
Zenith angle

0
-1

40
20
-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6
0.8
1
Zenith angle

0
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

Figure B.13: Basic distribution for MultiGeV multi-ring e-like other sample
reconstructed and selected by fiTQun.

0.4

0.6
0.8
1
Zenith angle

5

178

Appendix B. Performance check for FV expansion

SampleCC νe CC ν¯e CC νµ + ν¯µ CC ντ NC Data MC
Fully Contained (FC) Sub-GeV
e-like, Single-ring
0 decay-e
0.702 0.227
0.025
0.001 0.045 18901758.0
1 decay-e
0.712 0.015
0.208
0.001 0.063 222 210.2µ-like, Single-ring
0 decay-e
0.034 0.011
0.805
0.001 0.150 454 427.2
1 decay-e
0.001 0.000
0.968
0.000 0.031 14421351.5
2 decay-e
0.000 0.000
0.980
0.000 0.019 140 130.2
π 0 -likeTwo-ring
0.109 0.036
0.018
0.000 0.837 403 379.4
Fully Contained (FC) Multi-GeV
Single-ringνe -like
0.748 0.066
0.064
ν̄e -like
0.566 0.371
0.003µ-like0.000 0.000
0.995
Multi-ringνe -like
0.609 0.112
0.059
ν̄e -like
0.541 0.301
0.023µ-like0.016 0.002
0.946
Other
0.302 0.032
0.342

0.016 0.105
0.007 0.053
0.004 0.001

43 37.0
190 194.1
219 171.9

0.032 0.188
0.016 0.118
0.005 0.031
0.051 0.274

174 179.6
169 206.8
475 445.1
229 199.6

Table B.2: Sample purity broken down by neutrino flavor assuming neutrino
oscillations with ∆m232 = 2.52 × 10−3 eV2 and sin2 θ23 = 0.51. The right two columns
show the sample size of the data and MC for the entire SK data set, respectively.
Only fully contained events in the new region, within the range of the distance to thewall from 50cm to 200cm are shown.

Appendix B. Performance check for FV expansion

179

SampleCC νe CC ν¯e CC νµ + ν¯µ CC ντ NC Data MC
Fully Contained (FC) Sub-GeV
e-like, Single-ring
0 decay-e
0.722 0.238
0.007
0.000 0.032 76867180.4
1 decay-e
0.853 0.019
0.081
0.001 0.046 818 769.5µ-like, Single-ring
0 decay-e
0.017 0.006
0.798
0.001 0.179 14911439.1
1 decay-e
0.000 0.000
0.973
0.000 0.027 68376437.8
2 decay-e
0.000 0.000
0.983
0.000 0.017 633 623.8
π 0 -likeTwo-ring
0.064 0.020
0.012
0.000 0.903 17491683.9
Fully Contained (FC) Multi-GeV
Single-ringνe -like
0.729 0.075
0.059
ν̄e -like
0.555 0.378
0.003µ-like0.000 0.000
0.996
Multi-ringνe -like
0.592 0.116
0.055
ν̄e -like
0.529 0.300
0.021µ-like0.011 0.001
0.957
Other
0.286 0.027
0.342

0.026 0.111
0.008 0.056
0.003 0.001

260 245.1
11531169.3
11011079.0

0.036 0.201
0.019 0.130
0.004 0.027
0.053 0.292

933 953.0
813 861.4
24572388.2
12671224.1

Table B.3: Sample purity broken down by neutrino flavor assuming neutrino
oscillations with ∆m232 = 2.52 × 10−3 eV2 and sin2 θ23 = 0.51. The right two columns
show the sample size of the data and MC for the entire SK data set, respectively.
Only fully contained events are reconstructed by fiTQun and have a FV cut of 50 cm.
PC and Up-µ samples are reconstructed by APFit and no FV cut change for these
two samples.

Appendix C

Distribution for systematic error
evaluation
This section shows the distribution of ring counting (RC), particle identification (PID)
and two staged multi-GeV multi-ring e-like separation, which is introduced in Section
6.2.2. As has been stated in Section 7.5.2, the systematic error is evaluated by fitting
MC to data with smearing and shift parameters simultaneously, and this evaluation
method is proceed in both of conventional FV (Dwall > 200 cm) and new region (200cm > Dwall > 50 cm).
Those distributions indicate no issues in the fitting procedure and no abnormal distribution observed in both detector region.

700

Number of Events

Number of Events

800

300

1R like

MR like

1R like

MR like

250

600

200

500
400

150

300
100
200
50

100
0
-10

0

10

20

0
-10

30
40
50
Ring Counting likelihood

0

10

20

30
40
50
Ring Counting likelihood

Figure C.1: Ring counting likelihood distribution of sub-GeV e-like events with
momentum less than 400 MeV. Points show the distribution from data, while the redsolid line (blue dashed line) represents the distribution after (before) fitting. The
shaded histogram show events with only single-ring in the final states. Left figure
shows the distribution for conventional FV (Dwall > 200 cm), while right figure shows
the one for new region (200 cm > Dwall > 50 cm). All following figures in this section
have the same configuration.

181

Number of Events

Appendix C. Distribution for systematic error evaluation

Number of Events

182

800
1R like180

MR like

700

1R like

MR like

160

600

140
120

500

100

400

80
300
60
200

40

100

20

0
-10

0

10

20

0
-10

30
40
50
Ring Counting likelihood

0

10

20

30
40
50
Ring Counting likelihood

500

1R like

Number of Events

Number of Events

Figure C.2: Ring counting likelihood distribution of sub-GeV e-like events with
momentum larger than 400 MeV.

MR like

400

80
1R like

MR like

0

20

70
60
50

300
40
200

30
20

100
10
0

0

20

40

0

60
80
100
Ring Counting likelihood

40

60
80
100
Ring Counting likelihood

Number of Events

Number of Events

Figure C.3: Ring counting likelihood distribution of multi-GeV e-like events.

800

200

1R like

MR like

1R like

180

700

MR like

160

600

140

500

120

400

100

300

80
60

200
40
100
0
-10

20
0

10

20

30
40
50
Ring Counting likelihood

0
-10

0

10

20

30
40
50
Ring Counting likelihood

Figure C.4: Ring counting likelihood distribution of sub-GeV µ-like events with
momentum less than 400 MeV.

Number of Events

Number of Events

Appendix C. Distribution for systematic error evaluation

183

200

1000

1R like

MR like

1R like180

MR like

160

800

140
120

600
100
80

400

60
40

200

20
0
-10

0

10

20

0
-10

30
40
50
Ring Counting likelihood

0

10

20

30
40
50
Ring Counting likelihood

Number of Events

Number of Events

Figure C.5: Ring counting likelihood distribution of sub-GeV µ-like events with
momentum larger than 400 MeV.

220
200

1R like

MR like

180
160
140

50
1R like

MR like

0

20

40

30

120
100
20

80
60

10

40
20
0

0

20

40

0

60
80
100
Ring Counting likelihood

40

60
80
100
Ring Counting likelihood

Figure C.6: Ring counting likelihood distribution of multi-GeV µ-like events.

1R like

Number of Events

Number of Events

120

600

MR like

1R like

MR like

0

20

100

500
400

80

60
300
40

200

20

100
0

0

20

40

60
80
100
Ring Counting likelihood

0

40

60
80
100
Ring Counting likelihood

Figure C.7: Ring counting likelihood distribution of µ-like events with Evis > 600
MeV. This category has overlap with Figure C.5 and Figure C.6.

Number of Events

Appendix C. Distribution for systematic error evaluation

Number of Events

184

600

1800

µ-likee-like1600µ-likee-like

500

1400

400

1200
1000

300

800
200

600
400

100

200
0
-1500

-1000

-500

0

500

1000

0
-1500

1500
2000
PID likelihood

-1000

-500

0

500

1000

1500
2000
PID likelihood

Number of Events

Number of Events

Figure C.8: PID likelihood distribution for sub-GeV single-ring events. The shaded
histogram show νµ charged current interactions.

220µ-like200

e-like

180
160
140

45
µ-like40

e-like

35
30
25

120
100

20

80

15

60

10

40
5

20
0
-10000 -8000 -6000 -4000 -2000

00-10000 -8000 -6000 -4000 -2000

2000 4000 6000 8000 10000
PID likelihood02000 4000 6000 8000 10000
PID likelihood

Number of Events

Number of Events

Figure C.9: PID likelihood distribution for multi-GeV single-ring events.

700µ-like200

e-like600µ-likee-like

180
160

500

140

400

120
100

300
80
200

60
40

100
20
0
-1500

-1000

-500

0

500

1000

1500
2000
PID likelihood

0
-1500

-1000

-500

0

500

1000

Figure C.10: PID likelihood distribution for sub-GeV multi-ring events.

1500
2000
PID likelihood

600µ-likee-like500
400

Number of Events

Number of Events

Appendix C. Distribution for systematic error evaluation

100

185

µ-likee-like

80

60
300
40

200

20

100
0
-10000 -8000 -6000 -4000 -2000

00-10000 -8000 -6000 -4000 -2000

2000 4000 6000 8000 10000
PID likelihood02000 4000 6000 8000 10000
PID likelihood

Other

500

e-likeNumber of Events

Number of Events

Figure C.11: PID likelihood distribution for sub-GeV single-ring events.

Other

120

e-like

100

400

80
300
60
200
40
100

0
-10

20

-8

-6

0
-10

-4
-2
0
2
4
6
8
Multi-Ring e -like - other Separation Parameter

-8

-6

-4
-2
0
2
4
6
8
Multi-Ring e -like - other Separation Parameter400
ν e-like350

ν e-like

300
250

Number of Events

Number of Events

Figure C.12: Likelihood distribution used to separate SK-IV Multi-ring events withe-like most energetic ring into the e-like and unclassified samples. The shaded
histogram show νe and ν̄e charged current interactions.

100

ν e-likeν e-like

80

60

200
40

150
100

20
50
0
-4

-3

-2

-1
0
1
2
3
4
Multi-Ring ν e - ν e Separation Parameter

0
-4

-3

-2

-1
0
1
2
3
4
Multi-Ring ν e - ν e Separation Parameter

Figure C.13: Likelihood distribution used to separate SK-IV Multi-ring e-like events
into the neutrino-like and antineutrino-like samples. The shaded histogram show ν̄echarged current interactions.

Appendix D

Statistical Study for Oscillation
Analysis Result Validation
The previous Super-K atmospheric neutrino oscillation analysis paper published in 2018,
which shows a best fit value at the second octant [6], utilized 328 kton-year exposureof the detector including all four SK periods, while the data recorded until Apr. 2016of SK-IV are used. The analysis shown in this thesis, which uses two more years data:
data recorded until the end of Apr. 2018 are used, shows a different octant preference.
This appendix will validate the new result and discuss the reason of such change.
Figures D.1 and Table D.1 shows the normal hierarchy fit result with data until Apr.
2016 and data until Apr. 2018, assuming sin2 θ13 = 0.0210. Same systematic error areused in those two fits. The fit result for mass split |∆m232 | and CP phase δCP are foundto be in consistent for those two data set. However, the absolute χ2 value of best fit
result, increased from 551.6 to 575.2 when two more years data added. The preferenceon sin2 θ23 octant also changed from second to first.
10

10

8

10

8
99%

8
99%

99%

4

95%

∆ χ2

6

∆ χ2

6

∆ χ2

6

4

90%

95%

4

90%

2

90%

2
68%

0
0.0015

2
68%

0.002

0.00252|∆ M23| eV 20.003

0.0035

95%

0
0.3

68%

0.4

0.5
sin2θ23

0.6

0.7

0

0

2

4

6

δ CP

Figure D.1: Constraints on neutrino oscillation contours from SK-IV atmospheric
neutrino data with expanded FV (Dwall > 50 cm) assuming sin2 θ13 = 0.0210 ± 0.0011. Only normal hierarchy fit result is shown here. The solid line represents the resultwith data until Apr. 2016, while the dashed line show the result with data until Apr.
2018. Both of them are offset with their respective minimal χ2 value.

187

188

Appendix D. Statistical Study for Oscillation Analysis Result Validation
Fit within expanded FV
Apr. 2018
Apr. 2016
Hierarchy
NH
IH
NH
IH
2
χ
576.5
579.0
552.2
557.1
+0.055+0.047sin2 θ23 (1st oct.)
0.425+0.046
0.425
Excluded
in
1σ
0.413
−0.037−0.036−0.030sin2 θ23 (2nd oct.)|∆m232 | [×10−3 eV2 ]δCP+0.0220.600+0.013−0.030 0.588−0.037+0.22
2.53−0.12
2.53+0.14
−0.31
+2.67
3.14−1.35
4.89+1.51−3.46

0.613+0.035−0.036
2.63+0.13−0.36
3.49+1.76−1.42

0.613+0.027−0.0492.63+0.11−0.43
3.49+2.79−3.49

Table D.1: Summary of parameter estimates for analysis with data until Apr. 2016
and data until Apr. 2018.

To validate the fit result, MC ensembles were generated assuming statistical fluctuations of the pseudo data sets according to the current detector exposure, and Gaussian

Fraction of Pseudo Data Sets

fluctuations of the systematic errors.

10-1Best fit, SK θ13 Constrained
SK True NH
p-value: 0.103

-2

10

10-3

100

200

300

400

500

600

700

800

900 1000

χ2

Figure D.2: Distributions of the best fit χ2 values fits to pseudo data sets. The
pseudo data have been generated with the best fit value in NH expanded FV θ13
constrained fit shown in Table 7.13. Shaded portions of the histograms denote thefraction of pseudo data sets with more extreme values than that observed in the data,
χ2N H = 576.5.

Figure D.2 shows the distribution of the best fit χ2 values fits to pseudo data sets.
During the pseudo data generation, only the events corresponding to the livetime from
Apr. 2016 to Apr. 2018 were generated with fluctuations and then merged with theobserved data, which was taken until Apr. 2016. p-value for 576.5, the best fit χ2 ofexpanded FV, is 0.103. Figure D.3 shows the absolute χ2 distribution for each event
category. The error bar of pseudo data (blue box) represents one σ uncertainty fromstatistical fluctuations and systematic error. The observed data in this analysis is shownby red triangle. The top two categories which have largest deviation from pseudo data
center value are multi-GeV single-ring νe -like sample and multi-GeV multi-ring ν̄e -likeAppendix D. Statistical Study for Oscillation Analysis Result Validation

189

sample. The zenith angle distribution for both two samples are shown in Figure D.4
and Figure D.5. No abnormal is observed.

70
60
50
40
30
20
10
0

Sub Sub Sub Sub Sub Su Mult Mult Mult Mult Mult Mult Mult PC PC Up Non Sho-Ge -Ge -Ge -Ge -Ge b-Ge i-Ge i-G i-G i-Rin i-R i-R i-R Stop Thr Stop -sh weru
V e V e V µ V µ V µ V π 0 V e eV e eV µ g e ing ing ingµ ower ing µ-like -like -lik -lik -lik
-like e-lik µ -lik Unc-li -liing
lass
0-d 1-d e 0-d e 1-d e 2-dlike M ke ν ke ν likeµνe e ν ecy e cy e cy e cy e cy e -R
ified
e
e
e

Figure D.3: Distributions of the best fit χ2 of observed data (red triangle) andpseudo data sets (blue box). The error bar of pseudo data sets result represents the 1
σ uncertainty from statistical fluctuations and systematic errors. The pseudo data
have been generated with the best fit value in NH expanded FV θ13 constrained fit
shown in Table 7.13.

Similar study is done to validate the octant preference change after updating data.
Figure D.6 shows the distribution of the difference in the best fit χ2 values between fitswith sin2 θ23 fixed to 0.4 and 0.6 to pseudo data set. In the cyan (orange) histogram
the pseudo data have been generated assuming sin2 θ23 = 0.4 (0.6) hierarchy at normal
hierarchy. Pseudo data corresponding to two years livetime were generated and thenmerged with the observed data taken until Apr. 2016. The p-value to observed the firstoctant with ∆χ2 = −0.73 is 0.024 (0.161) when true sin2 θ23 = 0.6 (0.4). Figure D.7
shows the ∆χ2 distribution for each event category. The top two categories which havelargest deviation from pseudo data center value are sub-GeV single-ring µ-like 1 decaye sample and multi-GeV single-ring νe -like sample. The zenith angle distribution forboth two samples are shown in Figure D.8 and Figure D.9. This change mainly comesfrom less upward-going for sub-GeV muon events and multi-GeV e-like events, and noabnormal distribution is observed.

190

Appendix D. Statistical Study for Oscillation Analysis Result Validation

Apr 16 to Apr 18 Data
Data
SK4 Normal
SK4 Inverted

EventsEventsApr 16 Data
20

5

10

0
-1

1

0
cos (θ)10cos (θ)10cos (θ)10cos (θ)1

Events0cos (θ)Events0-1

Data
SK4 Normal
SK4 Inverted

5100cos (θ)0-1

1
EventsEvents

0
-1

10

2
5

0
-1

1
Events0cos (θ)Events

0
-1

5

0
-1

2

0
cos (θ)1

0-1

Figure D.4: Zenith angle distributions for of data (open circle) and best fit MC for
both NH (blue line) and IH (orange line). Multi-GeV single ring νe -like events areshown here. Left figures are for data taken before Apr. 2016, while right figures arefor data taken between Apr. 2016 to Apr. 2018. The momentum range are 1330 MeV∼ 2500 MeV, 2500 MeV ∼ 5000 MeV, 5000 MeV ∼ 10000 MeV, > 10000 MeV frommost top figures, respectively.

Appendix D. Statistical Study for Oscillation Analysis Result Validation

Apr 16 to Apr 18 Data
Data
SK4 Normal
SK4 Inverted

50EventsEventsApr 16 Data

191

20

Data
SK4 Normal
SK4 Inverted

100cos (θ)0-1

1

0cos (θ)10cos (θ)10cos (θ)1

EventsEvents

0
-1

40

10
20

0
cos (θ)0-1

1
EventsEvents

0
-1
40

5
20

0
-1

0
cos (θ)1

0-1

Figure D.5: Zenith angle distributions for of data (open circle) and best fit MC for
both NH (blue line) and IH (orange line). Multi-GeV multi ring ν̄e -like events areshown here. Left figures are for data taken before Apr. 2016, while right figures are fordata taken between Apr. 2016 to Apr. 2018. The momentum range are 1330 MeV ∼2500 MeV, 2500 MeV ∼ 5000 MeV, > 5000 MeV from most top figures, respectively.

Appendix D. Statistical Study for Oscillation Analysis Result Validation

10-1

Fraction of Pseudo Data Sets

192

10-2

Best fit, SK θ13 Constrained
sin2θ23=0.6
sin2θ23=0.4
p-value: 0.024
p-value: 0.839

10-3

10-4
-30

-20

-10

0

10

χ2 2

20

-χ2 2

sin θ23=0.4

sin θ23=0.6∆ χ2

Figure D.6: Distributions of the difference in best fit χ2 values between fits withsin2 θ23 fixed to 0.4 and 0.6 to pseudo data sets. In the cyan (orange) histogram thepseudo data have been generated assuming sin2 θ23 = 0.4 (0.6) hierarchy at normal
hierarchy. Shaded portions of the histograms denote the fraction of pseudo data setswith more extreme values than that observed in the data, ∆χ2data = −0.73.

4Prefer 2

nd

oct.

3
2
1
0

st

Prefer 1 oct.

-1
-2
-3
-4
Sub Sub Sub Sub Sub Su Mult Mult Mult Mult Mult Mult Mult PC PC Up Non Sho-Ge -Ge -Ge -Ge -Ge b-Ge i-Ge i-G i-G i-Rin i-R i-R i-R Stop Thr Stop -sh weru
V e V e V µ V µ V µ V π 0 V e eV e eV µ g e ing ing ingµ ower ing µ-like -like -lik -lik -lik
-like e-lik µ -lik Unc-li -liing
lass
0-d 1-d e 0-d e 1-d e 2-dlike M ke ν ke ν likeµνe e ν ecy e cy e cy e cy e cy e -R
ified
e
e
e

Figure D.7: Distributions of the best fit χ2 of observed data (red triangle) andpseudo data sets (blue box) generated with true sin2 θ23 = 0.6. The error bar ofpseudo data sets result represents the one σ uncertainty from statistical fluctuations
and systematic errors. The pseudo data have been generated with the best fit value in
NH expanded FV θ13 constrained fit shown in Table 7.13.

Appendix D. Statistical Study for Oscillation Analysis Result Validation

Apr 16 to Apr 18 Data
EventsEventsApr 16 Data
Data 2
SK4 sin2θ23 = 0.4
SK4 sin θ23 = 0.6
100

Data 2
SK4 sin2θ23 = 0.4
SK4 sin θ23 = 0.6
20

0
-1

1

0
cos (θ)10cos (θ)10cos (θ)10cos (θ)10cos (θ)1

Events0cos (θ)Events

0
-1

193

50
200

0
-1

1
Events0cos (θ)Events

0
-1

200

0
-1

1
Events0cos (θ)Events

0
-1

50

50
200

0
cos (θ)

200

20

100

0
-1

0
-1

1
EventsEvents0-10cos (θ)1

0-1

Figure D.8: Zenith angle distributions for of data (open circle) and best fit MC forsin2 θ23 = 0.4 (magenta line) and 0.6 (green dashed line). Sub-GeV single ring µ-like
1-decay electron events are shown here. Left figures are for data taken before Apr.
2016, while right figures are for data taken between Apr. 2016 to Apr. 2018. The
momentum range are 100 MeV ∼ 200 MeV, 200 MeV ∼ 300 MeV, 300 MeV ∼ 600
MeV, 600 MeV ∼ 1000 MeV, 1000 MeV ∼ 1330 MeV from most top figures,
respectively.

194

Appendix D. Statistical Study for Oscillation Analysis Result Validation

Apr 16 to Apr 18 Data
EventsEventsApr 16 Data
Data 2
SK4 sin θ23 = 0.4
SK4 sin2θ23 = 0.6

20

Data 2
SK4 sin θ23 = 0.4
SK4 sin2θ23 = 0.6
5100cos (θ)0-1

1

0cos (θ)10cos (θ)10cos (θ)10cos (θ)1

EventsEvents

0
-1

20

5
10

0
-1

1
Events0cos (θ)Events

0
-1

2

10

0
cos (θ)0-1

1
EventsEvents

0
-1
10

2
5

0
-1

0
cos (θ)1

0-1

Figure D.9: Zenith angle distributions for of data (open circle) and best fit MC for
both sin2 θ23 = 0.4 (magenta line) and 0.6 (green dashed line). Multi-GeV single ringνe -like events are shown here. Left figures are for data taken before Apr. 2016, whileright figures are for data taken between Apr. 2016 to Apr. 2018. The momentum
range are 1000 MeV ∼ 2500 MeV, 2500 MeV ∼ 5000 MeV, 5000 MeV ∼ 10000 MeV,
>10000 MeV from most top figures, respectively.

Appendix E

Improvement of decay electron
detection for fiTQun
Decay electrons (or positrons) from muons or charged pions are important in the atmospheric neutrino analysis at Super-K, therefore, a higher tagging efficiency is desirable.
The decay electron tagging efficiency is evaluated by taking the ratio of the number ofdetected Michel electrons and the number of the stopping cosmic ray muon events.
The event selection criteria for the stopping cosmic ray muon are as following:
1. Total charge in ID > 1000 p.e.
2. Goodness of stopping muon fit algorithm > -0.9, which means the algorithm fit
the stopping muon track successfully.
3. Estimated muon stopping point is within fiducial volume, which is defined as theregion located more than 2 m from the ID wall here.
Figure E.1 shows the distribution of the decay time of the tagged Michel electron, which
is defined as the time difference with the parent muon. The tagging efficiency, which
is defined as the ratio between the number of tagged electron and the expectation,is near 100% when the decay time is long (i.e. > 1200 ns) for both of APFit andfiTQun. However, the tagging efficiency decreases significantly for the Michel electrons
with short decay time, especially for multi-GeV muon, as shown in Figure E.2. The
decay e detection efficiency of the old version fiTQun (v4r2) is also lower than that of
APFit.
As introduced in Section 6.2.2, the decrease for multi-GeV muon in the short decay time
region is mainly to the assumption: all vertex positions are assumed to lie close to the195Appendix E. Improvement of decay electron detection

Number of tagged Michel electrons

Number of tagged Michel electrons

196

103

102

0

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Decay time [ns]

104

103

102

0

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Decay time [ns]Tagging efficiency

Tagging efficiency

Figure E.1: Distribution of the decay time of tagged Michel electron (or position)from cosmic ray muon. The left figure shows the sub-GeV cosmic ray muon while theright figure is for multi-GeV muon. Open circles are the electrons tagged by APFit,
and full triangles are for fiTQun. The solid line is the expectation based on thenumber of cosmic ray muons.

1

0.8

0.6

0.8

0.6

0.4

0.4

0.2

0.2

0
0

1

200

400

600

800

1000 1200 1400 1600 1800 2000
Decay time [ns]

0
0

200

400

600

800

1000 1200 1400 1600 1800 2000
Decay time [ns]Figure E.2: Tagging efficiency for the Michel electron from cosmic ray muon. The
left figure shows the sub-GeV cosmic ray muon while the right figure is for multi-GeV
muon. Open circles are the electrons tagged by APFit, and full triangles are for
fiTQun. The error bars indicate the statistical error.

pre-fit vertex when the peak-finding algorithm runs. This assumption is broken when
the primary particle travels a significant distance from the interaction vertex, as for highmomenta muons, which leads that the peak formed by the Michel electron smeared.
To resolve this issue, the vertex pre-fitting and peak-finding algorithm are rerun after
masking the hits caused by the primary particle to improve decay electron reconstruction
efficiency. The vertex position used in the goodness function x is then close to the vertexof the secondary particle, and the peak on goodness function becomes larger, as shownin Figure E.3.
Besides, the original time window, which is defined from -180 ns ∼ 800 ns around themain peak, will also be extended if many hits are around the edge during this process,to deal with the particle activity occurred near the time window boundary. The cut toextend the time window is 9hits / 70ns, which is consistent with the threshold definedin Equation 6.6.

Appendix E. Improvement of decay electron detection

197

Tagging efficiency

Tagging efficiency

Figure E.3: Goodness as a function of hit time for an event with a mutli-GeV muon
and a Michel electron. The black line shows the goodness as a function of hit timewith the vertex fixed to the pre-fix vertex (muon vertex). The red line is for the
goodness with the vertex at the decay electron position. Green curves denot thresholdfor identifying candidate peaks.

1

0.8

0.6

0.8

0.6

0.4

0.4

0.2

0.2

0
0

1

200

400

600

800

1000 1200 1400 1600 1800 2000
Decay time [ns]

0
0

200

400

600

800

1000 1200 1400 1600 1800 2000
Decay time [ns]Figure E.4: Tagging efficiency for the Michel electron from cosmic ray muon after
improvement. The left figure shows the sub-GeV cosmic ray muon while the rightfigure is for multi-GeV muon. Open circles are the electrons tagged by APFit, andfull triangles are for fiTQun.

The tagging efficiency of fiTQun for the Michel electron from cosmic ray muon is improved significantly for multi-GeV muon after revision, as shown shown in Figure E.4.
The efficiency are summarized in Table E.1.
The flat from 500 ns to 1000 ns on the fiTQun tagging efficiency is due to effect oftime window. It is required that each PMT hits at most once in one time window.
More hits at the same PMT within the same time window will be ignored. As shown in
Figure E.5, the tagging efficiency decreases as the increase of parent muon’s momentum.
More PMTs will be hit by the light from the muon with a higher momentum, whichmakes the hits from the decay electron ignored. The tagging efficiency is limited by thiseffect and keeps stable until the end of the time window.
In this appendix, all the study described until now uses a fiducial volume defined as

198

Appendix E. Improvement of decay electron detection
fiTQunfiTQunAPFit
After improvementBefore improvement
Sub-GeV Muon Data 89.31% ±0.16%
89.55% ±0.53% 88.93% ±0.35%
MC 88.43% ±0.39%
88.64% ±0.31% 88.53% ±0.39%
Multi-GeV MuonData 74.48% ±0.10%
69.01% ±0.33% 72.31% ±0.20%
MC 73.55% ±0.22%
67.86%±0.20% 71.38% ±0.23%
Reconstruction

Muon momentum [MeV/c]

Table E.1: Decay e tagging efficiency for cosmic ray muon.

10000

1

9000

0.9

8000

0.8

7000

0.7

6000

0.6

5000

0.5

4000

0.4

3000

0.3

2000

0.2

1000

0.1

0
0

200

400

600

800 1000 1200 1400 1600 1800 2000
Decay Time[ns]0Figure E.5: Tagging efficiency for the Michel electron from cosmic ray muon. The
horizontal axis shows the decay time, while the vertical axis represents the momentumof parent muon. The efficiency is denoted by the color.

the region located more than 2 m from the ID wall. As introduced Section 7.3, thefiducial volume used in atmospheric neutrino analysis has been extended to the region
located more than 50 cm. Comparing with the efficiency within the conventional FV(c.f. Table E.1), Table E.2 shows that the tagging efficiency decreased for multi-GeV
muon when the predicted stopping point of muon near the wall. Besides the effect of
the time window introduced before, the resolution of predicted stopping point will also
affect the tagging efficiency evaluation. The stopping point is predicted based on track
length, which is estimated by the momentum of muon and the energy loss in water (∼2.3
MeV/ cm), and the entrance position. The resolution of the predicted stopping point
for a high energy muon is not as good as the one for low energy muon, which makesthat the muon stopped out of the ID might be reconstructed within the region of 50 cm< Dwall < 200 cm. The Michel electron out of the ID is another reason of such decrease.

Appendix E. Improvement of decay electron detection

50 cm < Dwall < 200 cm
Sub-GeV Muon
Multi-GeV Muon

Data
MC
Data
MC

fiTQunAPFit
89.47% ±0.08% 88.79% ±0.08%
88.72% ±0.15% 88.69% ±0.15%
61.98% ±0.12% 60.59% ±0.12%
59.91% ±0.23% 58.76%±0.24%

Table E.2: Decay e tagging efficiency for cosmic ray muon in new region 50
cm< Dwall < 200 cm.

199

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