Full three flavor oscillation analysis of atmospheric neutrino data
observed in Super-Kamiokande

Chizue Ishihara
February 5, 2010

Abstract
This thesis presents the first-time neutrino oscillation studies considering entire oscillation
parameters, two mass differences ∆m 212 , ∆m223 , three mixing angles θ12 , θ23 , θ13 , and one CP
phase parameter (δcp ), by the atmospheric neutrino data observed in Super-Kamiokande.
The Super-Kamiokande, a 50 kt water Cherenkov detector, started taking data in 1996 andhas been observed a large number of atmospheric neutrino events. About 3,000 day neutrino
data is collected through the data taking phases, Super-Kamiokande-I, II and III. This analysis
is performed using the amount of this data for the first time.
The neutrino data selection and observed data quality for the analysis are summarized inthis thesis. The analysis examines for both the normal and inverted mass hierarchy cases. In
the normal (inverted) mass hierarchy case, ∆m 223 , sin2 θ23 and sin2 θ13 are constrained at 90 %C.L. to 1.88 (1.98) ×10−3 < ∆m223 < 2.75 (2.81) ×10−3 eV2 , 0.406 (0.426) < sin2 θ23 < 0.629(0.644), sin2 θ31 < 0.066 (0.122). No significant constraint on CP phase at 90 % C.L. is obtained
for both mass hierarchies and there is no clear preference of mass hierarchy.

** ACKNOWLEDGMENTS **First of all, I would like to express my gratitude to Prof. Takaaki Kajita for giving me theexcellent opportunity of studying atmospheric neutrino. I have learnt a lot of things and gained
invaluable experiences under his guidance.
I would like to thank Prof. Masato Shiozawa for his patience and guidance. He gave me
many valuable advice and support since I started the atmospheric neutrino analysis. This thesis
would never exist without his support and encouragement.
I would also like to thank Prof. Y. Suzuki for giving me the opportunity of studying inSuper-Kamiokande experiment.
I am grateful to all the members of the atmospheric neutrino analysis group, especiallyProf. K. Kaneyuki, Prof. S. Moriyama, Prof. Y. Hayato, Dr. K. Okumura, Dr. M. Miura,
Dr. Y. Obayashi, Dr. J. Kameda, Dr. C. Saji, Dr. S. Nakayama, Dr. I. Higuchi, Dr. N. Tanimoto,Dr. Y. Shimizu, Dr. H. Kaji, Dr. Y. Takenaga, Dr. G. Mitsuka, Dr. H. Nishino, Prof. Y. Itow,
Prof. E. Kearns, Prof. C.W. Waltar, Prof. K. Scholberg, Dr. S. Mine, Dr. J.L. Raaf, Dr. T. Kato,
Dr. F. Dufour, Dr. M. Litos Dr. P. Mijakowski and Dr. R. Wendell.
I would like to extend my gratitude to ICRR staffs, Prof. M. Nakahata, Prof. Y. Takeuchi,
Prof. M. Yamashita, Dr. Y. Koshio, Dr. K. Abe, Dr. A. Takeda, Dr. H. Sekiya, Dr. S. Yamada,Dr. H. Ogawa, Dr. K. Kobayashi and Dr. K. Hiraide.
I would like to thank my friends and colleagues who encouraged me for all the time, T. Iida,
M. Ikeda, K. Ueshima, N. Okazaki, D. Ikeda, T. Tanaka, D. Motoki T.F. McLachlan, S. Hazama,
K. Ueno, Y. Nakajima, T. Yokozawa, Y. Kozuma H. Nishiie, A. Shinozaki, K. Iyogi and Maggie.
The analysis in this thesis has also relied on the hard work by all members of SuperKamiokande and all the people supporting this experiment. This thesis is greatly indebted
to them.
Specially thanks to Y. Fukuda and K. Watanabe who supported me in many ways.
Last but not least, I appreciate to family for their continuous support, advice and encouragement.

Contents
1 Introduction
1.1 Neutrino Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Neutrino oscillation experiments . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Solar neutrino and Reactor neutrino . . . . . . . . . . . . . . . . . . . . .
1.2.2 Atmospheric neutrino and Accelerator neutrino . . . . . . . . . . . . . . .
1.2.3 Mixing angle θ13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.4 Summary for neutrino oscillation parameters and Remaining issues . . . .
1.3 Sub-dominant effects of atmospheric neutrino oscillation . . . . . . . . . . . . . .

1
1
3
3
4
5
6
7

2 The Super-Kamiokande Detector
2.1 Cherenkov Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Water Tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 photomultiplier tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 OD segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Water Purification and Air Purification System . . . . . . . . . . . . . . . . . . .
2.4 Electronics and Data Acquisition System . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13
14
15
15
17
20
20
21
22

3 Simulation of Atmospheric Neutrino
3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Atmospheric Neutrino Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Neutrino Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Elastic and Quasi-Elastic Scattering . . . . . . . . . . . . . . . . . . . . .
3.3.2 Single Meson Production . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Deep Inelastic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.4 Coherent Pion Production . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.5 Nuclear Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Detector Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26
26
26
32
32
33
38
38
39
40

4 Calibration
4.1 Detector Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Relative Gain Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Absolute Gain Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.3 Relative Timing Calibration . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.4 Light scattering measurement using a laser . . . . . . . . . . . . . . . . .

43
43
43
44
44
45

i

4.2

4.1.5 Water transparency measurement using cosmic ray muons . . . . . . . . .
Energy Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 High energy stopping muons . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Low energy stopping muons . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.3 Neutrino induced π 0 events . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.4 Decay electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.5 Time variation of energy scale . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.6 Summary of the energy scale calibration . . . . . . . . . . . . . . . . . . .
4.2.7 Uniformity of Energy Scale . . . . . . . . . . . . . . . . . . . . . . . . . .

47
50
50
51
51
55
55
55
59

5 Data Selection
62
5.1 Reduction for Fully Contained Events . . . . . . . . . . . . . . . . . . . . . . . . 64
5.1.1 First Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.1.2 Second Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.1.3 Third Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.1.4 Fourth Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.1.5 Fifth reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.1.6 FC Reduction Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.2 Reduction for Partially Contained Events . . . . . . . . . . . . . . . . . . . . . . 74
5.2.1 First Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.2.2 Second Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.2.3 Third Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.2.4 Fourth Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.2.5 Fifth Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.2.6 PC Reduction Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.2.7 PC OD stop/through separation . . . . . . . . . . . . . . . . . . . . . . . 91
5.3 Reduction for Upward-going Muon Events . . . . . . . . . . . . . . . . . . . . . . 92
5.3.1 Charge cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.3.2 Zenith angle cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.3.3 Scanning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.3.4 UPMU Reduction Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.3.5 Background estimation for Upward-going muon . . . . . . . . . . . . . . . 94
5.3.6 Upward through-going showering muon . . . . . . . . . . . . . . . . . . . 96
5.4 Event Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.4.1 Vertex fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.4.2 Ring counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.4.3 Particle Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.4.4 Precise Vertex Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.4.5 Momentum reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.4.6 Decay Electron Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.4.7 π 0 reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6 Neutrino Oscillation Analysis
114
6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.2 Data Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.2.1 Selection for Sub-GeV sample . . . . . . . . . . . . . . . . . . . . . . . . . 114
ii

6.3
6.4
6.5

6.6
6.7

6.2.2 Selection for Multi-GeV multi-ring e-like sample . . . . . . . . . . . . . . 115
6.2.3 Data set for the analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Oscillation effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Analysis Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
Systematic Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.5.1 Systematics in neutrino flux . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.5.2 Systematics in neutrino interaction . . . . . . . . . . . . . . . . . . . . . . 131
6.5.3 Systematics in event selection and reconstruction . . . . . . . . . . . . . . 135
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

7 Conclusion

159

A Oscillation analysis
166
A.1 Solar terms effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
A.2 Zenith angle distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
A.3 Best-fit systematic uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

iii

Chapter 1

Introduction
The present-day universe contains large number of baryons in the form of matter, but almost no antibaryons. According to the big bang model, quarks and untiquarks had existed
in a comparable number in the initial explosion of the universe. However, since there was asmall asymmetry of quarks and untiquarks at that time, quarks remain excessively throughthe universe expanding and cooling processes. This asymmetry could have developed with CP
violation [1]. In fact, CP invariance are violated in weak interaction process of quarks [2]. But
CP violation in lepton sector, which is considered to be related to the origin of the matter inthe universe directly, have not been observed yet. A viable way of observing CP violation in the
lepton sector is to use the phenomenon of neutrino oscillation. Thus, the discovery of neutrino
oscillations had carried the issue of leptonic CP violation onto one of the central agenda of theparticle physics.

1.1

Neutrino Oscillation

In the Standard Model of elementary particles, neutrinos are regarded as exactly masslessparticles [3, 4, 5]. However, there is no fundamental reason to forbid the finite masses of neutrinos. Some experiments set a limit of neutrinos’ mass. Electron neutrino ν e mass was measured
by beta spectra in Tritium β-decay. The electron neutrino mass is constrained to be less than
2.2 eV/c2 at 95% C.L. [6]. The upper limit of muon neutrino ν µ is set to be 190 keV/c2 fromthe measurement of two body decay of π + at rest π + → µ+ + νµ [7]. Tau neutrino ντ mass
was obtained to be less than 18.2 MeV/c 2 from the measurement of the hadronic final statesτ → nπ + ντ (n > 2) [8]. There had been no evidence of finite mass for neutrinos from theseexperiments.
On the other hand, a consequence of finite mass for neutrinos appears in neutrino oscillation because a weak eigenstate (flavor eigenstate) |ν α i produced in weak interaction is a linear
combination of the mass eigenstates |ν i i when neutrinos have mass. Neutrino oscillation is proposed by Maki, Nakagawa and Sakata in 1962 [9]. And see also Pontecorvo [10]. Their flavor
eigenstates can be written as the superpositions of the mass eigenstates as follows :|να i =

3X
i=1

Uαi |νi i,(1.1)

where |να i represents the flavor eigenstates ν e , νµ and ντ and |νi i represents the mass eigenstates1ν1 , ν2 and ν3 . U is a 3×3 unitary mixing matrix known as the MNSP matrix. The mixing matrix
U can be written as a product of three rotation matrices using the mixing angles θ 12 , θ23 andθ13 and a CP-violating phase δ as follows :




c12 s12 0c13
0 s13 e−iδ
1
0
0




(1.2)
U =  0 c23 s23  010  −s12 c12 0  ,−s13 eiδ 00 −s23 c23

c13

0

0

1

where cij ≡ cos θij , sij ≡ sin θij . The time evolution of the flavor eigenstates is represented as :X|να (t)i =Uαi exp(−iEi t)|νi (t = 0)i
(1.3)
i

XX

=

i

Uαi Uα∗0 i exp(−iEi t)|να0 (t = 0)i,

(1.4)

α0

where Ei is an eigenvalue of the mass eigenstate |ν i i. Therefore the probability for flavor
eigenstate να at t=0 to change the state νβ at time t, is calculated as :P (να → νβ ) = |hνβ (t)|να (0)i|2
=

X

(1.5)
2

∗
Uβi Uαiexp(−iEi t)(1.6)|Uαi Uβi |2 +

(1.7)

i

=

X
i

X
i6=j

∗ ∗
Uαi UβiUαj Uβj exp(−i(Ei − Ej )t).

Due to the flavor mixing of massive neutrinos, the flavor transition phenomenon, called neutrino
oscillation, could occur.
In the two-flavor mixing case, the mixing matrix U is simplified as follows :!
cos θ sin θ
U=
.
(1.8)
− sin θ cos θ
The transition probability for να → νβ (Eq. (1.7)) is expressed in much simpler form :22 (Ei − Ej )tP( να → νβ ) = sin 2θ sin
.
2

(1.9)

If neutrinos have finite masses, the following approximation can be used :qq
Ei − E j =m2i + p2i − m2j + p2j
!


m2j
m2i
− p+
'
p+
2p
2p
'

∆m2ij
,
2E

(1.10)

where ∆m2ij ≡ m2i − m2j is the mass-squared difference of neutrino mass eigenstates. Finallyusing the neutrino flight length Lν and the neutrino energy Eν , the survival probability of να in2the two-flavor mixing case is written as follows :P (να → να ) = 1 − sin2 2θ sin222

= 1 − sin 2θ sin∆m2ij Lν
4Eν!

1.27∆m2ij (eV 2 )Lν (km)Eν (GeV)(1.11)!

,(1.12)The oscillation probability is characterized by the mixing angle, the mass squared difference∆m2ij , the neutrino flight length Lν and the neutrino energy Eν . So the oscillation amplitude ismaximum when Lν [km]/Eν [GeV] ∼ π/(2.54 · ∆m2ij [eV2 ]). Putting experimentally conceivable
numbers in this relation, one easily finds that neutrino oscillation experiment can explore theneutrino mass square difference below the eV 2 range.

1.2

Neutrino oscillation experiments

There are many experiments which have observed neutrino oscillations and measured oscillation parameters; the mass-squared difference and the mixing angle. In this section, the current
status of our knowledge of neutrino oscillations is summarized.

1.2.1

Solar neutrino and Reactor neutrino

Electron neutrinos are produced via 4p → 4 He + 2e+ + 2νe + γ in the Sun. The Homestake
chlorine experiment [11], which was the first experiment to observe solar neutrinos, measured
neutrinos by using the reaction νe +37 Cl → e− +37 Ar. The produced Ar atoms were counted
by a chemical technique. The measured count rate of solar neutrinos is 2.56 per 10 36 target
atoms per second, 2.56 SNU (SNU: Solar Neutrino Unit), against a prediction of 7.6 SNU fromthe standard solar model (SSM) [13]. Other experiments SAGE [14] and GALLEX [15], using
71 Ga target, and Kamiokande [12], H O target, also observed this flux deficit which became2subsequently known as the “solar neutrino problem”. This flux deficit is mostly solved by theneutrino oscillation, but the analysis using only the flux deficit cannot produce a unique solution
for the neutrino oscillation.
Super-Kamiokande (SK) experiment [16, 18], which is a successor of Kamiokande experiment,
started in 1996, and SNO experiment [17], which uses the Cherenkov detector filled with heavy
water D2 O, started in 1999. SK and SNO measure solar neutrino flux from 8 B in different
neutrino reactions, charged current interaction (CC), ν e + D → e− + p + p, neutral current
interaction (NC), νx + D → νx + p + n, and elastic scattering (ES), ν x + e− → νx + e− . NC canbe initiated with equal cross sections by neutrinos of all active flavors. Therefore NC measuresφ(νe ) + φ(νµ , ντ ), where φ(νµ , ντ ) is the flux of νµ and/or ντ from the Sun, while CC only
measures the φ(νe ) flux from the Sun. From both NC and CC measurements, the ν e flux deficit
can be explained by νe oscillations into the other active neutrino flavors (ν µ and ντ ). On theother hand, ES can measure a neutrino of any flavors. But the cross-section σ(ν x ) is different
for each neutrino; σ(νµ ,τ e → νµ ,τ e) ' σ(νe e → νe e)/6.5. From the measurements of SNO
and SK, νe → νµ oscillation was discovered during 2001 and 2002. Combining all the resultsfrom solar neutrino experiments, the LMA-MSW solution [21] of the solar neutrino problem wasindicated [16].
3To confirm these results, the KamLAND experiment [20] played an important role. KamLAND is a liquid scintillator detector and searches for anti-electron-neutrino (ν e ) oscillations.
The ν e s are generated at distant power reactors. They are detected by the reaction ν e +p→
e+ +n, where the prompt signal from the positron and the delayed coincidence of characteristic
gammas from neutron capture allow an efficient reduction of backgrounds. The ν e flux at thedetector site is dominated by a number of reactors at an average distance of ∼ 180 km. Since theenergy of the produced ν e s is around 3 MeV, the experiment is sensitive to ∆m 2 down to several
10−6 eV2 . KamLAND observed a very clean energy dependent ν e disappearance, constraining
the oscillation parameters precisely. The combined analysis of the solar neutrino data and the
KamLAND data leads to the oscillation parameters [19]:7.38 × 10−5 < ∆m212 < 7.78 × 10−5 eV 2 ,0.30 < sin2 θ12 < 0.34 (1σ).

1.2.2(1.13)

Atmospheric neutrino and Accelerator neutrino

Atmospheric neutrinos are produced by the decays of pions and kaons which are generated
when primary cosmic rays interact with the nuclei in the Earth’s atmosphere :π + → µ+ + νµπ−→ e+ + νe + ν µ ,(1.14)→ e− + ν e + νµ .

(1.15)−→ µ + νµNeutrinos νe , ν e , νµ and ν µ are generated as the atmospheric neutrino. The energy spectrum of
the atmospheric neutrinos peaks around 40 MeV and extends to the energy range above TeV. The
neutrino flight length ranges from ∼ 10 to ∼ 10 4 km, corresponding to the production height ofabout 15 km and the diameter of the Earth (13000 km), respectively. Since typical atmospheric
neutrino experiments have enough event statistics for neutrino energies above several hundreds
MeV, the observation of atmospheric neutrinos is sensitive to ∆m 2 ∼ Eν /Lν = 10−1 ∼ 10−4 eV2 .
The predicted total flux of atmospheric neutrinos has ∼ 10 % uncertainty in the GeV energy
range, which mostly comes from the uncertainties on cross sections of hadronic interactions andon fluxes of primary cosmic ray components [22]. However, the flavor ratio (ν µ + ν µ )/(νe + ν e )is predicted within only 3 % uncertainty. This ratio has been measured by several undergroundexperiments (SK [23, 24], IMB [25, 26] and Soudan 2 [27]) and reported in terms of a double
ratio R≡ (Nµ /Ne )Data /(Nµ /Ne )M C , where Nµ (Ne ) is the number of muon (electron) events
produced by charged current interactions. The measured double ratio R was significantly lower
than 1.
In 1998, Super-Kamiokande reported that the zenith angle distributions of muon neutrino
events were asymmetric, and concluded that their results gave an evidence for neutrino oscillation [28]. In addition, MACRO [29] and Super-Kamiokande [30] have studied upward-going
muons produced by energetic νµ charged current interactions in the rock surrounding the detector and their data are consistent with the two-flavor neutrino oscillation ν µ ↔ ντ . SuperKamiokande also reported the observation of an oscillation signature with L/E dependence [33],which is well described by the hypothesis that ν µ oscillates to ντ with a nearly maximal mixing.
In the two-flavor mixing framework, the neutrino oscillation parameters from SK L/E result [35]4∆ m2 [10−3eV2]

4

3

SK(zenith) 90%C.L.
SK(L/E) 90%C.L.

2
MINOS 90%C.L.
K2K 90%C.L.

0.5

0.6

0.7

0.8

0.9

1

sin 2θ
2

Figure 1.1: 90 % confidence level allowed regions of oscillation parameters for ν µ ↔ ντ oscillation.
The horizontal axis shows sin2 2θ and the vertical axis shows ∆m2 . The results are taken fromlatest Super-Kamiokande [35] zenith angle analysis result (solid line), L/E result (dotted line),
K2K experiment [31] (dashed line) and MINOS experiment[32] (dash-dotted line).

are constrained to be :1.9 × 10−3 < ∆m223 < 2.6 × 10−3 eV 2 ,0.94 < sin2 2θ23 at 90 % C.L..

(1.16)

The atmospheric neutrino oscillation was confirmed by long-baseline experiments using themuon neutrino beam produced by accelerators, such as K2K [31] and MINOS [32]. These experiments measure the spectral distortion and the decreasing ν µ flux during the travel. Figure 1.1
shows the allowed parameter regions from the ν µ ↔ ντ oscillation measurement experiments.

1.2.3

Mixing angle θ13

The third mixing angle θ13 was tested by the CHOOZ reactor experiment. This experiment
looked for distortions in the energy spectrum of ν e produced by a nuclear reactor 1 km away
by using liquid scintillator in conjunction with ν e s from inverse β-decay process. ν e survival
probability can be expressed :22
2 1.27∆m13 L
P(ν e → ν e ) = 1 − sin 2θ13 sin.
(1.17)
E
5

Figure 1.2: Allowed region on 1-3 parameter is plotted for each experiment. Combined analysisof the global data is shown as filled region. This plot taken from [37]

The CHOOZ experiment ended without any evidence for ν e oscillation in large mass splittings∆m2 = 10−2 ∼ 10−3 . Thus, it implies that the νe fraction in ν3 mass eigenstate would have been
too small to be detected. The current upper limit on θ 13 is given by the CHOOZ experiment [36] :sin2 θ13 < 0.04

(1.18)

at 90 % C.L.(assuming ∆m223 = 2.0×10−3 eV2 ).
The atmospheric neutrino data have also been used to constrain θ 13 in the three-flavor
mixing framework with the one mass scale dominance approximation (|m 22 − m21 |  |m23 − m22 | ∼|m23 − m21 |). In this framework, oscillations of atmospheric ν e s are expressed by :22
2 1.27∆m23 L,(1.19)P(νe → νe ) = 1 − sin 2θ13 sin
EP(νµ → νe ) = P(νe → νµ )


2
2
2
2 1.27∆m23 L= sin θ23 sin 2θ13 sin.
(1.20)
E
Consistent results with CHOOZ experiment have been obtained [34, 35]. Detail of the analysis
with atmospheric neutrino data will be discussed in Section 1.3.

1.2.4

Summary for neutrino oscillation parameters and Remaining issues

In this way, neutrino oscillation has been well understood and consolidated the existing ofmassive neutrinos. The global best fit results of neutrino oscillation parameters are summarizedin Table 1.1. At the same time, there remain a few open questions.
Since oscillation experiments are measuring mass differences, ∆m 212 and ∆m223 , not theabsolute neutrino masses, it is not known what the mass ordering is. Figure 1.3 shows the6parameter∆m212|∆m213 |sin2 θ12
sin2 θ23
sin2 θ13

best-fit ± 1σ C.L.

−5 eV27.65 +0.23−0.20 × 10
−3 eV22.40 +0.12−0.11 × 10
0.304 +0.022−0.0160.50 +0.07−0.06
0.01 +0.016−0.011

Table 1.1: Summary table of the global best-fit to neutrino oscillation data from [37]normalinverted

m3

m2
m1

2

mass

m2
m1

νe
νµ
ντ

m3

Figure 1.3: A three-neutrino mass-squared spectrum, which accounts for the observed flavorchanges, for two mass hierarchy schemes

conceptual diagram for two types of the mass order. This is the first question which the mass
hierarchy is, normal or inverted.
Whether or not θ13 is zero is the important question. If θ 13 is found to be non-zero, it
becomes possible to address the question of CP-violation in the lepton sector, because a CPviolating phase δ appears as a combination with θ 13 parameter as shown in Eq. (1.2).

Typical atmospheric neutrino experiments cover a wide range of L/E ν and can detect νeinteractions. Therefore it is possible for those experiments to measure the effect of three-flavor
oscillation including θ13 and 1-2 parameters. In the next section, it is described how theseoscillation effects can be measured in Super-Kamiokande.

1.3

Sub-dominant effects of atmospheric neutrino oscillation

Atmospheric neutrino data are well fitted to pure ν µ → ντ 2-flavor oscillation. When 3-flavor
oscillation, electron neutrino oscillation, is considered, sub-dominant oscillation effects appear
due to θ13 mixing and/or the ∆m212 -induced oscillation wavelength. In these oscillation schemes,
matter effect [21] plays an important role.
For neutrino traversing the Earth, oscillation probability is calculated taking into account
Earth’s matter potential due to the forward scattering amplitude of charged current ν e and ν¯einteractions with electrons. The Earth contains a large number of electrons, electron neutrinos
7interact with them via W± boson while muon and tau neutrinos do not.
When taking into account the matter effect in the P (ν µ → νe ) oscillation with the approximation of ∆m212 ∼ 0 as shown in Eq. (1.20), it can described by replacing θ 13 and ∆m223 by
the corresponding mixing parameters in matter. For constant matter density, P (ν µ → νe ) (=P (νe → νµ )) is described as follows:!
2
1.27∆mL
ν
23,M,(1.21)P(νµ ↔ νe ) = sin2 θ23 sin2 2θ13,M sin2Eν
where∆m223,M
sin2 2θ13,M
ACC= ∆m223q(cos 2θ13 − ACC /∆m223 )2 + sin2 2θ13

sin2 2θ13(cos 2θ13 − ACC /∆m223 )2 + sin2 2θ13√
= 2 2GF Ne Eν

=(1.22)(1.23)(1.24)

ACC : the matter potential term
GF : the Fermi constant
Ne : the electron densities in the medium.
When |ACC /∆m2 | = cos 2θ13 , a resonant enhancement of the oscillation probability, called
Mikheyev-Smirnov-Wolfenstein (MSW) resonant enhancement, occurs and holds for either neutrinos or anti-neutrinos depending on the mass hierarchy. Figure 1.4 shows oscillation probabilityof νµ → νe for neutrinos traveling the Earth as a function of energy and zenith angle with amore realistic matter density distribution. The oscillation probability in vacuum is also shownin Fig. 1.4 for comparison. The resonance enhancement can be seen in the 2-10 GeV energy
region and the upward direction with some assumptions written in the caption. Note that theenhancement occurs in normal (inverted) hierarchy case for the neutrinos (anti-neutrinos).
When the oscillation driven by the LMA-MSW effect obtained from solar neutrino parameters(∆m212 and θ12 ) is considered, the oscillation effect of low energy electron neutrinos will beobservable even if θ13 = 0. In case of θ13 ∼ 0, the neutrino oscillation probabilities can be
written in the constant matter density as follows:
P (νe ↔ νe ) = 1 − Pex(1.25)2P (νe ↔ νµ ) = cos θ23 Pex(1.26)42p

P (νµ ↔ νµ ) = 1 − cos θ23 Pex − sin 2θ23 (1 − 1 − Pex cosφ) Lφ ∼ ∆m231 + s212 ∆m221
2Eν


2
2
2
2 ∆m21 Lν sin 2θ12Pex (νe → νx ) = sin 2θ12,M sin
4Eν sin2 2θ12,M
where
cos 2θ12,M=

cos 2θ12q.
(cos 2θ12 − ACC /∆m212 )2 + sin2 2θ12
8

(1.27)1

P(νµ → νe)

1

P(νµ → νe)

0.2

0.8

0.18

0.6

0.4

0.6

0.16

0.4

0.35

0.4

0.14

0.2

0.3

0.2

0.12

0

0.25

0

0.1

-0.2

0.2

-0.2

0.08

-0.4

0.15

-0.4

0.06

-0.6

0.1

-0.6

0.04

-0.8

0.05

-0.8

0.02

-1 -1
10

1

10

10

cosΘν

0.45

cosΘν

0.8

20

-1 -1
10

1

Eν (GeV)

10

10

20

Eν (GeV)Figure 1.4: The νµ to νe (or νe to νµ ) transition probability in the Earth (left) and in vacuum(right) for atmospheric neutrinos as a function of energy E ν and neutrino zenith angle. cos Θν =−1 and cos Θν = 0 correspond to upward-going and horizontal directions, respectively. The plots
assume the oscillation parameters (∆m 23 , sin2 θ23 , sin2 θ13 ) = (2.1 × 10−3 eV 2 , 0.5, 0.04 (Chooz
limit)).

Figure 1.5 shows the Pex as a function of energy and zenith angle with a more realistic matter
density distribution. The oscillated atmospheric ν e flux is expressed as :Feosc = Fe0 · P(νe → νe ) + Fµ0 · P(νµ → νe )= Fe0 1 + Pex (r cos223 −1)

Feosc− 1 = Pex (r cos2 θ23 − 1)Fe0(1.28)The observable νe oscillation is due to r cos 2 θ23 − 1 in Eq. (1.28). Since r in Eq. 1.28 is about 2
for low energy region, the νe oscillation effect can be seen by the flux deficit or excess in case ofcos2 θ23 6= 0.5. If cos2 θ23 is greater than 0.5 (θ23 < 45◦ ), enhancement of flux is expected. On
the other hand, if cos2 θ23 is less than 0.5 (θ23 > 45◦ ), deficit of flux is expected. Figure 1.5 also
shows the flux ratio for each case sin2 θ23 = 0.4, sin2 θ23 = 0.5 and sin2 θ23 = 0.6.
Two sub-dominant oscillation schemes are described above considering approximation of∆m212 ∼ 0 or θ13 ∼ 0. However, if we describe the effect due to the interference between θ 13 and ∆m212 -induced oscillations, full oscillation parameters should be taken into account without
any approximations. The full oscillation parameters mean all relevant oscillation parameters
including the CP violating phase. In constant matter density, ν µ ↔ νe transition probability
considering all oscillation parameters is described as follows:∼

P(νµ ↔ νe ) = cos2 θ 13 cos2 θ23 Pe2
∼

∼

∼

∼

+ sin2 θ 13 cos2 θ 13 sin2 θ23 (2 − Pe2 )− 2 sin θ 13 cos2 θ 13 sin θ23 cos θ23 (cos δR2 − sin δI2 )9Φe / Φe00P(νe → νx)

2

-0.5

0.8

0.9

0.6

0.8

0.4

0.7

0.2

0.6

0

0.5

-0.2

0.4

-0.4

0.3

-0.6

0.2

-0.8

0.1

-1
0

cosΘν

cosΘν

1

1.4
1.35
1.3
1.25
1.2
1.15
1.1
1.05
1

sin θ23 = 0.4

sin θ23 = 0.5
2

0.95
-0.5
1.0
1.05

-1 -1
10

1

10

10

-1
0

1
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0.6

sin θ23 = 0.6
2

-0.5

20

Eν (GeV)
-1
10

-1

1

Eν (GeV)Figure 1.5: The left side panel shows the ν e transition probability Pex for atmospheric
neutrinos as a function of energy Eν and neutrino zenith angle, assuming oscillations with(∆m12 , ∆m23 , sin2 θ12 , sin2 θ23 ) = (7.7 × 10−5 eV 2 , 2.1 × 10−3 eV 2 , 0.3, 0.5). The right side panels
show the electron neutrino flux ratio F eosc /Fe0 . An expected excess (deficit) for sin 2 θ23 in the first(second) octant is shown in the upper (bottom) panel while the center panel for sin 2 θ23 = 0.5
is seen with small both deficit and excess.

10wherePe2 = sin2 2θ12,M sin2

φm
2

R2 = − sin 2θ12,M cos 2θ12,M sin2

φm2

1
I2 = − sin 2θ12,M sin φm
2∼∼

φm is the phase oscillation in matter and θ 13 is basically the mixing angle in matter; θ 13 ≈
θ13,M [39]. If the full oscillation scheme is considered, the electron neutrino oscillations occur
with the CP violating phase. Figure 1.6 shows P(ν µ → νe ) as a function of energy and zenith
angle in case of the oscillations (∆m 12 , ∆m23 , sin2 θ12 , sin2 θ23 , sin2 θ13 , δcp ) = (7.7 × 10−5 , 2.1 ×
10−3 , 0.3, 0.5, 0.04, 0◦ ) with realistic matter density. It is obvious that the electron neutrino
oscillation may appear in both Sub- and Multi-GeV energy range. Here “Sub-GeV” (“MultiGeV”) is defined to be less (more) than 1.33 GeV in visible energy in a water Cherenkov detector.
In Super-Kamiokande, a particularly important observation is the change of expected electron
events. The expected electron events (N e ) as compared to no oscillated events (N e0 ) can be
written in three terms :Ne− 1 = (Pee − 1) + rPeµNe0
' ∆1 (θ13 ) + ∆2 (θ12 , ∆m212 ) + ∆3 (θ13 , θ12 , ∆m212 , δcp )where r is the ratio of atmospheric ν µ and νe fluxes, r ∼ 2 to ∼ 3.5 in a few MeV to 10 GeV.
∆1 (θ13 ) arises for θ13 > 0 and is independent of ∆m212 . ∆2 (θ12 , ∆m212 ) arises for ∆m212 > 0
and is independent of θ13 . Only ∆3 (θ13 , θ12 , ∆m212 , δcp ) depends on both ∆m212 and θ13 . It
is the only term which includes the effect from CP phase values. These three terms affectdifferent energy regions; ∆1 (θ13 ) induces the electron excess in Multi-GeV region due to θ 13 ;
∆2 (θ12 , ∆m212 ) induces an excess or deficit in Sub-GeV region due to an octant of sin 2 θ23 ;∆3 (θ13 , θ12 , ∆m212 , δcp ) occurs for both energy regions due to the interference between θ 13 - and∆m212 -induced oscillations. Figure 1.6 also shows the ν e flux ratio Ne /Ne0 assuming the same
oscillation parameters. Figure 1.7 shows the numerical examples of the effects from various CPphases at the observed zenith angle distributions. The number of observed events is expected tochange a few % due to CP phase. Therefore, looking for the interference sub-dominant effectsof the atmospheric neutrino enables us to examine the CP phase.

This thesis aims to examine the full neutrino oscillation and to obtain the parameter values,
which are ∆m212 , ∆m223 , θ23 , θ13 , and CP phase (δcp ), by considering sub-dominant effects inthe atmospheric neutrinos. This is the first analysis in the Super-Kamiokande experiment andthe first attempt to obtain the information of CP phase in the neutrino oscillation studies. The
thesis’s contents are as follows : Chapter 2 describes the Super-Kamiokande detector, Chapter 3
explains the simulation for the atmospheric neutrinos observed in Super-Kamiokande, Chapter 4
describes the detector calibrations which are important for precise measurements, Chapter 5
describes the measurement methods and the data of atmospheric neutrinos. The analysis andthe results are explained in Chapter 6.
11

P(νµ → νe)1

νe flux ratio Φosc / Φ0

1

0.7

0.8

1.4

0.8
0.6

0.6

1.3

0.6
1.2

0.5

0.2

0.4

0.4

0
0.3

-0.2
-0.4

0.2

1.1

0

1

-0.2

0.9

cosΘν

cosΘν

0.4

-0.4

0.2

0.8

-0.6

-0.6
0.1

-0.8
-1 -1
10

1

10

10

0.7

-0.8

20

-1 -1
10

1

Eν (GeV)

10

10

2 0.6

Eν (GeV)Figure 1.6: The left panel shows the probability ν µ → νe oscillation as a function of zenith
angle and energy in case of (∆m12 , ∆m23 , sin2 θ12 , sin2 θ23 , sin2 θ13 , δcp ) = (7.7 × 10−5 , 2.1 ×
10−3 , 0.3, 0.5, 0.04, 0◦ ). The right panel shows the νe flux ratio Ne /Ne0 assuming the same oscillation parameters.

1.2

1.2

N / N0N / N0MC(osc.)1.1

1

δ = 0o1.1

δ = 80oδ = 160oδ = 240o1δ = 320o

-1

-0.5

0

0.5

1

-1

cosΘ

-0.5

0

0.5

1

cosΘ

Figure 1.7: Zenith angle distributions of the expected MC events for Sub-GeV 1-ring elike (left) and Multi-GeV 1-ring e-like (right) samples assuming the oscillation parameters(∆m12 , ∆m23 , sin2 θ12 , sin2 θ23 , sin2 θ13 ) = (7.7 × 10−7 , 2.1 × 10−3 , 0.30, 0.55, 0.04) in normal hierarchy case. Each line indicates the different CP phase value. All plots are normalized by the
number of events without the oscillations.

12

Chapter 2

The Super-Kamiokande Detector
Super-Kamiokande is a cylindrical 50 kt water Cherenkov detector located at Kamioka Observatory of Institute for Cosmic Ray Research in Gifu Prefecture, Japan. A schematic view of
the Super-Kamiokande detector is shown in Figure 2.1. The geographic coordinates of the site
are 36◦ 25’N and 137◦ 18’E, and the altitude above sea level is 370 m. The detector lies in a zincmine under the top of Mt.Ikenoyama, where the mean rock overburden is ∼ 1000 m (2700 m
water equivalent). This overburden plays a role of a shield against cosmic ray muon backgroundin the detector. The cosmic ray muon flux at the Super-Kamiokande site is reduced by 5 ordersof magnitude compared to that on the surface of the earth. The detector construction is the water tank with the photomultiplier tubes(PMTs) which can detect weak lights due to Cherenkov
radiation.
The main scientific purposes of the Super-Kamiokande experiment are the searches for nucleon decays and the studies of various types of neutrinos: atmospheric neutrinos, solar neutrinos,
and the neutrinos from supernovae and the other astrophysical sources. The Super-Kamiokande
detector is also used as a target of the artificial neutrino beam in long-baseline neutrino oscillation experiments.
The Super-Kamiokande experiment started taking data in April, 1996 and continued the observation for five years within the running period referred to SK-I till the detector maintenancein July, 2001. During refilling water after the maintenance, an accident occurred in November,2001 in which more than a half of the PMTs were destroyed. The Super-Kamiokande detector
was rebuilt after the accident with the half of the original PMT density in the inner detector
and resumed observation from October, 2002, which is referred to the SK-II running period.
The SK-II continued the physics measurement for three years and finished in October 2005 forthe reconstruction work to put the PMT density back to the SK-I level. The Super-Kamiokande
detector has restarted observation in June, 2006, which is referred to the SK-III period. Meanwhile, new electronics module for the Super-Kamiokande detector had been developed. New
electronics was installed on the detector in September 2008, starting the SK-IV running period.
In this thesis, the data observed in the SK-I (1996-2001),the SK-II (2002-2005) and SK-III(2006-2008) running periods are used.
In this chapter, the construction and technical details of the Super-Kamiokande detector aredescribed.

13

Figure 2.1: Super-Kamiokande detector and its location.

2.1

Cherenkov Radiation

The Super-Kamiokande detector observes relativistic charged particles in water by detecting
the emitted Cherenkov light. The Cherenkov photons are radiated when the velocity of a charged
particle exceeds the light velocity in the medium :
v≥

c
,
n

(2.1)

where v is the velocity of the charged particle, n is the refractive index of the medium and c is thelight velocity in vacuum [40]. The momentum threshold of Cherenkov radiation is determinedby the refractive index of the medium and the mass of the particle. Since the refractive index
of water is about 1.34, the momentum thresholds of Cherenkov radiation for electrons, muons,charged pions and protons are 0.57, 118, 156 and 1051 MeV/c , respectively.
Cherenkov light is emitted on a cone with a characteristic half opening angle θ C along thedirection of the particle. The opening angle, called Cherenkov angle, is determined as follows :cos θC =

1
,
nβ

(2.2)

where β = v/c. For the particle with β ' 1 in water, the Cherenkov angle is about 42 ◦ .
The number of photons emitted by Cherenkov radiation is given as a function of the wavelength λ as follows :


d2 N
2πα
1
= 2 1− 2 2 ,
(2.3)
dxdλ
λ
n β
14

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

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

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

Comnt;

Figure 2.2: A visual display of a typical observed neutrino event in Super-Kamiokande. The sizeof small circles in the unrolled cylinder represents the amount of Cherenkov photons detectedin each photo-multiplier tube. The Cherenkov ring image is clearly visible.

where x is the path length of the charged particle and α is the fine structure constant. About
340 photons/cm are emitted between the wavelength of 300 nm to 600 nm, which is the sensitive
wavelength region to the PMTs used in the Super-Kamiokande detector.
Particles emitting Cherenkov light project ring images on the wall inside the detector.
Super-Kamiokande detects the Cherenkov photons by the PMTs arranged on the wall and theCherenkov rings can be recognized. Figure 2.2 shows a visual display of a typical neutrino eventin the Super-Kamiokande detector.

2.2

Detector

2.2.1

Water Tank

The whole size of the water tank is 41.4 m in the hight and 39.3 m in the diameter, in which
50 kt highly pure water is filled. The water tank is optically separated into two concentric cylindrical regions by a PMT support structure and a pair of opaque sheets as shown in Figure 2.3.
The detector part in the inner region of the tank is referred to the inner detector (ID). The
inner region contains 32 kt water with the size of 36.2 m in the hight and 33.8 m in the diameter.
11,146 inward-facing 20-inch PMTs are attached to the supporting frame uniformly at intervalsof 70 cm for SK-I. The effective photocathode coverage of the ID is about 40 %, and the rest of
the surface is covered with black polyethylene terephthalate sheet, called ”black sheet”. 5,182
20-inch PMTs are attached at one intervals to the supporting frame and the photocathodecoverage of the ID is about 20 % for SK-II. In the SK-III, the photocathode coverage is returned
to about 40% level with 11,129 PMTs.
15

Tyvek8" PMT

Top

Black sheet

Barrel20" PMT

BottomFigure 2.3: Supporting frame of PMTs.

16

The outer region completely surrounds the ID with the thickness of 2.05 m on top and bottom
and 2.2 m along the barrel wall. This region, called the outer detector (OD), is monitored by
1,885 outward-facing 8-inch PMTs attached to the outer side of the supporting frame. To
improve the light collection efficiency, a 60 cm × 60 cm wavelength shifting plate is attached toeach OD PMT and the walls are covered with reflective material called ”tyvek sheet”. The OD
is used as veto counter for entering cosmic ray muons and to tag the outgoing charged particles.
Furthermore, the 2 m thickness water layer itself serves as a shield to attenuate gamma ray andneutron fluxes from the rock. The inner and outer volumes are separated by a 55 cm thick deadregion. This region is not instrumented with any PMT. Stainless steel frames and signal andHV cables of the ID and OD PMTs are contained in this dead region.

2.2.2

photomultiplier tube

The photomultiplier tubes used in the ID, product name Hamamatsu R3600, have photocathode with a diameter of 50 cm (20 inch). This 20 inch PMT was originally developed byHamamatsu Photonics K.K. in cooperation with the Kamiokande collaborators [41]. Later,
the dynode structure and the bleeder circuit were modified for Super-Kamiokande to improvetiming response and photon collection efficiency [42]. A schematic view of the PMT is shown in
Figure 2.4 and the specifications are summarized in Table 2.1.
The photo-cathode of the PMT is coated by bi-alkali (Sb-K-Cs) owing to the high spectral
sensitivity to Cherenkov light and the low thermionic emission. The quantum efficiency is 22 %in the peak at 360-400 nm. The dynode structure and the bleeder circuit are optimized to achieve
high collection efficiency, fast timing response and good energy resolution. The averaged value ofthe collection efficiency at the first dynode is 70 %, which is uniform within a difference of ±7 %everywhere on the photocathode. The gain of the ID PMTs is 10 7 at a supply high voltage from1500 V to 2000 V. Figure 2.5 shows the charge distribution for the single photoelectron signal inwhich a clear 1 p.e. peak can be seen. The transit time spread of the single photoelectron signal
is about 2.2 nsec. The average dark noise rate at the 0.25 p.e.s threshold used in SK-I was about
3 kHz.
The magnetic field over 100 mG affects photoelectron trajectories in the PMT and makes
the timing resolution worse, while geomagnetic field at the detector site is about 450 mG. To
compensate for the magnetic field, 26 sets of horizontal and vertical Helmholtz coils are arranged
around the tank. As the result, the magnetic field inside the detector is reduced to about 50 mG.
On November 12th, 2001 while refilling the SK tank after completing the maintenance work,
one ID PMT on the bottom of the tank imploded, which triggered a cascade of implosions.
About 60 % of the ID and OD PMTs were destroyed. To avoid a chain reaction of implosion,
all of the inner PMTs are instrumented with acrylic covers from SK-II period. A clear 12 mm
thick UV-transparent acrylic dome is put over the photo-cathode area of each PMT and theside of the PMT is protected by the Fiber-Reinforced-Plastic(FRP) shield with holes, which letwater flow into the case freely as shown in Figure 2.6. The transparency of the acrylic cover forphotons with normal incidence in water is more than 96 % above 350 nm of wavelength and theeffect of the PMT case is small.
1,885 8-inch PMTs, Hamamatsu R1408, are used in the OD. The photocathode of the OD
PMT is fitted with 60 cm × 60 cm × 1.3 cm wavelength shifter plate. The wavelength shifterincreases the light collection efficiency by 60 %. The timing resolution at single photoelectron
is degraded to about 15 nsec from 13 nsec with the plate. However, since the OD is used as a
17

Figure 2.4: A schematic view of a 20-inch PMT.

Shape
Photocathode area
Window material
Photocathode 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
Pyrex glass (4 ∼ 5 mm)
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 2.1: Specifications of 20-inch PMT.

18

distribution (arbitrary unit)

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 a typical 20-inch PMT.

Figure 2.6: PMT case attached to the inner PMT in SK-II period and later

19

OD

ID

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)

OD segmentation

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

Comnt;

Comnt;

Figure 2.7: A schematic view of the OD segmentation (left). OD regions are separated by tyvec
sheet. The OD visual displays of the partially contained event are shown for SK-II (middle),
without segmentation, and SK-III (right), with segmentation.

veto counter, rather than a particle tracker, the extra photons are of important while the poor
timing resolution is of little consequence.

2.2.3

OD segmentation

The separations between top/bottom and barrel region of OD part were installed with tyvek
sheet in SK-III period. A schematic view of the segmentation is shown in Figure 2.7. ThisOD segmentation aims to distinguish the particles (mostly muons) as produced in the tank andgoing out around the edge of the tank from the corner clipping cosmic ray muons, since theformer typically generates the signal in only one of the OD regions (top, side or bottom) while
the later typically generates the signal in two OD regions.
The right two panels in Figure 2.7 show examples of the OD visual display for the simulated particle which produces in the tank and goes out around the bottom edge of the tank.
The left display and the right display are in case of SK-II detector configuration (without OD
segmentation) and SK-III detector configuration (with OD segmentation), respectively. Due tothe segmentation, OD signals for SK-III are seen in the side region of the detector and clearly
separated from signals in the bottom region, while that for SK-II is not. This feature helps to
identify the particle as a clipping muon or not. The OD segmentation is taken advantaged at
the event selection process for partially contained events, will be described in Section 5.2.

2.3

Water Purification and Air Purification System

The water for the Super-Kamiokande experiment is produced from abundant spring waterin the mine. The water in the tank is continuously circulated through the water purification
system to keep the high attenuation length. It is also an important purpose of this system toremove the radioactive materials, especially radon (Rn), which is a background source for thesolar neutrino observation in the MeV energy range. The water is constantly circulated throughthe purification system with flow rate of about 35 ton/hour.
It is also essential to have clean air in the detector and the experimental area to minimize
the radon level in the detector water.
20

to PMTSUM
HITOUTDISCRI.
CURRENT SPLITTER&AMPLIFIER

PMT INPUT

W:200ns

to HITSUM

ONE-SHOT HIT

W:300ns

SELF GATESTART/STOP

TAC-A

TAC-A

CLEAR
GATE

START/STOP

QAC-A
TAC-B

QAC-A

TAC-B

to ADCCLEAR
GATE

THRESHOLD

QAC-B

QAC-B

DELAY

GATE
TRIGGERPED_START

Figure 2.8: A block diagram of analog input part of ATM.

To keep the minimum radon level inside the detector, radon-free air is produced by the air
purification system in the mine and is continuously pumped into the space above the water
surface inside the tank at a positive pressure to prevent radon in the SK dorm air from entering
the detector and dissolving into the purified water [43]. The radon concentration of the radonfree air is less than 3 mBq/m3 . The air purification system consists of compressors, a buffertank, driers, and the filters. The air flow rate is about 18 m 3 /hour.

2.4

Electronics and Data Acquisition System

ID PMT signals are processed by custom built electronics modules called ATM (AnalogTiming-Module) of the TKO standard (TRISTAN KEK Online) [44, 45]. The ATM module
records the integrated charge and the arrival timing information of each PMT signal.
Figure 2.8 shows the block diagram of the analog input part of the ATM module. Each
PMT input channel of ATM has two switching pairs of QAC (Charge to Analog Converter) andTAC (Time to Analog Converter) to minimize the electronics dead time in the data taking fortwo successive events, such as a muon followed by its decay electron. Each PMT signal sent
to ATM is amplified 100 times, and then divided into four signals. One of them is sent to thediscriminator. When the pulse height of the PMT signal exceeds the threshold level, which is
set to 0.25 p.e.s , a 400 nsec gate signal for QAC and a start signal for TAC are generated. At

21the same time, rectangular signal called HITSUM(200 ns width and 15 mV pulse hight) is sent
to a global trigger module. Other two branches of the split PMT signal are fed to QAC. Whena global trigger is issued, a stop signal is sent to TAC and the information in QAC and TAC
is digitized by ADC. The digitized charge and timing information is stored in internal FIFO
memory in ATM. when a global trigger is not issued within 1.3 µsec, all the information in QAC
and TAC is cleared. ATM has 450 pC dynamic range with a resolution of 0.2 pC, and 1.3 µsec
dynamic range with a resolution of 0.4 nsec. The remaining one signal becomes an output signalof ATM called PMTSUM. Figure 2.9 shows a schematic view of the ID data acquisition system.
Signals from 12 PMTs are fed to an ATM board to be processed. There are in total 946 ATM
boards installed in 48 TKO crates. The digitized data in ATM FIFO memory are sent to VME
memory modules called SMP (Super Memory Partner) every 16 events. 48 SMP are installed in
8 VME crates, and one SMP module handles the data of 20 ATMs. The data in SMP memories
are read out by 8 slave computers and sent to the online host computer.
Figure 2.10 shows a schematic view of the OD data acquisition system [46]. The paddle
cards distribute high voltage from the main frame to the OD PMTs. A coaxial cable is used
to supply the high voltage to an OD PMT and to send a signal from the PMT. These cards
also pick off the PMT signals through a high voltage capacitor. Signals from the OD PMTs aresent to QTC (Charge to Time Converter) modules. A QTC module converts the PMT signal
to a rectangular pulse whose width is proportional to the input charge. At the same time, arectangular HITSUM signal is generated by QTC and sent to a global trigger module. The
threshold of QTC modules is set to 0.25 p.e. . When a global trigger is received, the leading edge
and the width of the rectangular pulse are converted to the timing and charge information by
a LeCroy 1877 multi-hit TDC module. The dynamic range is set to 16 µsec which starts from
10 µsec before the global trigger timing.

2.4.1

Trigger

Figure 2.11 shows a overview of the ID trigger scheme. An ATM module generates a rectangular HITSUM signal with 15 mV in pulse height and 200 nsec in width if an ID PMT signal
exceeds the threshold. These signals are analog-summed over all PMTs to generate an IDHITSUM signal. The pulse height of the ID-HITSUM signal is proportional to the number of
hit PMTs within 200 nsec time window. There are three types of trigger signals derived fromthe ID-HITSUM signal. In SK-I, the high energy (HE) trigger is generated when the pulse hightof ID-HITSUM signal exceeds a threshold of −340 mV, which corresponds to 31 hits within a
200 nsec time window. The threshold for the low energy (LE) trigger is set to −320 mV, whichcorresponds to 29 hits. This is equivalent to a signal expected from a 5.7 MeV electron assuming
50 % of trigger efficiency. The trigger rates for HE and LE triggers are ∼ 5 Hz and ∼ 11 Hz,
respectively. The super low energy (SLE) trigger was implemented in May 1997 in order tolower the solar neutrino analysis threshold. The threshold level for the SLE trigger was lowered
gradually with the increasing trigger CPU power. The threshold is set to −186 mV in July 2000,
which is equivalent to 4.6 MeV electron. The trigger rate is ∼ 1 kHz. In SK-II and SK-III, thesame trigger scheme with the proper thresholds for each period are used. Trigger conditions ineach period are summarized in Table 2.2.
The OD trigger is generated by a similar procedure. When an OD PMT signal exceeds athreshold, a QTC module generates a rectangular pulse with 20 mV in height and 200 nsec inwidth. These signals are also analog-summed to generate an OD-HITSUM signal. The threshold
2220-inch PMT
SCH
ATM
x 20
x 240

ATM
GONG

20-inch PMTSCHATMinterface
Ultra Sparc

x 20
x 240

SMP

ATM

SMP

GONG

online CPU(slave)

SMP
SMP
SMP
SMP

x6

Super Memory Partner

VME
20-inch PMT
SCHUltra Sparc

ATM

online CPU(slave)

x 20
x 240

Ultra Sparc

ATM
GONG

online CPU(host)

Analog Timing Module

online CPU(slave)
TKO

FDDI

Ultra Sparc

Ultra sparc

online CPU(slave)
Ultra Sparc

online CPU(slave)
Ultra Sparc

FDDI

online CPU(slave)
Ultra Sparc

online CPU(slave)

online CPU(slave)interface
Ultra Sparc
SMP
SMP

Ultra Sparc

online CPU(slave)

SMP
SMP
SMP
SMP

interface

Super Memory Partner

VME
VME

interrupt reg.
TRG

20-inch PMT
SCH
TRIGGERATM

TRIGGER

x 20
x 240

PROCESSORHIT INFORMATION
ATM
GONG

Analog Timing ModuleTKO

PMT x 11200

ATM x ~1000

SMP x 48

online CPU(slave) x 9

Figure 2.9: A schematic view of the inner detector data acquisition system.

23

Local Hitsum

QTC
Modulex40/quadx10/quad

TDC
x5
4 LSBsHigh Voltage

16 bit event #TDC

FASTBUSPMT

Paddle Card

StruckSupply~470/quadrant

FSCC

AUX

84 pin data ribbon cables

Ethernet FSCC Control

Figure 2.10: A schematic view of the outer detector data acquisition system.

periodnumber of ID PMTs
photocathod coverage
HE trigger threshold
LE trigger threshold
SLE trigger thresholdother features

SK-I

SK-II

SK-III

Apr.1996 - Jul.2001
11,14640 %-340mV-320mV-186mV

Oct.2002 - Oct.2005
5,182∗
20 %-180mV-152mV-110mV

Jun.2006 - Sep.2008
11,129∗40 %-320mV-302mV-186mVOD segmentation

Table 2.2: Summary of characteristic for each SK period. ∗ All inner PMTs are instrumented
with acrylic covers.

for the OD trigger is set to 19 hits within a 200 nsec time window.
These four types of trigger signals (HE, LE, SLE and OD) are fed to a hardware trigger
module called TRG. The TRG module generates a global trigger signal when any one of thetrigger signals is issued.
The specifications in each detector period are summarized in Table 2.2.

24

Analog signal
from PMT
ATM threshold

ADC gate
TDC startand stop
HITSUM

400 nsec width

stop

start

200 nsec width

Global triggerATM
Sum of HITSUM
Master threshold
Global trigger
CentralHut

Figure 2.11: An overview of the ID trigger scheme.

25Chapter 3

Simulation of Atmospheric Neutrino
3.1

Overview

Atmospheric neutrino events in Super-Kamiokande are simulated by a Monte Carlo method.
Monte Carlo simulation is made of three constituents : neutrino flux, neutrino interactions andparticle tracking in the detector. From products of neutrino flux and cross-sections, expected
events observed in the detector are calculated during a certain period. The produced particles
have the information such as their vertex, direction, momentum and so on. Using such vector
information, the detector simulation simulates the particle, Cherenkov photon emission, thephoton propagation and the response of detector hardware. In this way, the simulation events,
which are the same structure as a real observed data, are constructed. It is often called Monte
Carlo events in this thesis.
This chapter describes the neutrino flux model, NEUT library which is the simulator ofkinematics of neutrino interaction and the detector simulation.

3.2

Atmospheric Neutrino Flux

Nowadays, there are several atmospheric neutrino flux models calculated by M. Honda etal. [47, 48, 22] (Honda flux), G. Battistoni et al. [49] (Fluka flux) and G. Barr et al. [50] (Bartol
flux). In this analysis, the Honda flux is adopted as a default model, and the differences from theother two flux models are considered as systematic uncertainties in the flux calculation which
will be discussed in Section 6.5.1. The Volkova flux [51] is adopted to simulate the event which
has the energy above 10 TeV, because the Honda flux model supports the energy range up to
10 TeV.
The primary cosmic ray flux model which is an input of the neutrino flux calculation isdetermined by the experimental measurements. Current status of the measurements of cosmic
ray proton flux is shown in Figure 3.1 together with the model used in the Honda flux calculation.
The primary cosmic ray spectrum has been precisely measured by BESS and AMS experiments
up to 100 GeV [52, 53]. The cosmic ray flux changes depending on the turbulence of the solar
wind, which is higher when the solar activities are high (solar maximum) than when the solar
activities are low (solar minimum). The difference of the cosmic ray flux at solar maximum andsolar minimum is more than a factor of two for 1 GeV cosmic rays, while it decrease to ∼ 10 %for 10 GeV.

26

Figure 3.1: The primary cosmic ray proton flux for the model used in Honda flux calculation(line) and the observation results (points). The observation data are taken from Webber [54](crosses), LEAP [55] (upward triangles), MASS1 [56] (open circles), CAPRICE [57] (vertical diamonds), IMAX [58] (downward triangles), BESS98 [52] (circles), AMS [53] (squares), Ryan [59](horizontal diamonds), JACEE [60] (downward open triangles), Ivanenko [61] (upward open
triangles), Kawamura [62] (open squares) and Runjob [63] (open diamonds).

1.5

Data/Calculation

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

0

10

10

1

10

2

10

3

10

4

pµ (GeV/c)

Figure 3.2: Secondary cosmic ray muon fluxes normalized by the calculation [70]. Data areobserved at Tsukuba(open box) and Norikura(open triangle) with the BESS detector [71, 72],
and at CERN(open circle) with the L3 detector [69]. For reference, DEIS [73] and MUTRON [74]are plotted. Dashed lines indicate the sum of errors in data and calculation. This figure is takenfrom [70].

27In the atmosphere, primary cosmic ray protons and nuclei interact with air nuclei. Throughthe hadronic interactions, secondary particles, mostly pions and some kaons, are generated. In
the Honda flux the US Standard Atmosphere model [64] is employed for the density structureof the atmosphere and the structure gives the zenith angle dependence of the atmospheric
neutrinos. Geomagnetic filed effects are calculated based on the IGRF2005 model [65]. In the
hadronic interactions of the cosmic rays with air nuclei, the two theoretical models, NUCRIN [66]simulation for the primary cosmic ray energies < 5 GeV and DPMJET-III [67] for > 5 GeV havebeen employed. Generated mesons decay into muons and neutrinos. The flux of this cosmic raymuons have been measured by several experiments such as BESS [68] and L3+C [69]. DPMJETIII employed in the Honda flux is modified based on the measurements by BESS and L3+C.
Figure 3.2 shows the ratio of cosmic ray muon spectrum after the modification, and results in
the hadronic interaction model agreeing with the measurement within .10% in the 1∼100 GeV
muon energy range.
In the calculation of neutrino flux, interactions and propagation of particles are treated ina 3-dimensional way, considering the curvature of charged particles in geomagnetic field. The
features of the 3-dimensional calculation comparing to the 1-dimensional one are an enhancement of neutrino flux for near-horizontal direction and lower production height of neutrinos in
the atmosphere which is also prominent for near-horizontal direction. The first feature is explained by a difference of the effective area, generates horizontally incoming neutrinos, between
3-dimensional and 1-dimensional ways as illustrated in Figure 3.3. This effect is important for
low energy neutrinos (< 1 GeV), in which transverse momentum of the secondary particles inhadronic interactions and the bending of muons in the geomagnetic field are considerable. Figure 3.4 shows the calculated zenith angle distributions of the neutrino flux for four different
neutrino energy ranges and for the three models. The horizontal enhancement in the left panel
is mostly due to the 3-dimensional effect described in Figure 3.3, while the enhancement in
the higher energy regions is mostly due to muons reaching to the ground before decaying nearvertical.
The calculated energy spectrum of atmospheric neutrinos at the Super-Kamiokande site areshown in Figure 3.5 for the Honda flux, Fluka flux and Bartol flux, in which the flux of ν e +ν e
to νµ +ν µ is averaged over all of the direction. The flavor ratio is about 2 in the energy region
up to a few GeV, and it becomes larger than 2 as the neutrino energy increases because morecosmic ray muons reach the ground before decaying. The flavor ratio depends on the zenith
angle of incoming neutrinos as shown in Figure 3.6.

28

Figure 3.3: A schematic view of the effective areas of primary cosmic rays interacting with air
nuclei for 1-dimensional and 3-dimensional calculations. Arrows written by solid lines show theprimary cosmic rays and dotted lines show the neutrinos. The 3-dimensional calculation gives
larger areas for near-horizontal direction.

29

300

4000

Honda flux
Fluka flux
Honda-1D
flux

-2

-1

-1

Flux(m sec sr )0.1-0.3 GeV

1.0-3.2 GeV

–

νµ+νµ

200

2000
100
–

νe+νe
0
-1.0

0

1.0

0
-1.0

0

1.0

0

1.0

1200

>3.2 GeV

0.3-1.0 GeV-1-1

Flux(m sec sr )

40

800

-2

20
400

0
-1.0

0

1.0

cosΘ

0
-1.0

cosΘ

Figure 3.4: Zenith angle dependence of the atmospheric neutrino flux for four energy regions.
Boxes show the Honda flux, dots show the Fluka flux and solid lines show the Honda-1D flux.
The higher and lower plots in each panel show for ν µ + ν µ and νe + ν e , respectively. An
enhancement of the neutrino flux is seen for near-horizontal directions at energies below 1 GeV
in 3-dimensional calculations of the Honda flux and the Fluka flux, while no enhancement isseen in the Honda-1D flux.

30

( ν µ+ ν µ)/( ν e + ν e )102

νe + νe

Honda flux
Fluka flux
Bartol flux2−2

Flux*Eν (m sec −1sr −1GeV)

νµ + νµ10

Honda flux

10

Fluka flux
Bartol flux

1 −1
10

1

10

10
Eν (GeV)

1 −1
10

2

1

10

10
Eν (GeV)

2

Flux flavor ratio

Figure 3.5: Predictions of the direction averaged atmospheric neutrino flux (Left) and the flavor
ratio (Right). Solid curves show the Honda flux, dashed curves show the Fluka flux and dottedcurves show the Bartol flux.

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

3

0.5
co
s

1

1 -1
θ 10

10

10

2

10 GeV)E ν(

Figure 3.6: Flavor ratio (νµ + ν µ )/(νe + ν e ) as a function of neutrino energy and zenith angle.
Calculation is based on the Honda flux.

31

3.3

Neutrino Interaction

Atmospheric neutrinos interact with nucleons and electrons in water or the rock around thedetector. In the Monte Carlo simulation for this atmospheric neutrino analysis, the neutrino
events are generated through the simulation of the neutrino interaction with the nuclei of waterand the rock. The generation volumes are inside the ID detector, the region of OD detector
and the rock around the Super-Kamiokande which are assumed to spread spherically up to 4km
from the detector to cover all observable events.
For the simulation of neutrino interaction, a simulation program library NEUT [76, 77] isused. In the NEUT simulator, the following charged current (CC) and neutral current (NC)interactions are considered :CC/NC (quasi-)elastic scatteringν + N → l + N0

CC/NC single meson production

ν + N → l + N 0 + meson

CC/NC deep inelastic interaction

ν + N → l + N 0 + hadrons

CC/NC coherent pion production

ν + 16 O → l + 16 O + π

where N and N 0 are nucleons (proton or neutron) and l is a lepton. Those interactions are
explained briefly in the following sections.

3.3.1

Elastic and Quasi-Elastic Scattering

The charged quasi-elastic scattering for free protons (i.e. hydrogen atom) is, treated in thissimulator, described by Llewellyn-Smith [78]. For the free nucleon, The differential cross section
is :

2
M 2 G2F cos2 θc
dσ ν̄
2
2 s−u
2 (s − u)=A(q ) + B(q )+ C(q )
(3.1)
dq 2
8πEν2
M2
M4
where Eν is the neutrino energy, and s and u are Mandelstam variables [78]. The factors A, B
and C are given as :m2 − q 2
q2
q2
2
2
A(q ) =
4 − 2 |FA | − 4 + 2 |FV1 |2
4M 2
M
M


2
2
2
1
q
q
4q FV ξFV2
− 2 |ξFV2 |2 1 +
−
M
4M 2
M2

2

m
(3.2)
− 2 (FV1 + ξFV2 )2 + |FA |2Mq2
B(q 2 ) =FA (FV1 + ξFV2 )
(3.3)
2
M

1
q2
|ξFV2 |2
C(q 2 ) =|FA |2 + |FV1 |2 −
(3.4)
4
4M 2
where m is the lepton mass, ξ is defined by means of anomalous magnetic moment µ p −µn = 3.71.
The vector form factors, FV1 (q 2 ) and FV2 (q 2 ), and axial vector form factor, FA (q 2 ) are determined32

experimentally and given by :FV1 (q 2 )

=



q2
1−
4M 2

−1 

q2
GM (q 2 )GE (q ) −
4M 2
2

−1


q2
=
1−
GE (q 2 ) − GM (q 2 )
2
4M
−2

q2
FA (q 2 ) = −1.232 1 − 2
MA

−2
q2
2
−1
2
GE (q ) = (1 + ξ) GM (q ) = 1 − 2
MVξFV2 (q 2 )





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

where GE and GM are the electric and magnetic form factor, M V and MA are the vector andaxial vector masses.
For the scattering off bound nucleons in 16 O, nuclear effects such as the Fermi motion of
the nucleons or the Pauli exclusion principle are taken into account as described by the Smith
and Moniz [82] model. Since nucleons are fermions, the outgoing momentum of the nucleons in
the interactions is required to be larger than the Fermi surface momentum to allow quasi-elastic
scattering to occur. In this simulator, the Fermi surface momentum is assumed to be 225 MeV/c.
Taking into account the experimental results from K2K and MiniBooNe [79, 80, 81], M V
and MA are set to be 0.84 GeV and 1.21 GeV respectively in the NEUT simulation. Same M A
value is also used for single-meson productions. M A value affects the cross-section because it isa function of Q2 values. For larger MA values, interactions with higher Q 2 values (and therefore
larger scattering angles) are enhanced. In this neutrino oscillation analysis, the uncertainty of
MA value is considered (see Section 6.5.2) and it is estimated to be 10 % by the total uncertaintiy
adopted to the K2K and MiniBooNE results.
Various theoretical models have been developed to account for nuclear effects in the neutrino interactions at intermediate energies besides the Smith and Moniz model, for example,
Nieves et al . [83], and Nakamura et al . [84] and so on. These models are used for estimation
the systematic uncertainties of the neutrino interaction and nuclear effects (see Section 6.5.2).
Figure 3.7 shows the cross section of the quasi-elastic scattering for the experimental data
and the calculation from the NEUT.
The cross sections for neutral current elastic scatterings are estimated from the following
relations [85, 86] :σ(νp → νp) = 0.153 × σ(νn → e− p)+σ(ν̄p → ν̄p) = 0.218 × σ(ν̄p → e n)σ(νn → νn) = 1.5 × σ(νp → νp)σ(ν̄n → ν̄n) = 1.0 × σ(ν̄p → ν̄p).

3.3.2

(3.9)
(3.10)(3.11)(3.12)

Single Meson Production

Rein and Sehgal’s model [93] is used to simulate the resonance production of single π, η and
K productions. This method assumes one meson is generated by the decay of the intermediate33

σ (10-38 cm2)
σ (10-38 cm2)

2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

ANL
BNL
SKAT

-

(a) νµn→µ p

0

2

4

–

6
8
10
Eν (GeV)

2

4

12

14

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

Serpukhov
GGM (1977)
GGM (1979)

6
8
10
Eν (GeV)12

14

Figure 3.7: Charged current quasi-elastic interaction cross sections of (a) ν µ and (b) ν µ with theexperimental data from ANL [87], Gargamelle [88, 89], BNL [90], Serpukhov [91] and SKAT [92].
The solid line indicates for the scattering off a free proton. The dashed line indicates for the
scattering off bound nucleons in 16 O.

34

−38

cm2)

νµ p → µ− p π+σ(ν µ )(10

cm2)
−38

σ(ν µ )(10

1.2
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
0

σ(ν µ )(10

−38

cm2)

0

1
0.9

1

10

Eν (GeV)νµ n → µ− p π0

1

10Eν (GeV)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

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

0.4
BEBC Allasia Nucl.Phys.B 343, 285 (1990)

0.3
BNL Kitagaki Phys.Rev.D 34, 2554 (1986)

0.2

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

0.1
0

1

10

Eν (GeV)Figure 3.8: Cross sections for νµ charged current single pion productions. Solid lines are the
NEUT calculations. Points show the experimental data as shown in the bottom right panel.
resonances :ν + N → l + N ∗ (resonance production)
N ∗ → N 0 + meson (resonance decay)(3.13)

where N and N 0 are nucleons and N ∗ is a baryon resonance. In this simulator, the invariant
mass W of the intermediate baryon resonances is restricted less than 2 GeV/c 2 . For W larger
than 2 GeV, the interactions are simulated as a part of deep inelastic scattering as described in
the next section.
To determine the angular distribution of a pion in the final state, Rein’s method [94] isused for the P33 (1232) resonance. For the other resonances, the directional distribution of thegenerated pion is set to be isotropic in the resonance rest frame. The angular distribution of
π + has been measured for νµ p → µ− p π + [95] and the results agree well with the NEUT’s
prediction.
The Pauli exclusion principle in the decay of the baryon resonance is also considered byrequiring the momentum of the nucleon to be greater than the Fermi surface momentum.
A baryon resonance in a nucleus is known to be absorbed without producing mesons occasionally [96]. The NEUT simulation considers such phenomena, and 20% of the resonanceevents do not have the meson and only the lepton and nucleon are generated.
Figures 3.8, 3.9 and 3.10 show the cross sections of single meson production for the charged
current νµ , the charged current ν¯µ , and neutral current, respectively, for the calculations fromthe NEUT and the experimental data.
35

cm2)
−38

1

cm2)
−38

0.8
0.7

0.9

0.7
0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

1
0.9

1

10

νµ n → µ+ n π−

0.8

0.6

0

σ(ν µ )(10νµ p → µ+ p π−σ(ν µ )(10

cm2)
−38

σ(ν µ )(10

1
0.9

0
Eν (GeV)1

10Eν (GeV)νµ p → µ+ n π0
BEBC Allen Nucl.Phys.B 264, 221 (1986)

0.8
0.7

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

0.6
P. Schreiner, GGM, Proceedings of 1979 symposium at Fermilab

0.5
0.4

FNAL Barish Phys.Lett.B 91, 161 (1980)

0.3
0.2

BEBC Allisia, Z.Phys.C 20, 95 (1983)

0.1
0

1

10

Eν (GeV)Figure 3.9: Cross sections for ν µ charged current single pion productions. Solid lines are the
NEUT calculations. Points show the experimental data as shown in the bottom right panel.

36

cm2)
−38

0.35

σ(ν µ )(10

0.4

0.3

0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05
2

3

4

5

6

7

8

0

9
10
Eν (GeV)νµ(νµ) p → νµ(νµ) p π0

cm2)

1

0.4

−38

cm2)

0.35

σ(ν µ )(10

0.4

−38

0

0.3

0.35

0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05
1

2

3

4

5

6

7

8

9
10
Eν (GeV)

1

2

3

4

5

6

7

8

9
10
Eν (GeV)

6

7

8

9
10
Eν (GeV)νµ(νµ) n → νµ(νµ) n π0

0.3

0.25

0

νµ(νµ) p → νµ(νµ) p π−

0.3

0.25

σ(ν µ )(10cm2)+
0.35 νµ(νµ) p → νµ(νµ) n πσ(ν µ )(10

−38

0.4

0

1

2

3

4

5

Figure 3.10: Cross sections for neutral current single pion productions. Solid (dashed) lines are
the NEUT calculations for νµ (ν µ ). Points show the experimental data taken from [97] (dot)and [98] (box).

37

3.3.3

Deep Inelastic Scattering

The cross section of charged current deep inelastic scattering is calculated by [138] in therange of the invariant mass W > 1.3 GeV/c 2 :
G2F MN Eν
yy2d2 σ ν,ν̄2
2=+ C1 )F2 (x, q ) ± y(1 − + C2 )xF3 (x, q )(1 − y +
dxdy
π
2
2
2
2
2
yMl
m
ml
xyMN
C1 =
−
− l2 −
4MN Eν x
2Eν
4Eν
2MN Eν x
2
ml(3.14)
C2 = −
4MN Eν x
where x = −q 2 /(2M (Eν − El )) and y = (Eν − El )/Eν are Bjorken scaling parameters, MN is thenucleon mass, ml is the outgoing lepton mass, Eν and El are the energy of incoming neutrino
and outgoing lepton in the laboratory frame, respectively. The nucleon structure functions F 2and xF3 are taken from the Parton distribution function(PDF) of GRV98 [99]. The correction
function given by A. Bodek and U. K. Yang [100] are adopted to use the PDF in lower Q 2 region.
In the NEUT simulation, the cross section of deep inelastic scattering induced by the neutral
current interactions are assumed to have the relations which are estimated from experimental
results [101, 102].
Since the range of W ,1.3 < W < 2.0 GeV/c 2 , overlaps with that in single pion production,nπ ≥ 2 is required in this W region. So the kinematics of the hadronic system are simulatedby two different methods for those two range of invariant mass. In the region of 1.3 < W <2.0 GeV/c2 , only pions are considered as outgoing mesons. The mean multiplicity of pions isestimated from the result of Fermilab 15-foot hydrogen bubble chamber experiment [103]. The
number of pions in each event is determined by using the KNO (Koba-Nielsen-Olsen) scaling.
The forward-backward asymmetry of pion multiplicity in the hadronic center of mass system isincluded using the results from BEBC experiment [104].
In the region of W > 2.0 GeV/c2 , the kinematics of the hadronic system are calculated by
PHYTIA/JETSET [105]. This package treats not only π but also K, η, ρ and so on.
Cross section of the CC νµ and ν µ DIS interactions are shown in Figure 3.11.

3.3.4

Coherent Pion Production

The coherent pion production is a neutrino interaction with a oxygen nucleus, which remains
intact, and one pion with the same charge as the incoming weak current is produced. Since verysmall momentum is transferred to the oxygen nucleus, the angular distributions of the outgoing
leptons and pions are peaked in the forward direction. The Rein and Sehgal [106] model isadopted to simulate the interactions.
The measurement results by the K2K-SciBar detector set the upper limit of the cross sectionof CC coherent pion production [107]. This upper limit is significantly lower than the predictedcross section by the Rein and Sehgal model. Therefore the Rein and Sehgal [108], which ismodified to take into account the non-vanishing lepton masses in CC interactions, is used in
the NEUT simulator. In case of νµ , the cross section is suppressed about 25% at 1.3 GeV
due to the interference of the axial vector and pseudoscalar (pion-exchange) amplitudes by thismodification.

38

σ/Eν (10-38cm2 GeV-1)

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

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

CCFR 90
CDHSW 87
GGM-PS 79

0

10

CHARM 88
BNL 80
CRS 8020

30 50

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

CCFRR 84
SKAT

100

200

150

250

Eν (GeV)Figure 3.11: Cross sections for DIS charged current ν µ (upper lines and points) and ν µ (lower
lines and points) interactions. The NEUT treats GRV98 with Bodek-Yang correction as shownin the dashed-dotted lines. Each points indicate the experimental results as shown in the bottomon the figures.

3.3.5

Nuclear Effects

The secondary interactions of mesons and baryons or hadrons, produced in neutrino interactions, with nucleons inside the 16 O nuclei is also simulated in the NEUT. All of the mesons
produced within the 16 O nuclei are tracked from their production points to their exiting points
or points absorbed in the nuclei. This is simulated for π, K and η by using a cascade model.
Since the cross section of the pion production for neutrinos is large in E ν > 1 GeV andthe pion-nucleon interaction cross section is also large in this energy region, the interactionsof pions are important. The NEUT considers several pion interactions in 16 O nuclei: inelastic
scattering, charge exchange and absorption. The initial pion production point in the nucleus,
where neutrino-nucleon interactions occur, is determined by the Wood-Saxon density distribution [109]. The interaction mode of pions is determined by the calculated mean free path for eachinteraction, which is modeled by L. Salcedo et al. [110]. The mean free path of pions depends
on their momentum and positions in the nucleus. In the calculation of pion inelastic scatteringinteraction, the Fermi motion of the nucleus and the Pauli exclusion principle are considered.
If inelastic scattering or charge exchange occurs, the angular and momentum distributions of
the outgoing pions are determined by using the results of a phase shift analysis from π - N
scattering experiments [111].
The pion interaction simulation is compared with the experimental data for the following
three interactions : π - 12 C scattering, π -16 O scattering, and pion photo-production (γ+ 12 C
→ π − + X) [112, 113]. Figure 3.12 is shown for the π - 16 C scattering.
For kaons interactions, the elastic scattering and charge exchange interactions are consideredusing the results from the cross sections measured by the K ± -N scattering experiments [114,
39

+ 16

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

500
400

inela+abs+cex

300
200

inelastic

100

absorption

0

charge exchange

0

100

200

300

400

500

600

momentum of π (MeV/c)+

Figure 3.12: The cross sections of π + -16 O scattering as a function of π + momentum. The lines
are the cross sections calculated by the NEUT for each interaction mode, and the experimental
data points are taking from [113].

115, 116]. For η mesons interactions, the absorption (ηN → N ∗ → π(π)N ) is considered [117].
In the NEUT simulation, the nucleon re-scattering inside the oxygen nucleus is also considered using the cascade model and the measurements by Bertini et al. [118]. The pion production
caused by the decay of produced deltas is also taken into account as the isobar production model
by Lindenbaum et al. [119].

3.4

Detector Simulation

The produced particles in neutrino interactions are incorporated in a detector simulation
code, which simulates (1) the tracks of particles, (2) the generation and propagation of Cherenkov
photons in water, and (3) the PMT response and the readout electronics.
(1) Particle Tracking
The detector simulation has been developed based on the GEANT3 package [120]. Table 3.1 lists various processes which are considered in our simulation program. The hadronicinteractions in water are simulated using the CALOR package [121]. This package is known toreproduce the pion interactions well including low energy region (∼ 1 GeV/c). For still lower energy region (pπ ≤ 500 MeV/c), a custom program [122] based on experimental data from π− 16 O
scattering [123] and π − p scattering [124] is used in our simulation code.

40γ(e+ , e− ) pair production
Compton scattering
Photoelectric effecte±Multiple scattering
Ionization and δ-rays production
Bremsstrahlung
Annihilation of positron
Generation of Cherenkov radiationµ±Decay in flight
Multiple scattering
Ionization and δ-rays production
Bremsstrahlung
Direct (e+ , e− ) pair production
Nuclear interaction
Generation of Cherenkov radiation

Hadrons

Decay in flight
Multiple scattering
Ionization and δ-rays production
Hadronic interactions
Generation of Cherenkov radiation

Table 3.1: List of the processes considered in simulator.

41

(2) Cherenkov photon tracking in water
We take into account the dispersion of the refractive index. The group velocityvg =cn(λ) − λ ∂n(λ)∂λ(3.15)

where c is light velocity in vacuum, λ is light wavelength, is used for the light velocity in thewater. And the generated Cherenkov light is simulated to be scattered and absorbed in water.
The water transparency as a function of wavelength is determined as follows: in case of shortwavelength, we employed the relation C ∝ λ −4 where C is the scattering coefficient and λis wavelength. In case of long wavelength, we use the data taken from [125]. The absolute
water transparency is determined to be consistent with the direct measurement of the detector(Section 4.1.4).
(3) Response of the PMTCharge and timing response of PMTs is considered in the simulator. The detected charge
value of each hit PMTs is simulated by the random number distributed as the measured one
photon distribution as shown in Figure 2.5. And the timing distribution is dependent on thatcharge, so the relation between timing and charge is measured using the laser calibration (section 4.1.3). The timing of each PMT is a random variable distributed as Gaussian with one
sigma as shown in Figure 4.5.

42

Chapter 4

Calibration
4.1

Detector Calibration

Understanding and keeping high quality of the detector performance is important for an
accurate observation. There are several calibrations for detector charge and timing informationin the Super-Kamiokande detector. Water transparency is also checked continuously in thedetector.

4.1.1

Relative Gain Calibration

The high voltage value of each PMT is set to provide approximately uniform gain for all
PMTs in the detector. The uniformity of the PMT gain is important to determine the momentum
without systematic difference depending on its vertex position or direction.
Figure 4.1 shows a schematic view of the relative gain calibration system. Light generated
by a Xe lamp is passed through an ultraviolet filter and neutral density filter and is injected
into a scintillator ball. The diffused light from the scintillator ball is emitted uniformly. The
other outputs from the Xe lamp are used to monitor the intensity of the Xe light and to makea calibration trigger.
The high voltage value of each PMT is adjusted to give the same gain with the others. The
relative gain Gi of the i-th PMT is obtained by :
 
li
Qi
2
(4.1)
· li · exp
Gi =Q0 f (θ)
L
where Qi is the charge detected by the i-th PMT, l i is the distance from the light source tothe PMT, f (θ) is the PMT acceptance as a function of the photon incidence angle θ, L is theattenuation length of water and Q0 is the normalization factor. This measurement is performed
for several positions of the scintillator ball, changing the voltage. After adjusting high voltage
values, the relative gain spread is about 7 % for SK-I as shown in Figure 4.2. For SK-II and
III, the relative gain calibration method is improved by using standard PMTs whose gain are
adjusted within few % for each other in advance. Since the position dependence of the lightintensity from the Xe lamp can be reduced by the standard PMTs, the relative gain uncertainty
can be adjusted to 2 % for SK-II and III.

43

Number of PMTs in Each Bin

1400

1200

σgain = 7.0%
1000

800

600

400

200

0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Relative PMT Gain

Figure 4.1: A schematic view of the relative gain
measurement system using a Xe lamp.

4.1.2

Figure 4.2: The relative gain distribution of all the ID PMTs, measured in
1996.

Absolute Gain Calibration

The absolute gain calibration is necessary to convert the charge detected by each PMT inpico Coulomb (pC) into the number of photoelectrons. The absolute gain is determined using
the charge distribution of single photoelectron signals.
The low energy γ-ray generated from neutron capture on Nickel nucleus is used as a calibration source to measure the single photoelectron distributions. The schematic view of the Nickel
calibration source is shown in Figure 4.3. Neutrons produced by spontaneous fission of 252 Cf
are captured on surrounding Ni wires in polyethylene vessel, and low energy (6 ∼ 9 MeV) γ-rays
are generated simultaneously. The number of hit PMT is about 50 ∼ 80 in total, so that the
number of p.e. detected by each PMT is at most one. The charge distribution of a typical PMT
is also shown in Figure 4.3. The sharp peak near zero is caused by electrons that are emitted
from the photocathode but miss the first dynode, and the peak around 2 pC corresponds tothat of single photoelectrons. The mean value of this 1 p.e. charge distribution is used as aconstant to convert the PMT charge from pico Coulomb to the number of p.e.s.. The constantsof 2.044 pC/p.e, 2.297 pC/p.e and 2.243 pC/p.e are used for SK-I, SK-II and SK-III, respectively.

4.1.3

Relative Timing Calibration

The relative timing calibration is important for the vertex position reconstruction. The
timing response of the PMT depends on not only the length of the signal cable but also thedetected charge because of the slewing effect of discriminator. The large signal tends to exceed
the threshold earlier than the small one.
N2 laser is used in this calibration. Figure 4.4 shows the schematic view of the relative
44

mean

2.055pC

5103Ni wire+water

200mm
Cf

2

1

0

0

190mm

pC

Figure 4.3: Schematic view of an Nickel calibration source (left) and charge distribution of atypical ID PMT (right).

timing calibration system. N2 laser generator emits intense light with 337 nm wavelength withina time width of 0.4 nsec and this laser light is split into two. One of them is used to monitor thelight intensity and make event trigger. The other is injected into a diffuser ball in the ID via an
optical fiber. For injected light into the detector, the wavelength of the laser light is converted
to 398 nm from 337 nm by a dye laser module, since this wavelength is most sensitive to PMTs.
The PMT timing response is measured with various light intensity from 1 p.e. to a few hundredsof p.e. using an adjustable attenuation filter. The results are shown as a scatter plot of thetiming and the charge called TQ-map in Figure 4.5. Each dot corresponds one measurement,
and the open circles are the average timing with respect to charge for a PMT. The TQ-map ismade for all ID PMTs and used to correct the timing information.

4.1.4

Light scattering measurement using a laser

The understanding a status of detector water transparency is important to determine thenumber of photons which arrive to PMTs through water in the detector. Water transparency
represents the combined effect from absorption and scattering of the light intensity. So thelight attenuation length in water can be described as L = (α abs + αscat.sym + αscat.asym )−1 ,where αabs , αscat.sym and αscat.asym are the absorption, symmetric scattering and asymmetricscattering coefficient respectively. These coefficients are separately measured using a N 2 laser.
Figure 4.6 shows a schematic view of the measurement system. Each laser, wavelength of 337,
371(365 for SK-III), 400 and 420 nm, fires every 6 seconds during normal data taking. The lightfrom the laser is injected into the ID via an optical fiber. There are several points of the light
injection into the ID on the top, bottom and barrel of the tank. A typical event by the laser
450.CUGTPORWNUGYKFVJPUGE

980&[G.CUGTPORWNUGYKFVJPUGE 2/6

Linear Scale

Log Scale960

8CTKCDNGHKNVGT

950

T (nsec)

5-+&



970

940
930
920
910
900
890
880

   

1

5

10

100

Q (p.e.)N

Figure 4.4: A schematic view of thetiming measurement system using alaser.

Figure 4.5: TQ-map, a measured two-dimensional
plot of timing vs. charge distribution.

46

LASER
Optical Fiber

Top

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

Scattering

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

BottomFigure 4.6: A schematic view of the scattering and absorption parameter measurement using
laser systems, and a typical laser event.

light from the top is also shown in Figure 4.6. The PMT hits are clustered due to the direct(unscattered) photons.
In this calibration, the detector is separated into 6 regions, top and 5 in barrel, as shownin Figure 4.6. The total charge of hit bottom PMTs is used for the normalization of the lightintensity. Figure 4.7 shows the PMT hit timing distributions in each region for data and Monte
Carlo simulation. These PMT hits on the top and barrel wall are due to the photons scatteredin water, or the photons reflected by surfaces of bottom PMTs or black sheets. The first peaks
and slopes are characterized by the absorption and scattering coefficients and the second peaks
around 1100 nsec are due to the photons reflected by the PMTs or black sheets. For the Monte
Carlo simulation, the absorption and scattering coefficients are adjusted so that the PMT hit
time distributions are in agreement with data. The attenuation length in water is calculatedusing the measured absorption and scattering coefficients.
The attenuation coefficients (L−1 ) obtained by this method are plotted in Figure 4.8 with astar symbol. The lines shows a model used in the Monte Carlo simulation, which are determinedby fitting the measurements from this method.

4.1.5

Water transparency measurement using cosmic ray muons

Water transparency is also measured by using Cherenkov light from cosmic ray muons passing
through the detector. Since the energy deposit of such a energetic muon is almost constant(about 2 MeV/cm), cosmic ray muons can be used as a calibration source.
Only vertical downward muons are selected for this measurement. The muon track is reconstructed by connecting the entrance and the exit points in the ID. Under the assumptionthat the light detected by each PMT is not scattered, the detected charge (photoelectrons) is
47

2

Top

1
0
1

Barrel 1

PMTs / BottomQ

0

Barrel 2

Figure 4.7: PMT hit time distributions of 337 nm-laser events in eachdetector region for the data (dots)
and the Monte Carlo events which
is tuned by the scattering and absorption parameters (histograms).

1
0
2

Barrel 3

1
02

Barrel 4

1
0
2

Barrel 5

1
0

700

800

900

1000

1100

1200

PMT Hit Time (nsec)

attenuation coefficient(1/m)

Light scattering measurement

-1

10

-2

10

absorption

Rayliegh scattering-3

10
Mie scattering

300

400

500

600

700

wavelength(nm)Figure 4.8: Wavelength dependence of attenuation coefficients (L −1atten ) obtained by the lightscattering measurement (points) together with a model in the detector simulation.

48

20-inch
PMT

42o

li

Θi

log(Ql/f( θ)): Cherenkov
Photons

Inner Tank

500

RUN 3106

P1
P2

5.987
-0.9487E-04

1000

2000

3000

0.4609E-03
0.4444E-06

400
300
200

Cosmic Muon

i’th PMT

0

4000

5000
l (cm)

Figure 4.9: (Left) Schematic view of the attenuation length measurement using vertically penetrating cosmic ray muons. Arrows from muon track show the trajectories of Cherenkov photons
and li shows the flight length of the Cherenkov photons detected by the i-th PMT. (Right)
Effective observed charge (log(Ql/f (θ))) for through-going muons as a function of the photon
path length (l) where Q is the detected charge and f (θ) is the PMT acceptance.

expressed by :


l
f (θ)
· exp −Q = Q0 ·
l
L

(4.2)

where Q0 is a constant, f (θ) is the PMT acceptance, l is the photon path length and L is theattenuation length. The left panel on Figure 4.9 shows the schematic view of the measurement. The right panel on Figure 4.9 shows log (Q·l/f (θ)) as a function l in a typical run. The
attenuation length is estimated to be 95 m for this data set.
Since the cosmic ray muons are measured during the normal data taking, continuous checkof the water transparency is possible. Figure 4.10 shows the time variation of the attenuation
length of water in entire period from SK-I to III. This time variation is reflected in the calculationof a particle momentum (see Section 5.4.5).

49attenuation length [ m ]150using bottom PMT
140
130
120
110
100
90
80
70
60
50
01/01/1996 01/01/1998 01/01/2000 01/01/2002 01/01/2004 01/01/2006 01/01/2008 01/01/2010
date

Figure 4.10: Time variation of water attenuation length.

4.2

Energy Calibration

The momentum of a particle is determined based on the charge information of PMTs (seeSection 5.4.5). The accuracy of the energy scale is essential for the atmospheric neutrino analysis.
For the energy scale calibration in momentum range from a few tens of MeV/c to about 10 GeV/c,four independent calibration sources are used:• Track range of high energy stopping muon (1 ∼ 10 GeV/c)• Cherenkov angle of low energy stopping muon (200 ∼ 500 MeV/c)• Invariant mass of π 0 ’s produced by neutrino interactions (∼ 130 MeV/c)• Momentum of decay electron (∼ 50 MeV/c)
The accuracy of the absolute energy scale is checked by comparing observed data with theprediction data from the Monte Carlo simulation for each calibration source. Including the time
variation check of energy scale, the uncertainty on the energy scale is estimated (Section 4.2.6).
The detector uniformity of energy scale is also checked (Section 4.2.7).

4.2.1

High energy stopping muons

The momentum for high energy muons can be determined by its track length which isapproximately proportional to the momentum. The stopping cosmic ray muon track can be
estimated by the distance from the entrance position into the detector to the vertex positionof the subsequent decay electron. The resolutions for both stopping muon and decay electron
vertexes are better than 50 cm. The selections for the stopping muon events are follows:(1) The entrance point is on the top wall of the detector.
(2) The direction is downward (cos θ > 0.94).
(3) One decay electron event is detected.
(4) The reconstructed range of muon track L is 7 < L < 30 m.
50

Criteria (1)(2) require the straight down-going muon event entered at the OD top of the detector.
The left side panels in Figure 4.11 shows the averaged value of the ratio the momentum to track
range (momentum/range) as a function of the track range. The momentum loss per cm is about
2.3 MeV/c. The ratio MC to data of the averaged value of momentum/range is shown in theright side panels of Figure 4.11. They agree within 0.7 %, 1.1 % and 2.0 % for SK-I, SK-II and
SK-III, respectively.

4.2.2

Low energy stopping muons

The Cherenkov angle of charged particles is expressed as follows:s1m2
1
cos θC =
1+ 2
=
nβ
n
p

(4.3)

where θC , n, β, m and p are the Cherenkov angle, the refraction index of water, v/c, mass andmomentum. The Cherenkov angle depends on momentum when the momentum is comparable
to the mass. Thus, for low energy stopping muons (< 400 MeV/c), their momentum can be
estimated by the Cherenkov angle based on above equation. The selections for low energy
stopping muon events are follows:(1) The entrance point is on the top wall of the detector.
(2) The direction is downward (cos θ > 0.9).
(3) One decay electron event is detected.
(4) The total number of p.e.s in the ID is less than 1500 p.e.s (750 p.e.s for SK-II)
Criterion (4) selects low momentum muon event with momentum of < 380 MeV/c. The upper
left panel on Figure 4.12 shows the reconstructed momentum P p.e. distribution. The upper rightpanel shows the reconstructed opening angle θ C distribution. The scattered plots of P p.e. andopening angle θC is shown in the lower two panels for data and MC. The momentum dependence
on the opening angle is seen in both figures. Figure 4.13 shows the averaged P p.e./Pθ (Pθ isestimated momentum by θC ) and the ratio MC to data of it as a function of P θ . They agree
within 0.7 %, 1.3 % and 2.1 % for SK-I, SK-II and SK-III, respectively.

4.2.3

Neutrino induced π 0 events

π 0 events are produced by the interactions of the atmospheric neutrinos in the detector.
Since a π 0 decays immediately into two γ-rays, the invariant mass of π 0 can be calculated byusing reconstructed momentum of two γ-rays, P γ1 and Pγ2 as follows:q(4.4)
Mπ0 = 2Pγ1 Pγ2 (1 − cos θ)

where θ is the opening angle between two γ-rays. The NC single π 0 events are selected from theatmospheric neutrino sample by the following criteria :(1) Two Cherenkov rings are recognized and both of them are identified as electron-like.
(2) Electrons from muon decay is not detected.
51

2.6

SK-I

1.05

SK-I

1.03
1.02

2.4
MC/DATA

Momentum/Range(MeV/c/cm)

1.04
2.5

2.3
2.2
2.1

1.01
1
0.99

DATA

0.98

MC

0.97
0.96

2
0

2.6

0.95
0

500 1000 1500 2000 2500 3000 3500
Range (cm)

SK-II

1.05

500 1000 1500 2000 2500 3000 3500
Range (cm)

SK-II

1.03
1.02

2.4
MC/DATA

Momentum/Range(MeV/c/cm)

1.04
2.5

2.3
2.2
2.1

1.01
1
0.99

DATA

0.98

MC

0.97
0.96

2
0

2.6

0.95
0

500 1000 1500 2000 2500 3000 3500
Range (cm)

SK-III1.05

500 1000 1500 2000 2500 3000 3500
Range (cm)

SK-III

1.03
1.02

2.4
MC/DATA

Momentum/Range(MeV/c/cm)

1.04
2.5

2.3
2.2
2.1

1.01
1
0.99

DATA

0.98

MC

0.97
0.96

2
0

500 1000 1500 2000 2500 3000 3500
Range (cm)0.95
0500 1000 1500 2000 2500 3000 3500
Range (cm)Figure 4.11: Left side panels show the range v.s. the averaged momentum/range of stopping
muon events for the data (solid line) and the Monte Carlo events (dashed line), for each SK-I
(upper), SK-II (middle) and SK-III (lower). Right side panels show the ratio the Monte Carloevents to the data of the averaged momentum/range in each left panel.
52

500

200

number of events

number of events

250

150

100

400

300

200

100

50

0
100

150

200

250

300

350

400

450

0

500

24

26

28

32

34

36

38

40

42

44

500

reconstracted momemtum (MeV/c)

500

reconstracted momemtum (MeV/c)

30

opening angle (deg.)

reconstracted momentum (MeV/c)

DATA

450

400

350

300

250

200

Monte Carlo

450

400

350

300

250

200
30

32

34

36

38

40

42

44

opening angle (deg.)

30

32

34

36

38

40

42

44

opening angle (deg.)

Figure 4.12: Upper left panel shows the reconstructed momentum distribution P p.e. for the data(circle and error) and the Monte Carlo simulation (histogram). Upper right panel shows thereconstructed opening angle distribution. Lower two panels show the correlation between thereconstructed Cherenkov opening angle θ C and the reconstructed momentum Pp.e. for data (left)
and the Monte Carlo simulation (right). All distributions come from sample for SK-II.

53

SK-I

1.4

SK-I

1.05
1.04

1.3

1.03
1.2

MC/DATAP(p.e.)/P(θ)

1.02
1.1
1
0.9

1.01
1
0.99
0.98

0.8

DATA

0.6
150

0.97

MC

0.7

0.96
200

250

300

350

400

450

0.95
150

500

expected momentum P(θ) (MeV/c)

SK-II

1.4

200

250

300

350

400

450

500

expected momentum P(θ) (MeV/c)

SK-II

1.05
1.04

1.3

1.03
1.2

MC/DATAP(p.e.)/P(θ)

1.02
1.1
1
0.9

1.01
1
0.99
0.98

0.8

DATA

0.6
150

0.97

MC

0.7

0.96
200

250

300

350

400

450

0.95
150

500

expected momentum P(θ) (MeV/c)

SK-III

1.4

200

250

300

350

400

450

500

expected momentum P(θ) (MeV/c)

SK-III

1.05
1.04

1.3

1.03
1.2

MC/DATAP(p.e.)/P(θ)

1.02
1.1
1
0.9

1.01
1
0.99
0.98

0.8

DATA

0.6
150

0.97

MC

0.7

0.96
200

250

300

350

400

450

500

expected momentum P(θ) (MeV/c)

0.95
150

200

250

300

350

400

450

500

expected momentum P(θ) (MeV/c)

Figure 4.13: Left side panels show the averaged ratio of the momentum derived from the charge
to that from the opening angle (Pp.e. /Pθ ) as a function of the momentum Pθ for the data (solidline) and the Monte Carlo events (dashed line), for each SK-I (upper), SK-II (middle) and SK-III(lower). Right side panels show the ratio MC events to the data of P p.e. /Pθ in each left panel.
54

(3) The vertex position is reconstructed within the fiducial volume (more than 2 m away fromthe ID wall).
Criterion (2) rejects the contamination of π + π 0 or µ± π 0 events. Figure 4.14 shows the invariant
mass distribution of π 0 events compared with the prediction from the Monte Carlo simulation.
The peak positions of data spectrum agree with MC within 0.7 %, 1.3 % and 0.3 % for SK-I,
SK-II and SK-III, respectively.

4.2.4

Decay electrons

A large number of electron events is produced in the detector by the decay of stopping
cosmic ray muons. The decay electron’s energy spectrum is well understood and spreads around
50 MeV. The selection criteria for decay electrons are as follows:(1) The time interval from a stopping muon event is 2.0 µsec to 8.0 µsec.
(2) The number of hit PMT in a 50 nsec time window is larger than 60 (30 for SK-II).
(3) The goodness of the vertex fit is greater than 0.5 .
(4) The vertex position is reconstructed within the fiducial volume (more than 2 m away fromthe ID wall).
Criterion (1) is for the efficient timing for electron tagging. Criterion (2) rejects ∼6 MeV γ-rays
from µ− capture on the nucleon. In the simulation, measured µ + /µ− ratio of 1.37 [130] and theeffect of µ− capture by oxygen nuclei are considered. We use the vertex fitter which has beendeveloped for the low energy neutrino observations such as solar and supernova neutrinos [131].
Figure 4.15 shows the overlaid momentum spectra of decay electrons for the data and MC. The
mean values of data spectrum agree with MC within 0.6 %, 1.6 % and 0.8 % for SK-I, SK-II and
SK-III, respectively.

4.2.5

Time variation of energy scale

The stability of the energy scale is also confirmed by the stopping muons and the decay
electrons. In the reconstruction process, the particle momentum is corrected by the watertransparency to consider the time variation of it (see Section 5.4.5). In addition, the correction
to consider the time variation of trigger rate which affects the output charge of the electronics
is also used. Figure 4.16 show the time variation of the mean value of momentum/range forthe stopping muons, and the mean value of momentum for the decay electrons. The maximum
RMS of the variation among these calibration sources is given as 0.88 %, 0.55 % and 1.79 % for
SK-I, SK-II and SK-III, respectively. Since water quality is worse and not stable in SK-III (ref.
Figure 4.10), the stability of the energy scale is thought to be worse than the other SK run
periods.

4.2.6

Summary of the energy scale calibration

The absolute energy scale is checked by various methods for the momentum range from a few
tens of MeV/c to about 10 GeV/c. Figure 4.17 summarizes these absolute energy calibrations.
The uncertainty of the absolute energy scale is estimated to be less than 0.74 %, 1.60 % and
55

SK-I

SK-II
70

140

MC

60

MC

number of events

number of events

120
100
80
60

50
40
30

40

20

20

10

0

DATA

DATA

0

50

100
150
200
2
Mγγ (MeV/c )

250

300

0

0

50

100
150
200
2
Mγγ (MeV/c )

250

300

SK-III
50

DATA

number of events

MC

40
30
20
10
0

0

50

100
150
200
2
Mγγ (MeV/c )

250

300

Figure 4.14: Invariant mass distribution of neutrino induced π 0 events of the observed data (dot)and the atmospheric neutrino Monte Carlo events (boxes) for SK-I (upper left), SK-II (upper
right) and SK-III (lower). The peak position is fitted by a Gaussian distribution. Monte Carloevents are normalized by the livetime of the observed data.

56

SK-I

SK-II
500

200

DATA

175

MC

DATA

400
number of events

number of events

225

150
125
100
75
50

MC

300

200

100

25
0

0

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

80

0

0

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

80

SK-III
700
DATA

number of events

600

MC

500
400
300
200
100
0

0

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

80

Figure 4.15: The momentum distribution of decay electrons of the data (dot) and the Monte
Carlo (line) for SK-I (upper left), SK-II (upper right) and SK-III (lower). Monte Carlo events
are normalized by the number of observed data events.

57

e momentum(MeV/c)

0

43
42
41
40
39
38
37
36
2400

250

500

µ momentum/track(MeV/c/cm)±1%

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

±1%

0

250

500

SK-II

2600

2800

3000
3200
3400
days from 1996/Apr./1

SK-III

2.6
2.55
2.5
2.45
2.4
2.35
2.3
2.25
2.2
2400

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

SK-II

2600

2800

3000
3200
3400
days from 1996/Apr./1

SK-III

45
44
43
42
41
40
39
38
3800 3900 4000 4100 4200 4300 4400 4500
days from 1996/Apr./1µ momentum/track(MeV/c/cm)

e momentum(MeV/c)

SK-I

2.6
2.55
2.5
2.45
2.4
2.35
2.3
2.25
2.2

µ momentum/track(MeV/c/cm)

e momentum(MeV/c)

SK-I

45
44
43
42
41
40
39
38

2.6
2.55
2.5
2.45
2.4
2.35
2.3
2.25
2.2
3800 3900 4000 4100 4200 4300 4400 4500
days from 1996/Apr./1

Figure 4.16: The time variation of the reconstructed momentum of the decay electrons (left)
and the averaged momentum/range of stopping muons (right) as a function of elapse days foreach SK run period. Dotted (dashed) lines show ±1 %(±2 %) from mean value in each SK run
period.

58

2.08 % for SK-I, SK-II and SK-III, respectively. To consider the time variation of the energy
scale, the RMS of the variation is combined to the energy scale uncertainty. In summary, 1.1 %,1.7 % and 2.7 % are estimated as the energy scale uncertainties for SK-I, SK-II and SK-III,
respectively.

4.2.7

Uniformity of Energy Scale

The uniformity of the detector is measured using the decay electrons from the cosmic raymuons. They are good calibration sources to check the detector uniformity because the vertex is
distributed uniformly in the fiducial volume and the momentum distribution is almost uniform
in all directions. To take into account the muon polarization, only electrons whose direction isperpendicular to the parent muon direction are used. This condition is -0.25< cos Θ e↔µ < 0.25,
where cos Θe↔µ is the opening angle between the electron and muon directions. Figure 4.18
shows the averaged momentum of decay electrons for the Monte Carlo events normalized bythat for data as a function of the zenith angle of the electrons. From this figure, the detector
gains are uniform within ±0.6 %, ±0.6 % and ±1.3 % for SK-I, SK-II and SK-III, respectively.

59

(MC - DATA) / DATA (%)

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

(MC - DATA) / DATA (%)

10

2

3

4

10
10
10
momentum range (MeV/c)

10

5

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

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

3

4

10
10
10
momentum range (MeV/c)

10

5

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

µ rangeµ angle0π massdecay e
2

3

4

10
10
10
momentum range (MeV/c)

105Figure 4.17: The summary of the absolute energy scale calibration for each SK run period. The
horizontal axis shows the momentum range and the vertical axis shows the deviation of the datafrom the Monte Carlo predictions.

60

electron momentum

electron momentum

1.05
1.04
1.03
1.02
1.01
1
0.99
0.98
0.97
0.96
0.95

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 1cosΘ

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

Figure 4.18: The uniformity of the detector gain as a function of zenith angle for each SK period.
The vertical axes are the ratio MC to data of the averaged momentum of decay electron events.

61

Chapter 5

Data Selection
The Super-Kamiokande detector collects about 10 6 HE and LE trigger events per day. But
most of the events are background events such as cosmic rays and gamma rays from radioactivities. An efficient selection is required to select neutrino events in such a large amount of thedata.
The atmospheric neutrino events observed in Super-Kamiokande are categorized to 3 types:
fully contained (FC), partially contained (PC) and upward-going muon (UPMU). For FC andPC event types, the neutrino interacts in the fiducial volume of the detector (2m away fromthe ID wall). If all of the energy of induced lepton is deposited inside the ID, the event isclassified into FC. If high energy muon lepton exits the ID and deposits its energy in the OD
region, the event is classified into PC. UPMU events are the high energy muons produced by
the neutrino interactions with the rock surrounding the detector. Since these events cannot be
distinguished from the cosmic rays traveling in the downward direction, only muons travelingin upward direction through the detector are selected. They are separated into two categories:
those that come to rest in the detector (upward stopping muons), and those that traverse theentire detector volume (upward through-going muons). Figure 5.1 shows the scheme of these
event types. These neutrino mean energy is distributed ∼ 1GeV for FC, ∼ 10 GeV for PC
and stopping UPMU, and ∼ 100GeV for through-going UPMU as shown in Figure 5.2. The
procedures to select the neutrino events are performed separately to each event type.
The event selection consists of reduction and reconstruction processes. The reduction pro-

Fully contained

Partially contained

Upward stoppingmuon

Upward through-going
muon

Figure 5.1: Schematic view of observed atmospheric neutrino in Super-Kamiokande

62

1000

FC νe
FC νµ
PC

800

Events / 1000 days

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

Upward stopping µ
Upward through-going µ

1

10
10
Eν (GeV)

2

10

3

10

4

Figure 5.2: The expected neutrino spectra for each event category

cess is to search neutrino events by removing background events in the collected data. The
background events observed in Super-Kamiokande are categorized to:• Cosmic ray through-going muons which pass through the detector• Cosmic ray stopping muons which enter and stop inside the detector• Cosmic ray corner clipping muon which graze the corner of the detector• Flasher event which is accidentally happened when a PMT emit light by internal corona
discharge• Low energy event such as gamma-ray from radioactivities.
These background events are removed dramatically by the reduction processes. After the reduction, various quantities, such as the vertex position, are reconstructed for each neutrino
event.
In this chapter, the reduction methods for each event type FC, PC and UPMU are described.
(Since the author was most deeply involved in the SK-III data reduction and the analysis, various
distributions from SK-III are mainly shown.) The reconstruction methods and their performance
are also described.
When the distributions of Monte Carlo events compare with the data in the reduction sections, MC sample is assumed to be not oscillated unless specified in the caption. In the reconstruction sections, the distributions of MC sample are assumed to be 2-flavor mixing with(sin2 2θ, ∆m2 ) = (1.00, 2.5 × 10−3 eV2 ).
63

5.1

Reduction for Fully Contained EventsFirstly, FC events are distinguished from PC events using the number of OD hit PMTs in
the highest charge cluster (will be described in Section 5.1.6). Here describes the data reduction
procedure consisting of five steps for FC. Note that since the number of ID PMTs for SK-II isabout half of that for SK-I and III, the SK-II selection criteria relating to the number of hit andobserved charge of ID PMTs are different from that of SK-I and III.

5.1.1

First Reduction

An event passes the 1st reduction if it fulfills the following all criteria :(1)

PEtotal should be > 200 p.e.s (100 p.e.s for SK-II).
PEtotal is the number of total p.e.s observed by the ID PMTs.

and
(2)

NHITA800 should be ≤ 50 or OD trigger is off.
NHITA800 is the number of hit OD PMTs in a fixed 800 nsec time window from
500 nsec to 1300 nsec before and after the trigger timing.

Criterion (1) rejects the low energy background events due to radio activities. The 200 p.e.s(100 p.e.s for SK-II) corresponds to 22 MeV/c of electron momentum. Since the events whichhave visible energy below 30 MeV are not used in the analysis (see in Section 5.1.6), this cut issafe. Criterion (2) removes the cosmic ray muon events. The FC 1st reduction reduces the data
size from 106 events/day to ∼ 3000 events/day.

5.1.2

Second Reduction

In the second reduction, low energy events and cosmic ray muons are rejected. The selection
criteria are follows :(1)

NHITA800 should be ≤ 25
or OD trigger is off.

if PEtot < 100,000 p.e.s (50,000 p.e.s for SK-II)

and
(2)

PEmax /PE300 should be < 0.5.
PEmax is the maximum number of p.e.s observed by an ID PMT. PE 300 is the
maximum number of total p.e.s observed by the ID PMTs in a sliding 300 nsec
time window.

Criterion (1) is to reject cosmic ray muons by a tighter threshold than the 1st reduction. Figure 5.3 show the NHITA800 distributions for the data, FC atmospheric neutrino Monte Carloevents whose vertices 2m away from the ID wall, and the FC final sample events. Criterion (2)removes the low energy events and electrical noise events, which have one larger hit signal froma single PMT. Figure 5.4 shows the PE max /PE300 distributions.
The event rate is ∼200 events/day after the FC 2nd reduction.

64

Number of events

10

6

10

5

10

4

10

3

10

2

10

5

10

4

10

3

10

2

(a) data

rejected(b) MC

10
1
10

5

10

4

(c) final sample

10

3

10

2

data
MC

10
1
0

10

20

30

40

50

60

70

80

90

100

Figure 5.3:
(FC 2nd reduction)
NHITA800 distributions for SK-II. (a)shows the observed data after FC 1st
reduction. (b) shows FC Monte Carloevents after FC 1st reduction. (c)shows the final samples for the data
and the the Monte Carlo events. The
number of the Monte Carlo events in(c) is normalized to that of the data.
The selection criteria are shown by arrows. Dashed line is cut criteria in FC
1st reduction. Solid line is cut criteria
in FC 2st reduction.

NHITA800

6 (a) DATA

10
5
10
4
10
3
10
2
10

rejected

number of events(b) MC4103102

rejected

10101(c) final sample
3

data
10
MC
2
10
10
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
FC2 PEmax/PE300

65

Figure 5.4:
(FC 2nd reduction)
PEmax /PE300 distributions for SK-III.
(a) shows the data after the FC 1st reduction. (b) shows FC Monte Carloevents after the FC 1st reduction. (c)shows the final samples for the data
and the the Monte Carlo events. The
number of the Monte Carlo events in(c) is normalized to that of the data.
The selection criteria are shown by arrows.

5.1.3

Third Reduction

Remaining background events, mostly noise events and cosmic ray muons which have a small
number of OD hits, are rejected by following cuts.
Through-going muon cut
The through-going muons are very energetic and deposit a lot of charge in the ID. To
eliminate these events, a through-going muon fitter is applied if there are more than 1000 PMTs,
each detecting more than 230 p.e.s.. This fitter finds the point where the through-going muonenters into the ID by using earliest hit PMT with some neighboring hit PMTs. The exit point
where the through-going muon goes out the ID is defined the center of the saturated ID PMTs.
The goodness of through-going muon fit is defined as :
X 1(ti − Ti )2
1
×
exp −
(5.1)
goodness = X
12(1.5 × σi )2
σi2
i
σi2
i
where ti and σi are the observed hit time of the i-th PMT and its resolution, and T i is the hit
time expected from the entering time of muon and its track. Using a goodness-of-fit, the cut
criteria are following:(1)

goodness of through-going muon fit > 0.75

and
(2)

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

Events that satisfy all those criteria are rejected. Figure 5.5 show the scatter plots of NHITA inand NHITAout satisfying criterion (1) for the data after the 2nd reduction, the FC atmospheric
neutrino Monte Carlo events and the FC final samples.
Stopping muon cut
For rejection of stopping muons, a stopping muon fitter is applied to find the entrance pointby the similar way as the through-going muon fit. The goodness is defined as equation (5.1).
The events fulfilling the following criteria are removed as stopping muons:(1)

NHITAin ≥ 10 if goodness of stopping muon fit ≥ 0
NHITAin is the number of hit OD PMTs located within 8 m from the entrance
point in a fixed 800 nsec time window.
or NHITAin ≥ 5

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

The direction of muon is reconstructed to maximize the total number of p.e.s inside the conewith a half opening angle 42◦ . For SK-I, the additional criteria is included. Figure 5.6 shows
NHITAin distributions for the data after the 2nd reduction, the FC atmospheric neutrino Monte
Carlo events and the FC final events.
66

(a) DATA

30

30

25

25

20

20
rejected

15

10

5

5
0

20

30

(c) final sample DATA30NHITAout

10

0
30

25

25

20

20

15

15

10

10

5

5

0

0

rejected

15100(b) MC1020
30
FC3 NHITAin

0

0

10

20

30

(c) final sample MC

0

10

20

Figure 5.5: (FC 3rd reduction) Scatter plot of the number of hit OD
PMTs within 8 m of the entrance
point NHITAin (x-axis) and the exit
point NHITAout (y-axis) for SK-III. (a)shows the data after the FC 2nd reduction. (b) shows FC Monte Carlo events
after the FC 2nd reduction. (c) shows
the final samples for the data and thethe Monte Carlo events. The selection
criteria are shown by arrows.

30

number of events(a) DATA

10

4

10

3

4

rejected(b) MC

10
3
10
2
10
10
1

rejected(c) final sample
3

10
2
10
10
1
0

data
MC

5

10 15 20 25 30 35 40 45 50
FC3 NHITAent

67

Figure 5.6: (FC 3rd reduction) Number of hit OD PMTs near the entrance
point of muons NHITAin for SK-III. (a)shows the data after the FC 2nd reduction. (b) shows FC Monte Carlo events
after the FC 2nd reduction. (c) shows
the final samples for the data and thethe Monte Carlo events. The numberof the Monte Carlo events in (c) is normalized to that of the data. The selection criteria are shown by lines.

Cosmic ray µ

Veto Counter
Cable Bundle
from PMT
To
Electronics
Hut

Outer
Detector

Inner
Detector

Figure 5.7: A schematic view of a cable-hole muon and a veto counter.

Cable hole muons
On top of the detector tank, there are twelve cable holes to take the signal and HV supply
cables out. Four holes of those are covering the OD PMTs as shown in Figure 5.7. When cosmic
ray muons enter into the ID region through these holes, this event could be detected as contained
neutrino events because of no OD signal. In order to avoid such mis-identification, veto counters(2 m × 2.5 m plastic scintillation counters) were installed in April, 1997. The rejection criteriausing the veto counters of the cable hole muons are :(1)

One veto counter hit

and
(2)

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

Flasher event cut
Flasher events usually have a broad hit timing distribution compared with that of the neutrino events. By using this feature, following criteria are applied.
For SK-I :(1)

NMIN100 ≥ 15
or
NMIN100 ≥ 10
if the number of hit ID PMTs ≤ 800
NMIN100 is the minimum number of hit ID PMTs in a sliding 100 nsec time
window from +300 nsec to +800 nsec after the trigger.
68

30

(i)

20
10
0
200
(ii)

150
100

Figure 5.8: (FC 3rd reduction) The
timing distribution of (i) a typical
flashing PMT event and (ii) a typical FC neutrino event. The arrows
show the time window for counting
NMIN100 .

50
0
800

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

For SK-II and SK-III :(1)

NMIN100 ≥ 20

Figure 5.8 shows the timing distribution of (i) a typical flashing PMT event and (ii) a typicalFC neutrino event and the time window for NMIN 100 is shown by arrows. Figure 5.9 shows
NMIN100 distributions for the data after the 2nd reduction, the FC atmospheric neutrino Monte
Carlo events and the FC final events.
Accidental coincidence events cut
The accidental coincidence occurs when a low energy event forms the trigger and a cosmic ray
muon event follows in a single trigger gate. These events are not rejected in the former reduction
because of the absence of the OD activities on the trigger timing and the large number of total
p.e.s in the ID due to the muons. The accidental coincidence events are removed by the following
cuts :(1)

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

and
(2)

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

Low energy events cut
This is the rejection for the remaining low energy events such as the decay of radio isotopes
and the electrical noise. Events satisfying the following criteria are removed :(1)

NHIT50 < 50 (25 for SK-II)
NHIT50 is the number of hit ID PMTs in a sliding 50 nsec time window.
69

10

5

10

4

10

5

10

4

10

3

10

3

2

10

10

2

(a) data

rejected

10

10

5

10

4

10

3

10

2

Number of events

Number of events

10(b) MC10110(a) data

rejected

4103(c) final sample

10

2

data
MC

10

10

5

10

4

10

3

10

2

(b) MC

10
1
10

4

10

3

10

2

(c) final sample
data
MC

10

1

1
0

5

10

15

20

25

30

35

40

45

50

NMIN100 (NHITID > 800)

0

5

10

15

20

25

30

35

40

45

50

NMIN100 (NHITID < 800)

Figure 5.9: (FC 3rd reduction) NMIN 100 distributions for the events with more than 800 hit PMTs
in ID (left) and less than 800 hit PMTs (right) for SK-I. (a) shows the data after the FC 2nd
reduction. (b) shows FC Monte Carlo events after the FC 2nd reduction. (c) shows the final
samples for the data and the the Monte Carlo events. The number of the Monte Carlo events in(c) is normalized to that of the data. The selection criteria are shown by arrows.

70where NHIT50 is counted after subtracting the time of flight (TOF) of each observed photon
assuming all photons are generated at a same point. The vertex is determined as the position atwhich the timing residual distribution is peaked. NHIT 50 =50 corresponds to visible energy of
9 MeV. It is enough low to keep efficiency for contained neutrino events which is E vis >30 MeV.
After the FC third reduction, the event rate is ∼45 events/day.

5.1.4

Fourth Reduction

In the fourth reduction, an intelligent pattern matching algorithm is used to remove theremaining flasher events. Flasher events usually repeat with similar hit patterns in the detector.
These repeated events are not likely to be caused by neutrinos.
The algorithm of the pattern matching is as follows :(1)

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

(2)

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


A
B
B
1 X QAi − hQ i × Qi − hQ i
(5.2)
r=
N
σA × σ Biwhere N is the number of the patches, and hQ A(B) i and σA(B) are the averaged
charge and its standard deviation for event A and B, respectively.
(3)

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

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

(5)

If r exceeds the threshold(rth ), events A and B are recognized as matched
events. rth is defined as :Br > rth = 0.168 × log 10 ((PEA
tot + PEtot )/2.) + 0.130,(5.3)

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

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

(7)

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

The event rate after the fourth reduction is ∼18 events/day.

5.1.5

Fifth reduction

In the 5th reduction, the final rejection is applied by the several criteria specialized for eachbackground source.

71

(a) DATA
3

rejected

10
2
10
10
1

Figure 5.10:(FC 5th reduction)
NHITACearly
distribution satisfying criteria PEtot < 1000 p.e.s and
DISTclust > 500 cm for SK-III. (a)shows the data after the FC 4th
reduction. (b) shows FC Monte Carloevents after the FC 4th reduction. (c)shows the final samples for the data
and the the Monte Carlo events. The
number of the Monte Carlo events in(c) is normalized to that of the data.
The selection criteria are shown byarrows.

number of events(b) MC
3

10
2
10
10
1

rejected(c) final sample

10

2

data
MC

10
0

5

10

15 20 25 30
FC5 NHITACearly

35

40

Invisible muon cut
If a cosmic ray muon has momentum less than the Cherenkov threshold and the subsequent
decay electron is observed, this event could be mis-identified as neutrino event. Such event
is characterized by a low energy signal from decay electron in the ID and a signal in the OD
before the trigger timing. An event fulfilling the following cut criteria are rejected as invisible
muons :
(1)

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

and
(2)

NHITACearly + NHITAC500 ≥ 10
NHITACearly>9

if DISTclust < 500 cm
otherwise

NHITACearly is the maximum number of hit PMTs in the OD hit cluster in a
sliding 200 nsec time window from −8800 nsec to −100 nsec.
NHITAC500 is the number of hit PMTs in the OD hit cluster in a fixed 500 nsec
time window from −100 nsec to +400 nsec.
DISTclust is a distance between two OD hit clusters, which are used for the
NHITACearly and the NHITAC500 .
Figure 5.10 shows the NHITACearly distribution, satisfying criteria (1) and DIST clust > 500 cm,
for the data after the FC 4th reduction, Monte Carlo events and the final samples.
Coincidence muon cut
The remaining accidental coincidence muon events which are not rejected in the Forth reduction (Accidental coincidence events cut) are removed by following criteria :72

(a) DATA

10

3

10

2

rejected

10

Figure 5.11: (FC 5th reduction) PElate
distribution for SK-III. (a) shows thedata after the FC 4th reduction. (b)shows FC Monte Carlo events after the
FC 4th reduction. (c) shows the final samples for the data and the theMonte Carlo events. The number ofthe Monte Carlo events in (c) is normalized to that of the data. The selection criteria are shown by arrows.

number of events

14(b) MC

10
3
10
2
10
10
1
10

3

10

2

rejected(c) final sample
data
MC

10
1
0

(1)

10 20 30 40 50 60 70 80 90 100
FC5 PElate

PE500 < 300 p.e.s (150 p.e.s for SK-II)
PE500 is the total number of p.e.s observed in the ID in a fixed 500 nsec time
window from −100 nsec to +400 nsec.

and
(2)

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

Figure 5.11 shows the PElate distribution satisfying the criterion (1) for the data after the FC
4th reduction, Monte Carlo events and the final samples.
Long-tail flasher cut
This is a stricter version of the flasher cut in the FC 3rd reduction stage. Events satisfying
the following criterion are removed as flasher events :(1)

NMIN100 ≥ 6
if the goodness of Point-fit < 0.4
NMIN100 is the minimum number of the hit ID PMTs in a sliding 100 nsec
time window from +300 nsec to +800 nsec.

For SK-II and SK-III, additional cuts are applied :(2)

NMIN100 ≤ 5

if the goodness of Point-fit < 0.3

The Point-fit will be explained in Section 5.4.1.
After the fifth reduction, the FC event rate is ∼16 events/day.

73

PC

104

3

105

FC

Number of events

Number of events

Number of events

105 FC

PC

104

3

105

3

10

10

102

102

102

10

10

10

10

20

30

40

50

60

70 80
NHITAC

1
0

10

20

30

40

50

60

70 80
NHITAC

PC

104

10

1
0

FC

1
0

10

20

30

40

50

60

70 80
NHITAC

Figure 5.12: NHITAC distributions for FC and PC final samples for SK-I (left), SK-II (middle) and
SK-III(right). The points show the observed data, and histograms show the atmospheric neutrinoMonte Carlo events assuming no oscillation (dashed lines) and ν µ ↔ ντ 2-flavor oscillation with(sin2 2θ, ∆m2 ) = (1.00, 2.5 × 10−3 eV2 ) (solid lines).

5.1.6

FC Reduction Summary

Finally, the fully contained neutrino events are selected by following criteria :· Vertex of neutrino interactions is inside the fiducial volume (2 m from the ID PMT surface).
· The number of hit PMTs in the highest charge OD cluster (NHITAC) is less than 16 (10for SK-I).
· Visible energy (Evis ) is greater than 30 MeV.
Figure 5.12 shows the NHITAC distribution for FC and PC final sample for each SK run period.
The detection efficiencies in each reduction step are estimated by the atmospheric neutrinoMonte Carlo events as shown in Table 5.1. The number of events in the final sample and theevent rate are also summarized in Table 5.1. The event rate is consistent with constant for each
SK run period.
The background events contaminated in the final FC sample are mainly cosmic ray muonsand flasher events. These background events are largely rejected by a fiducial volume cut, thereconstructed vertex position > 2m from the ID PMT surface. In addition, background eventsare checked by eye-scan using a visual display. Since flasher events could be a background sourcefor e-like sample, and cosmic ray muons events could be a background source for µ-like sample,
the contaminations of the backgrounds for each sample are estimated as shown in Table 5.2.
The uncertainty of FC reduction efficiency is estimated by comparing the distributions ofeach cut variables of data and Monte Carlo. The uncertainties are estimated to be 0.2%, 0.2%
and 0.8% for SK-I, SK-II and SK-III, respectively.

5.2

Reduction for Partially Contained Events

PC events have OD activities while FC events do not, thus PC reduction processes are different from those of FC. Since OD segmentation was installed in SK-III period (see Section 2.2.3),
74

FC

SK-I

reduction step

SK-II

SK-III

selection efficiency (%)

1st
2nd
3rd
4th
5th
5th(FV)

99.95
99.94
99.85
99.17
99.15
97.59

99.92
99.89
99.71
99.39
99.32
99.17

100.0
99.97
99.79
98.62
98.55
98.55

number of events
Monte Carlo
DATA(event rate[/day])

13461.3
12232(8.21±0.07)

7222.56584(8.24±0.10)

4753.64356(8.41±0.12)

Table 5.1: The selection efficiencies and number of events for FC sample are summarized foreach SK period. The selection efficiencies are for events whose real vertex is in the fiducial
volume, the number of outer detector hits fewer than 16(10 for SK-I) and the visible energy
larger than 30 MeV. In the last line with FV, events with the fitted vertex in fiducial volume
are selected. Numbers of events for MC are normalized without oscillation to real data livetime,1489.2, 798.6 and 518.1 days for SK-I, SK-II and SK-III, respectively.

SK-I

SK-II

SK-III

BG

cosmic ray

flashercosmic ray

flashercosmic ray

flasher

sub-GeVmulti-GeV

0.1%
0.1%

0.5%
0.3%

0.1%
0.1%

0.2%
0.3%

0.1%
0.2%

0.1%
0.4%

Table 5.2: Estimated contamination for each background event.
Evis <1.3GeV ,and Multi-GeV is events with E vis >1.3GeV.

75

Sub-GeV is events withthe reduction process is substantially modified to adjust to it for SK-III. Here describes the data
reduction procedure consisting of 5 steps for PC.

5.2.1

First Reduction

For SK-I and -II, the 1st reduction rejects the through-going cosmic ray muons and low
energy events. An event satisfying all of the following criteria is selected.
For SK-I:(1)

PEtot should be ≥ 1000 p.e.s
PEtot is the number of p.e.s observed in the ID.

and
(2)

TWIDA should be ≤ 260 nsec.
TWIDA is the width of the hit timing distribution in the OD PMTs.

and
(3)

NCLSTA should be ≤ 1.
NCLSTA is the number of the hit clusters in the OD.

For SK-II:(1)

PEtot should be ≥ 500 p.e.s

and
(2)

TWIDA should be ≤ 170 nsec.

Exiting particles in the PC sample, mostly muons, must have at least 2 m track length in the ID,
which corresponds to the momentum loss of 500 MeV/c for muons. In criterion (1), PE tot should
be larger than 1000 p.e.s (500 p.e.s for SK-II), which corresponds to 310 MeV/c for muons. The
criteria (2) rejects through-going muons which have a broad hit timing distribution and two hit
clusters around the entrance and exit point in the OD. Since the reflected photon in the OD is
increased and the quantum efficiency of OD PMTs is increased in SK-II, TWIDA cut criterion
is tuned for SK-II.
For SK-III, the efficient reduction for the through-going muons is performed taking advantageof the OD segmentation. An event fulfilling following all criteria are selected :(1)

PEtot should be ≥ 1000 p.e.s

and
(2)

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

and
(3)

NHITAendcap < 25 or NHITAside < 70
NHITAendcap (NHITAside ) is the number of OD hits in top and bottom (side)
region.
76

(a) DATA

rejected

050
100 150NHITA(bottom)

200
180
160
140
120
100
80
60
40
20
0

(b) MCNHITA(top)

NHITA(top)

200
180
160
140
120
100
80
60
40
20
0

200

rejected

0(c) final sample
data
MC

2

200

(c) final sample

10
eventsevents1050
100 150NHITA(bottom)

10

1

data
MC

2

10

Figure 5.13: (PC 1st reduction)
The scatter plots of NHITAtop and
NHITAbottom . (a) shows the raw
data. (b) shows PC Monte Carloevents with the vertex in the fiducial volume.
(c) shows the 1dimensional plot of each NHITAtop
and NHITAbottom for final samples.
The number of the Monte Carloevents in (c) is normalized to thatof the data. The selection criteria
are shown by lines.

1
0

50
100 150NHITA(bottom)

200

0

50
100 150NHITA(top)

200

and
(4)

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

The criteria (2) rejects through-going muons which passes both top and bottom regions. Figure 5.13 shows the scatter plots of NHITA top and NHITAbottom for the observed data, PC
atmospheric neutrino Monte Carlo events, and the distributions of each variable for the final
sample. The criteria (3) rejects corner clipping muons. Figure 5.14 shows the scatter plots anddistributions for NHITAendcap and NHITAside . In the criteria (4), the ODRmean value for thethrough-going muons is expected lager than that of PC events. Figure 5.15 shows the scatterplots for ODRmean and the number of OD hits and the distribution of ODR mean .
The 1st reduction reduces the number of events to ∼ 2 × 10 4 event/day.

5.2.2

Second Reduction

In the second reduction, a clustering algorithm of OD hit is used to reject the through-going
muons and the stopping muons. The OD (ID) walls are divided into 11 × 11 (21 × 21) patches
and the charge observed in each patch is counted. The clusters are formed by looking for the
charge gradient to the neighboring patches. The algorithm is illustrated in Figure 5.16. Events
satisfying the following all criteria remain after the PC 2nd reduction. For both SK-I and -II,
(1)

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

and, for SK-I :77

(a) DATA

0

100
200
NHITA(side)(b) MC

250
225
200
175
150
125
100
75
50
25
0

rejected

NHITA(endcap)

rejected

NHITA(endcap)

250
225
200
175
150
125
100
75
50
25
0

300

0

(c) final sample

10
eventsevents

10

2

data
MC

10

1

1
0

100
200
NHITA(side)

300(a) DATA

300

0

300

50 100 150 200 250
NHITA(endcap)(b) MC

good ODHIT

250

good ODHIT

250
200

200

150

150

100

100

rejected

50
0

0

1000
2000
ODR

rejected

50
0

1000
2000
ODR

(c) final sample
10

2

data
MC

events300(c) final sample
data
MC

0

100
200
NHITA(side)

Figure 5.14: (PC 1st reduction)
The scatter plots of NHITAendcap
and NHITAside .
(a) shows theraw data. (b) shows PC Monte
Carlo events with the vertex in
the fiducial volume.
(c) showsthe 1-dimensional plot of each
NHITAendcap and NHITAside for final samples. The number of theMonte Carlo events in (c) is normalized to that of the data. The selection criteria are shown by lines.

10

1
0

1000
2000
ODR

78

Figure 5.15: (PC 1st reduction) The
scatter plots of the number of goodOD hits and ODRmean . (a) shows
the raw data. (b) shows PC Monte
Carlo events with the vertex in thefiducial volume. (c) shows the 1dimensional plot of ODRmean for final samples. The number of theMonte Carlo events in (c) is normalized to that of the data. The selection criteria are shown by lines.

more charge

Cluster 1

less charge

Cluster 2

Cluster 3

Figure 5.16: A schematic view of the algorithm to find hit clusters in the PC 2nd reduction.
The circles represent the charge observed in each patch. The size of the circle is proportional
to the number of p.e.s . The arrows represent the vector charge gradient, which point to thehighest charge among the neighboring patches.

(2)

NCLSTA2(2) should be ≤ 1
NCLSTA2(2) is the number of the 2nd OD hit clusters including more than 6
hit PMTs.

and(3)

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

and
(4)

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

for SK-II :(2)

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

and
(3)

NHITAendcap < 20 or NHITAendcap < MAX(NHITAside )MAX(NHITAside ) is defined as :MAX(NHITAside = exp(5.8-0.023× NHITAside )if NHITAside < 75
MAX(NHITAside = exp(4.675-0.008× NHITAside )if NHITAside ≥ 75
NHITAendcap is the number of OD hit PMTs in the top and bottom region.
NHITAside is the number of hit OD PMTs in the side region.
79

(a) DATA

number of events

rejected
10

6

10

5

10

Figure 5.17: (PC 2nd reduction) The
number OD hit PMTs in the 2nd OD
hit cluster. (a) shows the data after thePC 1st reduction. (b) shows PC Monte
Carlo events after the PC 1st reduction. (c) shows the final samples for thedata and the the Monte Carlo events.
The number of the Monte Carlo eventsin (c) is normalized to that of the data.
The selection criteria are shown by arrows.

4 (b) MC

10

3

10

2

rejected

101(c) final sample
10

2

data
MC

10
1
0

5

10

15

20

25

30

PC2 NCLSTA2(4)NHITAC2 < 12 + 0.085 × PE200
NHITAC2 is the number of the OD hit PMTs in the 2nd cluster.

Criteria (3) rejects corner clipping muons, which left the hit PMTs in both top (or bottom).
Criteria (4) rejects stopping muons by using ID hit cluster located behind the OD hit cluster.
For SK-III, since 1st reduction using the OD segmentation is very efficient, the 2nd reduction
consists of only criteria (2) and (3) of SK-II. Figure 5.17 shows the number OD hit PMTs in
the 2nd OD hit cluster for the data after 1st reduction, PC Monte Carlo events and the final
samples. The criteria (3) is more strict cut of criteria (3) in the 1st reduction. Figure 5.18 shows
the scatter plots of NHITAendcap and NHITAside for the data after 1st reduction, PC Monte
Carlo events and the final samples.
The event rate after the PC 2nd reduction is ∼8000 events/day.

5.2.3

Third Reduction

The flasher events are rejected by the same way as that in the FC 3rd reduction, in which
the broad timing distribution of the flasher events is used for the cut. Events which satisfy thefollowing criteria are rejected as the flasher event:(1)

NMIN100 > 14
or
NMIN100 > 9
if the number of hit ID PMTs < 800
NMIN100 is the minimum number of hit ID PMTs in a sliding 100 nsec time
window from +300 nsec to +800 nsec.

The stopping muon events are removed by the number of the hit OD PMTs near the entrance position. To find the entrance position, vertex and direction are fitted by the point-fit
80

(a) DATA(b) MC
300

250

NHITA(endcap)

NHITA(endcap)

300
rejected

200
150
100
50
0

0

100

200

250

150
100
50
0

300

rejected

200

0

NHITA(side)(c) final sample (DATA)

300

300

NHITA(endcap)

NHITA(endcap)

200(d) final sample (MC)

300
250
200
150
100
50
0

100

NHITA(side)

0

100

200

300

NHITA(side)

250
200
150
100

Figure 5.18: (PC 2nd reduction)
Scatter plots of NHITAendcap and
NHITAside . (a) shows the data after
the PC 1st reduction. (b) shows PC
Monte Carlo events after the PC 1st
reduction. (c) and (d) show the final
samples for the data and the Monte
Carlo events. The size of the boxes
shows the number of the events ineach bin. The cut criteria are shownby the lines.

50
0

0

100

200

300

NHITA(side)(see Section 5.4.1). The entrance position is estimated by a backward extrapolation from thereconstructed vertex. Events satisfying the following criteria are rejected as stopping muons for
SK-I and SK-II :(2)

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

For SK-III, this cut is not used but this cut variable is used in the 5th reduction.
The event rate after the PC 3rd reduction is ∼100 events/day.

5.2.4

Fourth Reduction

In the 4th reduction, the selection criteria are completely modified for SK-III. Here describes
two versions of the 4th reduction. The event rate after the PC 4th reduction is ∼150 events/day.
< For SK-I and SK-II >The 4th reduction rejects cosmic ray muons which have the relatively small OD activities.
The two types of event reconstruction tools are used in this reduction stage, the Point-fit anda though going muon fit. The through going muon fit determines the entrance point as the
position of the earliest hit cluster in the ID. For the PC events, the entrance point can not
be correctly determined by the through-going muon fit because they are generated inside theID, and the goodness tends to be worse compared with the through-going muon events. On
the other hand, the vertex position and the direction can be reasonably well estimated by the
Point-fit for both the PC events and the cosmic ray muons. An event fulfilling following all
criteria are selected:
81

(1)

d~pfit · d~PMT should be > −0.8d~pfit is the reconstructed direction by Point-fit, and d~PMT is the direction fromthe reconstructed vertex to the earliest saturated PMT.

and
(2)

DCORN should be > 150 cm
DCORN is the distance from the reconstructed vertex by the Point-fit to thenearest fringe of the ID.

and
(3)

TLMU should be > 30 m
if goodness of through-going muon fit > 0.85
TLMU is the track length of a muon estimated from the entrance and the exit
points by the through going muon fit.

The criterion (1) rejects the cosmic ray stopping muons which have the entrance point in opposite
direction to the reconstructed direction by Point-fit. The second criterion aims to reject cornerclipping muons. Through-going muons which have a long track length are rejected by the cut
criterion (3).
< For SK-III >The event reconstruction tool is used in this reduction stage. This fitter classifies each event
as either, stopping muon, through-going muons, multiple muons or corner clipping muons. 97%
of background events are categorized as stopping or through-going muons, whereas 96% PC
events categorized as the other muons types. The 5 selections based on the fitter results arefollows:(1)

anglemuon < 90oanglemuon is the angle between the fitted direction and the vector between the
Point-fit vertex and the center of the highest charge OD cluster.

(2)

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

(3)

lengthmuon < 1750 cm
lengthmuon is the length of the fitted muon track. Very long track length hints
toward cosmic ray muons.

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

(5)

cornermuon ≥ 300 cm
this is the distance between fitted entrance point and the corner of the tank.

If an event is classified as through-going muon, it should pass 4 of above 5 criteria to be selected. If an event is classified as stopping muon, it should pass 4 of above 5 criteria including
dotprodmuon pass necessarily. Then the stopping muon needs to pass the criteria goodness muon <82

x 10 3

2250
2000

data

selection

1750
1500

number of events

number of eventsx 10 3

MC

1250
1000
750
500
250
0

0

20

40

60

80

100

120

140

160

4000
2000

x 10

3500

6000

number of events

number of eventsx 10 3
MC

3000
2500
2000
1500
1000
500
0

1000

2000

3000

4000

5000

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

PC4 DOTPROD

3

data

selection

MC

5000
4000
3000
2000
1000
0

0

MC

6000

0
-1

180

data

selection

8000

PC4 ANGLE

selection

data

10000

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PC4 GOOD

PC4 LENGTH

Figure 5.19: (PC 4th reduction) The distributions of variables in 4th reduction for the data
(dashed line) and PC MC events (solid line) which are classified as through-going muons or
stopping muons after the 3rd reduction. The distributions, angle muon , dotprodmuon , lengthmuon
and goodnessmuon are shown in from the left top panel to the right bottom panel. The numberof the Monte Carlo events are normalized to that of the data. The selection criteria are shownby arrows.

0.5 or ehitmuon < 10; ehitmuon is the number of OD hits located within 8 m from the fitted entrance point in a fixed 500 nsec time window. For events classified as the other types, they have
only to pass 2 of above 5 criteria. Figures 5.19 shows the distributions for each cut variable incriteria (1)∼(4) for the data and MC events which are classified as the through-going muon or
the stopping muon by the fitter. Figures 5.20 shows the same distributions for the final samples
which are classified as the other muon types by the fitter from the final sample.

Finally, events satisfying the following criteria are removed as the low energy background
events:(1)

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

The requirement of total photoelectrons > ∼3000 p.e.s, which corresponds to muon momentumof 500 MeV/c, is safe for PC events because the exiting muons must have at least momentum of
700 MeV/c to reach the OD.

83

number of events

number of events

data
MC

10

1
0

20

40

60

80

100

120

140

160

data
MC

10

-1

180

-0.8

2

10

number of events

number of events

10

data
MC

10

1
1000

2000

3000

4000

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

5000

2

data
MC

10

1
0

-0.6

PC4 DOTPROD (final sample)

PC4 ANGLE (final sample)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PC4 GOOD (final sample)

PC4 LENGTH (final sample)

Figure 5.20: (PC 4th reduction) The distributions of variables in 4th reduction for the final
samples, data events (dot) and the PC MC events (line) which are classified as multiple muons or
corner clipping muons. The distributions, angle muon , dotprodmuon , lengthmuon and goodnessmuon
are shown in from the left top panel to the right bottom panel. The number of the Monte Carloevents is normalized to that of the data.

5.2.5

Fifth Reduction

The 5th reduction is the final rejection of the remaining background events. This is doneby some elaborate criteria specialized for each background source. In this stage, the SK-III
reduction criteria are different from the other SK period. So here also describes two versions of
5th reductions.
< For SK-I and SK-II >Low energy event cut
Events satisfying the following criteria are removed as the remaining low energy background
events :(1)

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

Through-going muon cut A
The remaining through-going muons are removed by investigating the existence of two OD
hit clusters and the existence of hit OD PMTs near the entrance and the exit points. The first
cut uses the OD hit cluster information obtained by the algorithm in the PC 2nd reduction.
Events satisfying the following criteria are removed as through-going muons :
84

(1)

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

and
(2)

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

and(3)

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

In criterion (3), the same clustering algorithm as that in the PC 2nd reduction with different
parameters is used. The OD wall is divided into 6 × 6 instead of 11 × 11 in the PC 5th reduction
to avoid the boundary effect of the patches.
The event passed above two cuts is reconstructed its vertex and so on. The reconstruction
methods will be described in Section 5.4. After applying the reconstruction, the final backgroundrejections are performed in follows.
Through-going muon cut B
Some cosmic ray muons enters from the edge on top, passes along the ID wall and exits
from the edge of the bottom. These through-going muon events tend to pass through the former
reduction criteria because the light collection efficiency around the edge of the OD is not good
and the event reconstruction is not so accurate for these events. To reject these events, thenumber of hit OD PMTs and the observed p.e.s in 8 m radius spheres centered at the top andthe bottom edges or fringes are used. Events satisfying the following criteria are rejected as
through-going muons :(1)

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

and
(2)

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

and
(3)

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

85

Through-going muon cut C
The remaining through-going muons are removed by the number of hit OD PMTs near theentrance and the exit points. The vertex position and the ring direction are reconstructed by aprecise fit (MS-fit) using the Cherenkov ring pattern (see Section 5.4.4). The entrance and theexit points on the detector wall are estimated by an extrapolation. The cut criteria are follows :(1)

NHITAin ≥ 5 and NHITAout ≥ 5
NHITAin (NHITAout ) is the number of hit OD PMTs within 8 m from theentrance (exit) point.

and
(2)

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

Stopping muon cut A
The number of hit OD PMTs near the entrance position is counted. The entrance positionof the stopping muons are estimated by reconstructing the vertex position and direction using
MS-fit and back extrapolating to the wall. Events satisfying the following criteria are rejected
as stopping muons :(1)

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

Stopping muon cut B
The opening angles between the OD hit cluster and the ring estimated by two different fitters,
TDC-fit (see Section 5.4.1) and MS-fit are compared. In case of the stopping muon event, theopening angle is expected to be large, while it is small for th PC event. The cut criterion is thefollowing :(1)

ΘTDC-fit > 90◦orΘMS-fit > 90◦

ΘTDC-fit (ΘMS-fit ) is the opening angle between the direction to the OD hit
cluster and the ring direction reconstructed by TDC-fit (MS-fit).
Stopping muon cut C
The charge inside a 42◦ cone in the ID is used. The vertex and direction are determined bystopping muon fit, which estimate the entrance as the position of the earliest hit cluster in the
ID. The entrance position can not be reconstructed correctly for the PC events since the vertexof the PC events are not on the wall. Events satisfying the following criteria are rejected as
stopping muons :(1)

goodness of stopping muon fit > 0
86

and
(2)

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

(3)

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

Cable hole muon cut
The veto scintillation counters are placed over the four cable holes on top of the detector. The
simple condition that only requiring veto counter hit is inapplicable for the PC event selection.
(1)

One veto counter hit.

and
(2)

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

Corner clipping muon cut A
The corner clipping muon events have a small hit cluster in the ID, then the vertexes areoccasionally mis-reconstructed inside the ID. As a result, the track length from the vertex tothe exit point reconstructed by MS-fit is large, while the track length which is estimated by
the visible energy using the energy loss of muons ∼2 MeV/cm is small. The cut criterion forrejecting corner clipping events is the following :(1)

Evis /2 (MeV/cm) < TRACK − 1500
if TRACK > 15 m
Evis /2 is the estimated track length by the visible energy and TRACK is the
track length from the vertex to the exit point estimated from the vertex point
and the direction by MS-fit.

< For SK-III >PC reduction for SK-III is modified to keep efficiency as SK-I and II. In this reduction, there
are two types of cut, Hard cut and Soft cut. PC events are required to pass all Hard cuts. And,the events may fail Soft cuts just once.
The Hard cuts are:(1)

Through-going muon cut A(2)Through-going muon cut B(3)

Stopping muon cut B(4)

Cable hole muon cut
87

(a) DATA
rejected

10

3

Figure 5.21: (PC 5th reduction) The
distribution of DCORN for SK-III. (a)shows the data after the PC 4th reduction. (b) shows PC Monte Carlo events
after the PC 4th reduction. (c) shows
the final samples for the data and thethe Monte Carlo events. The numberof the Monte Carlo events in (c) is normalized to that of the data. The selection criteria are shown by arrows.

number of events(b) MC102rejected

10(c) final sample
10

data
MC

1
0

200

400

600

800

1000

PC5 DCORN(5)Corner clipping muon cut B

Criteria (1), (2), (3) and (4) have already described in above reductions for SK-I and II.
Corner clipping muon cut B
This criterion aims to reject corner clipping muons.
(1)

DCORN should be > 150 cm
DCORN is the distance from the reconstructed vertex by the Point-fit to thenearest fringe of the ID.

Figure 5.21 shows the DCORN distributions for the data, PC atmospheric neutrino Monte Carloevents, and the PC final sample events.
The Soft cuts are:(1)

Through-going muon cut C(2)Through-going muon cut D(3)

Stopping muon cut A(4)

Stopping muon cut C(5)

Stopping muon cut D(6)Stopping muon cut E(7)Corner clipping muon cut A(8)

Decay electron cut

Criteria (1), (3), (4) and (7) have been already described in above reductions for SK-I and II.
88

(a) DATANCLASTA(5)2

NCLASTA(5)2

50
45
40
35
30
25
20
15
10
5
0

Soft cut

0

50

100

150

50
45
40
35
30
25
20
15
10
5
0

(b) MCSoft cut

0

50
45
40
35
30
25
20
15
10
5
0

(c) final sample DATA

0

50

100

50

100

150

NCLSTA(5)1

NCLASTA(5)2

NCLASTA(5)2

NCLSTA(5)1150

NCLSTA(5)1

50
45
40
35
30
25
20
15
10
5
0

(d) final sample MC

0

50

100

Figure 5.22: (PC 5th reduction)
The scatter plots of NCLSTA5(1)and NCLSTA5(2) for SK-III. (a)shows the data after the PC 4th
reduction. (b) shows PC Monte
Carlo events after the PC 4th reduction. (c) and (d) show the final samples for the data and the the Monte
Carlo events, respectively. The Softcut criteria are shown by lines.

150

NCLSTA(5)1

Through-going muon cut D
The remaining through-going muons are removed by the number of hit OD PMTs in 1st and2nd highest charge clusters. The cut criterion is following :(1)

NCLSTA5(1) ≥ 10 and NCLSTA5(2) ≥ 17
NCLSTA5(1) (NCLSTA5(2)) is the number of hit OD PMTs in 1st(2nd) highest charge cluster.

Figure 5.22 shows the scatter plots of NCLSTA5(1) and NCLSTA5(2) for the data, PC atmospheric neutrino Monte Carlo events, and the PC final sample events.
Stopping muon cut D
The stopping muon rejection is performed using the number of OD hits within 8 m of thereconstructed entry point (NHITAin ) as found in 3rd reduction. In the 3rd reduction, thereconstructed entry point is based on Point-fit. If the entry points between by Point-fit and by
MS-fit are not disagreed, this stopping muon rejection is applied.
→
−
→
−
(1)
NHITAin > 6
if | P point − P MS | < 1500cm
NHITAin is the number of hit OD PMTs within 8 m from the entrance position.
→−→−
P point ( P MS ) is the vertex position by the Point-fit (MS-fit).
Stopping muon cut E
The same criterion of anglemuon in 4th reduction is applied since the cut was looser for the
events which were classified as stopping or though-going muons.

89

(a) DATA
10

4

10

3

Soft cut

Figure 5.23: (PC 5th reduction) The
distribution of anglemuon for SK-III. (a)shows the data after the PC 4th reduction. (b) shows PC Monte Carlo events
after the PC 4th reduction. (c) shows
the final samples for the data and thethe Monte Carlo events. The numberof the Monte Carlo events in (c) is normalized to that of the data. The Softcut criteria are shown by arrows.

number of events(b) MC
10

3

10

2

Soft cut

10

10

2

(c) final sample
data
MC

10

1
0

20

40

60

80

100

120

140

160

180

PC5 ANGLE(1)anglemuon < 90◦anglemuon is the angle between the fitted direction and the vector between thepfit vertex and the center of the highest charge OD cluster.

Figure 5.23 shows the anglemuon distributions for the data, PC atmospheric neutrino Monte
Carlo events, and the PC final sample events.
Decay electron cut
High energy neutrinos mostly interact with nucleons in water via DIS interaction. andproduce hadrons including charged pions. Those pions decay into muons which in turn decay
into electrons. These electrons can be tagged by the method described in Section 5.4.6. If highenergy cosmic ray muon decays in the detector, no decay electron can be observed due to the
decay in flight. Therefore no decay electron event with high energy, E vis > 25 GeV, are rejected
as the cosmic ray muon.

5.2.6

PC Reduction Summary

Finally, the partially contained neutrino events are selected by following criteria :· Vertex of neutrino interactions is inside the fiducial volume (2 m from the ID PMT surface).
· The number of hit PMTs in the highest charge OD cluster (NHITAC) is larger than 15 (9for SK-I).
· Visible energy (Evis ) is greater than 350 MeV( The total observed charge of ID is > 3000 p.e.s (1500 p.e.s for SK-II) )The detection efficiency in each reduction step is estimated by the atmospheric neutrino Monte
Carlo events as shown in Table 5.3. The detection efficiency for final events is estimated to
90

PC

SK-I

reduction step
1st
2nd
3rd
4th
5th
5th(FV)

SK-II

SK-III

selection efficiency (%)
98.98
96.74
95.69
89.86
88.66
80.98

98.58
93.43
92.32
84.60
82.63
74.80

99.08
98.50
98.49
97.79
96.54
88.64

number of events
Monte Carlo
DATA(event rate[/day])

1185.5896(0.60±0.02)

589.6
430
(0.54±0.03)

437.7
343
(0.66±0.04)

Table 5.3: Selection efficiencies and number of events for PC sample are summarized for each
SK period. The selection efficiencies are for events whose real vertex is in the fiducial volume,the number of outer detector hits more than 15(9 for SK-I) and the total observed charge larger
than 3000p.e.s (1500p.e.s for SK-II). In the last line with FV, events with the fitted vertex infiducial volume are selected. Numbers of events for MC are normalized without oscillation to
the livetime for the data, 1489.2, 798.6 and 518.1 days for SK-I, SK-II and SK-III, respectively.

be 81.0 %, 74.8 % and 88.6 % for SK-I, SK-II and SK-III, respectively. The number of selected
events for each SK period are shown in Table 5.3.
The background events in final PC sample mainly come from cosmic ray muons. These background events are checked by eye-scan. Few background events contaminate PC final sample
but the most of background events are rejected by the fiducial volume cut. The contaminations
due to the background events is also estimated by extrapolating the distribution of background
events outside the fiducial volume. Figure 5.24 shows the number of background events outside
the fiducial volume for SK-II. Background events in the fiducial volume may lead to contamination as shown by the fitted exponential curve in this figure. As the results of eye-scan check andthe extrapolation estimation, the number of background events is expected to be 3 events for
SK-II. Table 5.4 summarizes the estimated contaminations due to background events for each
SK run period.
Uncertainties of the PC reduction efficiencies are estimated by the difference of cut variables
between data and Monte Carlo. The uncertainties are estimated to be 2.4%, 4.8% and 0.5%
for SK-I, SK-II and SK-III, respectively. The smaller uncertainty is obtained for SK-III due
to the improvement of the reduction efficiency and also better matching between data and MC
distributions for cut variables by introducing new cuts or modification of the existing cuts.

5.2.7

PC OD stop/through separation

The events stopping in the outer detector region contain in the PC sample. Such events canbe accurately reconstructed their energies rather than events going out through the OD. Thus

91

Figure 5.24: The number of background events outside of fiducial volume as a function of the reconstructedvertex position from the ID surface.
The line shows the fitted exponential
curve.

cosmic ray BG

SK-I

SK-II

SK-III

0.2%

0.7%

1.8%

Table 5.4: Estimated contamination of cosmic ray muons for PC sample.

PC sample is further separated to two categories, OD stopping and OD through-going, by theenergy deposit in OD region. The selection criteria for the OD stopping event are follows :• The maximum number of p.e.s observed in the OD in a sliding 500 nsec time window from−400 nsec to +600 nsec (PEanti ) is less than PEexp /1.5
(PEexp is the expected number of p.e.s in the OD from the potential track length in the
OD).
• The most energetic ring or the second one should be identified as µ-like.
Figure 5.25 shows the PEanti /PEexp distributions of the data and PC Monte Carlo events which
exit from ID barrel region to OD region.

5.3

Reduction for Upward-going Muon Events

The reduction for the upward-going muon (UPMU) events are briefly described in following
sections.

5.3.1

Charge cut

In the first reduction, the low energy events and extremely high energy events are rejected :(1)

PEtot should be > 6000 p.e.s (3000 p.e.s for SK-II)

and
(2)

PEtot should be < 1,750,000 p.e.s (800,000 p.e.s for SK-II)
PEtot is the total number of p.e.s observed in the ID.
92

90

45
40

70

35

60
50
40
30

20
15

10

5
0.2 0.4 0.6 0.81

1.2 1.4 1.6 1.8

PEanti/PEexp

2

OD stopping

0

OD through-going

30

25

10

0

OD through-going

30

20

0

OD stopping

Number of events

OD through-going

Number of events

Number of events

OD stopping
80

25
20
15
10
5

0

0.2 0.4 0.6 0.81

1.2 1.4 1.6 1.8

PEanti/PEexp

2

0

0

0.2 0.4 0.6 0.81

1.2 1.4 1.6 1.8

2

PEanti/PEexp

Figure 5.25: The distributions of PE anti /PEexp for the PC stop/through separation for the
events exiting from ID barrel region to the OD. The left, middle and right panels show for
SK-I, SK-II and SK-III, respectively. The points show the data, and histograms show the PC
atmospheric neutrino Monte Carlo events assuming no oscillation (black lines) and ν µ ↔ ντ 2flavor oscillation with (sin2 2θ, ∆m2 ) = (1.00, 2.5 × 10−3 eV2 ) (blue lines). Filled histograms are
the true number of the OD stopping MC events without the oscillation. The selection criteria
are shown by arrows.

PEtot =6000 p.e.s corresponds to muon momentum of 1 GeV/c and to track length of 3.5 m. The
requirement for final sample is track length longer than 7 m so this criterion is safe. At very
high ID charge corresponding to ∼ 1,750,000 p.e.s, the ID electronics is saturated and the muon
fitters cannot work.

5.3.2

Zenith angle cut

In order to reject the downward-going muon cosmic ray muons, seven different fitters specialized to fit stopping muons, through-going muon events and muon events with Bremsstrahlung
are used. The algorithm of the zenith angle cut is as following :(1) Apply a muon fitter.
(2) If the event is classified as upward and the goodness of fit is above the threshold, the event
is saved.
(3) If the event is classified as downward and the goodness of fit is above the threshold, theevent is rejected.
(4) If the event is classified as horizontal with the goodness of fit above the threshold, or if
the goodness of fit is below the threshold, the judgment is suspended.
(5) Go to (1) and apply the next muon fitter.
This sequence continues until the event has passed through all the fitters or has been classified.
If no fitter gives a goodness above the threshold, the event is rejected, while if at least one fitterclassifies the event as horizontal, the event is saved. Detailed description about the seven muon
fitters and the definition of the goodness can be found in [127].
93

All events from the output of the upward-going muon reduction are passed through theprecise fitter which is described in section 5.4.4. The fitter results are used to select upwardgoing events by the criterion that zenith angle is cosΘ <0.

5.3.3

Scanning

In order to eliminate the remaining misfitted backgrounds events, the selected events arescanned by physicists using a visual display. The upward-going muon events are selected one
by one. All events are checked by two independent scanners not to miss neutrino events. The
efficiency of scanning is estimated to be almost 100 %. About a half of the events remaining
after all the automated reduction steps are rejected by this final scan. We note that the scanners
only check that the reconstructed vertex and direction is not largely wrong.

5.3.4

UPMU Reduction Summary

The final selection criteria for UPMU stopping events are :• fitter classification is stopping event• the fitted momentum ≥ 1.6 GeV/c (corresponding to track length = 7 m)
• the number of hit OD PMTs within 8 m from the exit point (NHITEX) < 10 (16 for
SK-II).
The selection criteria for UPMU through-going events are :• fitter classification is through-going event• the distance from the ID entrance point to the ID exit point ≥ 7 m• the number of hit OD PMTs within 8 m from the exit point (NHITEX) ≥ 10 (16 for
SK-II).
Figure 5.26 shows the distributions of NHITEX. The detection efficiency for the final events is
estimated to be 98.0% (99.4%) for SK-I Stopping muons (Through-going muons), 97.0% (98.1%)
for SK-II Stopping muons (Through-going muons), 98.2% (99.4%) for SK-III Stopping muons(Through-going muons). The number of events in the final sample are summarized in Table 5.5.
The final sample is made by subtracting the expected background events. The backgroundestimation method is described in the next section.

5.3.5

Background estimation for Upward-going muon

The background events are mostly cosmic-ray muons, which are reconstructed as upwardgoing due to multiple Coulomb scattering of the muons or to the slight mis-reconstruction ofnear horizontally going muons. Therefore, they are mostly distributed in the most horizontalbin (−0.1 < cos θ < 0). The contaminations of the backgrounds are estimated by extrapolating
the distribution of downward events in 0 < cos θ. Figure 5.27 shows the zenith versus azimuth
directions for the upward going muon sample. Clusters of cosmic ray downward muons areseen in the regions zenith angle cosΘ > 0 (downward) and azimuth angle around φ = 120 ◦ ,180◦ and 270◦ . These clusters are caused by the thin rock covered over the detector. When
94

Thru

104

Stop

Entries

Entries

Entries

104 Stop

Thru

104 Stop

10

3

10

3

10

102

102

102

10

10

10

1
0

3

1
10

20
30
40
50
60
70
Numbers of hits within 8m of exit

0

Thru

1
10

20
30
40
50
60
70
Numbers of hits within 8m of exit

0

10

20
30
40
50
60
70
Numbers of hits within 8m of exit

Figure 5.26: The distributions of NHITEX for the UPMU stop/through separation for SK-I
(left), SK-II (middle) and SK-III (right). The points show the observed data, and histograms
show the PC atmospheric neutrino Monte Carlo events assuming no oscillation (dashed lines)and νµ ↔ ντ 2-flavor oscillation with (sin2 2θ, ∆m2 ) = (1.00, 2.5 × 10−3 eV2 ) (solid lines). The
selection criteria are shown by arrows.

UPMU

SK-I
stopping
throughSK-II
stopping
through

SK-III
stopping
throughfinal sample (MC)701.7

1850.7

355.2

915.4

273.8

715.6

final sample (data)
BG subtracted
event rate (/day)465
432
0.26±0.01

1879
1848.61.12±0.03223
210
0.25±0.02

871
833.4
1.01±0.30

210193
0.03±0.02

735
722.9
1.14±0.04

Table 5.5: Number of events for UPMU data sample are summarized for each SK period.
For the BG subtraction, see the text. Numbers of MC events and event rate are calculated
by normalization of the livetime, 1645.9, 827.7 and 635.6 days for SK-I, SK-II and SK-III,
respectively.

95

SK-I
BG event sys. error
stopping µthrough µ (non-shower)
through µ (shower)

1.97%
1.01%
0.81%

16%
11%
18%

SK-II
BG event sys. error
1.59%
2.02%
2.53%

21%
15%
14%

SK-IIIBG event sys. error
2.69%
0.65%
1.26%

20%
19%
24%

Table 5.6: The fraction of the estimated number of background events, contamination of horizontal muon to UPMU sample, in the zenith angle −0.1 < cos θ < 0. These numbers arenormalized by the livetime, 1645.9, 827.7 and 635.6 days for SK-I, SK-II and SK-III, respectively. The uncertainties for each estimated number of background events are also shown.

the contamination of the background events is estimated, the upward going muon samples are
divided into two azimuth angle regions. Figure 5.28 shows the azimuth angle distributions for
upward going muon samples, in which region (2) shows the thinner mountain direction, which
is defined as 60◦ < φ < 310◦ and region (1) shows the thicker mountain direction.
Figure 5.29 shows the zenith angle distributions for upward-going muon samples, in which
region (1) and (2) are normalized by the coverage of the azimuth angle. While the zenith angle
distributions are almost flat for upward going events (cosΘ <0), the number of events in theregion (2) exponentially increase with the cosine of zenith angle for downward-going (cosΘ >0).
The contaminations of the cosmic ray muons into the upward-going direction are estimated by
an extrapolation from the downward direction in the region (2) with (exponential + constant)function. In Figure 5.29, there are the best-fitted curves as the function which is defined asf (cos θ) = p0 + exp(p1 + p2(cos θ)). In the fitting, p0 is fixed to the value determined by
the average
R 0 of region (1) events. The contaminations of background events are estimated by
Nbg = cos θx (exp(p1 + p2(cos θ))). The numbers of fraction of background events in the range of−0.1 < cos Θ < 0 are summarized in Table 5.6. The uncertainties of these estimated numbers ofbackground events are evaluated by the maximum change on the number of background events
when fit parameters (p0, p1 and p2) shift to increase or decrease the background events withinthat parameter’s allowed regions. Table 5.6 also summarizes the systematic uncertainties foreach sample.

5.3.6

Upward through-going showering muon

Among the upward through-going muon events, some of them are accompanied with anelectromagnetic shower. The showering muons are observed as the high energy muons whichlose energy through radiative processes such as bremsstrahlung, e + e− pair production andphoto-nuclear interaction. Thus, the energy of the parent neutrino is approximately 1 TeV.
Therefore, the upward through-going muons is further separated into showering muon sample
and non-showering sample. Figure 5.30 shows the energy spectrum of the primary neutrino for
those UPMU samples.
The showering muons selection uses χ 2 test based on the observed charge and the expectedcharge of the non-showering muons [127]. The difference between the corrected observed charge
and the expected charge of non-showering muon is set as ∆(Q). The selection of the showeringmuon criteria are follows :96

Through−going Muon Zenith Angle vs. Azimuth

cos θ

cos θ

Stopping Muon Zenith Angle vs. Azimuth

0

0

−0.2

−0.2

−0.4

−0.4

−0.6

−0.6

−0.8

−0.8

−1
0

50

100

150

200

250

−1
0

300
350
φ (degrees)

50

100

150

200

250

300
350
φ (degrees)

Figure 5.27: The scatter plots of the zenith versus azimuth directions for stopping muon (left)
and through-going muons (right) samples for SK-III.

Azimuthal Angle Distribution of Through−going Muons

Number of events

Number of events

Azimuthal Angle Distribution of Stopping Muons

10

All stopping muons (cos θ < 0.1)
Upward stopping muons (cos θ < 0)

102

(1)

(2)

(1)

3

102

10

All thru−going muons (cosθ < 0.1)
Upward thru−going muons (cosθ < 0)

(1)

(2)

(1)

1
0

50

100

150

200

250

10
0

300
350
φ (degrees)

50

100

150

200

250

300
350
φ (degrees)

Figure 5.28: The azimuth angle distributions for stopping muon (left) and through-going muons(right) samples for SK-III. Region (1) is the thicker-mountain direction. Region (2) is thethinner-mountain direction. White histograms show the events of downward-going muons, while
the hatched histogram is upward-going muons.

97

10

Near−Horizon Distribution, Thru−going non−showering events

Events/day/sr

Events/day/sr

Near−Horizon Distribution, Stopping events
χ2 / ndf
Prob
p0
p1
p2

17.48 / 15
0.29070.1184 ± 0.0000−1.331± 0.132
42.16 ± 2.08

210

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

χ2 / ndf
Prob
p0
p1
p2

33.31 / 16
0.006727
0.2367 ± 0.0000−2.203± 0.144
75.26 ± 2.09

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

10
1
1
10−1
10−1
−0.1 −0.08 −0.06 −0.04 −0.02

00.02 0.04 0.06 0.08
cos θ

−0.1 −0.08 −0.06 −0.04 −0.02

00.02 0.04 0.06 0.08
cos θ

Figure 5.29: The zenith angle distributions for stopping muon (left) and through-going muons(right) samples for SK-III in both regions (1) and (2). The solid curves are fitted functions toestimate the background contaminations.

0.7

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

0.6

Stopping

0.5

Figure 5.30: The energy spectrum of
primary neutrino for upward stopping(dotted line), non-showering (dashedline) and showering (solid line) muon
events.

0.4
0.3
0.2
0.1
0
1

10

2

10
10
Eυ (GeV)

3

10

4

10

5

• The difference between the corrected charge and the expected charge for non-showeringmuon (∆(Q) =< Q > −ql ) > 0.5 when χ2 variable for showering selection ≥ 50.
• ∆(Q) > 4.5 - 0.08χ2 when χ2 < 50.
Figure 5.31 shows the scatter plot of ∆(Q) and χ 2 , and ∆(Q) distributions for the data and MC
events which are classified as through-going muon.

98∆ (corrected p.e.) [Sh_delta]∆ (corrected p.e.) [Sh_delta]

14
12
10
8
6
4
2

14
12
10
8
6
4
2

0

0

−2
0

−2
0

20

40

60

80

100 1202 140 160 180 200χ[Sh_chi1p]

20

40

60

Eventsshowering80

100 1202 140 160 180 200χ[Sh_chi1p]
showering

45
40
35
30
25
20
15
10
5
0

−5

0

5
10
15
∆ (corrected p.e.) [Sh_delta]

Figure 5.31: The distributions for the UPMU showering separation for SK-III. The scatter plotof ∆(Q) and χ2 of UPMU through-going events for the data (left) and MC (middle). ∆(Q)
distributions are shown in right panel with the data (point) and MC events (histogram) The
selection criteria are shown by red lines.

99

5.4

Event Reconstruction

Selected neutrino events are reconstructed to get their neutrino flavor type, directions momentum and so on. For FC and PC events, the reconstruction process consists of following
steps :
(1) Vertex Fitting
The vertex position is determined as the point where the timing residual distribution of
hit PMTs has the sharpest peak. The direction and the outer edge of the dominant ringis estimated.
(2) Ring Counting
The other possible rings are searched for using the vertex position and the direction of
the dominant ring. The ring candidates are tested whether they are true rings or not bya likelihood method, then the number of rings is determined.
(3) Particle Identification
Each ring is classified into two types, a showering type (e ± , γ) to be called as e-like andnon-showering type (µ± , π ± ) to be called as µ-like, based on a likelihood method using theCherenkov ring pattern and the Cherenkov opening angle.
(4) Precise Vertex Fitting (MS-fit)(only for single-ring event)More precise vertex position is determined using the Cherenkov ring pattern assuming theobtained particle type.
(5) Momentum Determination
The momentum of each ring is determined by the detected charge inside a Cherenkov cone.
The conversion from the charge to the momentum is based on a Monte Carlo simulationand the detector calibration.
(6) Decay Electron Search
Decay electrons produced by the primary events are searched.
(7) Ring Number Correction(only for FC multi-ring events)
Rings which have fitted by mistake are discarded using precise reconstructed information.
(8) π 0 Fitting(only for FC single-ring e-like events)
Candidates π 0 events are searched for among the events which are recognized as single-ringe-like by the above reconstruction processes.

100

UPMU events are also reconstructed by using some of these processes. Since PC and UPMU
events are basically treated as single-ring and µ-like events, information such as PID and thenumber of rings for these events are not used in the analysis.
In following sections, the reconstruction methods are described briefly. Detailed explanationof the reconstruction can also be found in [128]. The distributions and the numbers of Monte
Carlo events found in following sections are assumed to have undergone 2-flavor oscillation with(sin2 2θ, ∆m2 ) = (1.00, 2.5 × 10−3 eV2 ).

5.4.1

Vertex fitting

The vertex position is reconstructed using the timing information of hit PMTs in three steps.
In the first step, a simple fit, called Point-fit is applied. A vertex is roughly estimated under
an assumption that all photons are emitted at the same time from a point source. Subtracting
the time of flight from a tested vertex a distribution of residual PMT hit times is constructed.
The point which gives maximal goodness-of-fit, defined as a Gaussian fit to this distribution, istaken as the vertex of the event.
In the second step, the direction and the outer edge of the dominant ring is measured. A
pair of direction and opening angle is tested by a parameter defined as:

Q(θedge ) =Z θedgePE(θ)dθ0sin θedge×dPE(θ)dθ
θ=θedge!2(θedge − θexp )2
× exp −
2σθ2




(5.4)

where θexp and σθ are the Cherenkov opening angle expected from the charge within the
Cherenkov corn and its resolution, respectively. PE(θ) is the angular distribution of the observed charge as a function of the opening angle θ. Varying the direction around that given by
Point-fit, the direction and the ring edge which maximize Q(θ edge ) are estimated.
In the final step, more precise vertex position is determined, which is called TDC-fit. Unlike
Point-fit, photons are assumed to be generated along the particle’s track. Then the time residualfor PMTs inside of the Cherenkov ring is calculated considering this effect. For the photons
outside the ring, the time residual is still computed assuming all photons originated at thevertex and the goodness-of-fit is defined including the effect of scattering photons. Then the
precise vertex is determined as the position to maximize the goodness parameter.
The vertex position thus determined is used for multi-Cherenkov-ring events. Figure 5.37
and Table 5.9 show the estimated vertex position resolutions for multi-ring events. For single
ring events, a more precise fitter is used, see Section 5.4.4.

5.4.2

Ring counting

After an initial Cherenkov ring is detected and the vertex is determined, other ring candidates
are searched for by a Hough transform method [132]. Figure 5.32 illustrates the method tofind other possible rings, in witch a spherical coordinate centered on the vertex position isconsidered. The shaded circle in this figure represents the Cherenkov ring image. The dashedline circles centered on the hit PMTs are drawn with 42 ◦ half angle. The direction of theCherenkov ring is identified as a intersection point of those dashed line circles. The second ringcandidate is searched for choosing possible ring directions based on this method, and likelihood
technique is used to determine if the candidate ring is consistent with a ring. If the second ring101

42 deg. ringhit PMT(possible center)

centerCherenkov ring

(most probable)

Figure 5.32: A basic idea of finding ring candidates is shown. By drawing rings around the hit
PMT with Cherenkov opening angle of 42 ◦ from the vertex, the center of the actual Cherenkov
ring can be identified.

CCQE νe
CCQE νµSK-I
sub-GeV multi-GeV

SK-II
sub-GeV multi-GeV

SK-III
sub-GeV multi-GeV

97.1 %
97.2 %

95.7 %
97.2 %

97.3 %
97.2 %

94.5 %
95.9 %

93.6 %
96.8 %

95.6 %
95.8 %

Table 5.7: The efficiencies to select CCQE events as single-ring for each SK period. Sub-GeV
and Multi-GeV represent with Evis < 1.33 GeV and Evis > 1.33 GeV, respectively.

is determined to be existing, the same procedure is repeated to find other possible rings up to 5
rings. Figure 5.33 show the likelihood distributions between single-ring and multi-ring for both
the data and Monte Carlo. The cut threshold is 0 to separated single-ring events from multi-ring
events. The efficiencies to select CCQE events as single-ring are summarized in Table 5.7.
The distributions have slight difference between the data and MC. This difference is taken
as a source of systematic uncertainty in the event selection.

5.4.3

Particle Identification

The detected Cherenkov rings are identified as e-like or µ-like by Particle Identification (PID).
Figure 5.34 shows the event display of electron and muon neutrino MC events. An electron (anda gamma-ray) produces diffused ring pattern since electrons produce the electromagnetic showerand low energy electrons suffer multiple scattering whereas muon (and charged pion) produces
shaper ring edges. In addition, the Cherenkov rings from muons or charged pions can havesmaller Cherenkov angles if they are not highly relativistic (β = v/c < 1). These differences aretaken into account to classify the event into 2 types by the PID algorithm.
Figures 5.35 and 5.36 show the PID likelihood distribution for FC single-ring sample and102Super Kamiokande I 1489.2 days
450

single ring

Super Kamiokande I 1489.2 days
350

multi ring

400

CCQE

CC DIS

300

NC

multi ring
CCQE

300

CC Single π

350

single ring

CC Single π
CC DIS

CC Coherent π

CC Coherent π

250

NC

250

200

200

150

150

100

100
50

50
0
−15

−10

−5

0

5

10

0
−15

15

Ring counting likelihood sub−GeV(FC), stack

Super Kamiokande II 798.6 days
single ring

multi ring

220

CCQE

0

5

single ring

10

15

multi ring
CCQE

200

CC Single π
CC DIS

200−5

Ring counting likelihood multi−GeV(FC), stack

Super Kamiokande II 798.6 days
240

250

−10

CC Coherent π
NC

CC Single π

180

CC DIS

160

NC

CC Coherent π

140

150

120
100
100

80
60

50

40
20

0
−15

−10

−5

0

5

10

0
−15

15

Ring counting likelihood sub−GeV(FC), stack

Super Kamiokande III 518.1 days
220

single ring

−10

−5

0

5

10

15

Ring counting likelihood multi−GeV(FC), stack

Super Kamiokande III 518.1 days
single ring

multi ring

multi ring

120

200
CCQE

CCQE

CC Single π

180

CC Single π

100

CC DIS

160

CC DIS

CC Coherent π

CC Coherent π

NC

140

NC

80

120
60

100
80

40

60
40

20

20
0
−15

−10

−5

0

5

10

15

Ring counting likelihood sub−GeV(FC), stack

0
−15

−10

−5

0

5

10

15

Ring counting likelihood multi−GeV(FC), stack

Figure 5.33: The ring counting likelihood distribution for SK-I (upper), SK-II (middle) and
SK-III (lower). The left (right) side plots are in case of FC Sub-GeV (Multi-GeV) sample. Dots(solid lines) are the data (the MC events). The MC histograms assume the 2-flavor oscillation(sin2 2θ, ∆m2 ) = (1.00, 2.5×10−3 eV2 ). Hatched green histograms show the CCQE interactions.

103

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

NUM
195
RUN
999999SUBRUN
99
EVENT
1550
DATE **-Feb-22
TIME
19: 4:32
TOT PE:
19375.5
MAX PE:
52.5
NMHIT : 5622
ANT-PE:
59.0
ANT-MX:
5.1
NMHITA:
68

NUM
3
RUN
999999SUBRUN
99
EVENT
16
DATE **-Feb-22
TIME
19: 3:47
TOT PE:
13433.9
MAX PE:
52.8
NMHIT : 3839
ANT-PE:
65.3
ANT-MX:
8.5
NMHITA:66

90/00/00:NoYet:NoYet
RunMODE:MonteCarloTRG ID :00000011 90/00/00:NoYet:NoYet
90/00/00:NoYet:NoYet
T diff.:0.
us
90/00/00:NoYet:NoYetFEVSK :80000003 90/00/00:NoYet:NoYetnOD YK/LW: 2/ 1 90/00/00:NoYet:NoYet
90/00/00:;R=
0:NoYetSUB EV : 0/ 0
R :Z : PHI : GOOD
Dec-e: 0( 0/ 0/ 0)0.00:
0.00: 0.00:0.000*NO KEKGPS RED* CANG : RTOT : AMOM : MS

RunMODE:MonteCarloTRG ID :00000011
T diff.:0.
us
FEVSK :80000003
nOD YK/LW: 3/ 1
SUB EV : 0/ 1
Dec-e: 0( 0/ 0/ 0)*NO KEKGPS RED*

Comnt;

Comnt;

Figure 5.34: Event display of single-ring electron (left) and single-ring muon (right) neutrino
MC event.

νe mis-identified as νµνµ mis-identified as νe

SK-I

SK-II

SK-III

1.0 %
0.7 %

1.2 %
0.8 %

1.5 %
0.5 %

Table 5.8: The misidentification probabilities of PID for Sub-GeV 1-ring CCQE events for each
SK run period.

multi-ring sample, respectively. The separation of particle type for multi-ring events is not as
good as that for a single-ring event due to overlapping of Cherenkov photons from multiple
Cherenkov rings. The misidentification probabilities for single-ring events are summarized inTable 5.8.
The distributions have slight difference between the data and MC. This difference is taken
as a source of systematic uncertainty in event selection.

5.4.4

Precise Vertex Fitting

The FC/PC single-ring events are re-fitted to optimize their vertex position and direction
using expected light pattern such as e-like or µ-like. This fitter is called MS-fit. UPMU events
are also fitted by MS-fit with the assumption that the particle is a muon and the vertex is at
the inner detector surface.
The performance of fitting, MS-fit for single-ring events and TDC-fit for multi-ring events,
is represented by the resolution which is estimated using atmospheric neutrino Monte Carlosample. Figure 5.37 shows the distance between the true vertex and the reconstructed vertex
for various single-ring samples. Figures 5.38 and 5.39 show the angular difference between thetrue direction and the reconstructed direction. The estimated resolutions including those formulti-ring events are summarized in Table 5.9.
104

Super Kamiokande I 1489.2 days

Super Kamiokande I 1489.2 days160

e−like 3426

300

muon−like 3221

e−like 829

muon−like 694

140
250

120

200

100

CCQE muon

CCQE muon

80

150

60
100
40
50

20

0
−10 −8

−6

−4

−2

0

2

4

6

8

0
−30

10

PID likelihood sub−GeV 1ring (FC)Super Kamiokande II 798.6 days
220

e−like 1841

−20

−10

0

10

20

30

PID likelihood multi−GeV 1ring (FC)Super Kamiokande II 798.6 daysmuon−like 1686

120

200

e−like 392

muon−like 394

180
100

160
140

CCQE muon

CCQE muon

80

120
60

100
80

40

60
40

20

20
0
−10 −8

−6

−4

−2

0

2

4

6

8

10

PID likelihood sub−GeV 1ring (FC)0−30Super Kamiokande III 518.1 days120

e−like 1250

−10

0

10

20

30

PID likelihood multi−GeV 1ring (FC)Super Kamiokande III 518.1 days
60

muon−like 1166

e−like 282

muon−like 231

50

100
80

40

CCQE muon

60

30

40

20

20

10

0
−10 −8

−20

−6

−4

−2

0

2

4

6

8

10

PID likelihood sub−GeV 1ring (FC)0−30

CCQE muon

−20

−10

0

10

20

30

PID likelihood multi−GeV 1ring (FC)

Figure 5.35: The PID likelihood distribution for FC 1-ring sample for SK-I (upper), SK-II(middle) and SK-III (lower). The left (right) side plots are in case of FC Sub-GeV (Multi-GeV)
1-ring sample. Dots (solid lines) are the data (the MC events). The MC histograms assume
the 2-flavor oscillation (sin2 2θ, ∆m2 ) = (1.00, 2.5 × 10−3 eV2 ). Hatched histograms show the
CCQE νµ events.
105Super Kamiokande I 1489.2 days120

e−likeSuper Kamiokande I 1489.2 days
120

muon−like100

e−likemuon−like

100

80

80
nu_mu CC

nu_mu CC

60

60

40

40

20

20

0
−300

−200

−100

0

100

200

0
−600

300

PID likelihood for Pmax ring in sub−GeV multi−ring (FC)Super Kamiokande II 798.6 days
e−like

−400

−200

0

200

400

600

PID likelihood for Pmax ring in multi−GeV multi−ring (FC)Super Kamiokande II 798.6 daysmuon−likee−like160120muon−like

140
100
120
nu_mu CC80

nu_mu CC

100
80

60

60
40
40
20

20

0
−300

−200

−100

0

100

200

300

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

0
−600

−400

−200

0

200

400

600

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

Super Kamiokande III 518.1 days
50

e−like50muon−likee−likemuon−like40
40
nu_mu CC

nu_mu CC

30

30
20

20

10

10

0
−300

−200

−100

0

100

200

300

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

0
−600

−400

−200

0

200

400

600

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

Figure 5.36: The PID likelihood distribution for FC multi-ring sample for SK-I (upper), SK-II(middle) and SK-III (lower). The left (right) side plots are in case of FC Sub-GeV (Multi-GeV)
multi-ring sample. Dots (solid lines) are the data (the MC events). The MC histograms assume
the 2-flavor oscillation (sin2 2θ, ∆m2 ) = (1.00, 2.5 × 10−3 eV2 ). Hatched histograms show theCC νµ events.
106

1400

180
160

1200

FC sub−GeV 1ring e−like

FC multi−GeV 1ring e−like
140

1000
120
800

100

sigma= 27.1 cm

sigma= 53.8 cm

80

600

60

400

40
200
0
0

20
20

40

60

0
0

80 100 120 140 160 180 200∆ vertex (cm)

2200

20

40

60

80 100 120 140 160 180 200∆ vertex (cm)

600

2000
1800

500

FC sub−GeV 1ring mu−like

FC multi−GeV 1ring mu−like

1600
400

1400
1200

300

sigma= 30.6 cm

1000
800

sigma= 23.3 cm

200

600
400

100

200
0
0

20

40

60

0
0

80 100 120 140 160 180 200∆ vertex (cm)

220

20

40

60

80 100 120 140 160 180 200∆ vertex (cm)

900

200

800

180
160

700

FC multi−GeV multi−ring mu−like

PC

600

140

500

120
sigma= 77.8 cm

100

sigma= 54.6 cm

400

80

300

60
200

40

100

20
0
0

50

100

150

200

250

0
0

300 350 400∆ vertex (cm)

50

100

150

200

250

300 350 400∆ vertex (cm)Figure 5.37: Distance between the true vertex and the reconstructed of FC and PC Monte Carlosample for SK-III. Hatched region is where 68% of all events are indecated.

107

1200

4000
3500

1000

FC sub−GeV 1ring e−like

FC multi−GeV 1ring e−like

3000
800
2500
sigma = 3.1 deg

2000

sigma = 1.5 deg

600

1500

400

1000
200
500
0
0

1

2

3

4

5

0
0

6
7
8
9 10
∆ direction[degree]

1

2

3

4

5

6
7
8
9 10
∆ direction[degree]

4000
9000
3500

8000

FC sub−GeV 1ring mu−like

7000

3000

6000

2500

5000

sigma = 1.9 deg

4000

FC multi−GeV 1ring mu−likesigma = 0.8 deg

2000
1500

3000
1000

2000

500

1000
0
0

1

2

3

4

5

0
0

6
7
8
9 10
∆ direction[degree]

1

2

3

4

5

6
7
8
9 10
∆ direction[degree]

Figure 5.38: Angular difference between the true direction and the reconstructed direction of
FC Monte Carlo sample for SK-III. Hatched region is where 68% of all events are indecated.

108

800

5000

700

UPmu stopping

UPmu non−showering

4000

600
500

3000

sigma = 2.5 deg

400

sigma = 1.2 deg

2000

300
200

1000
100
0
0

1

2

3

4

5

0
0

6
7
8
9 10
∆ direction[degree]

1

2

3

4

5

6
7
8
9 10
∆ direction[degree]

1400
1200

UPmu showering

1000
800

sigma = 1.2 deg
600
400
200
0
0

1

2

3

4

5

6
7
8
9 10
∆ direction[degree]

Figure 5.39: Angular resolution of UPMU Monte Carlo sample for SK-III. Hatched region iswhere 68% of all events are indecated.

109

Vertex resolution (cm)
SK-I

SK-II

SK-III

27
26

31
31

27
31

FC Multi-GeV
single-ringe-likeµ-likemulti-ringµ-like

49
24

47
27

54
23

67

75

78

PC

56

64

55

SK-I

SK-II

SK-IIIFC Sub-GeV
single-ringe-likeµ-like

3.1
1.9

3.2
2.2

3.1
1.9

FC Multi-GeV
single-ringe-likeµ-like

1.5
0.8

1.5
0.9

1.5
0.8

PC

1.0

1.2

0.9

UPMU stop
UPMU through non-showering
UPMU through showering

2.5
1.2
1.3

2.5
1.6
2.3

2.5
1.2
1.2

FC Sub-GeV
single-ringe-likeµ-likeAngular resolution (degree)

Table 5.9: Vertex and angular resolutions for each event category and each SK period aresummarized.

1105.4.5

Momentum reconstruction

The momentum reconstruction for each particle is estimated by the observed p.e.s inside a
Cherenkov cone with the half opening angle of 70 o . In order to determine the momentum forindividual rings, the observed p.e.s in hit PMTs are separated to the contribution from eachring. The separation of the observed p.e.s is carried out based on the expected p.e. distribution
from each ring as a function of the opening angle θ, assuming a uniformity in azimuthal angle
φ. The observed p.e.s in the i-th PMT from the n-th ring are estimated as :
exp
qi,n
obs
qi,n
= qiobs × P exp
n0 qi,n0(5.5)

obs is the fractional p.e.s from the n-th ring in the i-th PMT, q obs is the observed p.e.s
where qi,ni
exp
in the i-th PMT and qi,nis the expected p.e.s .
To calculate the total number of p.e.s inside the 70 ◦ cone, the number of p.e.s in each PMT
is corrected considering the light attenuation in the water and the acceptance of the PMT as
follows : r  cos Θ 
X
X

GMC 
i
i
obs
 α×
×
−
q
×
exp
(5.6)
Si 
RT OTn =
i,n


Gdata
L
f (Θi )◦◦
θi,n <70−50nsec<ti <250nsec

θi,n <70

whereα: normalization factor

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

θi,n: opening angle between the n-th ring direction and the i-th PMT
directionti: photon time-of-flight subtracted hit timing of the i-th PMT positionL: light attenuation length in waterri: distance from the vertex position to the i-th PMT

f (Θi )

: correction function for the PMT acceptance as a function of the photon incidence angle ΘiSi: expected p.e.s for the i-th PMT from scattered photons

The summation is restricted inside the time window from -50 nsec to +250 nsec around the peakof the photon time-of-flight subtracted hit timing distribution to reject the effect from muon
decay electrons. The attenuation length in water L is measured continuously using cosmic ray
through-going muons (Section 4.1.5). Figure 5.40 shows the momentum resolution for electrons
and muons estimated from the Monte Carlo simulation as a function of the true momentum foreach SK geometry. The resolution is defined as 1σ width of the Gaussian fit.
The absolute energy scale is calibrated using the four independent calibration sources as
described in Section 4.2.
111

Momentum resolution (%)(i) electron
20
SK-I
SK-II
SK-III

15
10
5
0

0

250 500 750 1000 1250 1500 1750 2000

Momentum resolution (%)

Momentum (MeV/c)(ii) muon
6

SK-I
SK-II
SK-III4200

250 500 750 1000 1250 1500 1750 2000
Momentum (MeV/c)

Figure 5.40: Momentum resolutions for electron (top) and muon (bottom) estimated by theMonte Carlo simulation as a function of the true momentum.

112

5.4.6

Decay Electron SearchElectrons may be produced from the decay of muons which stopped in the detector. Also
π + produced by a neutrino interaction may decay into µ + , then to e+ . Therefore, for example,
if a muon decay is observed for single-ring e-like event, the event is likely to be a CC-nonQE ν eevent that produced π + along with an electron. Thus the information from the decay-electrons
is useful to estimate the type of neutrino interactions. Decay electrons are observed in a later
independent time window (900 nsec) from that of the primary event or in a same time windowof the primary event. If the decay occurs around 900 nsec, the electron event may be observed
separately in the primary trigger and subsequent event trigger.
The detection efficiency of decay electron is 80 % for µ + and 63 % for µ− , since approximately
20 % of µ− is captured on 16 O nuclei [129].

5.4.7

π 0 reconstruction

π 0 reconstruction is applied to only single-ring e-like events. In the FC sub-GeV single-ringe-like event sample, the contamination of NC events is estimated to be about 9 %. These NC
are mostly originated from NC single π 0 production. π 0 immediately decays two gamma-rays
and produces two e-like Cherenkov rings. If one of these two rings was not detected due totoo low energy or overlapping rings, the event can be identified to single-ring e-like. To findthese π 0 events, the π 0 fitter reconstructs the second gamma-ray on the assumption of theexistence of two Cherenkov rings by comparing the observed photoelectron distribution with the
expected photoelectron distribution of two gamma-rays. Then the likelihood technique is used
to determine the best-fit configuration of two gamma-rays. The results from π 0 reconstruction
are used in the analysis sample as described in Section 6.2.1.

113

Chapter 6

Neutrino Oscillation Analysis
6.1

Overview

Neutrino oscillation is examined using the information of detected neutrino momenta anddirections. The observed data are compared against the MC expectation using a χ 2 method. In
this chapter, data samples for this analysis and the expected effects from the full oscillation onthat samples are explained. The analysis method together with detail systematics and results
are described.

6.2

Data Set

For this analysis, all data samples, FC, PC and UPMU, are used. In order to separate
CCQE, CC non-QE and NC events from the 1-ring sample, FC sub-GeV 1-ring samples arefurther separated to sub-samples by using information such as the number of decay electrons.
Additionally, in order to use the high energy electron sample in the analysis, Multi-GeV multiring e-like sample which contains inelastic interaction events mostly is selected by the likelihood
method. These additional selection methods are described in the following sections.

6.2.1

Selection for Sub-GeV sample

The FC sub-GeV single-ring e-like sample contains background events which are mainly
NC π 0 events producing two gamma-rays. If one of the two γ rays is missed by the event
reconstruction, the NC π 0 event is identified as single-ring e-like since the electromagnetic showerfrom the γ gives a light pattern similar to that of an electron. To reduce this type of backgrounda specialized π 0 fitter is used [136]. This fitter enforces a second ring on the 1-ring data event andthen predicts a light pattern that would result from γ rays propagating through the tank withthe vertex of the fitted ring and with the best fit direction. The left five panels in Figure 6.1 show
the invariant mass distributions from this π 0 fitter for CCQE and NC events in the FC sub-GeV
single-ring e-like Monte Carlo in five energy regions. NC events tend to form a peak close to
the π 0 mass while CCQE events do not. For events with electron momentum below 250 MeV/c,
a cut at 100 MeV/c2 is used to create a π 0 -like sample. However, this cut is not sufficient forevents in higher momenta. Thus a likelihood selection is additionally used incorporating threevariables: the fraction of the event’s reconstructed momentum carried by the second ring, theπ 0 invariant mass distribution, and the difference of two likelihood variables which result from114

a π 0 -fit and electron-fit. The distribution of these variables is also shown in Figure 6.1 for eachenergy region. The π 0 -like selection likelihood functions are defined as:
Lπ 0 =3X
i=1

log(ΓSi (xi )) − log(ΓBi (xi )),

(6.1)

where ΓSi (xi ) (ΓBi (xi )) represents the CCQE (NC) events’ probability distribution function(PDF) for the ith variable with observable xi . After separating the π 0 -like sample, the remaining e-like events are divided into two categories, 0-decay which has no decay electrons and1-decay which has one or more decay electrons. Since ν e CCQE events are not expected to
produce decay electrons, there is a high fraction of CCQE interactions in the 0-decay sample.
For the FC sub-GeV single-ring µ-like sample, there are three categories using the numberof decay electrons: 0-decay, 1-decay, and 2-decay (2 and more), corresponding to the numberof decay electrons reconstructed in the event. Since these CCQE events produce a muon they
are expected to have one decay electron. The fraction of CCQE events is increased in 1-decayµ-like samples.
Details of the event composition by interaction mode after these event selections are shownin Table 6.1.

6.2.2

Selection for Multi-GeV multi-ring e-like sample

In order to make a νe -enriched Multi-GeV multi-ring e-like sample, an additional selection
is applied based on the likelihood method. Four variables are used in the selection for eachfive energy region, PID likelihood, momentum fraction of the most energetic ring, number ofdecay electrons, and distance between the neutrino vertex and any muon decay electrons. The
distribution of these variables is shown in Figure 6.2 for each energy region. The likelihood
functions for the Multi-GeV Multi-ring e-like selection are defined as,
LM M E =4X
i=1

log(ΓSi (xi )) − log(ΓBi (xi )),

(6.2)

th
where ΓSi (xi ) (ΓBi (xi )) represents the CCνe (NC+CCνµ ; background) events’ PDF for the ivariable with observable xi . Details of the event composition by interaction mode after thisselection are shown in Table 6.2. The likelihood selection increases CC events and decreases
background events significantly.

6.2.3

Data set for the analysis

In summary, data sets for the analysis are specified as follows :• FC samples(1) FC sub-GeV single-ring e-like samples∗ single-ring which is identified as e-like∗ pe > 100 MeV/c∗ Evis < 1.33 GeV
115

(a)
0.2

Variable distributions for π -likelihood
0π0-like

π0 mass

energy fraction

0

(b)

0.2

∆-likelihood

0.2

0.2

0.1

0.1

0
0.2

0
0.2

0.1

0.1

0
0.2

0
0.2

0.1

0.1

0
0.2

0
0.2

0.1

0.1

0.2

0.1
0

0

(c)

0.2

0.2

0.1
0

0

(d)

0.2

0.2

0.1
0
0.2

0

(e)
0.2
0.1
0

0
0

200

0

0
100

0.5

150

200

0
-200

.
100

400

π invariant mass (MeV/c )02

Figure 6.1: The distributions used in the π 0 selection for five momentum regions: (a)P e <250 MeV/c, (b)250 MeV/c ≤ Pe < 400 MeV/c, (c)400 MeV/c ≤ Pe < 630 MeV/c, (d)630
MeV/c ≤ Pe < 1000 MeV/c and (e)1000 MeV/c ≤ Pe . Solid (dashed) lines represent
CCQE (NC) events in the FC sub-GeV single-ring e-like Monte Carlo in SK-I. Events with
π 0 mass above 100 MeV/c2 are selected as π 0 -like. To separate π 0 -like and electron-likemore efficiently, an additional likelihood selection is applied for events with momentum
above 250 MeV/c. The distributions of the three likelihood variables are shown: thefraction of energy carried by the second fitted ring (E 2 /(E1 + E2 )), the π 0 mass and∆-likelihood (described in the text). All distributions have been normalized to unit area.

116

FC sub-GeV single-ring e-like
0-decay 1-decay
π 0 -likeMC events

FC sub-GeV
single-ring e-like

2663.2

210.9

191.8

2996.477.7 %
12.4 %
1.0 %
1.3 %

3.8 %
50.3 %
9.7 %8.5 %10.6 %
7.0 %
1.8 %
0.5 %

70.6 %
15.2 %
1.7 %1.7 %CC νµ + ν̄µ0.6 %15.2 %

7.0 %

2.0 %

NC

6.8 %

11.2 %

72.0 %

8.7 %CC
νe + ν̄eQ.E.
single meson
multi πcoherent π

FC sub-GeV single-ring µ-like
0-decay 1-decay
2-decay
MC events

FC sub-GeV
single-ring µ-like

1412.42745.4164.3

4297.871.3 %
12.9 %
1.1 %
0.8 %78.5 %
15.5 %
1.5 %
1.5 %

5.8 %
65.7 %
14.9 %
8.6 %

74.7 %
16.7 %
1.9 %1.6 %CC νe + ν̄e

1.8 %<0.1 %<0.1 %

0.7 %NC

11.8 %

2.6 %

3.3 %

4.3 %

CC
νµ + ν̄µQ.E.
single meson
multi πcoherent π

Table 6.1: The number of FC Sub-GeV events and their fractional composition by neutrino
interaction mode for the SK-I MC scaled to 1489 days without oscillations. The upper(lower) table shows the e-like (µ-like) sample. The most right side of the table shows theresult before separation into sub-samples. After separation the CCQE purity is increased
and the NC backgrounds are reduced in the 0-decay e-like and 1-decay µ-like sub-samples.

CC νe + ν̄e
No LL

Percentage (%)

472.1
53.0

331.0
74.5

CC νµ + ν̄µ
No L
L

201.7
22.6

39.2
8.8

NC
No L
L

218.1
24.5

74.4
16.7

Total
No L
L

891.9
100.0

444.6
100.0

Table 6.2: The expected number of events for each interaction component of the MultiGeV multi-ring e-like sample before (No L) and after (L) likelihood selection for the SK-I
MC scaled to 1489 days. Two-flavor neutrino oscillations ν µ ↔ ντ are assumed withfull-mixing (∆m2 = 2.1 × 10−3 eV2 , sin2 2θ = 1.0).

117

> 20 GeV

102

102

10

10

10

1

1

1
-200

10 - 20 GeV

102
10

-100

0

0

5

10

1
0

0.5

1

102

102

1

1

1
-100

0

0

0.5

1

0

0.5

1

0

0.5

1

0

0.5

1

10

0

5

10

1
0

0.5

1

3

10

5 - 10 GeV

1

102

10

10

3

10

102

102
102

102

10

10

10
1
0

10

1

1
-50

0

5

10

1
0

0.5

1

3

2.5 - 5 GeV

0.5

10

-200

3

10

102

10
102

102

102

10

10

10
1
-40

1.33 - 2.5 GeV

0

-20

0

102

10

1

1
0

5

10

1
0

0.5

1

3

3

10

10

102

102

10

102
10

10
1
-20

10
1

1
-10

PID of MER

0

0

5

10

number of decay electrons

1
0

0.5

momentum fraction of MER

1

distance from vertex to decay electron

Figure 6.2: Variables used in the likelihood definition to create the SK-I multi-GeV multiring e-like sample. The energy bins correspond to the energy of the most energetic ring.
Signal events (CC νe + ν̄e ) are shown the solid lines and background events (CC ν µ + ν̄µand NC) are shown the dashed lines.

118

(1-a) 0-decay electron sample· without any decay electrons(2-b) 1-decay electron sample· with one decay electrons(3-c) π 0 -like sample· fitted as π 0 -like by π 0 -fitter and π 0 -likelihood(2) FC sub-GeV single-ring µ-like samples∗ single-ring which is identified as µ-like∗ pµ > 200 MeV/c∗ Evis < 1.33 GeV(2-a) 0-decay electron sample· without any decay electrons(2-b) 1-decay electron sample· with one decay electrons(2-c) 2-decay electron sample· with two or more decay electrons(3) FC Multi-GeV single-ring e-like sample∗ single-ring which is identified as e-like∗ Evis > 1.33 GeV(4) FC Multi-GeV single-ring µ-like sample∗ single-ring which is identified as µ-like∗ Evis > 1.33 GeV(5) FC Multi-GeV multi-ring e-like sample(described in Section 6.2.2)∗ The most energetic ring is identified as e-like and E vis > 1.33 GeV∗ Events that satisfy the Multi-GeV, multi-ring e-like likelihood cut(6) FC Sub- and Multi-GeV multi-ring µ-like sample∗ The most energetic ring is identified as µ-like and the momentum p µ > 600 MeV/cand Evis > 600 MeV(7) FC 2-ring π 0 -like sample(described in Section 4.2.3)∗ Two rings are recognized and both of them are identified as e-like∗ without any decay electrons• PC samples(8) PC OD stopping sample(described in Section 5.2.7)

119∗ The maximum number of p.e.s observed in the OD in a sliding 500 nsec time
window from −400 nsec to +600 nsec (PE anti ) is less than PEexp /1.5 (see Section 5.2)
∗ The most energetic ring or the second one should be identified as µ-like(9) PC OD through-going sample∗ (rest of the events after stopping selection)• UPMU samples(10) Upward stopping muon sample∗ Classified as stopping event by the fitter∗ The number of hit OD PMTs within 8 m from the exit point (NHITEX) < 10
for SK-I and -III (16 for SK-II)∗ The fitted momentum ≥ 1.6 GeV/c(11) Upward through-going muon sample∗ Classified as through-going event by the fitter∗ The number of hit OD PMTs within 8 m from the exit point (NHITEX) ≥ 10
for SK-I and -III (16 for SK-II)∗ The distance from the entrance point to the exit point ≥ 7 m(11-a) showering muon sample(described in Section 5.3.6)· The difference between the average corrected charge and the expected charge
for non-showering muon (∆(Q) =< Q > −q l ) > 0.5 when χ2 valuable forshowering selection ≥ 50.
· ∆(Q) > 4.5 - 0.08χ2 when χ2 < 50.
(11-b) non-showering muon sample· (rest of the events after showering selection)
In this analysis, FC, PC and UPMU data during SK-I (1489.2 days for FC and PC, 1645.9
days for UPMU), SK-II (798.6 days for FC and PC, 827.7 days for UPMU) and SK-III (518.1
days for FC and PC, 635.6 days for UPMU) period are used. The live time is determined by
the requirement of the good detector condition. Since the reconstruction of long path length
muon for UPMU events is less affected by the detector condition, the live time is longer thanthat of FC and PC. The amount of simulated MC atmospheric neutrino events are equivalent
to an exposure of 500 years for each SK run period.
Table 6.3 summarizes the number of events in each categorized sample for the data and MC
events.

6.3

Oscillation effect

As discussed in Section 1.3, the electron neutrino oscillation is observable by the subdominant oscillation effects. Figure 6.3 shows the expected flux ratio at the detector site for eachelectron and muon neutrino as a functions of the zenith angle and momentum. The expected
120

Data

SK-II
MC (osc.)

SK-IIIData
MC (osc.)2655.9 (2652.4)
204.4 (194.3)
159.1 (155.2)

1605
155
81

1405.8 (1403.4)
113.5 (107.1)
81.3 (79.1)

1098
106
46

935.7 (934.7)
69.6 (66.7)
45.6 (44.4)

1036
2035
150
497

1385.6 (973.0)
2760.6 (1846.8)
163.7 (114.6)
460.0 (456.1)

563
1043
80
267

765.9 (537.2)
1429.4 (957.3)
82.8 (57.7)
237.3 (235.1)

346
759
61
178

497.5 (350.4)
999.8 (668.3)
58.5 (41.0)
157.8 (156.5)

829
694

777.8 (777.7)
1027.4 (744.4)

392
394

409.9 (411.1)
550.2 (399.0)

282
231

279.3 (278.4)
352.8 (255.7)

433
617

457.9 (458.9)
882.4 (660.9)

260
361

252.3 (251.9)
459.6 (344.1)

149
226

159.2 (159.1)
313.8 (234.4)

PC
OD stopping
OD through-going

163
735

222.7 (167.3)
965.4 (755.0)

116
314

105.8 (80.9)
482.5 (374.7)

63280

75.1 (55.7)
334.9 (262.8)

Upward-going muon
stopping
non-showering
showering

435.9
1577.4271.6

701.7 (419.4)
1548.0 (1343.9)
302.7 (292.2)

207.6
725.3
108.1

355.2 (212.5)
767.6 (668.7)
147.8 (143.6)

193.7
612.9
110

273.8 (163.5)
599.4 (520.0)
116.2 (112.3)

FC Sub-GeV
single-ringe-like0-decay
1-decay
π 0 -likeµ-like0-decay
1-decay
2-decay
2-ring π 0 -like
FC Multi-GeV
single-ringe-likeµ-likemulti-ringe-likeµ-likeData

SK-I
MC (osc.)

2984
275
167

Table 6.3: Summary of the number of neutrino events for the data and MC events. The numberof MC events is normalized by the livetime of the data, and the number of oscillated MC eventsis shown in (osc.) assuming 2-flavor full-mixing at (∆m 223 = 2.1 × 10−3 eV2 , sin2 θ23 = 0.5).

121

1

νe flux ratio Φosc / Φ0

0.8

1.4

1

1.3

0.8

0.6

νµ flux ratio Φosc / Φθ13=0

1.4
1.3

0.6
1.2

1.2
0.4

0.2

1.1

0

1

-0.2

0.2

1.1

0

1

-0.2

0.9

cosΘν

cosΘν

0.4

0.9

-0.4

-0.4
0.8

0.8

-0.6

-0.6
0.7

-0.8
-1 -1
10

1

10

10

0.7

-0.8

2 0.6

-1 -1
10

Eν (GeV)

1

10

10

2 0.6

Eν (GeV)Figure 6.3: The flux change of electron neutrino events (left) and muon neutrino events (right) for
oscillations with the parameters (∆m 12 , ∆m23 , sin2 θ12 , sin2 θ23 , sin2 θ13 , δcp ) = (7.7 × 10−5 , 2.1 ×
10−3 , 0.30, 0.60, 0.04, 180◦ ). The plots are shown relative to null oscillation for the electron
neutrino, and no νe oscillation (sin2 θ13 = 0) for the muon neutrino. The intensity scale of theplots is restricted to ± 40 %.
events are calculated by using both disappearance and appearance oscillation probabilities. Figure 6.3 show the expected events assuming full oscillations, (∆m 12 , ∆m23 , sin2 θ12 , sin2 θ23 , sin2 θ13 ,δcp ) = (7.7 × 10−5 , 2.1 × 10−3 , 0.30, 0.50, 0.04, 0◦ ), which are normalized to the non-oscillated
events and the events after 2-flavor full mixing (∆m 23 = 2.1 × 10−3 , sin2 θ23 = 0.50) for the electron neutrino and for the muon neutrino, respectively. For the electron neutrino, the oscillation
effect due to θ13 is dominant from a few GeV to 10 GeV. In Multi-GeV region, resonance enhancement of νµ → νe (ν¯µ → ν¯e ) oscillation due to non-zero θ13 is expected for normal hierarchy(inverted hierarchy) case, while there is no enhancement for inverted hierarchy (normal hierarchy) case. According to the solar neutrino oscillation parameters (LMA-MSW), the electron
neutrino flux change should be seen in Sub-GeV energy region. The muon neutrino flux shows
alternating bands of event excess and deficit. Since the energy reconstruction is not perfect, it
is difficult to see such flux changes. However, the wide regions of excess and deficit are appearedin Multi-GeV region due to the electron neutrino oscillation. Thus, the muon neutrino events
could be somewhat useful to detect the sub-dominant oscillation effects.
The CP phase changes the sub-dominant effects intricately. Figures 6.4 and 6.5 show thezenith angle distributions for sub-GeV 1-ring e-like 0-decay sample (will be discussed in Section 6.2.1) and multi-GeV 1-ring e-like sample. Each plot is described relative to non-oscillation
events. Each colored line indicates the different CP phase value. If θ 13 is zero, there is nodifference in the sub-dominant effect among the CP phase values. When θ 13 is not zero, thesmall difference appears due to the CP phase.
Since the enhancement only occurs for neutrinos or anti-neutrinos depending on the hierarchy, and since the absolute cross section and dσ/dy (where y = (E ν − Elepton )/Eν ) are
different between neutrinos and anti-neutrinos, the excess for single and multi-ring multi-GeV

122

Sub-GeV e-like 0-DK (>400MeV/c)

Sub-GeV e-like 0-DK (>400MeV/c)(sin θ13=0.00)

1

2

1.2

N / N0

1.2

1.1

1.1

1

-1

-0.5

0

0.5

(sin θ13=0.04)

2

N / N0N / N01.2

Sub-GeV e-like 0-DK (>400MeV/c)(sin θ13=0.006)

2

1

1.1

1

-1

-0.5

cosΘ

0

0.5

1

-1

-0.5

cosΘ

0

0.5

1

cosΘ

Figure 6.4: Zenith angle distributions for Sub-GeV 1-ring e-like 0-decay ( > 400M eV /c ) sample
with the data (dots) and the expected MC events (lines) assuming the oscillation parameters(∆m12 , ∆m23 , sin2 θ12 , sin2 θ23 ) = (7.7 × 10−7 , 2.1 × 10−3 , 0.30, 0.55) in normal hierarchy case.
The left, middle and right panels show the case of sin 2 θ13 = 0.0, 0.006 and 0.04, respectively.
Each line indicates the different CP phase value as described in Fig. 6.6. All plots are normalized
by the number of events without the oscillations.

Multi-GeV 1-R e-like

Multi-GeV 1-R e-like(sin θ13=0.00)

1

2

1.2

N / N0

1.2

1.1

1.1

1

-1

-0.5

0

cosΘ

0.5

(sin θ13=0.04)

2

N / N0N / N0

1.2

Multi-GeV 1-R e-like(sin θ13=0.006)

2

1

1.1

1

-1

-0.5

0

cosΘ

0.5

1

-1

-0.5

0

0.5

1

cosΘ

Figure 6.5: Zenith angle distributions for Multi-GeV 1-ring e-like sample with thedata (dots) and the expected MC events (lines) assuming the oscillation parameters(∆m12 , ∆m23 , sin2 θ12 , sin2 θ23 ) = (7.7 × 10−3 , 2.1 × 10−3 , 0.30, 0.55) in normal hierarchy case.
The left, middle and right panels show the case of sin 2 θ13 = 0.0, 0.006 and 0.04, respectively.
Each line indicates the different CP phase value as described in Fig. 6.6. All plots are normalized
by the number of events without the oscillations.

123

e-like events depends on the hierarchy. About multi-ring Multi-GeV e-like sample will be morediscuss in Section 6.2.2. Figure 6.6 shows the same zenith angle distributions for single andmulti-ring Multi-GeV e-like samples in each mass hierarchy case as Fig. 6.5. It is seen that anelectron excess for the inverted hierarchy case is suppressed compared with that for the normal hierarchy case. Due to the difference in the number of excess events between normal andinverted hierarchy, mass hierarchy can be examined.
Note, in this analysis a algebraically computation for the neutrino oscillation probabilities
described in [144] is used. The Earth density is measured by the seismic velocity observation.
The density described in [135] is used in this analysis.

6.4

Analysis Method

To examine the neutrino oscillation parameters, the data are compared against the MC
expectation using a pulled χ2 method based on a Poisson probability distribution:


!
obs
X
X
N
iP
N exp (1 +
χ2 ≡ 2
fji · j ) − Niobs  + Niobs ln
exp
i
i
N
(1
+
i
j fj ·  j )
i
j
!2
X
j
+
,
(6.3)
σjsys
j

where Niobs is the number of observed events in the i-th bin and N iexp is the expected numberof events in the i-th bin from MC simulation. In the fitting process, N iexp is recalculated toaccount for neutrino oscillations and systematic errors, represented by 123 error parameters  j(j : the systematic error index). These  j are varied to minimize χ2 for each choice of oscillation
parameter sets. fji is the fractional change in the predicted MC events in i-th bin due to 1σchange of j-th systematic error. The 1σ value of a systematic error is labeled as σ jsys . Detail
explanation about systematic errors is given in next section.
At the minimum χ2 location for each choice of oscillation parameters, each of the parametersj is estimated according to ∂χ2 /∂j = 0. So the minimization of χ2 in the equation (6.3) isequivalent to solving the following equations for every  j [140]:



!
X
X
X
X
1
1 + (
fji j )2 − ... Niobs − Niexp  fki . (6.4)
δjk +Niobs fji fki j =(σjsys )2
j=1

i=1

j=1

i=1

Since these equations have nonlinear terms of  j , an approximate solution by an iteration method
is obtained.
To ensure a proper function in equation (6.3) the binning is defined so that there are 6
expected MC events for each bin after scaling to the SK-I live time. The binning is definedby the zenith angle and momentum information as shown in Figure 6.7. All sub-GeV samples
are divided into 5 momentum bins. Furthermore, e-like 0-decay sample and µ-like 0- and 1decay samples are divided into 10 zenith angle bins. All multi-GeV samples including PC and
UPMU are divided into 10 zenith angle bins. Their momentum bin divisions are set to eachsample properly as shown in Figure 6.7. The number of bins is 420 for each SK-I, SK-II and
SK-III. Since there is the low statistics data in the later SK run periods, it is expected to have
124

Multi-GeV 1-R e-like

Multi-GeV 1-R e-like(sin θ13=0.04)(sin θ13=0.04)

2

2

1.2

N / N0N / N0

1.2

1.1

1

1.1

1

MC(osc.)δ = 0o

-1

-0.5

0

0.5

1

-1

-0.5

cosΘ

0

1

cosΘ

Multi-GeV M-R e-like

2

1.2

o

δ = 240o(sin θ13=0.04)

2

δ = 80oδ = 160

Multi-GeV M-R e-like(sin θ13=0.04)

1.2

0.5

δ = 320o

N / N0N / N0

data

1.1

1

1.1

1

-1

-0.5

0

0.5

1

-1

cosΘ

-0.5

0

0.5

1

cosΘ

Figure 6.6: Zenith angle distributions for Multi-GeV 1-ring e-like sample (upper) and MultiGeV multi-ring e-like sample (lower) in case of normal (left) and inverted (right) hierarchy cases.
The points show the data and lines show the expected MC events assuming the oscillation
parameters (∆m12 , ∆m23 , sin2 θ12 , sin2 θ23 , sin2 θ13 ) = (7.7 × 10−3 , 2.1 × 10−3 , 0.30, 0.55, 0.04).
Each line indicates the different CP phase value as shown in the right side of the figures. Allplots are normalized by the number of events without the oscillations.

125

FC Multi-GeV FC Multi-GeV FC Mulit-GeV FC Multi-ring
1-ring e-like mulit-ring e-like 1-ring µ-likeµ-like(4 x 10 bins)(4 x 10 bins)(2 x 10 bins)(4 x 10 bins)

PC stop

PC throughUpmu stop(2 x 10 bins)(4 x 10 bins)(3 x 10 bins)

Upmu shower(1 x 10 bins)

log10P (MeV/c)

5.0

Upmu non-shower

4.0
3.7

3.7

3.4

3.4

3.0

3.0

FC Sub-GeV
1-ring e-like
0-decay

1-decay(5 x 10 bins)(5 x 1 bins)(1 x 10 bins)

FC Sub-GeV 1-ring µ-like

FC 1-ring0
π-like(5 x 1 bins)

FC Sub-GeV
02-ring π-like

0-decay

1-decay(5 x 10 bins)(5 x 10 bins) (5 x 1 bins)

2-decay(5 x 1 bins)

3.2

3.2

3.0

3.0

2.8

2.8

2.8

2.6

2.6

2.6

2.4

2.4

2.4

2.2

2.2

3.2

2.0

2.0

Figure 6.7: The definition of the binning for the neutrino oscillation analysis : 310 bins for
the FC samples, 60 for the PC samples, 50 for the UPMU samples. The bins shown in whiteboxes are divided into 10 bins equally spaced between cos Θ = −1 and cos Θ = +1 for FC andPC, cos Θ = −1 and cos Θ = 0 for UPMU. The FC single-ring e-like 1-decay, π 0 -like, the µ-like
2-decay samples and 2-ring π 0 sample are not divided by the zenith angle bins.

instabilities in the χ2 value. To avoid this problem, SK-II and SK-III bins are merged with thoseof SK-I after evaluating the systematic error parameters by Eq. (6.4) :X
obs
Niobs →
Ni,SKnnNiexp (1 +Xjfji · j )

→

X
n

exp
Ni,SKn(1 +X
jfji · j )Since the systematic error coefficients for each SK-I, II and III are computed in advance, thesystematic error effects specific to each SK run period are properly taken into account.
Details of the systematic error terms are described together with the best fit results of theerror parameter j in the next section.

6.5

Systematic Uncertainties

As discussed in the previous section, 123 systematic error parameters are used in this analysis.
Among them, 21 are for the errors related to the neutrino flux model, 18 are for the neutrino
interaction, 28 × 3 are for the detector response for each SK run period. All the systematic
errors are assumed to be uncorrelated.
The best-fit systematic uncertainty parameters  j for the fit in normal hierarchy case aresummarized together with the estimated 1σ uncertainty (σ jsys ) in Tables from 6.4 to 6.10 foreach systematic error term. Their summary can be also found in Appendix A.3.

126

Relative Uncertainty

0.30

δ tot

0.25

δπ
δK
δσ
δ air

0.20
0.15
0.10
0.05
0

0

Absolute normalization (%)

10

10

1

10

2

10
E ν (GeV)

3

35
30
25
20
15
10
5
0
10−1

1

10

102

103
104
Eν (GeV)Figure 6.8: The contribution of each error source for atmospheric neutrino flux [22] is shownin the upper panel. In the lower panel, absolute normalization uncertainty due only to δ π(atmospheric muon data (pion production)) and δ σ (hadronic interaction model) as a functionof neutrino energy is shown.

6.5.1

Systematics in neutrino flux

• Absolute normalization
The uncertainty in the absolute atmospheric neutrino flux is calculated by Honda et al . [22],taking into account the atmospheric muon data (δ π , pion production in the hadronic interaction model), Kaon production (δ K ), hadronic interaction cross section (δ σ ) and atmospheric density profile (δair ) as shown in the upper panel of Figure 6.8. The sum of
dominant uncertainties, δπ and δσ , is considered as the absolute normalization uncertainty
which is shown in the solid line in the lower panel of Figure 6.8. The uncertainty due toKaon production is separately included, as discussed later. This systematic uncertainty is
divided into two terms for two energy region, E ν < 1 GeV and Eν > 1 GeV.
• Flavor ratio
The systematic uncertainty in the flavor ratio (ν µ + ν µ )/(νe + ν e ) of the atmospheric
neutrino flux is estimated by comparing of the Honda flux [22] with FLUKA [49] andBartol [50] flux models. Figure 6.9 shows the flavor ratio normalized by Honda flux model.
The uncertainty is estimated 2 % for E ν < 1 GeV, 3 % for 1 < Eν < 10 GeV, and 5 %for 10 < Eν < 30 GeV. Above 30 GeV, the uncertainty increases along with the function
logEν from 5 % to 30 % up to 1TeV. Due to the difference of error sources in each energy
127

Figure 6.9: The flavor ratio (νµ +ν µ )/(νe + ν e ) of the atmospheric neutrino flux. Vertical axis stands forthe double ratio, FLUKA flux/Honda
flux(solid) and Bartol flux/Honda
flux(dashed). A kink in the FLUKA
flux around 8 GeV due to the technicaltreatment in the original calculation iscorrected in the application.

1.3
1.2
1.1
1
0.9
0.8
0.7

νe/ νe double ratio

0.6 −1
10

1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5 −1101103Eν(GeV)102

10νµ/ νµ double ratio(νµ+νµ)/(νe+νe) double ratio

1.4

1

10

102

103
Eν(GeV)

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

1

10

102

103
Eν(GeV)

Figure 6.10: Anti-neutrino/neutrino double ratio for ν e /νe (left panel) and ν µ /νµ (right panel).
Double ratio is defined as (ν/ν)Fluka or Bartol /(ν/ν)Honda . Solid(dashed) line indicates the double
ratio with the Fluka(Bartol) flux.

region, this uncertainty is divided into three terms, E ν < 1 GeV, 1 < Eν < 10 GeV, 10< Eν .
• Anti-neutrino/neutrino ratio
The systematic uncertainty in the anti-neutrino to neutrino ratio comes from π + /π − andK + /K − ratio in hadronic interaction of the flux calculation. The uncertainty is estimatedby comparing the three flux models [22, 49, 50]. Figure 6.10 shows anti-neutrino to neutrino
ratio normalized by Honda flux model for each ν e and νµ case. In case of ν e /νe , theestimated uncertainty is 5 % in Eν < 1 GeV, 5 % for 1 < Eν < 10 GeV, 8 % for 10 < Eν <100 GeV, and above that, the error increases with logE ν upto 30 % (1 TeV). In case ofν µ /νµ , the estimated uncertainty is 2 % in E ν < 1 GeV, 6 % for 1 < Eν < 10 GeV, 6 % for
10 < Eν < 50 GeV, and above that, the error increases with logE ν upto 60 % (1 TeV).
• Up/down ratio
The neutrino flux has an up/down asymmetry due to the rigidity cutoff by the geomagnetic
field. However, the uncertainty in low energy neutrino flux is not expected to affect thezenith angle distributions substantially, because the up/down asymmetry in low energy islargely smeared by the poor angular correlation in neutrino interactions. The systematic
uncertainties of the up/down ratio are estimated by the comparison of the zenith angle
distributions using the three flux calculation models for each neutrino sample, which are128

summarized in Table 6.4.
• Horizontal/vertical ratio
The systematic uncertainty of horizontal/vertical ratio is caused mainly by the differencein the 3-dimensional calculation method in each flux calculation model below 3 GeV ofneutrino energy. For the neutrinos above 3 GeV, the uncertainty is caused by the differencein the predicted K/π ratio in hadronic interactions in the atmosphere. The systematic
uncertainties of horizontal/vertical ratio are estimated by the comparison of the zenith
angle distributions using the three flux calculation models for each neutrino sample, which
are summarized in Table 6.5.
• K/π ratio
Atmospheric neutrinos source is mainly π + or π − in the neutrino energy region below
10 GeV. K’s contribute to the neutrino production above a few tens of GeV. The uncertainty of K/π ratio production ratio in cosmic ray interactions in atmosphere could be thesystematic error. According to the SPY experiment [133], the measurement of the K/πratio is obtained with an accuracy of 3%, where K and π momentum ranges from 7 GeV/c
to 135 GeV/c. Using the SPY experiment result and the correspondence between hadron
momentum and neutrino energy, the uncertainty of K/π ratio is estimated to be 5% below
Eν =100GeV and 20% above Eν =1TeV. The uncertainty is assumed to increase linearly
from 5% to 20% between 100GeV and 1TeV.
• Neutrino pass length
Since the atmospheric neutrino flight length is used in the calculation of the neutrino oscillation probabilities, the uncertainty in the altitude of the neutrinos production hight isneeded to be estimated, especially for downward-going and horizontally-going neutrinos.
The uncertainty of the neutrino production hight is related to the structure of the atmosphere. The systematic uncertainty is estimated using the flux calculation which is carriedout by changing the density structure of atmosphere by 10%. This 10% uncertainty isgiven by the comparison between US-standard’76 and MSISE90 [134]. Figure 6.11 shows
the comparison of the flight length in the different density structures. The difference of theneutrino flux due to the change of the path length is taken into account in the systematic
uncertainty.
• Solar activity
The primary cosmic ray flux is affected by the solar activity, which changes the cosmic
ray flux in every 11 year period. ± 1 year of uncertainty is assigned to the modulation of
the solar activity, which corresponds to 20 % uncertainty in the activity for SK-I, 50 % for
SK-II and 20 % for SK-III.
• Sample-by-sample normalization
The energy dependence of the absolute normalization is different for each flux modeland the difference can not be explained by a simple spectral index uncertainty as seenin Figure 6.12. From a comparison of the predicted number of events among three flux
models [22, 49, 50], 5 % is assigned as the relative normalization uncertainty for FC multiGeV sample and PC+UPMU-stop sample.
• Matter effect
The electron density of the Earth affects to the calculation of the neutrino oscillation129

path length (km)

10

4

10

3

10

Figure 6.11: Neutrino flight length asa function of zenith angle. Solid curveshows the flight length used in our analysis and dashed curve shows that forthe compressed density structure of atmosphere by 10%.

2

10

1

-0.2

0

0.2

0.4

0.6

0.8

1

Flux ratio

Zenith angle (cosΘ)

1.4
1.3
1.2

Figure 6.12: The ratio of the calculated neutrino flux in which thefluxes calculated in Bartol (solidline) and FLUKA (dashed line) arenormalized by the Honda flux.

1.1
1
0.9
0.8
0.7
0.6 −1
10

1

10

102

103
Eν(GeV)130ν / ν double ratio(Nieves/Smith-Moniz)

Ratio(Nieves/Smith-Moniz)

2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2

1.4
1.2
1
0.8
0.6
0.4
0.2

µ/e double ratio(Nieves/Smith-Moniz)

0

50

100

150

200

250

300

350

400

450
500
Eν (MeV)

0

1.5

50

100

150

200

250

300

350

400

450
500
Eν (MeV)

1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
0

50

100

150

200

250

300

350

400

450
500
Eν (MeV)Figure 6.13: Top left: The ratio of the
CCQE cross section, the Nieves model
normalized by the Smith and Monitz
model, for νe + ν e (solid line) and forνµ + ν µ (dashed line). Top right: (ν̄/ν)ratio. Bottom: (νµ + ν µ )/(νe + ν e ) ratio.

probabilities. The density of matter is measured well by [135], but the electron density
depends on the chemical composition in the Earth. Since the core region is expected
to consist of heavy elements (Fe) mostly, the electron density is less than that of lightelements by about 6.8 %. The difference of number of oscillated neutrino events from thereduced electron density is taken into account in the uncertainty of the matter effect.

6.5.2

Systematics in neutrino interaction• CCQE cross-section
The cross-section of CCQE for the bound nucleon is calculated based on the Smith andMonitz Fermi-gas model. The difference from the Nieves et al. model [83] is set to 1σ
uncertainty as the CCQE total cross-section. The uncertainties in the CCQE cross-sectionof (ν̄/ν) ratio and flavor ratio (νµ + ν µ )/(νe + ν e ) are also set by the difference from the
Nieves model. Figure 6.13 shows the calculated CCQE cross-section ratio, the Nievesmodel to the Smith and Monitz model, as a function of the neutrino energy for the total
cross-section, the ν̄/ν ratio and the flavor ratio.
• MA in CCQE and single meson production
For MA in CCQE and single meson production, the difference of Q 2 spectrum between
MA = 1.2 and MA = 1.1 is set to the systematic uncertainty in order to take into account
10 % uncertainty of MA parameter as described in Section 3.3.1.
• Single meson production cross-section
The uncertainty in the cross-section of the single-meson production is estimated to be
20 % according to the difference between the expectation and experimental data of theνµ p → µ− pπ + observation. The νµ n → µ− pπ 0 interaction is poorer understood due tofew experimental results. So the additional uncertainty is estimated by the comparison
131

Systematic error in neutrino flux (1)Flux normalization

fit value %σ (%)

Eν < 1GeV
Eν > 1GeV

37.20
9.51

25a
15b

Eν < 1GeV
1 < Eν < 10GeV
Eν > 10GeV

-0.68
-2.31
-2.97

2
3
5c

Eν < 1GeV
1 < Eν < 10GeV
Eν > 10GeV

3.43
0.73
-0.76

5
5
8d

Eν < 10GeV
1 < Eν < 10GeV
Eν > 10GeV< 400 MeV

-0.17
-1.25
-4.38
-0.05
-0.52
-0.14
-0.38
-0.88
-0.24
-0.33
-0.10
-0.10
-0.15
-0.10
-0.10

2
6
15e
0.1
1.1
0.3
0.8
1.7
0.5
0.7
0.2
0.2
0.3
0.2
0.2

νµ /νeν̄e /νeν̄µ /νµUp/down ratio> 400 MeV

Multi-GeV
Multi-ring Sub-GeV
Multi-ring Multi-GeV
PC

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

Uncertainty linearly decreases with log Eν from 25 %(0.1 GeV) to 7 %(1 GeV).
Uncertainty is 7 % upto 10 GeV, and it linearly increases with log Eν from 7 %(10 GeV) to 12 %(100 GeV) andthen from 12 %(100 GeV) to 20 %(1 TeV)c
Uncertainty linearly increases with log Eν from 5 %(30 GeV) to 30 %(1 TeV).
dUncertainty linearly increases with log Eν from 8 %(100 GeV) to 20 %(1 TeV).
e
Uncertainty linearly increases with log Eν from 6 %(50 GeV) to 40 %(1 TeV).
b

Table 6.4: Systematic errors in neutrino flux calculation, that are common between all the SK
geometries. The second column shows the best fit value of the systematic errors parameter  i inpercent and the third column shows the estimated 1σ error size in percent.

132

Systematic error in neutrino flux (2)Horizontal/Vertical ratio< 400 MeV

> 400 MeV

Multi-GeV
Multi-ring Sub-GeV
Multi-ring Multi-GeV
PC
K/π ratio in flux calculation
Neutrino path length
Sample-by-sample
FC Multi-GeV
PC + Up-stop µ
Matter effectSolar Activity
SK-I
SK-II
SK-IIIabe-like 0-decayµ-like 0-decayµ-like 1-decay
e-like 0-decayµ-like 0-decayµ-like 1-decay
e-likeµ-likeµ-likee-likeµ-likefit value %

σ (%)

0.04
0.11
0.04
0.52
0.52
0.71
1.19
0.86
0.50
1.04
0.56
0.63
-9.43
-0.71
-5.39
-3.93
-0.002
-0.11
23.32
2.88

0.1
0.3
0.1
1.4
1.4
1.9
3.2
2.3
1.3
2.8
1.5
1.7
10 a
10
5
5
1.0b
20
50
20

Uncertainty increases linearly from 5% to 20% between 100GeV and 1TeV.
Difference of the electron density in the core region from heavy element case is set to 1.0

Table 6.5: Systematic errors in neutrino flux calculation, that are common between all the SK
geometries. The second column shows the best fit value of the systematic errors parameter  i inpercent and the third column shows the estimated 1σ error size in percent.

133

0

CC ν/ ν double ratio

CC νµ cross section ratio

2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0.5

1

1.5

2

2.5

3

3.5

4

4.5 5
Eν (GeV)

2.2
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0

0.5

1

1.5

2

2.5

3

3.5

4

4.5 5
Eν (GeV)Figure 6.14: the predicted cross-section(left panel) and the ν/ν ratio(right panel) of the Hernandez model normalized by that of the Rein and Sehgal model for ν µ p → µ− pπ + (solid line),νµ n → µ− pπ 0 (dashed line) and νµ n → µ− pπ + (dotted line).
with the Hernandez [141] model. The cross-section ratio of ν µ n → µ− pπ 0 is shown by
the dashed line in the left panel of Figure 6.14. It is considered that π 0 production
interaction has 40 % uncertainty relative to the π ± emitting interaction. The uncertainty
in the single meson production cross-section of the ν̄/ν ratio is also set by the difference
from the Hernandez model. The right panel of Figure 6.14 shows the ν̄/ν ratio of the CC
cross-section normalized by the Rein and Sehgal model.
• Deep inelastic scattering cross section
The calculation of cross-section for DIS agrees with the experimental measurement within
5 %. But, since the uncertainty is larger in the lower energy region (< 10 GeV) , theadditional error is estimated by the difference from the CKMT model [142].
• Q2 spectrum in deep inelastic scattering
The uncertainty of the Q2 spectrum is taken into account by the comparison with original
GRV98 model. This uncertainty is considered to be separated for W < 1.3 GeV/c 2 andW > 1.3 GeV/c2 .
• Coherent pion production cross-section
The uncertainty is obtained as 50 % by the difference between the Rein and Sehgal modeland experimental measurement for NC and CCν e interactions. Since CCνµ coherent pion
production is not measured, 100 % uncertainy is considered.
• NC/CC ratio
The uncertainty of NC/CC ratio is estimated to be 20 %.
• Nuclear effect in 16 O nucleus
The uncertainty in the mean free path of hadrons produced via neutrino interaction in the
16 O nucleus is estimated to be 30 % by the comparison with experimental data.
• Nuclear effect in π spectrum
The difference in the predicted pion spectrum between two simulator, NEUT [76, 77] andNUANCE [137], is taken to be the systematic uncertainty.

134• CCντ interaction cross section
The systematic uncertainty for CCν τ cross-section is estimated to be 30 % by the comparison with Hagiwara et al. model [143].

6.5.3

Systematics in event selection and reconstruction• FC/PC separation
The FC and PC events are separated by the number of OD hit PMT clusters (NHITAC).
The systematic uncertainty is estimated to be 0.6 % for SK-I, 0.5 % for SK-II and 0.9 %for SK-III by the difference between observed data and MC NHITAC distributions (Figure 5.12).
• FC reduction related
The uncertainty in FC reduction efficiency is estimated to be 0.2 % for SK-I, 0.2 % for SK-II
and 0.8 % for SK-III by comparing the distributions of each cut valuable for observed data
and MC. The contamination of the background is estimated as described in Section 5.1.
Error values are summarized in Table 6.7.
• PC reduction related
The uncertainty in PC reduction efficiency is estimated to be 2.4 % for SK-I, 4.8 % for SK-II
and 0.5 % for SK-III by comparing the distributions of each cut valuables for observed data
and MC. The contamination of the background is estimated as described in Section 5.2.
• PC stop/through separation
PC OD stopping and OD through-going events are separated by the energy deposit in
OD region (PEanti /PEexp ). The systematic uncertainty is estimated by the difference
between observed data and MC PEanti /PEexp distributions (Figure 5.25). The estimationis separately performed for OD bottom, barrel and top region since the OD calibration
has the difference depending on the OD position. The error values are summarized inTable 6.7.
• UPMU reduction related
The uncertainty in UPMU reduction efficiency is estimated by comparing the distributionsof each cut valuables for observed data and MC. For stopping muon, the estimated uncertainties are 0.7 % for SK-I, 0.7 % for SK-II and 0.7 % for SK-III. For through-going muon,
the estimated uncertainties are 0.5 % for SK-I, 0.5 % for SK-II and 0.5 % for SK-III. These
uncertainties for stopping and through-going muons are assumed to be correlated. The
stopping muon and through-going muon are separated by the number of hit OD PMTs
within 8m from the projected exit point (NHITEX). The separation uncertainty is estimated to be 0.4 % for SK-I, 0.4 % for SK-II and 0.6 % for SK-III by the difference between
observed data and MC NHITEX distributions (Figure 5.26). Additional systematic uncertainty for stopping muon from energy cut in the final selection is estimated to be 0.8 %for SK-I, 1.4 % for SK-II and 2.1 % for SK-III. Other additional systematic uncertainty forthrough-going muon from path length cut in the final selection is estimated to be 1.8 % for
SK-I, 2.1 % for SK-II and 1.6 % for SK-III. Finally, the contamination of the background in
UPMU sample is estimated as described in Section 5.3. The error values are summarizedin Table 6.7.
135

Systematic error in neutrino interaction

fit value %σ (%)

CCQE cross section
CCQE ν̄i /νi (i=e,µ) ratio
CCQE µ/e ratio
MA in CCQE and single meson production
Single meson production cross section
Single π production, π 0 /π ± ratio
Single π production, ν̄i /νi (i=e,µ) ratio
DIS cross section (Eν < 10 GeV)DIS cross section
Q2 spectrum in DIS (W < 1.3 GeV/c2 )Q2 spectrum in DIS (W > 1.3 GeV/c2 )Coherent π production
NC/(CC)
Nuclear effect in 16 O nucleus
Nuclear effect in pion spectrumντ contamination
NC in FC µ-like (hadron simulation)
Decay-e tagging (π + interaction)f Sub-GeV 1-ring

0.57
0.92
0.68
0.59
9.87
-29.21
-0.01
0.05
2.02
0.24
0.39
-30.990.40-12.92
0.85
-5.59
-2.40
-0.2∼-0.5
4.0∼4.2-0.5∼-0.8
0.5∼0.9
5.2∼5.8

1.0a
1.0a
1.0a
10
20
40
1.0b
1.0c
5
1.0d
1.0d
100
20
30
1.0 e
30
10
0.2∼0.5-3.9∼-4.1
0.5∼0.8-0.4∼-0.8-5.1∼-5.7

e-like 0-decay
e-like 1-decayµ-like 0-decayµ-like 1-decayµ-like 2-decayaDifference from the Nieves model is set to 1.0.
Difference from the Hernandez model is set to 1.0.
c
Difference from CKMT parametrization is set to 1.0.
dDifference from original GRV98 model is set to 1.0.
eDifference between NEUT and NUANCE is set to 1.0.
f
They are assumed to be identical and full correlated for each SK run period.
b

Table 6.6: Systematic errors in neutrino interactions, that are common between all the SK
geometries. The second column shows the best fit value of the systematic errors parameter  i inpercent and the third column shows the estimated 1σ error size in percent.

136• UPMU non-showering/showering separation
The UPMU non-showering and showering events are separated by the difference between
the average corrected charge and expected charge for non-showering events (∆(Q)). The
systematic uncertainty is estimated by the comparison of ∆(Q) distribution (Figure 5.30)
for data and MC. The error values are set to be 9 % for SK-I, 13 % for SK-II and 6 % for
SK-III.
• Ring separation
Ring separation is based on the ring-counting algorithm (Section 5.4.2). The uncertainties
are estimated by the difference of likelihood distributions between observed data and MC
for each sample. The error values are summarized in Tables 6.8, 6.9 and 6.10.
• Particle identification
The uncertainties in the particle identification are estimated by the difference of PID
likelihood distributions (Section 5.4.3) between observed data and MC for each sample.
The error values are summarized in Tables 6.8, 6.9 and 6.10.
• Energy calibration
The energy scale uncertainties are estimated in Section 4.2. The systematic uncertainty
for absolute energy scale is set to 1.1 % for SK-I, 1.7 % for SK-II and 2.7 % for SK-III. The
uncertainty in the upward/downward symmetry of energy calibration is estimated to be
0.6 % for SK-I, 0.6 % for SK-II and 1.3 % for SK-III.
• Multi-GeV Multi-ring e-like sample selection
The FC Multi-GeV Multi-ring e-like events are selected as described in Section 6.2.2. The
uncertainties in the selection efficiency are estimated by comparing observed data with
MC for the likelihood distributions. The MC likelihood distribution is re-scaled to get thebest-fit to the data distribution. The maximal change on the number of events after shifted
distributions by ±1σ from the best-fit parameter is taken to the systematic uncertainty.
The change on the signal event (CCνe ) is set to the selection efficiency uncertainty, andthe change on the background event (NC and CCν µ ) is set to the uncertainty for thecontamination of the background in this sample. The estimated systematic uncertainties
for the selection efficiencies are 6.4 % for SK-I, 11.1 % for SK-II and 5.3 % for SK-III. The
estimated systematic uncertainties for the BG contamination are 35.6 % for SK-I, 22.3 %for SK-II and 38 % for SK-III.
• Single-ring π 0 -like sample selection
The FC Sub-GeV single-ring π 0 -like events are selected as described in Section 6.2.1. The
uncertainties in the selection efficiency are estimated by comparing observed data with
MC for the π 0 mass and likelihood distributions. Since likelihood distribution is different
for each momentum bin, the systematic uncertainties are estimated for each momentum
region. The uncertainties in the selection efficiency also come from the absolute energy
scale uncertainty. The combined uncertainties, from the difference of distribution and theshift of energy scale, are summarized in Tables 6.8, 6.9 and 6.10.
• Two-ring π 0 selection
The FC Sub-GeV two-ring π 0 -like events are selected as described in Section 4.2.3. The
uncertainty in the selection efficiency comes from the ring-counting and PID. The uncertainty is estimated to be 2 % for SK-I, 2 % for SK-II and 1 % for SK-III.
137

• Decay electron tagging
The systematic error in the decay electron detection efficiency is estimated to be 1.5 % using
cosmic ray muons [139]. This systematic error is assumed to be identical for each SK run
period. In addition, π + decay probability uncertainty is considered for that systematic
error. Since π + ’s are interacted in the water depending on the energy, the differences ofthe decay probability are 3∼13 % due to the π + momentum. So it affects the uncertainty
for decay electron tagging by 0.3∼0.6 % for FC Sub-GeV single-ring e-like 0-decay sample.
The error values are summarized in Tables 6.6 for each sub-sample. This systematic error
is assumed to be correlated for each sub-sample and it is a common error value for all SK
run periods.

138

SK-I
fit
σ

Systematic error in event reduction
FC/PC separation
FC reduction
PC reduction
Non-ν BG ( e-like )Sub-GeV e-like
Multi-GeV e-like
Non-ν BG ( µ-like )Sub-GeV µ-like
Multi-GeV µ-likePC
PC-stop/PC-through separation -TOPPC-stop/PC-through separation -BARRELPC-stop/PC-through separation -BOTTOMFiducial volume
Upward-going muon reduction a Stopping
Through-going
Upward stopping/through-going µ separation
Upward through-going µ showering separation
Energy cut for upward stopping µ
Path length cut for upward through-going µ
BG subtraction for upward µ b
Stopping
Non-showering
Showering

-0.03
0.003
-1.64
0.03
0.02
-0.01
-0.01
0.03
13.29
-1.57
-5.08
-0.88
0.02
0.02
0.006
13.28
-0.006
0.27
5.00
-6.19
-0.47

0.6
0.2
2.4
0.5
0.3
0.1
0.1
0.2
14
7.5
11
2
0.7
0.5
0.4
9
0.8
1.8
16
11
18

SK-II
fit
σ
0.08
0.006
-3.33
-0.001-0.001
0.001
0.001
0.04
-16.00-33.62
-9.88
0.06
-0.11
-0.08
0.01
5.65
-0.07
-1.13
-7.38
3.47
-19.58

0.5
0.2
4.8
0.2
0.3
0.1
0.1
0.7
21
17
12
2
0.7
0.5
0.4
13
1.4
2.1
21
15
14

SK-III
fit
σ
-0.24
0.03
0.02
0.002
0.009
0.03
0.05
0.47
-15.19
3.49
0.48
0.53
0.19
0.14
0.05
8.30
1.02
0.55
0.79
2.30
23.84

0.9
0.8
0.5
0.1
0.4
0.1
0.2
1.8
31
23
11
2
0.7
0.5
0.6
6
2.1
1.6
20
19
24

a

The uncertainties in the upward-going stopping and through-going muons are assumed to be correlated.
Therefore there is only one error term in each run period.
b
The uncertainties in BG subtraction for upward-going muons are only for the most horizontal events which
are in the bin of −0.1 < cos θ < 0.

Table 6.7: Systematic errors in event reduction for each SK run period. The column of “fit”
show the best fit values of the systematic errors parameter  i in percent and the column of “σ”
show the estimated 1σ error sizes in percent.

139

(SK-I) Systematic error in event reconstruction and selection
Ring separation< 400 MeV
> 400 MeV
Multi-GeV
Multi-ring sub-GeV
Multi-ring multi-GeV

PID (single-ring)

Sub-GeV
Multi-GeVPID (multi-ring)

Sub-GeV
Multi-GeV

e-likeµ-likee-likeµ-likee-likeµ-likeµ-likee-likeµ-likee-likeµ-likee-likeµ-likeµ-likee-likeµ-likeEnergy calibration
Up/Down asymmetry energy calibration
Multi-GeV Single-ring e-like non CCν e BG
Multi-GeV Multi-ring e-like non CCν e BG
Multi-GeV Multi-ring e-like selection (likelihood)
Sub-GeV 1-ring π 0 selection
100 < Pe < 250 MeV/c
250 < Pe < 400 MeV/c
400 < Pe < 630 MeV/c
630 < Pe < 1000 MeV/c1000 < Pe < 1330 MeV/c
0
Sub-GeV 2-ring π
Decay-e tagging (µ decay)

fit value %

σ(%)

1.23
0.37
0.22
0.37
1.98
0.91
-2.41
-1.66
-2.19
-0.01
0.01
0.02
-0.02
-0.33
0.15
-0.26
0.10
-0.33
5.21
-5.16
-1.56
-3.35
-3.44
-7.01
-5.72
-3.89
-0.18
0.26

2.3
0.7
0.4
0.7
3.7
1.7
-4.5
-3.1
-4.1
0.1
-0.1
-0.2
0.2
-3.9
1.7
-2.9
1.1
0.6
16.3
35.6
6.4
11.2
11.5
23.4
19.1
13
2
1.5

Table 6.8: Systematic errors in event reconstruction and selection for SK-I. The second column
shows the best fit value of the systematic errors parameter  i in percent and the third column
shows the estimated 1σ error size in percent.

140

(SK-II) Systematic error in event reconstruction and selection
Ring separation< 400 MeV
> 400 MeV
Multi-GeV
Multi-ring sub-GeV
Multi-ring multi-GeV

PID (single-ring)

Sub-GeV
Multi-GeVPID (multi-ring)

Sub-GeV
Multi-GeV

e-likeµ-likee-likeµ-likee-likeµ-likeµ-likee-likeµ-likee-likeµ-likee-likeµ-likeµ-likee-likeµ-likeEnergy calibration
Up/Down asymmetry energy calibration
Multi-GeV Single-ring e-like non CCν e BG
Multi-GeV Multi-ring e-like non CCν e BG
Multi-GeV Multi-ring e-like selection (likelihood)
Sub-GeV 1-ring π 0 selection
100 < Pe < 250 MeV/c
250 < Pe < 400 MeV/c
400 < Pe < 630 MeV/c
630 < Pe < 1000 MeV/c1000 < Pe < 1330 MeV/c
0
Sub-GeV 2-ring π
Decay-e tagging (µ decay)

fit value %

σ(%)

-1.53
-2.70
-2.00
-0.82
-3.06
-2.00
9.64
2.23
0.94
0.06
-0.05
0.01
-0.01
0.44
-0.36
0.68
-1.19
-0.18
-5.77
-1.39
0.24
-3.67
-4.35
-8.56
-5.23
-5.43
-0.03
0.47

1.3
2.3
1.7
0.7
2.6
1.7
-8.2
-1.9
-0.8
0.5
-0.4
0.1
-0.1
2.2
1.8
-3.4
1.7
0.6
23.4
22.3
11.1
7.5
8.9
17.5
10.7
11
2
1.5

Table 6.9: Systematic errors in event reconstruction and selection for SK-II. The second column
shows the best fit value of the systematic errors parameter  i in percent and the third column
shows the estimated 1σ error size in percent.

141

(SK-III) Systematic error in event reconstruction and selection
Ring separation< 400 MeV
> 400 MeV
Multi-GeV
Multi-ring sub-GeV
Multi-ring multi-GeV

PID (single-ring)

Sub-GeV
Multi-GeVPID (multi-ring)

Sub-GeV
Multi-GeV

e-likeµ-likee-likeµ-likee-likeµ-likeµ-likee-likeµ-likee-likeµ-likee-likeµ-likeµ-likee-likeµ-likeEnergy calibration
Up/Down asymmetry energy calibration
Multi-GeV Single-ring e-like non CCν e BG
Multi-GeV Multi-ring e-like non CCν e BG
Multi-GeV Multi-ring e-like selection (likelihood)
Sub-GeV 1-ring π 0 selection
100 < Pe < 250 MeV/c
250 < Pe < 400 MeV/c
400 < Pe < 630 MeV/c
630 < Pe < 1000 MeV/c1000 < Pe < 1330 MeV/c
0
Sub-GeV 2-ring π
Decay-e tagging (µ decay)

fit value %

σ(%)

0.10
0.03
0.02
0.03
0.15
0.07
-0.19
-0.17
-0.17
0.002
-0.002-0.004
0.004
-0.29
0.13
-0.22
-0.23
0.007
3.92
-19.18-0.39-5.09-17.47-8.27-17.66-17.66
0.02
1.1

2.3
0.7
0.4
0.7
3.7
1.7
-4.5
-3.1
-4.1
0.1
-0.1
-0.2
0.2
-3.9
1.7
-2.9
2.7
1.3
41.4
38
5.3
7.7
26.4
12.5
26.7
26.7
1
1.5

Table 6.10: Systematic errors in event reconstruction and selection for SK-III. The second
column shows the best fit value of the systematic errors parameter  i in percent and the thirdcolumn shows the estimated 1σ error size in percent.

142

6.6

Results

In this analysis, the zenith angle and momentum distributions of the various samples are
simultaneously fitted with the simulated distributions including all oscillation parameters, twomass differences (∆m212 , ∆m223 ), three mixing angles (θ12 , θ23 , θ13 ), and CP phase (δcp ). A χ2 iscalculated in the fitting process by four dimensional oscillation parameter spaces of ∆m 223 , θ23 ,θ13 and δcp . The parameters ∆m212 and sin2 θ12 are fixed to the values determined by the solar
neutrino analyses ([37, 146, 147, 18, 19, 20]), since they are already well constrained. It is foundthat changes of these parameters in the allowed region give negligible effect on the fitting (refer
Appendix A.1).
The examined oscillation parameter spaces are:• (∆m212 , sin2 θ12 ) = (7.65 × 10−5 , 0.304) (fixed)• ∆m223 = 10−2.875 to 10−2.450 (18 points)• sin2 θ23 = 0.3 to 0.7 (17 points)• sin2 θ13 = 0.0 to 0.4 (65 points)• δcp = 0◦ to 340◦ (18 points)
for both mass hierarchy cases.
In the normal hierarchy case, the best-fit is obtained at (∆m 223 , sin2 θ23 , sin2 θ13 , δcp ) =(2.1×10−3 , 0.525, 0.006, 220◦ ) with χ2min /dof = 469.94/416. In the inverted hierarchy case,the best-fit is obtained at (∆m223 , sin2 θ23 , sin2 θ13 , δcp ) = (2.5×10−3 , 0.575, 0.044, 220◦ ) with
χ2min /dof = 468.34/416. Figures 6.15 and 6.16 (6.17 and 6.18) show the best-fit zenith angle
distributions together with the distributions of the observed data and null oscillation for thenormal (inverted) mass hierarchy case. These distributions were merged from that of each SK
run period, and divided into momentum bins as shown in the figures. The distributions divided
by SK run period can be found in Appendix A.2. The best-fit oscillations reproduce the observed
data well for both the normal and inverted mass hierarchy cases.
Figures 6.19 and 6.20 show the χ2 −χ2min distributions as a function of oscillation parameter.
When finding the minimum χ2 against one oscillation parameter dimension, the other threeparameters are chosen so that the χ2 is minimized. The 90 % confidence interval is located at
χ2 = χ2min + 2.7. The parameter allowed regions are estimated using linear interpolating andsummarized in Table 6.11.
Normal hierarchy

Inverted hierarchy∆m223 (eV2 ) 90 % C.L.
sin2 θ23 90 % C.L.
sin2 θ13 90 % C.L.

(1.88 < ∆m223 < 2.75) ×10−3
0.406 < sin2 θ23 < 0.629
sin2 θ13 < 0.066

δcp 90 % C.L.

—

(1.98 < ∆m223 < 2.81) ×10−3
0.426 < sin2 θ23 < 0.644
sin2 θ13 < 0.122
121.4◦ < δcp < 319.1◦

Table 6.11: Results of the allowed regions for each neutrino oscillation parameters at
90 %C.L.. 90 % C.L. regions are defined to be χ 2 − χ2min < 2.7.

143

Figures 6.21 and 6.22 show the allowed regions drawn in the two dimensions at 68 %, 90 %and 99 % C.L. defined as χ2 − χ2min = 2.3, 4.6 and 9.2, respectively.

144

Sub-GeV 1-ring e-like 0-decay
100-250(MeV/c)200

250-400

400-6301000 < P

630-1000

150
100
50
0

-1

0

1-1

0

1-1

0

1-1

0

1-1

0

1

cosΘ

Sub-GeV 1-ring µ-like 0-decay
200-250(MeV/c)100

250-400

400-630630-1000

200-250(MeV/c)250-400

400-630630-1000

1000 < P

150

75

100

50

50

25
0

Sub-GeV 1-ring µ-like 1-decay

2001000 < P

-1

1-1

1-1

1-1

1-1

1

0

-1

cosΘ

1-1

1-1

1-1

1

Multi-GeV 1-ring µ-like

Multi-GeV 1-ring e-like1 - 2.5 (GeV/c)

100

1-1

cosΘ

2.5 - 5 (GeV/c)

5 - 10 (GeV/c)

150

10 (GeV/c)< P

2.5 (GeV/c) < P

1 - 2.5 (GeV/c)

100

75
50

50
25
0

-1

1-1

1-1

1-1

1

0

-1

0

cosΘ

Multi-GeV multi-ring e-like1 - 2.5 (GeV/c)

60

1-1

0

1

cosΘ

2.5 - 5 (GeV/c)

5 - 10 (GeV/c)

multi-ring µ-like

100
10 (GeV/c)< P

P< 1(GeV/c)

1 - 2.5 (GeV/c)

2.5 - 5 (GeV/c)

5 (GeV/c)< P

80
60

40

40

20
20

0

-1

1-1

1-1

1-1

1

cosΘ

0

-1

1-1

1-1

1-1

1

cosΘ

Figure 6.15: SK-I + II + III zenith angle distributions for FC events for data (line with error bar)and MC assuming no oscillation (dashed line) and the best-fit results (solid line). Normal hierarchy
is assumed.

145

PC stop40
P < 2.5 (GeV/c)

PC through150

2.5 (GeV/c) < P

P< 1(GeV/c)

1 - 2.5 (GeV/c)

2.5 - 5 (GeV/c)

5 (GeV/c)< P

30
100

20
50

10
0

-1

0

1-1

0

1

0

-1

cosΘ

1-1

1-1

1-1

1

cosΘ
Upmu stoppingP < 2.5 (GeV/c)

100

2.5 - 5 (GeV/c)

5 (GeV/c)< P

75
50
25
0

-1

0-1

0-1

0

cosΘ
Upmu showering

Upmu non-showering

600

100
400

75
50

200

25
0

-1

-0.8

-0.6

-0.4

-0.2

0

0

-1

-0.8

-0.6

-0.4

-0.2

0

cosΘ

cosΘ

Figure 6.16: SK-I + II + III zenith angle distributions for PC and UPMU events for data (linewith error bar) and MC assuming no oscillation (dashed line) and the best-fit results (solid line).
Normal hierarchy is assumed.

146

Sub-GeV 1-ring e-like 0-decay
100-250(MeV/c)200

250-400

400-6301000 < P

630-1000

150
100
50
0

-1

0

1-1

0

1-1

0

1-1

0

1-1

0

1

cosΘ

Sub-GeV 1-ring µ-like 0-decay
200-250(MeV/c)100

250-400

400-630630-1000

200-250(MeV/c)250-400

400-630630-1000

1000 < P

150

75

100

50

50

25
0

Sub-GeV 1-ring µ-like 1-decay

2001000 < P

-1

1-1

1-1

1-1

1-1

1

0

-1

cosΘ

1-1

1-1

1-1

1

Multi-GeV 1-ring µ-like

Multi-GeV 1-ring e-like1 - 2.5 (GeV/c)

100

1-1

cosΘ

2.5 - 5 (GeV/c)

5 - 10 (GeV/c)

150

10 (GeV/c)< P

2.5 (GeV/c) < P

1 - 2.5 (GeV/c)

100

75
50

50
25
0

-1

1-1

1-1

1-1

1

0

-1

0

cosΘ

Multi-GeV multi-ring e-like1 - 2.5 (GeV/c)

60

1-1

0

1

cosΘ

2.5 - 5 (GeV/c)

5 - 10 (GeV/c)

multi-ring µ-like

100
10 (GeV/c)< P

P< 1(GeV/c)

1 - 2.5 (GeV/c)

2.5 - 5 (GeV/c)

5 (GeV/c)< P

80
60

40

40

20
20

0

-1

1-1

1-1

1-1

1

cosΘ

0

-1

1-1

1-1

1-1

1

cosΘ

Figure 6.17: SK-I + II + III zenith angle distributions for FC events for data (line with errorbar) and MC assuming no oscillation (dashed line) and the best-fit results (solid line). Inverted
hierarchy is assumed.

147

PC stop40
P < 2.5 (GeV/c)

PC through150

2.5 (GeV/c) < P

P< 1(GeV/c)

1 - 2.5 (GeV/c)

2.5 - 5 (GeV/c)

5 (GeV/c)< P

30
100

20
50

10
0

-1

0

1-1

0

1

0

-1

cosΘ

1-1

1-1

1-1

1

cosΘ
Upmu stoppingP < 2.5 (GeV/c)

100

2.5 - 5 (GeV/c)

5 (GeV/c)< P

75
50
25
0

-1

0-1

0-1

0

cosΘ
Upmu showering

Upmu non-showering

600

100
400

75
50

200

25
0

-1

-0.8

-0.6

-0.4

-0.2

0

0

-1

-0.8

-0.6

-0.4

-0.2

0

cosΘ

cosΘ

Figure 6.18: SK-I + II + III zenith angle distributions for PC and UPMU events for data (linewith error bar) and MC assuming no oscillation (dashed line) and the best-fit results (solid line).
Inverted hierarchy is assumed.

148

10

9

9

8

8

7

7

χ - χ min
2

6
5
4

2

2

2

χ - χ min

10

2

2

1

1
0
0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7

1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5

2

x 10

sin θ23
2

-3

10

10

9

9

8

8

7

χ - χ min

7

2

6
5

2

2

4
3

∆m 23

2

5

3

0

χ - χ min

6

4

6
5
4

3

3

2

2

1

1

0

0

0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

0

50

100

150

200

250

300

δcp (degree)

sin θ132

Figure 6.19: The ∆χ2 = χ2 −χ2min distributions of each ∆m223 , sin2 θ23 , sin2 θ13 and δcp parameterfor the normal mass hierarchy case. The three horizontal lines at ∆χ 2 = 1.0, 2.7 and 6.6 indicate
68 % C.L., 90 % C.L. and 99 % C.L.

149

10

9

9

8

8

7

7

χ - χ min
2

6
5
4

2

2

2

χ - χ min

10

2

2

1

1
0
0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7

1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5

2

x 10

sin θ23
2

-3

10

10

9

9

8

8

7

χ - χ min

7

2

6
5

2

2

4
3

∆m 23

2

5

3

0

χ - χ min

6

4

6
5
4

3

3

2

2

1

1

0

0

0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

0

50

100

150

200

250

300

δcp (degree)

sin θ132

Figure 6.20: The ∆χ2 = χ2 −χ2min distributions of each ∆m223 , sin2 θ23 , sin2 θ13 and δcp parameterfor the inverted mass hierarchy case. The three horizontal lines at ∆χ 2 = 1.0, 2.7 and 6.6 indicate
68 % C.L., 90 % C.L. and 99 % C.L.

150

Normal hierarchy

0.00350.0035

0.003

0.003

∆ m223∆ m223

Normal hierarchy

0.00250.00250.002

0.001500.002

0.0015
0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7
sin2 θ23

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
sin2 θ13

Normal hierarchy

0.4

0.4

0.35

0.35

0.3

0.3

0.25

0.25

sin2 θ13

sin2 θ13

Normal hierarchy

0.2
0.15

0.2
0.15

0.1

0.1

0.05

0.05

0
0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7
sin2 θ23

0
0

50

100

150

200

250

300

δcp

Figure 6.21: Contours for the normal mass hierarchy case. The three lines indicate the allowed
regions at 68 % C.L., 90 % C.L. and 99 % C.L. defined as ∆χ 2 = 2.3, 4.6 and 9.2. The stars
show the best-fit parameter point.

151

Inverted hierarchy

0.00350.0035

0.003

0.003

∆ m223∆ m223Inverted hierarchy

0.00250.00250.002

0.001500.002

0.0015
0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7
sin2 θ23

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
sin2 θ13

Inverted hierarchy

0.4

0.4

0.35

0.35

0.3

0.3

0.25

0.25

sin2 θ13

sin2 θ13

Inverted hierarchy

0.2
0.15

0.2
0.15

0.1

0.1

0.05

0.05

0
0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7
sin2 θ23

0
0

50

100

150

200

250

300

δcp

Figure 6.22: Contours for the inverted mass hierarchy case. The three lines indicate the allowed
regions at 68 % C.L., 90 % C.L. and 99 % C.L. defined as ∆χ 2 = 2.3, 4.6 and 9.2. The stars
show the best-fit parameter point.

152

6.7

Discussion

The obtained 90 % C.L. limits are compared with the other experiments. The ∆m 223 andsin 2θ23 parameter allowed regions from 2-flavor analyses, SK zenith and L/E with SK-I to-III data, and MINOS [32], are shown in Figure 6.23, together with the results of this analysis.
90 %C.L. 1-dof allowed regions from global-fit [37] are (2.22 < ∆m 223 < 2.60) ×10−3 and 0.401< sin2 θ23 < 0.615 while this full 3-analysis gave (1.88 < ∆m 223 < 2.75) ×10−3 and 0.406< sin2 θ23 < 0.629 for normal mass hierarchy case. Therefore, this new analysis considering
the oscillation scheme without any approximations was confirmed to obtain consistent resultswith the 2-flavor analyses. Since the change in the oscillation probability from CP phases has
small but non-negligible effects on the measured distributions, this analysis gives slightly wider
allowed regions compared with the previous 2-flavor analyses. Figure 6.24 shows χ 2 distributions
for CP phase values fixed at 0◦ , 140◦ , 220◦ and 300◦ . The χ2 distribution at δcp ∼ 220◦ shows
the largest allowed region.
In Figure 6.25, new results of ∆m223 −sin2 θ13 parameter allowed regions are overlaid with thepast 3-flavor analysis with one-mass-scale-dominant scheme and the excluded region by Chooz
experiment [36]. The results are consistent with that of the Chooz experiment. sin 2 θ13 < 0.066
for normal mass hierarchy and sin2 θ13 < 0.122 for inverted mass hierarchy in 90 %C.L. 1-dof
allowed region while global-fit [37] shows sin 2 θ13 < 0.035.
The sensitivity of this experiment on θ 13 is studied for the normal hierarchy case. The bestfit point of oscillation parameters with sin2 θ13 = 0.006 were used as true values. The probability
for sin2 θ13 < 0.066 was calculated by “Toy Monte Carlo” experiments which were determinedbased on the Poisson distribution. Figure 6.26 shows the histogram of the 90 %C.L. upper limit
on sin2 θ13 by toy MC calculations. The probability for sin2 θ13 < 0.066 is estimated to be 38.5 %.
This test suggests that the constraint from the real SK data is almost as good as expected.
The best-fit parameter among all oscillation parameter sets in both mass hierarchies is incase of inverted mass hierarchy. The χ 2 distributions of the two cases are overlaid in Figure 6.27.
The difference of the minimum χ2 (∆χ2 = χ2min (normal) − χ2min (inverted)) is found to be 1.6.
The significance is still low to draw a conclusion the mass hierarchy.
As for as CP phase, this is the first time that the SK atmospheric neutrino data gives
information. The best-fit value appeared at δ cp = 220◦ for both mass hierarchies. For inverted
hierarchy case, 90 % C.L. can be barely set as 121.4 < δ cp < 319.1, as shown in Figure 6.20. But
the χ2 value is become flat just above the limit so no strong conclusion can be drawn for the
CP phase. More statistics is expected to make better sensitivity to the CP phase as shown in
Figure 6.28. Meanwhile, a large θ13 is needed to enable the SK analysis to constrain CP phasein the future. The current global-best θ 13 value remains consistent to zero, although non-zeroθ13 topic has been discussed in many literatures [37, 38]. On the other hand, there is a tension
between solar [145, 146, 147, 18, 150] and KamLAND [20] experiments and the preliminary
MINOS νe appearance results [149], which may indicate the value of sin 2 2θ13 could be near the
CHOOZ bound [148]. In this analysis, although the significance is still insufficient, the bestvalues are obtained at sin2 θ13 = 0.04 (sin2 2θ13 = 0.15) with ∆χ2 = 2.76 from sin2 θ13 = 0.0
for inverted hierarchy case. At this moment, the uncertainty in sin 2 2θ13 is still large to draw
any conclusion. However, if new T2K [151] and reactor [152, 153] experiments confirm the large
sin2 2θ13 , the present and future results from the atmospheric neutrino analysis could be used
to constrain the δcp value depending on the actual θ13 value.
2

153

Normal hierarchy
0.0035∆ m2230.003

0.00250.002

SK(zenith) 90%C.L.
SK(L/E) 90%C.L.
MINOS 90%C.L.

0.00150.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98
sin2 2θ23

1

Inverted hierarchy
0.0035∆ m2230.003

0.00250.002

SK(zenith) 90%C.L.
SK(L/E) 90%C.L.
MINOS 90%C.L.

0.00150.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98
sin2 2θ23

1Figure 6.23: Contours of ∆m223 and sin2 2θ23 parameter allowed regions for normal hierarchy (upper panel) and inverted hierarchy (lower panel). Solid lines show the new results
obtained by this full 3-flavor analysis; each colored lines indicate the allowed regions at
68 % C.L., 90 % C.L. and 99 % C.L. defined as ∆χ 2 = 2.3, 4.6 and 9.2, respectively. The
stars show the best-fit parameter points. 90 % C.L. allowed regions from other analysis
are also overlaid. Dotted lines show SK 2-flavor analysis using the zenith angle, dashedlines show SK L/E analysis and dash-dotted lines show MINOS results [32].

154

10

10

9

9

8

8
7

6

6

2

5

∆χ

∆χ

2

7

5

4

4

3

3

2

2

1

1

0

0
0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7

1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5

x 10

∆m 23
2

-3

sin θ23
2

10
9
8
7

∆χ

2

6
5
4
3
2
1
0

0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

sin θ132

Figure 6.24: The ∆χ2 distributions for CP phase values fixed at 0 ◦ (dashed line), 140◦ (dottedline), 220◦ (solid line) and 300◦ (dash-dotted line) for the normal hierarchy case. The threehorizontal lines indicate 68 % C.L., 90 % C.L. and 99 % C.L. (∆χ 2 = 1.0, 2.7 and 6.6).

155

Normal hierarchy
0.0035∆ m2230.003

0.0025

0.002
full 3−flavor allowed(One−mass−dominance ana.)

0.00150Chooz excluded 90%C.L.

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
sin2 θ13

Inverted hierarchy
0.0035∆ m2230.003

0.0025

0.002
full 3−flavor allowed(One−mass−dominance ana.)

0.00150Chooz excluded 90%C.L.

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
sin2 θ13

Figure 6.25: Contours of ∆m223 and sin2 θ13 parameter allowed regions for normal hierarchy(upper panel) and inverted hierarchy (lower panel). Solid lines show the new results
obtained by this full 3-flavor analysis; each colored lines indicate the allowed regions at
68 % C.L., 90 % C.L. and 99 % C.L. defined as ∆χ 2 = 2.3, 4.6 and 9.2, respectively. The
solid stars show the best-fit parameter points. The results from the past 3-flavor analysis
with one-mass-scale-dominant scheme are shown by dashed lines and hollow stars. The
shaded regions show Chooz 90 % C.L. excluded region.

156

number of events

22.5
20
17.5
15
12.5
10
7.5
5
2.5
0

Figure 6.26: sin2 θ13 90 % C.L. upper limit distribution made by 200
toy MC calculations. The hatchedhistogram indicates sin2 θ13 < 0.066.

38.50 %

00.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
2
sin θ13 90%C.L. upper limit

480

478

478

476

476

474

474

2

χ

χ

2

480

472
470

470

468

468

466

466

1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7∆m 23

sin θ232

2

x 10-3

480

480

478

478

476

476

474

474

χ

2

2

χ

472

472

472

470

470

468

468

466

466
0

0.05

0.1

0.15

0.2

0.25

0.3

0

sin θ13
2

50

100

150

200

250

300

350

CPδ

Figure 6.27: The χ2 distributions of each ∆m223 , sin2 θ23 , sin2 θ13 and δcp parameter for bothnormal (dashed line) and inverted (solid line) mass hierarchy cases.

157

10

9

9

8

8

7

7

6

6

2

5

∆χ

∆χ

2

10

4

4

3

3

2

2

1

1

0

1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.50
0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7∆m 23

sin θ232

2

x 10-3

10

10

9

9

8

8

7

7

6

6

2

2

5

∆χ

∆χ

5

5

4

4

3

3

2

2

1

1

0

0

0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

sin θ13
2

0

50

100

150

200

250

300

350

CPδ

Figure 6.28: The expected ∆χ2 distributions of each ∆m223 , sin2 θ23 , sin2 θ13 and δcp parameterusing the oscillated Monte Carlo events of a 10 years (solid line), 20 years (dashed line) and 30
years (dotted line) exposure with the true oscillation parameters of the best-fit by this analysis,(∆m223 , sin2 θ23 , sin2 θ13 , δcp ) = (-2.5×10−3 , 0.575, 0.043, 220◦ ). The two horizontal lines at∆χ2 = 2.7 and 6.6 indicate 90 % C.L. and 99 % C.L., respectively.

158

Chapter 7

Conclusion
The full neutrino oscillation analysis was performed for the first time with the atmospheric
neutrino data observed in Super-Kamiokande. All data observed in SK-I (1489.2 days exposure
for FC and PC, 1645.9 days for UPMU), SK-II (798.6 days exposure for FC and PC, 827.7 daysfor UPMU), and SK-III (518.1 days exposure for FC and PC, 635.6 days for UPMU) periods
through Apr. 1996 to Sep. 2008 were used.
In the normal mass hierarchy case, ∆m 223 , sin2 θ23 and sin2 θ13 are constrained to 1.88 ×10−3< ∆m223 < 2.75 ×10−3 eV2 , 0.406 < sin2 θ23 < 0.629, sin2 θ13 < 0.066 at 90 % C.L.. In
the inverted mass hierarchy case, ∆m 223 , sin2 θ23 and sin2 θ13 are constrained to 1.98 ×10−3< ∆m223 < 2.81 ×10−3 eV2 , 0.425 < sin2 θ23 < 0.644, sin2 θ13 < 0.122 at 90 % C.L.. This
full 3-flavor analysis gives consistent results on the oscillation parameters with the past 2-flavor
analysis and show that the allowed parameter regions are less restricted taking account of theeffects from the parameters such as δ cp sin2 θ13 parameter is also consistent with the results
obtained in the past reactor experiment.
Although it was shown that the analysis has sensitivity, no significant constraint on CP
phase at 90 % C.L. was obtained for both mass hierarchies and there is no clear preference ofmass hierarchy. However, the present analysis demonstrated the usefulness of high statistics ofthe atmospheric neutrino data for the understanding of θ 13 , δcp and mass hierarchy as well as
the measurement of ∆m223 and sin2 2θ23 .

159

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165

Appendix A

Oscillation analysis
A.1

Solar terms effect

The results of the oscillation analysis with different solar terms are checked. Since the globalbest of ∆m212 and sin2 θ12 parameters is obtained by [37] with 1σ allowed region, two parameter
sets, which are selected in the allowed region, are used for this check :(∆m212 , sin2 θ12 ) = (7.45 × 10−5 , 0.288) · · · A= (7.89 × 10−5 , 0.326) · · · B.

The analysis fitting process is performed in the same way described in Section 6 for both of theabove two sets of solar parameters. Figure A.1 shows the ∆m 223 and sin2 θ13 parameters allowed
region obtained by the analysis using the two solar parameters together with the global-bestsolar parameters, (∆m212 , sin2 θ12 ) = (7.65 × 10−5 , 0.304). These allowed regions are exactly thesame for these analyses with different solar parameter sets. It is obvious that the results of thisanalysis are not affected by the solar parameters within the parameter error size.

A.2

Zenith angle distributions

From Figure A.2 to Figure A.7 the zenith angle distributions together with the best-fit
distributions in the normal hierarchy case for each SK run period are shown.

A.3

Best-fit systematic uncertainties

The systematic error parameters ( j ) for the best fit point are summarized in a bar graphwhich indicates the error parameters ( j ) in units of the estimated 1σ error (σ jsys ) value for eachsystematic error term. From Figure A.8 to Figure A.10 the best-fit results in normal hierarchy
case are shown and from Figure A.11 to Figure A.13 the best-fit results in inverted hierarchy
case are shown.

166

Inverted hierarchy

0.00350.0035

0.003

0.003

∆ m223∆ m223

Normal hierarchy

0.00250.002

0.001500.00250.002

0.001500.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
sin2 θ13

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
sin2 θ13

Figure A.1: The allowed region contour of sin 2 θ13 and ∆m223 parameters obtained by the oscillation analyses with the different solar parameters, global-best (black-solid lines), parameter
set A (red-dashed lines) and parameter set B (green-dotted lines), in normal hierarchy (left)
and inverted hierarchy (right). A contour indicates 68 %, 90 % and 99 % C.L. allowed regions by
three lines. Since there is no difference on the results among the three analyses, all lines overlap.

167

SK1 Sub-GeV 1-ring e-like 0-decay
100-250(MeV/c)250-400

400-6301000 < Pl

630-1000

100

50

0

-1

1-1

1-1

1-1

1-1

1

cosΘ
60

SK1 Sub-GeV 1-ring µ-like 0-decay

SK1 Sub-GeV 1-ring µ-like 1-decay

200-250(MeV/c)200-250(MeV/c)250-400

400-6301000 < Pl

630-1000100

40

250-400

400-630630-1000

1000 < Pl

75
50

20
25

0

-1

1-1

1-1

1-1

1-1

1

0

-1

cosΘ
SK1 Multi-GeV 1-ring e-like

80

1-1

1-1

1-1

1-1

1

cosΘ

1 - 2.5 (GeV/c)

2.5 - 5 (GeV/c)

5 - 10 (GeV/c)

SK1 Multi-GeV 1-ring µ-like

100
10 (GeV/c)< Pl

2.5 (GeV/c) < Pl

1 - 2.5 (GeV/c)

80

60

60

40
40

20
0

20

-1

1-1

1-1

1-1

1

0

-1

0

cosΘ
SK1 Multi-GeV multi-ring e-like

50

1-1

0

1

cosΘ

1 - 2.5 (GeV/c)

2.5 - 5 (GeV/c)

5 - 10 (GeV/c)

SK1 multi-ring µ-like60

Pl< 1(GeV/c)

10 (GeV/c)< Pl

1 - 2.5 (GeV/c)

2.5 - 5 (GeV/c)

5 (GeV/c)< Pl

40
40

30
20

20

10
0

-1

1-1

1-1

1-1

1

cosΘ

0

-1

1-1

1-1

1-1

1

cosΘ

Figure A.2: SK-I zenith angle distributions for FC events for data (line with error bar) and MC
assuming no oscillation (dashed line) and the best-fit results (solid line). Normal hierarchy isassumed.

168

SK1 PC stop25
Pl < 2.5 (GeV/c)

SK1 PC through80

2.5 (GeV/c) < Pl

Pl< 1(GeV/c)20

1 - 2.5 (GeV/c)

2.5 - 5 (GeV/c)

5 (GeV/c)< Pl

60

15
40
10
20

5
0

-1

0

1-1

0

1

0

-1

cosΘ

1-1

1-1

1-1

1

cosΘ
SK1 Upmu stopping60

Pl < 2.5 (GeV/c)

2.5 - 5 (GeV/c)

5 (GeV/c)< Pl

40

20

0

-1

0-1

0-1

0

cosΘ
SK1 Upmu non-showering

SK1 Upmu showering

80

300

60
200

40

100

0

20
0
-1

-0.8

-0.6

-0.4

-0.2

0

-1

-0.8

-0.6

-0.4

-0.2

0

cosΘ

cosΘ

Figure A.3: SK-I zenith angle distributions for PC and UPMU events for data (line with error bar)and MC assuming no oscillation (dashed line) and the best-fit results (solid line). Normal hierarchy
is assumed.

169

SK2 Sub-GeV 1-ring e-like 0-decay

80100-250(MeV/c)250-400

400-6301000 < Pl

630-1000

60
40
20
0

-1

1-1

1-1

1-1

1-1

1

cosΘ
40

SK2 Sub-GeV 1-ring µ-like 0-decay

SK2 Sub-GeV 1-ring µ-like 1-decay

200-250(MeV/c)200-250(MeV/c)250-400

400-6301000 < Pl

630-100060

250-400

400-630630-1000

1000 < Pl

30
40
20
20

10
0

-1

1-1

1-1

1-1

1-1

1

0

-1

cosΘ
SK2 Multi-GeV 1-ring e-like

40

1-1

1-1

1-1

1-1

1

cosΘ

1 - 2.5 (GeV/c)

2.5 - 5 (GeV/c)

5 - 10 (GeV/c)

SK2 Multi-GeV 1-ring µ-like

60
10 (GeV/c)< Pl

2.5 (GeV/c) < Pl

1 - 2.5 (GeV/c)

30
40
20
20
10
0

-1

1-1

1-1

1-1

1

0

-1

0

cosΘ
SK2 Multi-GeV multi-ring e-like

25

1-1

0

1

cosΘ

1 - 2.5 (GeV/c)

2.5 - 5 (GeV/c)

5 - 10 (GeV/c)

SK2 multi-ring µ-like40

Pl< 1(GeV/c)

10 (GeV/c)< Pl

20

1 - 2.5 (GeV/c)

2.5 - 5 (GeV/c)

5 (GeV/c)< Pl

30

15
20
10
10

5
0

-1

1-1

1-1

1-1

1

cosΘ

0

-1

1-1

1-1

1-1

1

cosΘ

Figure A.4: SK-II zenith angle distributions for FC events for data (line with error bar) and MC
assuming no oscillation (dashed line) and the best-fit results (solid line). Normal hierarchy isassumed.

170

SK2 PC stop15
Pl < 2.5 (GeV/c)

SK2 PC through

2.5 (GeV/c) < Pl

Pl< 1(GeV/c)40

10

1 - 2.5 (GeV/c)

2.5 - 5 (GeV/c)

5 (GeV/c)< Pl

30
20

5
10

0

-1

0

1-1

0

1

0

-1

cosΘ

1-1

1-1

1-1

1

cosΘ
SK2 Upmu stopping30

Pl < 2.5 (GeV/c)

2.5 - 5 (GeV/c)

5 (GeV/c)< Pl

20

10

0

-1

0-1

0-1

0

cosΘ
SK2 Upmu non-showering

SK2 Upmu showering

30

150

20
100

10

50

0

0
-1

-0.8

-0.6

-0.4

-0.2

0

-1

-0.8

-0.6

-0.4

-0.2

0

cosΘ

cosΘ

Figure A.5: SK-II zenith angle distributions for PC and UPMU events for data (line with errorbar) and MC assuming no oscillation (dashed line) and the best-fit results (solid line). Normal
hierarchy is assumed.

171

SK3 Sub-GeV 1-ring e-like 0-decay

60

100-250(MeV/c)250-400

400-6301000 < Pl

630-1000

40

20

0

-1

1-1

1-1

1-1

1-1

1

cosΘ
SK3 Sub-GeV 1-ring µ-like 0-decay
200-250(MeV/c)30

250-400

400-630

SK3 Sub-GeV 1-ring µ-like 1-decay

50

200-250(MeV/c)1000 < Pl

630-1000

250-400

400-630630-1000

1000 < Pl

40
30

20

20
10
10
0

-1

1-1

1-1

1-1

1-1

1

0

-1

cosΘ
SK3 Multi-GeV 1-ring e-like

30

1-1

1-1

1-1

1-1

1

cosΘ

1 - 2.5 (GeV/c)

2.5 - 5 (GeV/c)

5 - 10 (GeV/c)

SK3 Multi-GeV 1-ring µ-like40
10 (GeV/c)< Pl

2.5 (GeV/c) < Pl

1 - 2.5 (GeV/c)

30
20
20
10
10
0

-1

1-1

1-1

1-1

1

0

-1

0

cosΘ
SK3 Multi-GeV multi-ring e-like

20

1-1

0

1

cosΘ

1 - 2.5 (GeV/c)

2.5 - 5 (GeV/c)

5 - 10 (GeV/c)

SK3 multi-ring µ-like25

Pl< 1(GeV/c)

10 (GeV/c)< Pl

1 - 2.5 (GeV/c)

2.5 - 5 (GeV/c)

5 (GeV/c)< Pl

20

15

15
10
10
5
0

5

-1

1-1

1-1

1-1

1

cosΘ

0

-1

1-1

1-1

1-1

1

cosΘ

Figure A.6: SK-III zenith angle distributions for FC events for data (line with error bar) andMC assuming no oscillation (dashed line) and the best-fit results (solid line). Normal hierarchy isassumed.

172

SK3 PC stop15
Pl < 2.5 (GeV/c)

SK3 PC through

2.5 (GeV/c) < Pl

Pl< 1(GeV/c)30

1 - 2.5 (GeV/c)

2.5 - 5 (GeV/c)

5 (GeV/c)< Pl

10
20

5

0

10

-1

0

1-1

0

1

0

-1

cosΘ

1-1

1-1

1-1

1

cosΘ
SK3 Upmu stopping30

Pl < 2.5 (GeV/c)

2.5 - 5 (GeV/c)

5 (GeV/c)< Pl

20

10

0

-1

0-1

0-1

0

cosΘ
SK3 Upmu showering

SK3 Upmu non-showering

150

30
100

20
50

0

10

0
-1

-0.8

-0.6

-0.4

-0.2

0

-1

-0.8

-0.6

-0.4

-0.2

0

cosΘ

cosΘ

Figure A.7: SK-III zenith angle distributions for PC and UPMU events for data (line with errorbar) and MC assuming no oscillation (dashed line) and the best-fit results (solid line). Normal
hierarchy is assumed.

173

Figure A.8: (Normal hierarchy, No.1) The best-fit value of the systematic error parameter  j inunits of the estimated 1σ error value for each systematic error term shown in the right column.

174

Figure A.9: (Normal hierarchy, No.2) The best-fit value of the systematic error parameter  j inunits of the estimated 1σ error value for each systematic error term shown in the right column.

175

Figure A.10: (Normal hierarchy, No.3) The best-fit value of the systematic error parameter  j inunits of the estimated 1σ error value for each systematic error term shown in the right column.

176

Figure A.11: (Inverted hierarchy, No.1) The best-fit value of the systematic error parameterj in units of the estimated 1σ error value for each systematic error term shown in the rightcolumn.

177

Figure A.12: (Inverted hierarchy, No.2) The best-fit value of the systematic error parameterj in units of the estimated 1σ error value for each systematic error term shown in the rightcolumn.

178

Figure A.13: (Inverted hierarchy, No.3) The best-fit value of the systematic error parameterj in units of the estimated 1σ error value for each systematic error term shown in the rightcolumn.

179