Atmospheric neutrino oscillation analysis with solar terms in
Super-Kamiokande

Yumiko Takenaga
February 15, 2008

Abstract
The atmospheric neutrino data have been well explained in the two-flavor νµ ↔ ντ oscillation scheme and the ∆m223 and sin2 2θ23 parameters have been measured. No evidence of theatmospheric νe oscillation has been observed and the mixing angle θ13 is consistent with zero by
the results of θ13 search experiments. On the other hand, thanks to the precise measurements
by solar neutrino observations and the KamLAND experiment, the LMA-MSW solution of thesolar neutrino problem is established and the mixing angle θ12 and mass difference ∆m212 havebeen measured accurately.
If these 1-2 parameters are considered into the atmospheric neutrino oscillation analysis, theoscillation of low energy atmospheric νe might be observed even in case of zero-θ13 . The effectof the νe oscillation induced by the 1-2 parameters depends on the octant of θ23 (θ23 > 45◦ orθ23 < 45◦ ). Therefore the atmospheric neutrino oscillation analysis including the 1-2 parameters
has a possibility to determine the octant of θ23 for the non-maximal sin2 2θ23 . This information
cannot be obtained by the standard νµ ↔ ντ two-flavor oscillation analysis because the oscillation
probability in that framework depends on sin2 2θ23 , not on sin2 θ23 .
We have observed large number of atmospheric neutrino events in Super-Kamiokande. In
this thesis, the atmospheric neutrino data from the Super-Kamiokande-I (1996-2001) and SuperKamiokande-II (2003-2005) are summarized and used in the oscillation analysis. The analysis
has been done in the two schemes, pure two-flavor νµ ↔ ντ analysis and three-flavor analysis
with the 2-3 and 1-2 parameters. There is no significant discrepancy of θ23 from 45◦ , though theeffect of the 1-2 parameters is seen in the χ2 distribution. The stronger constraint on sin2 θ23
has been observed in the analysis with the 1-2 parameters. The 1σ arrowed region for sin2 θ23 is
0.43 < sin2 θ23 < 0.58, which is equivalent to sin2 2θ23 > 0.973.

Acknowledgments
First of all, I would like to express my great gratitude to Prof. Takaaki Kajita for givingme the opportunity of studying atmospheric neutrino. I have learned a lot of things under his
excellent guidance. This thesis would never exist without his invaluable support.
I would like to be appreciative to Prof. Yoichiro Suzuki, the spokesperson of the SuperKamiokande experiment. He encouraged me in many cases.
I further owe my sincere thanks to Prof. M.Shiozawa and K.Okumura. They were willing togive me many valuable advice and support since I entered graduate school.
I would be grateful to all the members of the atmospheric neutrino analysis group, especially Prof. K.Kaneyuki, Prof. Y.Itow, Prof. S.Moriyama, Prof. Y.Hayato, M.Miura,
Y.Obayashi, J.Kameda, C.Saji, S.Nakayama, M.Ishitsuka, I.Higuchi, G.Mitsuka, H.Nishino,
C.Ishihara, S.Hazama, E.Kearns, H.W.Sobel, C.K.Jung, R.J.Wilkes, J.G.Learned, C.W.Walter,
K.Scholberg, A.Habig, S.Mine, N.Tanimoto, J.L.Raaf, S.Desai, W.Wang, J.S.Jang, H.K.Seo,
T.Kato, R.Terri, R.Wendell, C.Regis, F.Dufour, E.Thrane, M.Litos, and C.Simon.
I would like to extend my gratitude to ICRR staffs and members, Prof. M.Nakahata,
Prof. Y.Takeuchi, Y.Koshio, K.Abe, A.Takeda, H.Sekiya, S.Yamada, H.Ogawa, K.Kobayashi,
H.Watanabe, A.Minamino, T.Iida, M.Ikeda, K.Ueshima, K.Ueno, and Y.Nakajima.
The analysis in this thesis has also relied on the hard work by all members of SuperKamiokande. This thesis is greatly indebted to them. I gratefully acknowledge the cooperationof Kamioka Mining and Smelting Company.
I would like to thank all the people who supported and encouraged me during my time in
graduate school.

Contents
1 Introduction
1.1 Neutrino Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Neutrino Oscillation Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Solar Neutrino and Reactor Neutrino Observations . . . . . . . . . . . . .
1.2.2 Atmospheric Neutrino and Accelerator Neutrino Observations . . . . . . .
1.2.3 Constraint on the Mixing Angle θ13 . . . . . . . . . . . . . . . . . . . . .
1.3 Sub-dominant Oscillation Effects driven by the Solar Terms . . . . . . . . . . . .

1
1
3
3
3
4
5

2 The Super-Kamiokande Detector
2.1 Cherenkov Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Water Tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Inner detector photomultiplier tube . . . . . . . . . . . . . . . . . . . . .
2.2.3 Outer detector photomultiplier tube . . . . . . . . . . . . . . . . . . . . .
2.3 Water Purification System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Radon Hut and Air Purification System . . . . . . . . . . . . . . . . . . . . . . .
2.5 Electronics and Data Acquisition System . . . . . . . . . . . . . . . . . . . . . .
2.5.1 Inner detector electronics and data acquisition . . . . . . . . . . . . . . .
2.5.2 Outer detector electronics and data acquisition . . . . . . . . . . . . . . .
2.5.3 Trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.1 Relative Gain Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.2 Absolute Gain Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.3 Relative Timing Calibration . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.4 Water Transparency Measurement . . . . . . . . . . . . . . . . . . . . . .
2.6.5 Absolute Energy Calibration . . . . . . . . . . . . . . . . . . . . . . . . .

10
10
12
12
14
14
18
18
21
21
22
22
24
24
25
25
28
31

3 Simulation
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 . . . . . . . . . . . . . . . . . . . . . . . . . . .

41
41
41
46
46
48
52
53

i

3.3.5 Nuclear Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Upward-Going Muon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Detector Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53
55
56

4 Data Reduction
4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Reduction for Fully Contained Sample . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 First Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Second Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.3 Third Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.4 Fourth Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.5 Fifth Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.6 FC Reduction Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Reduction for Partially Contained Sample . . . . . . . . . . . . . . . . . . . . . .
4.3.1 First Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Second Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.3 Third Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.4 Fourth Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.5 Fifth Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.6 PC Reduction Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Reduction for Upward-Going Muon Sample . . . . . . . . . . . . . . . . . . . . .
4.4.1 Charge cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.2 Zenith angle cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.3 Scanning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.4 Efficiency and Background for Upward-Going Muon Sample . . . . . . . .

57
57
59
59
60
61
70
70
74
76
76
76
78
81
82
87
88
88
89
90
90

3.4
3.5

5 Event Reconstruction
94
5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.2 Vertex Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.2.1 Point-Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.2.2 Ring Edge Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.2.3 TDC-Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.3 Ring Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.3.1 Ring Candidate Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.3.2 Ring Candidate Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.4 Particle Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.4.1 Expected Charge Distributions . . . . . . . . . . . . . . . . . . . . . . . . 101
5.4.2 Estimation of Particle Types . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.5 Precise Vertex Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.6 Momentum Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.7 Ring Number Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.8 Decay Electron Finding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.9 Event reconstruction for single-ring π 0 sample . . . . . . . . . . . . . . . . . . . . 111
5.9.1 Reconstruction algorism for π 0 fitter . . . . . . . . . . . . . . . . . . . . . 112
5.9.2 Best-fit Ring Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.10 Event Reconstruction for Upward-Going Muon Sample . . . . . . . . . . . . . . . 116ii

6 Data set
118
6.1 Event Samples for FC Sub-GeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.1.1 FC Sub-GeV single-ring e-like events . . . . . . . . . . . . . . . . . . . . . 120
6.1.2 FC Sub-GeV single-ring µ-like events . . . . . . . . . . . . . . . . . . . . . 122
6.1.3 FC Sub-GeV two-ring π 0 -like events . . . . . . . . . . . . . . . . . . . . . 127
6.2 Vertex Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.3 Number of Rings and Momentum Distributions . . . . . . . . . . . . . . . . . . . 132
6.4 Flavor Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.5 Zenith angle Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7 Oscillation analysis with solar terms
138
7.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.2 Analysis Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.3 Systematic Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.3.1 Systematic Uncertainties in Atmospheric Neutrino Flux . . . . . . . . . . 145
7.3.2 Systematic Uncertainties in Neutrino Interaction . . . . . . . . . . . . . . 149
7.3.3 Systematic Uncertainties in Event Selection . . . . . . . . . . . . . . . . . 152
7.3.4 Systematic Uncertainties in Event Reconstruction . . . . . . . . . . . . . 154
7.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
8 Future Sensitivity on the θ23 Determination

168

9 Conclusion

171

iii

Chapter 1

Introduction
1.1

Neutrino Oscillation

In the Standard Model of elementary particles, neutrinos are regarded as exactly masslessparticles [1, 2, 3]. However, there is no fundamental reason to forbid the finite masses of neutrinos. Neutrino oscillation is proposed by Maki, Nakagawa and Sakata in 1962 as a consequenceof the finite masses of neutrinos [4].
If neutrinos have finite masses, their flavor eigenstates can be written as the superpositionsof the mass eigenstates as follows :|να i =

3X
i=1

Uαi |νi i,(1.1)

where |να i represents the flavor eigenstates νe , νµ and ντ and |νi i represents the mass eigenstates
ν1 , ν2 and ν3 . Uαi is a 3×3 unitary mixing matrix known as the MNS matrix. The mixing matrix
U can be written as a product of three rotation matrices using the mixing angle θ12 , θ23 and θ13
for each flavor 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∗′ i exp(−iEi t)|να′ (t = 0)i,

(1.4)

α′

where Ei is an eigenvalue of the mass eigenstate |νi i. Therefore the probability for flavor

1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

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

Thus, due to the flavor mixing of massive neutrinos, the flavor transition phenomenon,
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 (Equation (1.7)) is expressed in much simpler form :22 (Ei − Ej )t.
(1.9)
P( να → νβ ) = sin 2θ sin
2
If neutrinos have finite masses, the following approximation can be used :qq
Ei − Ej =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 in
the two-flavor mixing case is written as follows :
!
2L
∆m
ν
ij
P (να → να ) = 1 − sin2 2θ sin2
(1.11)4Eν
!
2 (eV2 )L (km)1.27∆mνij(1.12)= 1 − sin2 2θ sin2
Eν (GeV)Lν22= 1 − sin 2θ sin π,(1.13)lwhere l ≡ 4πE/∆m2ij is the oscillation length. The oscillation probability is characterized by themixing angle θ, the mass squared difference ∆m2ij , the neutrino flight length Lν and the neutrinoenergy Eν . The oscillation amplitude is maximum when Lν [km]/Eν [GeV] = π/2.53 · ∆m2ij [eV2 ].

2

1.2

Neutrino Oscillation Experiments

There are a lot of experiments which have observed neutrino oscillations and measured
oscillation parameters, such as 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 Observations

Electron neutrinos are produced by nuclear reactions in the Sun. Since the energy range ofthe solar neutrinos is from several hundred keV to 15 MeV and the distance between the Sun
and the Earth is ∼ 108 km, the observation of νe disappearance is sensitive to ∆m2 down to∼ 10−11 eV2 . Homestake [5] experiment is the oldest solar neutrino experiment, which used
the interaction νe +37 Cl → e− +37 Ar. The produced Ar atoms were counted by a chemical
technique. Kamiokande [6] experiment used electrons in H2 O molecules for targets of elastic
scattering νl +e− → νl +e− (l=e, µ, τ ). SAGE [7] and GALLEX [8] experiments used νe +71 Ga→ e− +71 Ge interaction and the produced Ge atoms were counted by a chemical technique.
Super-Kamiokande [9] is a successor of Kamiokande experiment. SNO experiment [10] started
in 1999, which uses the Cherenkov detector filled with heavy water D2 O. They measures theflux of νe via νe +D → e− +p+p, as well as the total neutrino flux via νx +D → νx +p+n.
All the results except for the neutral current interaction measurement at the SNO observed
the significant deficit in the solar neutrino flux compared with the standard solar model [11].
Combining the results of SNO and SK, the LMA-MSW solution of the solar neutrino problem
was indicated [12]. The result was confirmed by the KamLAND experiment, which is a liquid
scintillator detector and searches for ν e oscillations. ν e ’s generated at distant power reactors aredetected by the interaction ν e +p→ e+ +n, where the prompt signal from the positron and thedelayed coincidence of characteristic gammas from neutron capture allow an efficient reduction of
backgrounds. The ν e flux at the detector site is dominated by a number of reactors at an average
distance of ∼ 180 km. Since the energy of the produced ν e s is around 3 MeV, the experiment issensitive to ∆m2 down to several 10−6 eV2 . The combined analysis of the solar neutrino data
and the KamLAND data leads to the oscillation parameters [13]:7.3 × 10−5 < ∆m212 < 8.1 × 10−5 eV 2 ,0.28 < sin2 θ12 < 0.37 (2σ).

1.2.2(1.14)

Atmospheric Neutrino and Accelerator Neutrino Observations

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 ,(1.15)→ e− + ν e + νµ ,(1.16)−→ µ + νµThe energy spectrum of the atmospheric neutrinos peaks around 40 MeV and extends to theenergy range above TeV. The neutrino flight length ranges from about 15 km for neutrinos
produced above the detector to 13000 km for neutrinos produced on the opposite side of the3Earth. Since typical atmospheric neutrino experiments have enough event statistics for neutrinoenergies above several hundred MeV, the observation of atmospheric neutrinos is sensitive to∆m2 down to 10−4 eV 2 . The predicted total flux of atmospheric neutrinos has ∼ 20 % uncertainty
which mostly comes from the uncertainties on cross sections of hadronic interactions and on
fluxes of primary cosmic ray components [14]. However, the flavor ratio (νµ + ν µ )/(νe + ν e )is predicted with only 3 % uncertainty. This ratio has been measured by several undergroundexperiments 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 double ratio R obtained by underground Cherenkov detectors, Kamiokande [15, 16] andIMB [17, 18] and an iron calorimeter detector Soudan 2 [19] was significantly lower than 1. In
1998, Super-Kamiokande reported that the zenith angle distributions of muon neutrino eventswere asymmetric and concluded that their measurement results gave an evidence for neutrino
oscillation [20]. MACRO [21] and Super-Kamiokande [22] have studied about upward-going
muons produced by energetic νµ charged current neutrino interactions in the rock surrounding
the detector and their data are consistent with the two-flavor neutrino oscillation νµ ↔ ντ .
Super-Kamiokande also reported the observation of an oscillation signature with L/E dependence [26], which is well described by the hypothesis that νµ oscillates to ντ with a probabilityof the nearly maximal mixing.
Long-baseline experiments using the muon neutrino beam produced by accelerators, suchas K2K [23] and MINOS [24] also confirmed the atmospheric neutrino oscillation by measuring
the spectral distortion and the decreasing νµ flux during the travel. In the two-flavor mixing
framework, the neutrino oscillation parameters are constrained to be :1.9 × 10−3 < ∆m223 < 3.1 × 10−3 eV 2 ,0.92 < sin2 2θ23

(1.17)

at 90 % C.L.. Figure 1.1 shows 90 % confidence level allowed parameter regions of the several
experiments.

1.2.3

Constraint on the Mixing Angle θ13

The atmospheric neutrino data have also been used to constrain θ13 in the three-flavor
mixing framework with the one mass scale dominance approximation (|m22 − m21 | ≪ |m23 − m22 | ∼|m23 − m21 |). In this framework, oscillations of atmospheric νe s are expressed by :P(νe → νe ) = 1 − sin2 2θ13 sin2 (1.27∆m2 L),
E(1.18)P(νµ → νe ) = P(νe → νµ )1.27∆m2 L),(1.19)
E
So far, no evidence for the atmospheric νe oscillation has been observed and the mixing angle
θ13 is consistent with zero as a result of the oscillation analysis using the atmospheric neutrino
data. Non-zero θ13 has also been searched for by reactor neutrino experiments or accelerator
neutrino experiments and their results are also consistent with zero-θ13 . The current upper limiton θ13 comes from the CHOOZ experiment [27]:= sin2 θ23 sin2 2θ13 sin2 (sin2 2θ13 < 0.15
at 90 % C.L.(assuming ∆m223 = 2.5×10−3 eV2 ).
4(1.20)

-222∆m (eV )10

SK
SK L/E
K2K
MINOS
-3

10

0.5

0.6

0.7

0.8

0.9

1

2

sin 2θ

Figure 1.1: 90 % confidence level allowed regions of oscillation parameters for νµ ↔ ντ oscillation.
The horizontal axis shows sin2 2θ and the vertical axis shows ∆m2 . The results are taken fromSuper-Kamiokande[25] (solid line), SK L/E [26] (dotted line), K2K experiment [23] (dashed line)
and MINOS experiment[24] (dash-dotted line).

1.3

Sub-dominant Oscillation Effects driven by the Solar Terms

As described in the previous section, no evidence for the atmospheric νe oscillation induced
by non-zero θ13 has been found. However, if the LMA-MSW oscillation parameters are taken
into account, the oscillation of low energy atmospheric νe s is expected to appear even in thecase of θ13 =0.
If the transition probability from νe to νx (x = µ or τ ) in matter driven by ∆m212 is denotedby P2 , the oscillation probabilities of νe and νµ are expressed as [28, 29]:P(νe → νe ) = 1 − P2 ,P(νe → νµ ) =(1.21)P(νµ → νe ) = c223 P2 ,P(νµ → νµ ) = 1 − c423 P2 − 2s223 c223 1 −φ ∼∆m231 + s212 ∆m221(1.22)
p

 L
.
2Eν


1 − P2 cos φ ,(1.23)(1.24)

Using these oscillation probabilities, the oscillated atmospheric electron and muon neutrino
fluxes are expressed by :Feosc = Fe0 · P(νe → νe ) + Fµ0 · P(νµ → νe )= Fe0 1 + P2 (rc223 − 1) ,
Fµosc

=

=

Fe0 · P(νe → νµ ) + Fµ0 · P(νµ → νµ )pc223 2102
Fµ 1 −(rc23 − 1)P2 − sin 2θ23 (1 − 1 − P2 cos φ) ,r

2

5

(1.25)(1.26)

cosΘν

0

1

-0.1

0.9

-0.2

0.8

-0.3

0.7

-0.4

0.6

-0.5

0.5

-0.6

0.4

-0.7

0.3

-0.8

0.2

-0.9

0.1

-1 -1
10

0
1
Eν (GeV)10

Figure 1.2: The calculated νe transition probability P2 for atmospheric neutrinos with an energyEν (horizontal axis) and a direction of cosΘν (vertical axis). The 1-2 oscillation parameters are
assumed to be ∆m212 = 7.9 × 10−5 eV 2 and sin2 θ12 = 0.31. The Earth’s matter effect is taken
into account.

where Fe0 and Fµ0 are the atmospheric neutrino fluxes without oscillations. r is the ratio of theoriginal muon and electron neutrino fluxes Fµ0 /Fe0 , which is ∼ 2 for neutrinos in the low energy
region.
Figure 1.2 shows the calculated P2 for atmospheric neutrinos with an energy Eν and a
direction Θν assuming ∆m212 = 7.9 × 10−5 eV 2 and sin2 θ12 = 0.31. Since the oscillation effectis larger in the lower energy region as shown in Figure 1.2, the Sub-GeV atmospheric neutrino
events play an important role to observe the sub-dominant oscillation effect.
The atmospheric electron neutrino flux decreases by P2 , on the other hand, it increases by themuon neutrino oscillation (Equation (1.22)). As a result, the atmospheric electron neutrino fluxchanges by the effect of oscillations driven by the 1-2 parameters and the size of the oscillation
effect depends on cos2 θ23 as seen in Equation (1.25). Equation (1.25) can be rewritten as follows :Feosc− 1 = P2 (r cos2 θ23 − 1).
Fe0(1.27)

Since the flavor ratio r is approximately 2 in the Sub-GeV neutrino energy region, in the caseof cos2 θ23 = 0.5 (i.e. sin2 θ23 = 0.5, θ23 = 45◦ ), the increasing effect and the decreasing
effect on the electron neutrino flux are expected to be canceled out. If cos2 θ23 > 0.5 (i.e.
sin2 θ23 < 0.5, θ23 < 45◦ ), an excess of the Sub-GeV e-like sample is expected. If cos2 θ23 < 0.5(i.e. sin2 θ23 > 0.5, θ23 > 45◦ ), the Sub-GeV e-like sample is expected to be reduced.
Therefore the atmospheric neutrino experiments have a possibility to determine the octant of
θ23 for the non-maximal sin2 θ23 using the effect due to the solar neutrino oscillation parameters.
The information on the octant of θ23 cannot be obtained by the standard νµ ↔ ντ two-flavor
oscillation analysis because the oscillation probability depends on sin2 2θ23 , not on sin2 θ23 inthat framework.
In this thesis, the precise oscillation analysis of the Super-Kamiokande atmospheric neutrino
data considering the sub-dominant oscillation effect induced by the 1-2 parameters is described.
6

Neosc / Ne2flavor full-mixing

1.1
Sub-GeV e-like

1.08

sin θ23=0.4
2
sin θ23=0.5
2
sin θ23=0.6
2

1.06
1.04
1.02
1

0.98
0.96
-1

-0.5

0

0.5

1

cosθ

Figure 1.3: Zenith angle dependence of the ratio of the number of FC Sub-GeV single-ringe-like events including sub-dominant oscillations due to the 1-2 oscillations (Neosc ) and withpure νµ ↔ ντ two-flavor full-mixing oscillations (Ne2f lavorf ull−mixing ). For calculation of Neosc ,∆m212 = 7.9 × 10−5 eV 2 , sin2 θ12 = 0.31, ∆m223 = 2.5 × 10−3 eV 2 and three different values ofsin2 θ23 , sin2 θ23 = 0.4 (dashed line), 0.5 (solid line) and 0.6 (dotted line), are assumed. For
calculation of Ne2f lavorf ull−mixing , ∆m223 = 2.5 × 10−3 eV 2 and sin2 θ23 = 0.5 are assumed.
Figure 1.3 shows the zenith angle dependence of relative effect due to 1-2 oscillations on thenumber of FC Sub-GeV single-ring e-like events for different values of sin2 θ23 . Here “Sub-GeV”
is defined to be less than 1.33 GeV in visible energy in a water Cherenkov detector. The 1-2
oscillation parameters are assumed as ∆m212 = 7.9 × 10−5 eV 2 and sin2 θ12 = 0.31. In the figure,the number of e-like events with sub-dominant 1-2 oscillations (Neosc ) is normalized by the number of e-like events with pure νµ ↔ ντ two-flavor full-mixing oscillations (Ne2f lavorf ull−mixing ).
For sin2 θ23 = 0.4 (dashed line) and 0.6 (dotted line), the sub-dominant 1-2 oscillation effectappears as a ∼2 % excess and deficit of Sub-GeV e-like events, respectively. The sub-dominant
1-2 oscillation effect is expected to be larger in the lower energy region (see Figure 1.2).
Figure 1.4 shows the corresponding plot for Sub-GeV single-ring µ-like events. The subdominant 1-2 oscillation effect for the µ-like sample is smaller in size and opposite. This isunderstood from Equations (1.25) and (1.26).
However, due to the large uncertainty in the absolute neutrino flux (∼20 %), it should not be
possible to extract the 1-2 oscillation effect by comparing the total number of e-like (or µ-like)events of the data and Monte Carlo. On the other hand, we notice that the (νµ + ν µ )/(νe + ν e )flux ratio is accurately predicted (∼3 %). Furthermore, the sub-dominant 1-2 oscillation effect is
opposite for e-like and µ-like as seen in Figures 1.3 and 1.4 . Therefore, by taking the µ-like/e-likeratio, one might be able to observe the 1-2 oscillation effect.
Figure 1.5 shows the zenith angle dependence of the ratio between the number of FC SubGeV single-ring µ-like events and the number of FC Sub-GeV single-ring e-like events. For
upward-going events, the difference on the µ/e ratio between the sin2 θ23 = 0.4 and sin2 θ23 = 0.6
cases reaches about 5 %, which is comparable to the size of the estimated systematic error on
the µ/e ratio measurement.
7

Nµosc / Nµ2flavor full-mixing

1.1
Sub-GeV µ-like

1.08

sin θ23=0.4
2
sin θ23=0.5
2
sin θ23=0.6
2

1.06
1.04
1.02
1

0.98
2

0.96

2flavor (sin 2θ23=0.96)

-1

-0.5

0

0.5

1

cosθ

Figure 1.4: Zenith angle dependence of the ratio of the number of FC Sub-GeV single-ring elike events including sub-dominant oscillations due to the 1-2 oscillations (Nµosc ) and with pure
νµ ↔ ντ two-flavor full-mixing oscillations (Nµ2f lavorf ull−mixing ). For the dash-dotted line, thenumerator is the number of µ-like events in the case of pure νµ ↔ ντ two-flavor oscillations withsin2 2θ23 = 0.96, which corresponds to sin2 θ23 = 0.4 or sin2 θ23 = 0.6. The other line definitions
are same as Figure 1.3.

Figure 1.5 suggests that the observation of the 1-2 oscillation effect might be possible, if wecontrol the Nµ /Ne ratio accurately. Therefore, in this thesis, we discuss the systematic error inthe (νµ + ν µ )/(νe + ν e ) flux ratio carefully.
Experimentally, the fraction of charged-current quasi-elastic (CCQE) events in the standard
Sub-GeV single-ring e-like (µ-like) sample in Super-Kamiokande is about 71 % (74 %). The
remaining events are due to charged-current non-quasi-elastic (CC non-QE) events and neutralcurrent (NC) events. However, the understanding of the details of the cross-section and thekinematics are rather poor. Therefore, in this thesis, we try to improve the purity of theevents by eliminating CC non-QE and NC events by several methods. Figure 1.6 compares thesensitivity of this experiment based on the standard Super-K event samples [25] and the special
event samples which the CC non-QE and NC events are reduced. It is clear that the sensitivity
in the octant of θ23 due to the 1-2 oscillation effect is improved by this work.
In this thesis, details of the improvements in the analysis of the 1-2 oscillation effect together
with the related systematic errors are described. Finally the results from this work using theSuper-K-I and II data are described.

8

(Nµ/Ne)osc / (Nµ/Ne)2flavor full-mixing

1.1
Sub-GeV1.08

sin θ23=0.4
2
sin θ23=0.5
2
sin θ23=0.6
2

1.06
1.04
1.02
1

0.98
2

0.96

2flavor (sin 2θ23=0.96)

-1

-0.5

0

0.5

1

cosθ

Figure 1.5: Zenith angle dependence of the Nµ /Ne ratio for FC Sub-GeV single-ring events
including sub-dominant oscillations due to the 1-2 oscillations normalized by the Nµ /Ne withpure νµ ↔ ντ two-flavor full-mixing oscillations. The line definitions are same as Figure 1.4.

5
4.5
4
3.5
∆χ2

3
2.5
2
1.5
1
0.5
0
0.4

0.5
sin2θ23

0.6

Figure 1.6: χ2 -χ2min distribution as a function of sin2 θ23 using the oscillated Monte Carlo eventsof a 6 year exposure with the true oscillation parameters of θ13 = 0, sin2 θ12 = 0.32, ∆m212 =7.8×10−5 eV2 , ∆m223 = 2.5×10−3 eV2 and sin2 θ23 = 0.4. Details of the definition of χ2 will
be described in Section 7.2. The solid line shows the sensitivity based on the standard Super-K
event samples, the dotted (blue) line based on the event samples which are reduced the CC
non-QE and NC interactions to half and the dashed (red) line based on the event samples where
all the CC non-QE and NC interactions are eliminated.

9

Chapter 2

The Super-Kamiokande Detector
Super-Kamiokande is a cylindrical 50 kt water Cherenkov detector located at Kamioka Observatory in Gifu Prefecture, Japan. The geographic coordinates of the site are 36◦ 25’N and137◦ 18’E, and the altitude above sea level is 370 m. The detector lies in a zinc mine under the
peak of Mt.Ikenoyama, where the mean rock overburden is ∼ 1000 m (2700 m water equivalent).
This overburden serves as a shield against cosmic ray muon background. The cosmic ray muon
flux at the Super-Kamiokande site is reduced by 5 orders of magnitude compared to that on thesurface of the earth.
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 thedetector maintenance in 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 SuperKamiokande detector was rebuilt after the accident with the half of the original PMT density inthe 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 for the reconstruction work to put the PMT density back to the SK-I level. The
Super-Kamiokande detector has restarted observation in June, 2006.
In this thesis, the data observed in the SK-I (1996-2001) and the SK-II running periods(2002-2005) are used.

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 [30]. The momentum threshold of Cherenkov radiation is determined
10

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 2.1: A typical observed neutrino event in Super-Kamiokande. The size of small circles
in the unrolled cylinder represents the amount of Cherenkov photons detected in each photomultiplier tube. The Cherenkov ring image is clearly visible.

by 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, muonsand charged pions are 0.57, 118 and 156 MeV/c , respectively.
Cherenkov light is emitted on a cone with a characteristic half opening angle θC 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
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 of 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 2.1 shows a visual display of a typical neutrino eventin the Super-Kamiokande detector.

11

Figure 2.2: Super-Kamiokande detector and its location.

2.2

Detector

2.2.1

Water Tank

A schematic view of the detector is shown in Figure 2.2. The whole size of the water tankis 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 inner region, this part of the detector is referred to the inner detector (ID), 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 intervals of 70 cm for SK-I.
The effective photocathode coverage of the ID is about 40 %, and the rest of the surface iscovered with black polyethylene terephthalate sheet, called ”black sheet”. For SK-II, 5,182 20inch PMTs are attached at one intervals to the supporting frame and the photocathode coverageof the ID is about 20 %.
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, each PMT in the OD is attached to a 60 cm × 60 cm
wavelength shifting plate, and the walls are covered with reflective material called ”tyvek sheet”.
The OD is used to veto 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 dead region. This region is not
12Tyvek8" PMT

Top

Black sheet

Barrel20" PMT

BottomFigure 2.3: Supporting frame of PMTs.

13

instrumented with any PMT. Stainless steel frames and signal and HV cables of the ID and OD
PMTs are contained in this dead region.

2.2.2

Inner detector 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 [31]. Later, thedynode structure and the bleeder circuit were modified for Super-Kamiokande to improve timing response and photon collection efficiency [32]. A schematic view of the PMT is shown in
Figure 2.4 and the specifications are summarized in Table 2.1.
The photocathode of the PMT is coated by bialkali (Sb-K-Cs) owing to the high spectral
sensitivity to Cherenkov light and the low thermionic emission. The quantum efficiency of the
PMT together with the Cherenkov spectrum is shown in Figure 2.5, in which the peak is 22 % at
360-400 nm (Figure 2.6). 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 107 at a supply high voltage from1500 V to 2000 V. Figure 2.7 shows the charge distribution for the single photoelectron signal
in which 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 SuperKamiokande is 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 upgrade 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 theinner PMTs are instrumented with acrylic covers in SK-II. A clear 12 mm thick UV-transparent
acrylic dome is put over the photo-cathode area of each PMT and the side of the PMT isprotected by the fiberglass shield with holes, which let water flow into the case freely as shownin Figure 2.8. Figure 2.9 shows the transparency of the acrylic cover for photons with normal
incidence in water, which is mode than 96 % above 350 nm of wavelength and the effect of the
PMT case is small. 11,146 and 5,182 PMTs are used in SK-I and SK-II, respectively.

2.2.3

Outer detector photomultiplier tube

1,885 8-inch PMTs, Hamamatsu R1408, are used in the OD. For SK-I, the OD PMTs werereused from the IMB experiment [33] after finishing that experiment. The photocathode of theOD 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 isabout 13 nsec without the wavelength shifter and about 15 nsec with the plate. However, since
the OD is used as a veto counter, rather than a particle traceker, The extra photons are ofimportande while the poor timing resolution is of little consequence.

14

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.

15

Quantum efficiency

Figure 2.5: The spectrum shape of Cherenkov light through pure water and the quantum efficiency of 20-inch PMT as a function of wavelength.

0.2

0.1

0

300

400

500

600
700
Wave length (nm)Figure 2.6: the quantum efficiency of ID 20-inch PMT as a function of wavelength.

16

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 2.7: Single photoelectron distribution of a typical 20-inch PMT.

Figure 2.8: PMT case attached to the inner PMT after SK-II experiment

1
0.98
0.96
0.94
0.92
0.9
0.88
0.86

300

350

400

450

500

550

600

Figure 2.9: The transparancy of the acylic case as a function of wavelength

17

2.3

Water 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. The outline of the system is shownin Figure 2.10. The system consists of the following components.
• 1 µm mesh filter

removes dust and small particles which reduce the water transparency.

• Heat exchanger

cools water to suppress the growth of bacteria. The water temperature in the detector
is kept at 14 ◦ C.

• Cartridge polisher

eliminates ions which also reduce water transparency.

• Ultra-Violet sterilizer

kills bacteria.

• Radon-reduced air dissolving tankdissolves radon-reduced air into the water toincrease radon removal efficiency at the vacuum degasifier stage which follows.

• Reverse osmosis filter

removes particulates.

• Vacuum degasifier

removes dissolved gases in water, such as theoxygen and radon.

• Ultra filter

removes minute particles of the order of 10 nm.

• Membrane degasifier

removes radon dissolved in water.

The typical number of particles larger than 0.2 µm is reduced to 6 particles/cc and the lightattenuation length is achiedved to be ∼100 m after purification. The resistivity of the water
entering the purification system from the detector is about 11 MΩ·cm. After the purification,
the water has an average resistivity of 18.20 MΩ·cm, approching the chemical limit.

2.4

Radon Hut and Air Purification System

It is also essential to have clean air in the detector and the experimental area to minimize
the radon level in the detector water. The radon concentration of the mine air in the accesstunnel to the experimental site has a strong seasonal variation of 2,000 ∼ 3,000 Bq/m3 during
summer and 100 ∼ 300 Bq/m3 during winter as shown in Figure 2.11. This is caused by theseasonal variation of the air flow inside the mine.
Fresh air from outside the mine is continuously pumped into the SK dome area at the rate of
10 m3 /minute through an air duct along the 1.8 km Atotsu access tunnel to the SK experimental
area. As a result, the typical radon concentration in the SK dome air is 20 ∼ 30 mBq/m3 .
18

Figure 2.10: The water purification system for Super-Kamiokande.

2000 Radon monitor readings at SuperK
3500
Mine Air (at sink)
Control Room Air
3000

radon level [Bq/m^3]

2500

2000

1500

1000

500

0
01/01/00

03/01/00

05/01/00

07/01/00

09/01/00

11/01/00

01/01/01

Figure 2.11: The upper histogram (solid curve) shows the measured radon concentration levelsin the mine air [34]. The lower histogram (dashed curve) shows the corresponding radon levelsin the SK dome.

19

COMPRESSORCARBONCOLUMNBUFFERTANK

AIR FILTER(0.3mm)AIR DRIERCARBONCOLUMNHEATEXCHANGER

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

COOLEDCHARCOAL(-40 oC)

SUPER-KAMIOKANDE AIR PURIFICATION SYSTEM

Figure 2.12: A schematic view of the radon-reducing air system.

Furthermore, to keep the radon level inside the detector absolute minimum, 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 [35]. A schematic view of
the air purification system is shown in Figure 2.12. The radon concentration of the radon-free
air is less than 3 mBq/m3 . The air purification system consists of compressors, a buffer tank,driers, and the filters. The air flow rate is about 18 m3 /hour. The process of the air purification
system is described as follows:• Compressor:
Compresses air to 7 ∼ 8.5 atmospheric pressure.
• Air Filter:
Removes dusts in sizes of ∼ 0.3 µm.
• Buffer Tank:
Stores the air.
• Air Drier:
Dries the air and removes CO2 gas to improve the radon removal capability in Carbon
Columns.
• Carbon Columns:
Removes radon gas using activated charcoal.
• Air Filter:
Further Removes small dust and particles of ∼ 0.01 µm.
• Cooled Charcoal Columns:
Further removes the remaining radon gas with the charcoal cooled down to –40◦ C.

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

QAC-A

QAC-A

to ADCSTART/STOP

TAC-B

TAC-B

CLEAR
GATE

THRESHOLD

QAC-B

QAC-B

DELAY

GATE
TRIGGERPED_START

Figure 2.13: A block diagram of analog input part of ATM.

2.5

Electronics and Data Acquisition System

2.5.1

Inner detector electronics and data acquisition

ID PMT signals are processed by custom built electronics modules called ATM (AnalogTiming-Module) of the TKO standard (TRISTAN KEK Online) [36, 37]. The ATM module
records the integrated charge and the arrival timing information of each PMT signal.
Figure 2.13 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
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 splitted PMT signal are fed to QAC. If
a 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. If 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

21

dynamic range with a resolution of 0.4 nsec. The remaining one signal becomes an output signalof ATM called PMTSUM.
Figure 2.14 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.

2.5.2

Outer detector electronics and data acquisition

Figure 2.15 shows a schematic view of the OD data acquisition system [38]. 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. . If 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. TDC module can record up to 8 QTC pulses with aresolution of 0.5 nsec. The dynamic range is set to 16 µsec which starts from 10 µsec before
the global trigger timing. The digitized data stored in TDC are read by a slave computer
through VME memory module called DPM (Dual Port Memory) and then sent to the online
host computer.

2.5.3

Trigger

Figure 2.16 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 is set to 4.6 MeV equivalent. The
SLE triggered events are not used in atmospheric neutrino analysis. In SK-II, the same trigger
scheme with the different thresholds is used. The threshold energies for the SK-II HE and LE
triggers are 10 MeV and 8 MeV, respectively.
The OD trigger is generated by a similar procedure. When an OD PMT signal exceeds
a threshold, a QTC module generates a rectangular pulse with 20 mV in height and 200 nsec
in width. These signals are also analog summed up to generate an OD-HITSUM signal. The
threshold for the OD trigger is set to 19 hits within a 200 nsec time window.
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

Super Memory Partnerinterface

VME
VME

interrupt reg.
TRG

20-inch PMT
SCH
TRIGGERATM

TRIGGER

x 20
x 240

PROCESSORHIT INFORMATION
ATM
GONG

Analog Timing ModuleTKO

PMT x 11200

ATM x ~1000

SMP x 48

online CPU(slave) x 9

Figure 2.14: 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 2.15: A schematic view of the outer detector data acquisition system.

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.

2.6

Calibration

2.6.1

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 necessary to determine the momentum
without systematic difference depending on its vertex position or direction.
Figure 2.17 shows a schematic view of the relative gain calibration system. Light generated
by a Xe lamp is passed through an ultraviolet (UV) filter and neutral density (ND) filter andthen split into four. One is injected into a scintillator ball via an optical fiber. The others areused to monitor the intensity of the Xe light and to make a calibration trigger. The scintillatorball is made of acrylic resin mixed with BBOT wavelength shifter and MgO powder diffuser.
BBOT wavelength shifter absorbs UV light and emits light with a peak at 440 nm, which is atypical wavelength of Cherenkov light.
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
· li · exp
(2.4)
Gi =
Q0 f (θ)
L
where Qi is the charge detected by the i-th PMT, li 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 isthe attenuation length and Q0 is the normalization factor. This measurement is performed forseveral positions of the scintillator ball, changing the voltage.
Figure 2.18 shows the relative gain distribution of all the PMTs after adjusting high voltage
values. The relative gain spread is about 7 %. The remaining gain difference is corrected insoftware.

24

Analog signal
from PMT
ATM threshold
400 nsec width

ADC gate
TDC startand stop

stop

start

HITSUM

200 nsec width

Global triggerATM
Sum of HITSUM
Master threshold
Global trigger
CentralHut

Figure 2.16: An overview of the ID trigger scheme.

2.6.2

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 gamma ray generated from neutron capture on Nickel nucleus is used as acalibration source to measure the single photoelectron distributions. The schematic view of the
Nickel calibration source is shown in Figure 2.19. Neutrons produced by spontaneous fission of
252 Cf are captured on surrounding Ni wires in polyethylene vessel, and low energy (6 ∼ 9 MeV)
gamma 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 ofa typical PMT is also shown in Figure 2.19. 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 to that of single photoelectrons. The mean value 2.055 pC is used as a constant toconvert the PMT charge from pico Coulomb to the number of p.e.s .

2.6.3

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.
Figure 2.20 shows the schematic view of the relative timing calibration system. N2 laser gen25

Number of PMTs in Each Bin

Figure 2.17: A schematic view of the relative gain measurement system using a Xe lamp.

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

26

mean

2.055pC

5103Ni wire+water

200mm
Cf

2

1

0

0

190mm

pC

Figure 2.19: Schematic view of an Nickel calibration source (left) and charge distribution of atypical ID PMT (right).

variable
attenuation
filteroptical fiber
optical fiber

N 2 laser
generator
λ=384nmSuper Kamiokande
inner tankmonitor
PMT
sig.

20’PMT
diffuserball
trigger
elec.
sig.
sig.

LUDOX

diffuser tip(TiO2 )20’PMT

sig.

DAQ
elec.

Figure 2.20: A schematic view of the timing measurement system using a laser.

27

980
970

Linear Scale

Log Scale960

T (nsec)

950
940
930
920
910
900
890
880
1

5

10

100

Q (p.e.)Figure 2.21: TQ-map, a measured two-dimensional plot of timing vs. charge distribution.

erator emits intense light with 337 nm wavelength within a time width of 3 nsec. The wavelengthof the laser light is converted to 384 nm by a dye laser module. The PMT is most sensitive tothis wavelength. Then the light is split into two. One is injected into a diffuser ball in the ID
via an optical fiber and the other is used to monitor the light intensity. The schematic viewof the diffuser ball is also shown in Figure 2.20. The laser light injected to the diffuser ball isfirst diffused by a TiO2 diffuser tip at the center and then further diffused by the surrounding
LUDOX silica gel made of 20 nm glass fragments. The PMT timing response is measured withvarious light intensity from 1 p.e. to a few hundreds of p.e. using an adjustable attenuation
filter. The results are shown as a scatter plot of the timing and the charge called TQ-map in
Figure 2.21. Each dot represents one measurement, and the open circles are the average timingwith respect to charge for a PMT.The TQ-map is made for all ID PMTs and used to correct
the timing information.

2.6.4

Water Transparency Measurement

Water transparency is the length scale of the optical attenuation in water, which represents
the combined effect of absorption and scattering on the intensity of the light. Water transparency in Super-Kamiokande is measured using N2 laser beam injected into the detector. The
wavelength dependence of the attenuation length can be measured by changing the dye of thelaser. The absorption and scattering coefficients are separately measured by this method. Further, the water transparency is independently measured using Cherenkov light from cosmic raymuons. The time variation of the water transparency can be monitored by this method without
28

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

disturbing the normal data taking.
Light scattering measurement using a laser
The light attenuation length in water can be described as L = (αabs + αscat )−1 , where αabs
and αscat are the absorption coefficient and scattering coefficient, respectively. The absorption
and scattering coefficients are separately measured using a N2 laser [39]. Figure 2.22 shows aschematic view of the measurement system. Each laser, wavelength of 337, 371, 400 and 420 nm,
fires every 6 seconds during normal data taking. The light from the laser is injected into the
ID via an optical fiber toward the bottom of the tank. A typical event by the laser light is also
shown in Figure 2.22. The PMT hits clustered at the bottom of the tank are due to the direct(unscattered) photons.
In the analysis, the detector is separated into 6 regions, top and 5 in barrel, as shown in
Figure 2.22. The total charge of bottom PMTs is used for the normalization of the light intensity.
Figure 2.23 shows the PMT hit timing distributions in each region for data and Monte Carlosimulation. These PMT hits on the top and barrel wall are due to the photons scattered inwater, or the photons reflected by surfaces of bottom PMTs or black sheets. The first peaks
and slopes are characterised 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 2.24 with astar symbol. The lines shows a model used in the Monte Carlo simulation, which are determined
29

2

Top

1
0
1

Barrel 1

PMTs / BottomQ

0

Barrel 2
1
02

Barrel 3

1
02

Barrel 4

1
0
2

Barrel 5

1
0

700

800

900

1000

1100

1200

PMT Hit Time (nsec)

Figure 2.23: PMT hit time distributions of 337 nm-laser events in each detector region forthe data (dots) and the Monte Carlo events which is tuned by the scattering and absorption
parameters (histograms).

by fitting the measurements from this method.
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) isexpressed by :


l
f (θ)
· exp −
(2.5)
Q = Q0 ·l
Lwhere Q0 is a constant, f (θ) is the PMT acceptance, l is the photon path length and L isthe attenuation length. Figure 2.25 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 by this method. Figure 2.26 shows the time variation of
the attenuation length. This time variation is corrected in event reconstruction processes.

30

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 2.24: Wavelength dependence of attenuation coefficients (L−1atten ) obtained by the lightscattering measurement (points) together with a model in the detector simulation (lines, see
Section 3.5).

2.6.5

Absolute Energy Calibration

The momentum of a particle is determined by the charge information of PMTs. Since thesystematic uncertainty in the absolute energy scale affects the atmospheric neutrino analysis,
it is essential to understand the absolute energy scale as accurte as possible . Four kind of
methods are employed in momentum range from a few ten MeV/c to about 10 GeV/c by using
independent calibration sources: decay electrons from stopping muons, π 0 events, low and highenergy stopping muons.
Decay electrons
Many electron events produced by the decay of cosmic ray muons are observed in SuperKamiokande. These electrons have the well known energy spectrum extending up to ∼53 MeV
and are used to determine the absolute energy scale by comparing the energy spectrum between
the observed data and the Monte Carlo simulation. Decay electrons are selected by the following
criteria :1. The time interval from a stopping muon event is 1.5 µsec to 8.0 µsec.
2. The number of hit PMT in a 50 nsec time window is larger than 60(30) for SK-I (SK-II).
3. The goodness of the vertex fit is greater than 0.5 .
4. The vertex position is reconstructed more than 2 m away from the ID wall.

31

log(Ql/f( θ))500

RUN 3106

P1
P2

5.987
-0.9487E-04

1000

2000

3000

0.4609E-03
0.4444E-06

400
300
200
0

4000

5000

atteunuation length (m)Figure 2.25: 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.

140
120
100
80
60
40

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

20
0

0

1000

2000
3000
elapse days from 1996/Jan./1

Figure 2.26: Time variation of water attenuation length.
Criterion (2) rejects ∼6 MeV gammma rays from µ− capture on the nucleon. Figure 2.27 shows
the momentum spectra of decay electrons compared with the prediction from the Monte Carlosimulation. In the Monte Carlo, measured µ+ /µ− ratio of 1.37 [40] is used and the effect ofµ− capture by oxygen nuclei is also considered. The mean values of data agree with the Monte
Carlo prediction within 0.4 %(2.4 %) for SK-I (SK-II).
Neutrino induced π 0 events
The π 0 events produced in the interactions of the atmospheric neutrino are used for theenergy scale calibration in the several hundred MeV energy range. Since a π 0 decays immedeately
into two γ-rays, the invariant mass of π 0 is obtained by the reconstructed momentum of two
γ-rays, Pγ1 and Pγ2 , and the opening angle θ as follows :q(2.6)
Mπ0 = 2Pγ1 Pγ2 (1 − cos θ)
The π 0 events are selected from the atmospheric neutrino sample (see Chapter 6) by the following
criteria :1. Two Cherenkov rings are recognized and both of them are identified as electron-like.
32

400

350

350
Number of events

Number of events

300

300

250

250

200

200

150

150

100

100

50

50

0

0

10 20 30 40 50 60 70 80 90
Momemtum (MeV/c)00

10 20 30 40 50 60 70 80 90
Momentum(MeV/c)

Figure 2.27: The momentum distribution of decay electrons of the data (dot) and the Monte
Carlo (line) for SK-I (left) and SK-II (right).

2. Electrons from muon decay is not detected.
3. The vertex position is reconstructed more than 2 m away from the ID wall.
Criterion (2) rejects the contamination of π + π 0 or µ± π 0 events. Figure 2.28 shows the invariant
mass distribution of π 0 events, which is compared with the prediction from the Monte Carlosimulation. A peak from π 0 is clearly observed at around 135 MeV/c2 . The peak position of thedata is 1.6 %(1.4 %) higher than that of the Monte Carlo simulation for SK-I (SK-II).
Low energy stopping muons
The Cherenkov angle of charged particles is expressed as a function of the momentum :
s
m2
1
1
1+ 2
=
(2.7)
cos θC =nβnpwhere θC , n, β, m and p are the Cherenkov angle, the refraction index of water, v/c, mass andmomentum. Since the Cherenkov angle has a large dependence on the momentum for low energy
stopping muons (< 400 MeV/c), the momentum can be estimated by measuring the Cherenkov
angle. The momentum estimated from the detected charge (Pp.e. ) is used to check the systematic
uncertainty of the momentum determination by comparing it with the momentum derived fromthe Cherenkov angle (Pθ ). On the other hand, this method is not useful for electrons or highenergy muons because the Cherenkov angle approaches a limit when the momentum is highcompared to its mass. Low momentum stopping muon events are selected for this study by thefollowing criteria :1. The total number of p.e.s in the ID is less than 1500 p.e.s for SK-I (750 for SK-II)
2. One cluster of hit PMTs in the outer detector.
33

160

70

140
60
Number of events

Number of events

120
100
80
60

40
30
20

40

10

20
0

50

0

50

100

150

200

250

300

2

0

0

50

100

150

200

250

300

2

Invariant mass (MeV/c )Invariant mass (MeV/c )Figure 2.28: 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 (left) and SK-II (right). The
peak position is fitted by a Gaussian distribution.

3. The entrance point is on the top wall.
4. The direction is downward (cos θ > 0.9).
5. One decay electron event is detected.
Criterion (1) corresponds to the approximate muon momentum of < 380 MeV/c. Criterion (2)requires an entrance point of a muon in the OD and criterion (3) and (4) select the muon events
which are straight downgoing.
Figure 2.29 shows the scattered plots of the reconstructed Cherenkov opening angle and themomentum Pp.e. for the data and the Monte Carlo simulation. The momentum dependence of
the Cherenkov opening angle is seen in both figures. The energy scale of the data is comparedwith that of the Monte Carlo simulation by comparing the ratios of Pp.e. /Pθ . Figure 2.30 shows
the averaged Pp.e. /Pθ for the data and the Monte Carlo simulation and the MC/data ratio as afunction of the expected momentum Pθ . They agree within 1.5 % for SK-I and 2.3 % for SK-II.
High energy stopping muons
The momentum for high energy stopping muons can be estimated from its track length
because the range of the muon track is well understood and approximately proportional to themomentum. Since the measurement of the range can be independent of the measurement of
the momentum that is based on the observed photoelectrons, the estimated momentum fromits range is used for checking the energy scale from 1 GeV/c up to 10 GeV/c. Stopping muon
events are selected by the following criteria :1. The entrance point is on the top wall.
2. The direction is downward (cos θ > 0.94).
34

400

400

(a) DATA

Momentum(MeV/c)

375(b) MC

375

350

350

325

325

300

300

275

275

250

250

225

225

200

200

175

175

150

150

30

32

34

36

38

40

42

30

32

Openning angle(deg.)
500

36

38

40

42

500
(a)DATA

Reconstructed momemtum (MeV/c)

Reconstucted momemtum (MeV/c)

34

Openning angle(deg.)

450

400

350

300

250

200
30

32

34

36

38

40

42

44

Opening angle (deg.)

(b)MC
450

400

350

300

250

200
30

32

34

36

38

40

42

44

Opening angle (deg.)

Figure 2.29: Scatter plot of the reconstructed Cherenkov opening angle and the reconstructed
momentum Pp.e. for (a) the data and (b) the Monte Carlo simulation. The momentum dependence of the Cherenkov opening angle is seen in both figures. The top two panels show for SK-I
and the bottom two panels show for SK-II.

35

1.1
1.075

1.1

(a)

1.075

1.025

1.025

M.C./DATA

1.05

pp.e./pθ

1.05

1

0.975

1
0.975

0.95

0.95

data
MC

0.925
0.9
200

(b)

220

240

0.925

260

280

300

320

0.9
200

340

220

240

260

280

300

320

340

Estimated momentum (MeV/c)

Estimated momentum (MeV/c)
1.2

1.1

(a)
1.075

(b)

1.15
1.05
1.1

MC/DATA

Pp.e./Pθ

1.025

1.05

1

0.975

1
0.95
0.95
0.925
0.9

200 225 250 275 300 325 350 375 400 425 450

0.9

200 225 250 275 300 325 350 375 400 425 450

Estimated momentum (MeV/c)

Estimated momentum (MeV/c)

Figure 2.30: Figure (a) shows the averaged ratio of the momentum derived from the charge tothat from the opening angle (Pp.e. /Pθ ) as a function of the momentum Pθ for the data (black
points) and the Monte Carlo events (white points), and figure (b) shows the ratio of the data
to the Monte Carlo events. The top two panels show for SK-I and the bottom two panels show
for SK-II.

36

3. One decay electron event is detected.
4. The reconstructed range of muon track is greater than 7 m.
The range of the muon is determined by the entrance position of the stopping muon and thevertex position of the subsequent decay electron. Figure 2.31 shows the averaged value of theratio momentum/range as a function of the range for the data and the Monte Carlo simulation.
The momentum loss per cm is about 2.3 MeV/c. The comparison of the momentum/range
between the data and the Monte Carlo simulation is also shown in Figure 2.31. Although themomentum dependence is seen, the deviations from unity are less than 2 %(1 %) for SK-I (SK-II).
Time variation of energy scale
The stability of the energy scale is also monitored by the stopping muons and the decay
electrons. Figure 2.32 shows the time variation of the averaged momentum/range of stoppingmuons and the reconstructed momentum of the decay electrons as a function of elapse days from
April 1st, 1996. The RMS of the energy scale variation is 0.8 %(0.9 %) for SK-I (SK-II).
Uniformity of Energy Scale
The uniformity of the detector is measered 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 Θe↔µ < 0.25,
where cos Θe↔µ is the opening angle between the electron and muon directions. Figure 2.33
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
gain is uniform within ±0.6 % for SK-I and SK-II.
Summary of the absolute energy calibration
Figure 2.34 shows the summary of the absolute energy calibration. The absolute energy scale
is checked by various methods over a wide energy range. The uncertainty of the energy scaleis estimated to be less than 1.8 % and 2.4%̇ for the momentum range from a few ten MeV/c toabout 10 GeV/c for SK-I and SK-II, respectively. Combined with the RMS of the time variationof the energy scale, 2.0 % (2.5 %) is adopted to the systematic error for SK-I (SK-II).

37

2.8

2.8

(a)

(a)

2.7
Momentum/Range(MeV/c/cm)

Momentum/Range(MeV/c/cm)

2.7
2.6

2.6

2.5

2.5
2.4

2.4

2.3

2.3

2.2

2.2

2

2
0

data
MC

2.1

data
MC

2.1

500 1000 1500 2000 2500 3000 3500 4000

0500

Range (cm)

1500

2000

2500

3000

3500

Range (cm)

1.1

1.1

(b)

1.075
1.05

1.05

1.025

1.025

1

1

0.975

0.975

0.95

0.95

0.925

0.925
0.9

0.9
0

(b)

1.075

M.C./DATA

M.C./DATA

1000

500 1000 1500 2000 2500 3000 3500 4000

Range (cm)05001000 1500 2000 2500 3000 3500
Range (cm)Figure 2.31: Figure (a) shows the range v.s. averaged momentum/range of stopping muon events
for the data (black points) and the Monte Carlo events (open circles), and figure (b) shows theratio of the Monte Carlo events to the data for SK-I (left) and SK-II (right).

38

µ momentum/track
e 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

500

1000
1500
2000
2500
Elapsed days from Apr.-1-1996

3000

3500

3000

3500

±1%

0

500

1000
1500
2000
2500
Elapsed days from Apr.-1-1996

Figure 2.32: The time variation of the averaged momentum/range of stopping muons (upper)
and the reconstructed momentum of the decay electrons (bottom) as a function of elapsedays
from April 1st, 1996.

electron momentum

1.05
1.04
1.03
1.02
1.01
1
0.99
0.98
0.97
0.96
0.95

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

Figure 2.33: The uniformity of the detector gain as a function of zenith angle for SK-I (left) and
SK-II (right). The vertical axes in the two figures are the averaged momentum of decay electron
events.

39

(MC-data)/data (%)

10
8
6
4
2
0
-2

decay-e0πlow stop µhigh stop µ

-4
-6
-8
-10
10

10

2

10

3

10

4

10

5

Momentum range (MeV/c)(MC-data)/data (%)

10
8
6
4
2
0
-2

decay-e0πlow stop µhigh stop µ

-4
-6
-8
-10
10

10

2

10

3

10

4

10

5

Momentum range (MeV/c)

Figure 2.34: The summary of the absolute energy scale calibration for SK-I (top) and SK-II(bottom). The horizontal axis shows the momentum range of each source and the vertical axis
shows the deviation of the data from the Monte Carlo predictions.

40

Chapter 3

Simulation
3.1

Overview

Expectation of atmospheric neutrino events in Super-Kamikande is calculated by a Monte
Carlo simulation. The Monte Carlo simulation consists of four components : atmospheric neutrino flux, neutrino interactions, particle tracking in the detector and the detector hardware
simulation.

3.2

Atmospheric Neutrino Flux

There have been several atmospheric neutrino flux models calculated by M.Honda et al. [14](Honda flux), G.Battistoni et al. [41] (Fluka flux) and G.Barr et al. [42] (Bartol flux). In ourMonte Carlo simulation program, the Honda flux is adopted and two other flux calculations areused to estimate the systematic uncertainties in the flux calculation. The primary cosmic ray flux
model as an input of the flux calculation is determined based on the experimental measurements.
Current status of the measurements of cosmic ray proton flux is shown in Figure 3.1 with themodel used in the Honda flux calculation. The primary cosmic ray spectrum has been precisely
measured by BESS and AMS experiments up to 100 GeV [43, 44]. The cosmic ray flux changes
depending on the turbulence of the solar wind, which is higher when the solar activities arehigh (solar maximum) than when the solar activities are low (solar minimum). The differenceof the cosmic ray flux at solar maximum and solar minimum is more than a factor of two for
1 GeV cosmic rays, while it decrease to ∼ 10 % for 10 GeV. The effect of geomagnetic field isrepresented as rigidity (= momentum/charge) cutoff. The geomagnetic field works as a shield,
by which cosmic ray with lower momentum than the cutoff cannot arrive at the Earth. Cosmic
ray flux above 100 GeV, which are responsible for ≥ 10 GeV neutrinos, is not affected by thesolar activity nor the geomagnetic field.
Primary cosmic ray protons and nuclei interact with air nuclei, and secondary particles,
mostly pions and some kaons, are generated in the atmosphere. The US Standard Atmospheremodel [55] is employed for the density structure of the atmosphere and the structure gives thezenith angle dependence of the atmospheric neutrinos.
For the hadronic interactions of the cosmic rays with air nuclei, the two theoretical models,
NUCRIN [56] for the primary cosmic ray energies < 5 GeV and DPMJET-III [57] for > 5 GeV.
Through the interactions, mostly pions and some kaons are generated. Then they decay into

41

Figure 3.1: Measurements of primary cosmic ray proton flux and the model used in Honda
flux calculation. The data are taken from Webber [45] (crosses), LEAP [46] (upward triangles),MASS1 [47] (open circles), CAPRICE [48] (vertical diamonds), IMAX [49] (downward triangles),
BESS98 [43] (circles), AMS [44] (squares), Ryan [50] (horizontal diamonds), JACEE [51] (downward open triangles), Ivanenko [52] (upward open triangles), Kawamura [53] (open squares) andRunjob [54] (open diamonds).

muons and neutrinos. The flux of the secondary cosmic ray muons have been measured byseveral experiments such as BESS [58]. The hadronic interaction model used in the Honda
flux calculation agree with the measurement in the 1∼10 GeV muon energy range as shown in
Figure 3.2.
The calculated energy spectrum of atmospheric neutrinos at the Super-Kamiokande site for
the Honda flux, Fluka flux and Bartol flux is shown in Figure 3.3, in which the flux of νe +ν e
to νµ +ν µ is averaged over all of the direction. The flavor ratio is about two up to a few GeV
energy regions, but it becomes larger than two as the neutrino energy increases because morecosmic ray muons reach the ground before decaying. The systematic uncertainty in absolute
flux is estimated to be about 20 % due to the uncertainty of the absolute primary cosmic ray
flux and the uncertainties in the cosmic ray interaction, while the uncertainty in the flavor ratio
is smaller than 3 %. The systematic error in the flavor ratio is one of the dominant errors to
observe the 1-2 oscillation effect.
Figure 3.4 shows the calculated flux ratios of νe /ν e and νµ /ν µ . The calculations agree toabout 5 % for both below 10 GeV, while above 10 GeV the disagreement is larger as a functionof the neutrino energy to 10 % and 25 % at 100 GeV for νe and νµ , respectively.
Figure 3.5 shows the zenith angle distributions of the atmospheric neutrino flux, where cosθ
= -1 refers to upward-going neutrinos and cosθ = 1 refers to downward-going neutrinos. The
enhancement of the Sub-GeV flux near horizon is due to the 3 dimensional nature of the earth
geometry and the poor angular correlation between the primary cosmic ray and the neutrino
directions. Details of the horizontal enhancement in discussed in [61]. The enhancement of theMulti-GeV flux toward the horizontal directions is due to the larger decay volume of the cosmic
ray muons. The cosmic ray muons coming in the horizontal direction travel for a longer distance42

Figure 3.2: Secondary cosmic ray muon flux at Lynnlake observed by the BESS experiment [58](points) and the prediction from the Honda flux calculation [60] (lines).

10
9
8
7

–

–

νe+νe

5
4

–

-2

Honda flux
Fluka flux
Bartol flux

6

–

νµ+νµ

2

(νµ+νµ)/(νe+νe)

10

3

-1

-1

Flux×Eν2(m sec sr GeV)

10

10

Honda flux

3

Fluka flux
Bartol flux

2
1
10

-1

1

10

10

2

10

-1

1

10

10

2

Eν (GeV)Eν (GeV)Figure 3.3: Predictions of the direction averaged atmospheric neutrino flux and the flavor ratio.
Solid lines show the Honda flux, dashed lines show the Fluka flux and dotted lines show the
Bartol flux.

43

1.6
1.5

Flux ratio

1.4
1.3
1.2
1.1
1
–

0.9
10

–

νe/νe
-1

1

νµ/νµ
10

10

Eν (GeV)

-1

1

10

10

2

Eν (GeV)Figure 3.4: The calculated flux ratios of νe /ν e and νµ /ν µ as a function of the neutrino energy
for the Honda flux (solid line), the Bartol flux (dashed line), and the Fluka flux (dotted line).

than those in the vertical direction and have a higher probability to decay into neutrinos. For
lower energy netrinos, the flux shows up-down asymmetry. This is caused by the geomagnetic
field. The promary cosmic rays of the lower energy neutrinos are deflected by the rigidity cutoff.
This up-down asymmetry for low energy neutrinos largely washed out due to the poor angularcorrelation between neutrinos and leptons.

44Integrated neutrino flux (m-2sec-1sr-1)

700

200

600

175

30

25

150

500

20

125
400
–

300

νµ+νµ

100

–

νµ+νµ

–

15

75
200

–

50

νe+νe
100
0
-1

10
–

νe+νe

–

5

25

0.3-0.5 GeV
00.9-1.5 GeV
1

νµ+νµ

0
-1

0

νe+νe
3.0-5.0 GeV

1

0
-1

0

1

cosΘ
Figure 3.5: The atmospheric neutrino flux as a function of zenith angle for the Honda flux (solidline), the Bartol flux (dashed line), and the Fluka flux (dotted line). Zenith angle cosΘ = −1(+1)
indicates upward-going (downward-going).

45

3.3

Neutrino Interaction

Atmospheric neutrinos interact with nucleons and electrons in water. The neutrino interactions are treated using NEUT program library [62] in our Monte Carlo simulation. This library
was first developed to study atmospheric neutrinos as a background of nucleon decay analysis
in the Kamiokande experiment, and then it was inherited to the Super-Kamiokande experimentwith various modifications. In the NEUT code, the following charged current (CC) and neutral
current (NC) interactions are considered :CC/NC (quasi-)elastic scatteringν + N → l + N′

CC/NC single meson production

ν + N → l + N ′ + meson

CC/NC deep inelastic interaction

ν + N → l + N ′ + hadrons

CC/NC coherent pion production

ν + 16 O → l + 16 O + π

where N and N ′ are nucleons (proton or neutron) and l is a lepton.
Since the cross-section of neutrino-electron elastic scattering is about 103 times smaller thanthat of the neutrino-nucleon interactions at a neutrino energy of ∼1 GeV, this interaction modeis neglected in our simulation.

3.3.1

Elastic and Quasi-Elastic Scattering

The differential cross-section of the charged current quasi-elastic scattering for free protons(i.e. hydrogen atom in water) is given by [63, 64] :2
M 2 G2F cos2 θcdσ ν(ν̄)22 s−u
2 (s − u)=A(q ) ∓ B(q )+ C(q )
dq 2
8πEν2
M2
M4

(3.1)

where Eν is the neutrino energy, M is the mass of the target nucleon, G is the Fermi coupling
constant, θc is the Cabbibo angle, q is the four-momentum transfered of the lepton, and s andu are Mandelstam variables [63]. The factors A, B and C are :m2 − q 2
q2
q2
2
A(q ) =
4 − 2 |FA | − 4 + 2 |FV1 |2
4M 2
M
M


2
1
4q FV ξFV2
q2
q2−
− 2 |ξFV2 |2 1 +
M
4M 2
M2


m2
− 2 (FV1 + ξFV2 )2 + |FA |2M2
q
FA (FV1 + ξFV2 )B(q 2 ) =
2
M

1
q2
2
2 2
2
1 2
C(q ) =
|ξFV ||FA | + |FV | −
4
4M 2
2

46

(3.2)
(3.3)
(3.4)

o
events / 10

Data(stat.+syst.)

5000

MC
4000

3000

2000

1000

0

0

50

100

150

q m [deg.]

Figure 3.6: The scattering angle distribution by neutrino interactions off the H2 O target fromthe K2K experiment (dot) [65] together with our Monte Carlo events (histgram).

where m is the lepton mass, ξ ≡ µp − µn = 3.71. The vector form factors, FV1 (q 2 ) and FV2 (q 2 ),and the axial vector form factor, FA (q 2 ) are determined experimentally and are given by :FV1 (q 2 )=ξFV2 (q 2 ) =FA (q 2 ) =GE (q 2 ) =


−1 
q2
2
2
GM (q )GE (q ) −
4M 2
−1



q2
GE (q 2 ) − GM (q 2 )
1−
2
4M

−2
q2
− 1.23 1 − 2
MA

−2
q2
−1
2
(1 + ξ) GM (q ) = 1 − 2
MV


q2
1−
4M 2

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

where GE and GM are the electric and magnetic form factor, the vector mass MV is set to be
0.84 GeV and the axial vector mass MA is set to be 1.11 GeV from experimental data [65], whichis common parameter for single meson production model.
For larger MA values, interactions with higher Q2 values (and therefore larger scattering
angles) are enhanced. The MA value is tuned using the K2K near detector data. The uncertaintyof the value is estimated to be 10 % comparing the K2K data with our Monte Carlo. Figure 3.6
shows the K2K 1 kton water Cherenkov data on the scattering angle for single-ring events [65]together with our Monte Carlo events. The scattering angle agrees well between the data andMonte Carlo overall, although the suppression of events at small angle is being studied by several
groups[66].
For scattering off nucleons in 16 O, the Fermi motion of the nucleons and Pauli exclusion
principle must be considered [67]. Since nucleons are fermions, the outgoing momentum of thenucleons in the interactions is required to be greater than the Fermi surface momentum to allow
quasi-elastic scattering to occur. In NEUT, the Fermi surface momentum is set to be 225 MeV/c.
47

1.4

2
-

(a) νµn→µ p

1.75

ANL

Serpukhov

BNL

GGM (1977)

SKAT

GGM (1979)–

1.5

GGM (1977)

Serpukhov

GGM (1979)

SKAT

1

σ (10-38 cm2)

1.25

σ (10-38 cm2)+(b) νµp→µ n

1.2

1

0.75

0.8
0.6
0.4

0.5
free nucleon

0.25

0.2

free nucleon
bound nucleon

bound nucleon

0

0

2

4

6
8
Eν (GeV)

10

12

0

14

0

2

4

6
8
Eν (GeV)

10

12

14

Figure 3.7: cross-sections of (a) νµ and (b) ν µ with the experimental data from ANL [70],Gargamelle [71, 72], BNL [73], Serpukhov [74] and SKAT [75].

The cross-section for neutral current elastic scattering are estimated from the following
relations [68, 69] :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)

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

Figure 3.7 shows the cross-section of the quasi-elastic scattering for the experimental data andthe calculation by NEUT. Figure 3.8 shows the calculated cross-sections with free nucleon andbound nucleon in oxygen nuclei in the low energy region.
The systematic uncertainty of the CCQE cross-sections with bound nucleons is estimatedfrom the difference between the model of Nieves et al . [125] and that of Smith and Monitz [67].
The uncertainties in the ν µ + ν e to νµ + νe ratio and the νµ + ν µ to νe + ν e ratio are also taken
into account (see Section 7.3). The systematic errors in MA and in the CCQE cross-sections areimportant to observe the 1-2 sub-dominant oscillation effect.

3.3.2

Single Meson Production

The resonant single meson production of π, K, and η is simulated based on the model ofRein & Sehgal [76]. This method assumes an intermediate baryon resonance :ν + N → l + N∗
N ∗ → N ′ + meson(3.13)

where N and N ′ are nucleons and N ∗ is a baryon resonance. The differential cross-section of
single-meson production is a product of the amplitude of each resonance production and the
48

2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

-

(a) νen→e p
free nucleon

0

0.2

0.4

0.6 0.8Eν (GeV)σ (10-38 cm2)

bound nucleon

1

1.2

1.4

-

(c) νµn→µ p
free nucleon
bound nucleon

0

0.2

0.4

0.6 0.8Eν (GeV)σ (10-38 cm2)

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

1

1.2

1.4

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

–

+

(b) νep→e nfree nucleon
bound nucleon

0

0.2

0.4

–

0.6 0.8Eν (GeV)

1

1.2

1.4

1

1.2

1.4

+

(d) νµp→µ n
free nucleon
bound nucleon

0

0.2

0.4

0.6 0.8Eν (GeV)Figure 3.8: cross-sections of quasi-elastic scattering in the low energy region for (a) νe , (b) ν e ,(c) νµ and (d) ν µ . The solid lines show the cross-sections for free nucleon and the dashed lines
show that for bound nucleon.

49σ (10-38 cm2)
σ (10-38 cm2)

1.2
1
0.8
0.6
0.4
0.2
0
1.2
1
0.8
0.6
0.4
0.2
0

(a) νµp → µ pπ+(b) νµn → µ pπ(c) νµn → µ nπ

+

1

-

-

-

10
Eν (GeV)

ANL
BNL

1

+

BEBC 86
GGM

10Eν (GeV)Figure 3.9: Cross-sections for charged current single pion productions of νµ . (a) νµ p → µ− pπ + ,(b) νµ n → µ− pπ 0 , (c) νµ n → µ− nπ + . Solid lines show our calculations. Experimental data
are from ANL [80], BNL [81], BEBC [82] and Gargamelle [84].

probability of the baryon resonance decay to this meson. For negligible decay width of a baryon
resonance (N ∗ ), the differential cross-section is :
1
1 X
d2 σ
2
T (νN → lNj∗ ) δ(W 2 − Mj2 )=·22
dq dEν
32πM Eν 2(3.14)

j,spinwhere M is the mass of the target nucleon, Eν is neutrino energy, W is the invariant mass ofthe hadronic system (or the mass of the intermediate baryon resonance), Mj is the mass of thebaryon resonance (Nj∗ ), and T (νN → lN ∗ ) is the amplitude of resonance production, which is
calculated using the FKR (Feynman-Kislinger-Ravndal) model [77]. The invariant mass, W , isrestricted to be less than 2 GeV/c2 . For W larger than 2 GeV, the interactions are simulated
as deep-inelastic scattering as described in Section 3.3.3. The differential cross-section for the
resonance with finite decay width Γ can be derived by replacing the δ-function with a BreitWignar factor :Γ1·(3.15)
δ(W 2 − Mj2 ) →2π (W − Mj )2 + Γ2 /4

For a single-meson production, the axial vector mass MA is also set to be 1.11 GeV experimentally [65]. Total of 18 resonances are simulated in our simulation. The Pauli blocking effect inthe decay of the baryon resonance is considered by requiring the momentum of the nucleon to begreater than the Fermi surface momentum. Pion-less decay of ∆ resonance in 16 O nuclei, where
about 20 % of the events don’t have a pion in the decay, is also simulated [79]. Figures 3.9, 3.10
and 3.11 show the cross-sections of charged current and neutral current resonant single-meson
productions for our calculations and the experimental data.
50

σ (10-38 cm2)
σ (10-38 cm2)

1.2
1
0.8
0.6
0.4
0.2
0
1.2
1
0.8
0.6
0.4
0.2
0

–

+

–

+

–

(a) νµn → µ nπ(b) νµp → µ nπ

-

+

0

1

(c) νµp → µ pπ-

10Eν (GeV)
BEBC 83
BEBC 86
FNAL

1

10Eν (GeV)Figure 3.10: Cross-sections for charged current single pion productions of ν µ . (a) ν µ n →µ+ nπ − , (b) ν µ p → µ+ nπ 0 , (c) ν µ p → µ+ pπ − . Solid lines show the result of our calculations.
Experimental data are from BEBC [85, 82] and FNAL [86].

σ (10-38 cm2)0.25

νp → νpπ
 0
νp → νpπ(b)νn → νnπ
 0
νn → νnπ

(d)

0

(a)



0.2

νp → νnπ +νp → νnπ
+



0.15
0.1

0.05
0

σ (10-38 cm2)

0.25

0

(c)



0.2

νn → νpπ
 νn → νpπ
-



0.15
0.1

0.05
0
1

10

1

Eν (GeV)

10Eν (GeV)Figure 3.11: Cross-sections for neutral current single pion productions of νµ and ν µ .
Solid(dashed) lines show the result of our calculations for νµ (ν µ ) interactions.

51

The uncertainty in total cross-sections for single pion productions is estimated from thedifferent between the FKR model [77] and the model calculation based on [78] to be 20 %. The
uncertainty in the single pion production cross-section for each π + , π 0 and π − production is
also estimated from the model difference to be 20 %.

3.3.3

Deep Inelastic Scattering

The cross-section of charged current deep inelastic scattering is calculated by integrating thefollowing equation in the range of the invariant mass W > 1.3 GeV/c [126] :
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
m2l(3.16)
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 F2
and xF3 are taken from GRV94 [88].
In our simulation, the cross-section of deep inelastic scattering induced by neutral current are
assumed to have the following relations which are estimated from experimental results [89, 90] :
0.26( Eν < 3 GeV )σ(νN → νX)(3.17)=
0.26 + 0.04 (Eν /3 − 1) ( 3 GeV ≤ Eν < 6 GeV )σ(νN → µ− X)0.30( Eν ≥ 6 GeV )
0.39( Eν < 3 GeV )σ(ν̄N → ν̄X)
(3.18)=
0.39 − 0.02 (Eν /3 − 1) ( 3 GeV ≤ Eν < 6 GeV )σ(ν̄N → µ+ X)
0.37( Eν ≥ 6 GeV )The kinematics of the hadronic system is simulated by two different methods according
to the range of invariant mass. In the region of 1.3 GeV/c2 < W < 2.0 GeV/c2 , only pions are
considered as outgoing mesons. The mean multiplicity of pions is estimated from the result of
Fermilab 15-foot hydrogen bubble chamber experiment [91] :hnπ i = 0.09 + 1.83 ln(W 2 )(3.19)The number of pions in each event is determined by using KNO (Koba-Nielsen-Olsen) scaling.
Since the range of W overlaps with that in single pion production, nπ ≥ 2 is required in this Wregion. The forward-backward asymmetry of pion multiplicity in the hadronic center of mass
system is included using the results from BEBC experiment [92] :0.35 + 0.41 ln(W 2 )nFπ=nB
0.5 + 0.09 ln(W 2 )π
52(3.20)

In the region of W > 2.0 GeV/c2 , the kinematics of the hadronic system are calculated byusing JETSET/GRV94 package [93]. This package treats not only π but also K, η, ρ and so on.
For neutrino energy above a few ten GeV, the cross-section calculation agrees with theexperimental measurement within 5 %, but in the lower energy region, the uncertainty is large.
The difference between the model calculation based on [126] and A. Bodek and U. K. Yang [127]are considered as a systematic error. The difference in these two models depends on the squareof 4-momentum transfer q 2 and is quite large in a few GeV energy region.

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 very
little momentum is transferred to the oxygen nucleus, the angular distributions of the outgoing
leptons and pions are peaked in the forward direction. The formalism developed by Rein and
Sehgal [94] is used to simulate the interactions, and the differential cross-section is given by :G2F M 2 2
1
fπ A Eν (1 − y)(σ πN )222π
16π total
M2
×(1 + r 2 )( 2 A 2 )2 e−b|t| Fabs
MA + Qr = Re(fπN (0))/Im(fπN (0))

d3 σ
dQ2 dydt= β×(3.21)(3.22)

where β is the axial vector couping constant and is 1 (2) for neutral current (charged current)interactions, GF is the weak coupling constant, M is the nucleon mass, fπ is pion decay constantand is 0.93 mπ , A is the atomic number ( = 16 for oxygen), Eν is the neutrino energy, y is theπN is the averaged pion-nucleon cross-section, b is in the order
lepton fractional energy loss, σtotalof the nucleus transverse dimensions and is 80 GeV−2 , MA is the axial-vector mass, Q2 is thesquare of the four–momentum transfer of the lepton, and t is the square of the four-momentum
transfer to the nucleus, Fabs is a factor to account for the absorption of pions in the nucleus.
fπN (0)) in r is the πN scattering amplitude. However, the K2K experiment set an upper limiton the cross-section of coherent CC pion production [96]. The upper limit was significantly lower
than the predicted cross-section by Rein and Sehgal. Therefore some modification is necessary
to the Monte Carlo coherent pion production cross-sections. The calculated cross-sections of thecoherent pion production for the CC and NC interactions by Kartavtsev and Paschos [95], agree
better with the several experimental data than that of Rein and Sehgal as shown in Figure 3.12.
In this thesis, events due to coherent pion production are re-weighted by the cross-section ratio,
σKartavtsevandPaschos (Eν )/σReinandSehgal (Eν ). Namely, we use the cross-section by Kartavtsev andPaschos in this thesis. The difference between the experimental data and the model proposedby Kartavtsev and Paschos is taken into account as a systematic uncertainty of 100 %.

3.3.5

Nuclear Effects

It is also important to simulate the secondary interactions of mesons produced in neutrino
interactions with nucleons inside the 16 O nuclei. All of the mesons produced within the 16 O
nuclei are tracked from their production points until they exit or are absorbed in the nuclei.
This is done for π, K and η by using a cascade model in our simulation. The interactions of pions53

σ (10-40cm2/Carbon Nucleous)σ(10-40cm2/Carbon Nucleous)

140

Rein and Sehgal
Kartavtsev et al.

120
100
80
60

250
225
200

CHARM(93)MiniBooNE(05)Aachen-Padova(83)Gargamelle(84)

175
150

Rein and Sehgal
Kartavtsev et al.

125
100
75

40

50
20
0

25
1

2

3

4

5

6

7

8

9

10

Eν(GeV)

0 -1
10

1

10

10

2

Eν(GeV)

Figure 3.12: The cross-sections of coherent pion productions off the carbon nucleus for CC
interaction(left) and NC interaction(right) by two models with the experimental data. The
solid lines are calculation by Rein and Sehgal, while the dashed lines are by Kartavtsev andPaschos. On the left figure, the arrow shows the experimental upper limit by K2K [96] and on
the right figure, experimental data are from CHARM [97], MiniBooNE [98], Aachen-Padove [99]and Gargamelle [100].

are especially important since the cross-section for pion productions are large for Eν > 1 GeV,
and the pion-nucleon interaction cross-section is also large. The pion interactions in 16 O nuclei
considered in our simulation are : inelastic scattering, charge exchange, and absorption. First,
the initial pion production point in the nucleus, where neutrino-nucleon interactions occur, isdetermined by the Wood-Saxon density distribution [101] :ρ(r) =1Z
ρ0
A 1 + exp( r − c )a(3.23)

where ρ0 is the average density of the nucleus, a and c are the density parameters, Z is theatomic number, and A is the mass number. For 16 O nucleus : ρ0 = 0.48 m3π , a = 0.41 fm, c =2.69 fm, Z = 8 and A = 16. The pion interaction is determined from the calculated mean-free
path for each interactions, which is modeled by L. Salcedo et al. [102]. The mean-free path ofpions depends on their momenta and positions in the nucleus. In the interactions, the Fermi
motion of the nucleus and the Pauli blocking are considered, and the outgoing nucleon musthave the energy above the Fermi surface momentum defined by :pF (r) =



1
3
3 2
π ρ(r) .
2(3.24)

The angular and momentum distributions of the outgoing pions are determined by using theresults of a phase shift analysis from π - N scattering experiments [103]. The pion interaction
simulation is tested using 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) [104, 105] as shownin Figure 3.13. For kaons, the elastic scattering and charge exchange interactions are considered
54

+ 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 3.13: The cross-sections of π + -16 O scattering as a function of π + momentum. Thelines are the cross-sections calculated by our simulation for each interaction mode, and theexperimental data points are taking from [105].

using the results from the cross-sections measured by the K ± -N scattering experiments [106,107, 108]. For η mesons, the absorption (ηN → N ∗ → π(π)N ) is considered [109]. These pions
are tracked as described above.
In our simulation, the nucleon re-scattering inside the oxygen nucleus is also consideredusing the cascade model, which is similar to that for the pion simulation. With the crosssections obtained from nucleon-nucleon scattering experiments [110], the interactions of elastic
scattering and resonance production processes for nucleon propergating through the 16 O nucleus
are simulated in NEUT [111]. The pions from delta decay are traced by the method describedabove.

3.4

Upward-Going Muon

The prediction of upward-going muon signals is also calculated by Monte Carlo method
using the identical tools to that for the vertex contained events. The energy range and thetarget volume are extended for upward-going muons. The energy range of the atmosphericflux is extended up to 100 TeV, which is calculated by M.Honda et al. [14] up to 7 TeV, andby L.V.Volkova [112] above that. The target volume of neutrino interactions is extended to
the rock around the detector. The composition of the rock is set to be SiO2 with the density
2.7 g/cm3 . Neutrino interactions occur in the water of the OD are also considered. The outputof the Monte Carlo simulation are passed through the same reduction and event reconstruction
process as that for the observed data.

55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.

3.5

Detector Simulation

The produced particles in neutrino interactions are fed into a detector simulation code, which
simulates the tracks of particles, the generation and propagation of Cherenkov photons in water,
and the PMT response and the readout electronics.
The detector simulation has been developed based on GEANT package [113]. Table 3.1 lists
various processes which are considered in our simulation program. The hadronic interactions
in water are simulated using CALOR package [114]. This package is known to reproduce thepion interactions well including low energy region (∼ 1 GeV/c). For still lower energy region(pπ ≤ 500 MeV/c), a custom program [115] based on experimental data from π−16 O scattering [116] and π − p scattering [117] is used in our simulation code.
The uncertainty in the hadron simulation is estimated by comparing the model used in our
detector simulation, CALOR [114], and the FLUKA model. The uncertainty affects the contamination of NC interactions in single-ring µ-like, i.e. hadrons (mostly pions) in NC interactions are
identified as µ-like events. The difference from the FLUKA model is assumed to be a systematic
error.

56

Chapter 4

Data Reduction
4.1

Overview

The Super-Kamiokande detector collects about 106 events per day (except for the super
low energy triggered events) and most of the events are cosmic ray muons and low energy
background from radioactivities such as radon decay. The efficient event selection is required to
select neutrino events from a large quantity of data.
The atmospheric neutrino events observed in Super-Kamiokande are classified into the following four categories; fully contained (FC), partially contained (PC), upward stopping muonsand upward through-going muons as shown in Figure 4.1. For FC and PC events, the verticesof neutrino interactions are required to be within a fiducial volume, 2 m from the ID wall. If
the tracks of entire particles are contained inside the ID, the event is classified into FC. If oneof the particles exits the ID and deposits energy in the OD, the event is classified into PC.
The upward-going muons are produced by the atmospheric muon neutrinos via charged-current
interactions in the rock surrounding the detector. Since the downward-going neutrino-induced
muons cannot be distinguished from the cosmic ray muons, only muons traveling in the upwardgoing direction are selected. The upward going stopping muons enter from outside the detector
and stop inside the ID, while the upward through-going muons enter the detector and exit theID.
The neutrino energy range for each event class are, ∼ 1 GeV for FC, ∼ 10 GeV for PC, ∼10 GeV for upward stopping muons, and ∼ 100 GeV for upward through-going muons. Figure 4.2
shows the expected number of neutrino events in each category as a function of neutrino energy.
A data sample for each event class goes through a different reduction process. To separate
FC and PC events, the number of hit PMTs in the OD hit cluster (NHITAC, see Section 4.3.2)
is used, where hit cluster means the spatial cluster of neighboring hit PMTs. Figure 4.3 shows
the NHITAC distribution for both FC and PC final samples in the fiducial volume. FC andPC events are clearly separated at NHITAC = 10 (16) for SK-I (SK-II). Since the reflective
sheet covered on the OD wall is exchanged to a new one improving the reflectivity in SK-II, thenumber of the OD hits is increased and the cut criterion is tuned. The systematic uncertainties
for FC/PC separation are estimated to be 0.9 % for SK-I and 0.5 % for SK-II by comparing the
NHITAC distributions for the data and the Monte Carlo events.
The reduction processes are basically automated and are the same for SK-I and SK-II except
for some of the event selection criteria because of the difference in the number of the ID PMTs.

57

Fully contained

Partially contained

Upward stoppingmuon

Upward through-going
muon

Figure 4.1: Categories of atmospheric neutrino events in Super-Kamiokande. Dashed lines show
trajectories of primary neutrinos, and solid arrows show those of secondary charged particles.

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 4.2: The expected parent neutrino energy distribution for each event class.

58

10

5

10

4

10

3

10

2

10

1

0

10

4

10

3

10

2

PC
Number of events

Number of events

FC

10

20

30

40

50

60

70

80

90

PC

10

20

10

1

100

FC

0

30

NHITAC

40

50

60

70

80

90

100

NHITAC

Figure 4.3: NHITAC distributions for FC and PC final samples for SK-I (left) and SK-II(right). The points show the observed data, and histograms show the atmospheric neutrinoMonte 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).
In this section the procedures of the data reduction for FC, PC and upward-going muon eventsare described.

4.2

Reduction for Fully Contained Sample

There are five steps in FC reduction process. The main background sources are cosmic raymuons, low energy events from radio isotopes and electrical noise events.

4.2.1

First Reduction

The selection criteria of the FC 1st reduction are the following :(1)

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

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−400 nsec to +400 nsec before and after the trigger timing.

and
(3)

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

Criterion (1) rejects the low energy background events from radio isotopes. Figure 4.4 shows
the PE300 distributions for the raw data and the FC final events. The 200 p.e.s (100 p.e.s for
SK-II) corresponds to 22 MeV/c of electron momentum. Since the events which have visibleenergy below 30 MeV are not used in the analysis, this cut is safe. Criterion (2) removes the
59

6

10

5

10

4

10

4

10

3

10

3

10

2

10

2

6

10

5

rejected(a) data

Number of events

Number of events

10

10

10
1
10

3

(b) final sample
10

data
MC

2

rejected(a) data

10
1
10

3

10

2

(b) final sample
data
MC

10

10

1

1
0

50

100

150

200

250

300

350

400

450

500

0

50

100

150

200

250

300

350

400

450

500

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

Figure 4.4: PE300 distribution for (a) the raw data and (b) the final FC data and atmospheric
neutrino Monte Carlo events (no oscillation) in FC final samples for SK-I (left) and SK-II (right).
The number of the Monte Carlo events in (b) is normalized to that of the data. The selection
criterion is shown by arrows.

cosmic ray muon events. Figure 4.5 shows the NHITA800 distributions for the raw data, FC
atmospheric neutrino Monte Carlo events and the FC final events. Criterion (3) removes theelectron events from the decay of the cosmic ray muons stopped in the ID. The events within
30 µsec after the selected events by the above criteria are selected to keep the decay electrons.
These events are attached to the fully contained event candidates as sub-events and not counted
as a primary atmospheric neutrino event. These cuts reduce the data size from 106 events/day
to 3000 (2200) events/day for SK-I (SK-II).

4.2.2

Second Reduction

In the second reduction, the low energy events and cosmic ray muons are rejected. The
selection criteria of the FC 2nd reduction are the following :(1)

NHITA800 should be ≤ 25
OD trigger is off.

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

and
(2)

PEmax /PE300 should be < 0.5.
PEmax is the maximum number of p.e.s observed by an ID PMT.

Criterion (1) is to reject cosmic ray muons by tighter threshold than the 1st reduction
as shown in Figure 4.5 by solid lines. The second cut removes the low energy events andelectrical noise events, which have one larger hit signal from a single PMT. Figure 4.6 shows
the PEmax /PE300 distributions for the data after the 1st reduction, FC atmospheric neutrinoMonte Carlo events and the FC final events. A PMT sometimes flashes because of a discharge
around the dynode structure. The flashing PMT tends to record a very large charge. The flasher
60

6

10

5

10

4

3

10

3

10

2

10

2

10

5

10

5

10

4

10

4

10

3

10

3

10

2

10

2

6

10

5

10

4

10

(a) data

rejected

Number of events

Number of events

1010(b) MC101(a) data

rejected(b) MC

10
1
10

5

10

5

10

4

(c) final sample

104(c) final sample

10

3

10

3

10

2

data
MC

10

2

data
MC

10

10

1

1
0

10

20

30

40

50

60

70

80

90

100

0

10

20

NHITA800

30

40

50

60

70

80

90

100

NHITA800

Figure 4.5: NHITA800 distributions for (a) the raw data, (b) fully contained atmospheric neutrinoMonte Carlo events whose vertex position is more than 2 m away from the ID wall and (c) final
samples of the data and the Monte Carlo events for SK-I (left) and SK-II (right). The numberof the Monte Carlo events in (c) is normalized to that of the data. The arrows written by dotted(solid) lines show the selection criteria in the FC 1st (2nd) reduction.

events due to such a noisy PMT are rejected further in the next step. The event rate is 200(280) events/day for SK-I (SK-II) after the FC 2nd reduction.

4.2.3

Third Reduction

After the 1st and 2nd reduction steps, remaining background events are mostly noise eventsand cosmic ray muons which have a small number of OD hits.
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 special through-going muon fitter is applied if the number of p.e.s in
any single PMT is larger than 230 p.e.s. This fitter selects the entrance point, which is thepoint of the earliest hit PMT with some neighboring hit PMTs and the exit point, which isdefined the center of the saturated ID PMTs. Then, if the goodness of the fit is greater than
0.75, and the number of hit OD PMTs around the reconstruted muon entrance or exit point ismore than 9, the events are rejected as through-going muons. In SK-II, the condition to apply
a special through-going muon fitter is modified to additionally require that number of hit ID
PMTs exceeds 1000. The rejection criteria are summarized :(1)

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

and

61

10

5

10

4

3

10

3

2

10

2

10

5

10

4

10
10

rejected(a) data(a) data

10

10

1

10

5

10

4

10

3

10

2

Number of events

1

Number of events

rejected(b) MC

10
1

10

5

10

4

10

3

10

2

(b) MC

10
1

10

5

10

5

10

4

(c) final sample

104(c) final sample

10

3

10

3

10

2

data
MC

10

2

data
MC

10

10

1

1
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

PEmax/PE300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PEmax/PE300

Figure 4.6: PEmax /PE300 distribution for (a) the data after the FC 1st reduction, (b) fully
contained atmospheric neutrino Monte Carlo events and (c) final samples of the data and thethe Monte Carlo events for SK-I (left) and SK-II (right). The number of the Monte Carlo eventsin (c) is normalized to that of the data. The selection criteria are shown by arrows.

(2)

goodness of through-going muon fit > 0.75

and
(3)

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

The goodness of through-going muon fit is defined as :goodness =



X 1
1
(ti − Ti )2
exp −
X1 ×
2(1.5 × σi )2
σi2
i
2
σi
i

(4.1)

where ti and σi are the observed hit time of the i-th PMT and its resolution, and Ti is the hit
time expected from the entering time of muon and its track. Figure 4.7 and 4.8 show the numberof hit OD PMTs near the entrance and the exit points for the data after the 2nd reduction,
the FC atmospheric neutrino Monte Carlo events and the FC final events satisfying the above
criteria (1) and (2).
Stopping muon cut
To eliminate stopping muons, a stopping muon fitter is applied, which finds the entrance
point in the similar way as through-going muon fit. The events satisfying the following criteria
are classified as stopping muons and removed :
62

10

5

10

4

10

5

10

4

10

3

10

3

10

2

10

2

rejected(a-in) data

10

4

10

3

10

2

Number of events

Number of events

1
5

10

(b-in) MC

10
10
10

(a-out) data

10

1
10

rejected

3

10

5

10

4

10

3

10

2

10
10

3

10

2

(c-in) final sample

2

data
MC

10

(b-out) MC(c-out) final sample
data
MC

10

1

1
0

5

10

15

20

25

30

35

40

45

50

Nhitain

0

5

10

15

20

25

30

35

40

45

50

Nhitaout

Figure 4.7: Number of hit OD PMTs within 8 m of the entrance point (left) of muons and within
8 m of the exit position (right) for (a) the data after FC 2nd reduction, (b) fully contained
atmospheric neutrino Monte Carlo events and (c) final samples for SK-I. The number of theMonte Carlo events in (c) is normalized to that of the data. The selection criteria are shown byarrows.

63

4

10

3

10

2

rejected(a-in) data

10

4

10

3

10

2

10

10

1

1

10

3

Number of events

Number of events

10(b-in) MC
10

2

10

10
10

3

10

3

10

2

data
MC

10

(a-out) data(b-out) MC

10

10

3

10

2

(c-in) final sample

2

rejected(c-out) final sample
data
MC

10

1

1
0

5

10

15

20

25

30

35

40

45

50

Nhitain

0

5

10

15

20

25

30

35

40

45

50

Nhitaout

Figure 4.8: Number of hit OD PMTs within 8 m of the entrance point (left) of muons and within
8 m of the exit position (right) for (a) the data after FC 2nd reduction, (b) fully contained
atmospheric neutrino Monte Carlo events and (c) final samples for SK-II. The number of theMonte Carlo events in (c) is normalized to that of the data. The selection criteria are shown byarrows.

64

10

5

10

4

10

3

10

2

rejected

4

10

3

10

2

Number of events

Number of events

4

10

3

10

2

rejected(a) data

10

10
10

10

(a) data(b) MC

10

10

4

10

3

10

2

(b) MC10(c) final sample
102(c) final sample
10

data
MC

10

2

data
MC

10

1

1
0

5

10

15

20

25

30

35

40

45

50

Number of hit OD PMTs

0

5

10

15

20

25

30

35

40

45

50

Number of hit OD PMTs

Figure 4.9: Number of hit OD PMTs near the entrance point of muons for (a) the data after
FC 2nd reduction, (b) fully contained atmospheric neutrino Monte Carlo events and (c) final
samples satisfying a condition, goodness > 0.5 for SK-I (left) and SK-II (right). The number ofthe Monte Carlo events in (c) is normalized to that of the data. The selection criteria are shownby arrows.

(1)

NHITAin ≥ 10
or
NHITAin ≥ 5
if goodness of stopping muon fit > 0.5
NHITAin is the number of hit OD PMTs located within 8 m from the entrance
point in a fixed 800 nsec time window.

The direction of muon is reconstructed to maximize the total number of p.e.s inside the conewith a half opening angle 42◦ . The goodness definition is same as that of through-going muon fit.
Figure 4.9 shows NHITAin distributions for the data after the 2nd reduction, the FC atmospheric
neutrino Monte Carlo events and the FC final events satisfying the condition, goodness > 0.5.
Cable hole muons
On top of the detector tank, there are twelve cable holes to take signal and HV supply cables
out. Four holes out of twelve are directly above the ID and block the OD. Cosmic ray muons
going through these holes would not leave any OD signal and could be detected as contained
neutrino events. To eliminate this mis-identification possibility, a set of veto counters (2 m ×
2.5 m plastic scintillation counters) were installed in April, 1997 (see Figure 4.10). The rejection
criteria for cable hole muons are :(1)

One veto counter hit

and65Cosmic ray µ

Veto Counter
Cable Bundle
from PMT
To
Electronics
Hut

Outer
Detector

Inner
Detector

Figure 4.10: A schematic view of a cable-hole muon and a veto counter.

(2)

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

Figure 4.11 shows the reconstructed vertex distribution for the FC 1-ring µ-like events before
and after the installation of veto counters. Cable hole muons are well eliminated by veto counters
as shown in this figure.
Flasher event cut
Flasher events usually have a broad hit timing distribution compared with that of the neutrino events. The cut criteria to eliminate these flasher events for SK-I are :(1)

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

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

NMIN100 ≥ 20

Figure 4.12 shows the timing distribution of (i) a typical flashing PMT event and (ii) a typicalFC neutrino event and the time window for NMIN100 is shown by arrows.
Figure 4.13 and 4.14 shows NMIN100 distributions for the data after the 2nd reduction, theFC atmospheric neutrino Monte Carlo events and the FC final events.
66

2000

2000

(i)

1000

1000

500

500

0

Y(cm)

-500

-500

-1000

-1000

-1500

-1500

-2000
-2000 -1500 -1000 -500

(ii)

1500

Y(cm)

1500

0
500 1000 1500 2000
X(cm)

0

-2000
-2000 -1500 -1000 -5000500 1000 1500 2000
X(cm)Figure 4.11: Reconstructed vertex distribution for the fully contained 1-ring µ-like events (i)before the installation of the veto counters and (ii) after installation. The outer solid circle
shows the ID wall and the inner solid circle shows the fiducial volume (2 m from the wall).
Small dashed circles indicate the positions of the veto counters. Four clusters are clearly seenaround cable holes for data without veto counters, while no cluster is visible for data after the
installation.

30

(i)

20
10
0
200
(ii)

150
100
50
0
800

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

Figure 4.12: 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 .

67

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(a) data

rejected(b) MC

10
1
10

4

10

3

(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 4.13: NMIN100 distributions (left) for the events with more than 800 hit PMTs in the
ID and (right) less than 800 hit PMTs for SK-I. Three figures show (a) the data after FC 2nd
reduction, (b) fully contained atmospheric neutrino Monte Carlo events and (c) final samples,
respectively. The number of the Monte Carlo events in (c) is normalized to that of the data.
The selection criteria are shown by arrows.

68

10

4

10

3

10

2

rejected(a) data

Number of events

10
10

5

10

4

10

3

10

2

(b) MC

10
10

4

10

3

10

2

(c) final sample
data
MC

10
1
0

5

10

15

20

25

30

35

40

45

50

NMIN100

Figure 4.14: NMIN100 distributions for SK-II. Three figures show (a) the data after FC 2nd
reduction, (b) fully contained atmospheric neutrino Monte Carlo events and (c) final samples,
respectively. The number of the Monte Carlo events in (c) is normalized to that of the data.
The selection criteria are shown by arrows.

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

Low energy events cut
The remaining low energy events are from the decay of radio isotopes and the electrical noise.
Events satisfying the following criteria are removed as low energy background events :(1)

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

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

4.2.4

Fourth Reduction

In the fourth reduction, an intelligent pattern matching algorithm is used to further remove
the remaining flasher events. Flasher events usually repeat with similar hit patterns in thedetector in the course of hours and days. These repeated events are not likely to be caused byneutrinos. Figure 4.15 shows the scatter plots of Nmatch and the maximum value of the estimatorr for the data after the FC 3rd reduction and atmospheric neutrino Monte Carlo events. For
the Monte Carlo events, the data which were removed as flasher events in this reduction stage
are mixed, and the estimator r is calculated. The cut criteria are also shown in Figure 4.15.
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
(4.2)
r=
N
σA × σBiwhere N is the number of the patches, and hQA(B) i and σA(B) are the averaged
charge and its standard deviation for event A and B, respectively.

(3)

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

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

(5)

If r exceeds the threshold(rth ), events A and B are recognized as matched
events. rth is defined as :Br > rth = 0.168 × log10 ((PEA
tot + PEtot )/2.) + 0.130,(4.3)

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 for both SK-I and SK-II.

4.2.5

Fifth Reduction

The remaining background events are removed by several criteria specialized for each kindof background events.
70

1.2

1

1

0.8

0.8

Maximum Correlation

Maximum Correlation

1.2

0.6

rejected

0.4
0.2

(a)

0
1.2
1
0.8

rejected

0.4
0.2

(a)

0
1.2
1
0.8
0.6

0.6

rejected

0.4

rejected

0.4

0.2

0.2
0

0.6

(b)
0

10

20

30

40

50

60

0

70

(b)
0

10

20

30

40

50

60

70

Number of matched events

Number of matched events

Figure 4.15: Scatter plots of Nmatch and the maximum value of the estimator r for (a) the data
after the FC 3rd reduction and (b) atmospheric neutrino Monte Carlo events for SK-I (left) and
SK-II (right). The cut criteria are shown by arrows.

Stopping muon cut
The remaining stopping muons are rejected by tighter criteria than those in the third reduction stage. Events satisfying the following criteria are rejected :(1)

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

The entrance position in the OD is estimated by a backward extrapolation from the reconstructedvertex determined by TDC-fit (see Section 5.2.1). Figure 4.16 shows NHITAin distributions forthe data after the 4th reduction, the FC atmospheric neutrino Monte Carlo events and the FC
final events.
Invisible muon cut
Invisible muon events are caused by cosmic ray muons with momenta less than the Cherenkov
threshold and the subsequent decay electrons being observed. These events are characterized
by a low energy signal from decay electron and a signal in the OD before the trigger timing.
Events which satisfy the following cut criteria are rejected as invisible muons :(1)

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

and
(2)

NHITACearly > 4
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.
71

10

5

10

4

10

3

10

2

5

10
4
10
3
10
2
10
10
1

(a) data

rejected10

5

Number of events

Number of events

1(b) MC

10
4
10
3
10
2
10
10
1
10

5

10

4

(c) final sample

10

3

10

2

data
MC

10

4

10

3

10

2

rejected(a) data(b) MC

10
1
10

4

10

3

10

2

(c) final sample
data
MC

10

10

1

1
0

5

10

15

20

25

30

35

0

40

NHITAin

5

10

15

20

25

30

35

40

NhitainFigure 4.16: NHITAin distribution for (a) the data after FC 4th reduction and (b) fully contained
atmospheric neutrino Monte Carlo events, and (c) final samples. The number of the Monte Carloevents in (c) is normalized to that of the data for SK-I (left) and SK-II (right). The selection
criteria are shown by arrows.

and
(3)

NHITACearly + NHITAC500 > 9
NHITACearly>9

if DISTclust < 500 cm
otherwise

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 4.17 shows the NHITACearly distribution for the data after the FC 4th reduction, fully
contained atmospheric neutrino Monte Carlo events and final samples, satisfying criterion (1)and DISTclust > 500 cm. Although the number of hit PMTs in off-timing window for data islarger than that for the Monte Carlo simulation, the cut is safe for FC sample.
Coincidence muon cut
The remaining accidental coincidence muon events are removed by :(1)

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

and
72

10

5

10

4

(a) data

rejected

10

5

10

4
3
2

10

3

10

10

2

10

rejected(a) data

10

10

1

5

Number of events

Number of events

1(b) MC

10
4
10
3
10
2
10
10
1
10

4

10

3

(c) final sample

10

2

data
MC

10

4

10

3

10

2

(b) MC

10
10

3

10

2

(c) final sample
data
MC

10

10

1

1
0

5

10

15

20

25

30

35

40

0

5

10

15

20

25

30

35

40

Nhitaearly

NHITACearly

Figure 4.17: NHITACearly distribution for (a) the data after FC 4th reduction and (b) fully
contained atmospheric neutrino Monte Carlo events, and (c) final samples, satisfying criterion (1)and DISTclust > 500 cm for SK-I (left) and SK-II (right). The number of the Monte Carlo eventsin (c) is normalized to that of the data. The selection criteria are shown by arrows.

(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 4.18 shows the PElate distribution for the data after the FC 4th reduction, fully contained
atmospheric neutrino Monte Carlo events and final samples, satisfying the criterion (1). The
cut is safe for FC sample.
Long-tail flasher cut
This is a stricter version of flasher cut in the FC 3rd reduction stage. Events satisfying thefollowing criterion are removed as flasher events :(1)

NMIN100 > 5 if the goodness of point fit < 0.4
NMIN100 is the minimum number of the hit ID PMTs in a sliding 100 nsec
time window from +300 nsec to +800 nsec.

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

Goodness of point fit < 0.3

and
(4)

NHITMIN100 < 6

See Section 5.2.1 for the explanation of point fit.
After the fifth reduction, the FC event rate is 16 events/day for both SK-I and SK-II.
73

10

4

10

3

10

2

5

(a) data

rejected

10
4
10
3
10
2
10
10
1

10

5

(b) MC

10
4
10
3
10
2
10
10
1
10

Number of events

Number of events

1

2

10

5

10

4

10

3

10

2

(a) data

rejected(b) MC101
102(c) final sample(c) final sample
data
MC

10

data
MC

10

1

1
0

20

40

60

80

100

120

140

160

180

200

0

20

40

60

80

100

120

140

160

180

200

PElatePElateFigure 4.18: PElate distribution for (a) the data after FC 4th reduction and (b) fully contained
atmospheric neutrino Monte Carlo events, and (c) final samples, satisfying criterion (1) for SK-I
(left) and SK-II (right). The number of the Monte Carlo events in (c) is normalized to that ofthe data. The selection criteria are shown by arrows.

4.2.6

FC Reduction Summary

Finally, the fully contained neutrino events are selected by applying the FC event cuts :• Vertex of neutrino interactions should be inside the fiducial volume (2 m from the ID PMT
surface).
• The number of hit PMTs in the highest charge OD cluster (NHITAC) should be less than
10 (16 for SK-II).
• Visible energy (Evis ) should be greater than 30 MeV.
The detection efficiencies in each reduction step are estimated by the atmospheric neutrino
events as shown in Table 4.1. The detection efficiency for final events is estimated to be 97.6 %and 99.2 % for SK-I and SK-II, respectively. The systematic uncertainty of FC reduction isestimated to be 0.2 % for both SK-I and SK-II. The event rate of FC events for real data isshown in Figure 4.34 and is 8.18±0.07 (8.26±0.10) events/day for SK-I (SK-II).
The main sources of the background for FC events are cosmic ray muons, flasher events andneutrons from the rock around the detector. These background events are largely rejected byrequiring fiducial volume cut. The contaminations of the backgrounds are estimated as shownin Table 4.2.

74

Reduction step

SK-I (%)

SK-II (%)

1st reduction
2nd reduction
3rd reduction
4th reduction
5th reduction
Final cut

99.95
99.94
99.85
99.17
99.15
97.59

99.92
99.89
99.71
99.39
99.32
99.17

Table 4.1: The detection efficiency in the each reduction step for SK-I and SK-II for events
whose true vertices are in the fiducial volume, NHITAC less than 10 (16 for SK-II) and Evislarger than 30 MeV. In the last line, the fitted vertex is used.

SK-I

Cosmic ray µ
Flashing PMT
Neutron event
SK-II
Cosmic ray µ
Flashing PMT
Neutron event

Sub-GeV(Evis < 1330M eV /c)e-like(%) µ-like(%)
—
0.42
0.1

0.07
—
—

Sub-GeV
e-like(%) µ-like(%)
—
0.27
0.1

0.01
—
—

Multi-GeV(Evis ≥ 1330M eV /c)e-like(%) µ-like(%)
—
0.16
0.1

0.09
—
—

Multi-GeV
e-like(%) µ-like(%)
—
0.65
0.1

0.07
—
—

Table 4.2: Summary of the estimated upper limit of the contamination of each background.

75

4.3

Reduction for Partially Contained Sample

Since PC events have the OD activities by its definition, the reduction scheme is different
from that that of FC events and it is rather difficult to eliminate cosmic ray muon backgrounds.
There are five steps in PC reduction process.

4.3.1

First Reduction

The aim of the 1st reduction is to reject the through-going cosmic ray muons and low energy
events. The selection criteria of the PC 1st reduction are as follows :(1)

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

and
(2)

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

and
(3)

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

Exiting particles in the PC sample, mostly muons, must have at least 2 m track length in theID, which corresponds to the momentum loss of 500 MeV/c for muons. In criterion (1), PEtot
should be larger than 1000 p.e.s (500 for SK-II), which corresponds to 310 MeV/c for muons.
The second criteria rejects the through-going muons. The through-going muon events have abroad hit timing distribution and two hit clusters around the entrance and exit point in the OD.
Figure 4.19 shows the TWIDA distribution for the raw data, partially contained atmospheric
neutrino Monte Carlo events and final samples. Since the reflected photon in the OD is increased
and the quantum efficiency of OD PMTs is increased in SK-II, the hit timing becomes broad.
Therefore TWIDA is redefined by the counted hit OD PMTs with the threshold of 2 p.e.s in
SK-II (it was 0 p.e. in SK-I) and the cut criterion for TWIDA is tuned. The hit cluster means
the spatial cluster of neighboring hit PMTs. A hit cluster is formed around the PMT which
detects more than 8 p.e.s and the clusters within 8 m are merged.
The event rate after the PC 1st reduction is 14000 (36000) events/day for SK-I (SK-II).

4.3.2

Second Reduction

In the second reduction, another clustering algorithm of OD hit is used to reject the remaining
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 formedby looking for the charge gradient to the neighboring patches. The algorithm is illustrated in
Figure 4.20. Events satisfying the following criteria remain after the PC 2nd reduction for SK-I :(1)

NCLSTA2 should be ≤ 1
NCLSTA2 is the number of the OD hit clusters including more than 6 hit
PMTs.
76

9

rejected(a) data

10

9

10

7

10

7

10

5

10

5

10

3

10

3

10

5

10

5

10

4

10

4

10

3

10

3

10

2

10

2

Number of events

Number of events

10(b) MC

10
1
10

4

10

3

(c) final sample

10

2

data
MC

(a) data

rejected(b) MC

10
1
10

4

10

3

(c) final sample

10

2

data
MC

10

10

1

1
0

50

100

150

200

250

300

350

400

TWIDA (nsec)

0

50

100

150

200

250

300

350

400

TWIDA (nsec)

Figure 4.19: TWIDA distributions for (a) the raw data, (b) partially contained atmospheric
neutrino Monte Carlo events whose vertex position is more than 2 m away from the ID wall and(c) final samples of the data and the Monte Carlo events for SK-I (left) and SK-II (right). The
number of the Monte Carlo events in (c) is normalized to that of the data. The selection criteria
are shown by arrows.

more charge

Cluster 1

less charge

Cluster 2

Cluster 3

Figure 4.20: 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.

77

and
(2)

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

and
(3)

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, the criteria are tuned to keep the reduction efficiency.
(1)

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

and
(2)

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.

(3)

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

Criteria (2) for both SK-I and SK-II reject corner clipping muons, which left the hit PMTs inboth top (or bottom) and side regions in the OD. Figure 4.21 shows the NHITACmin distributions
for the data, atmospheric neutrino Monte Carlo events after the PC 1st reduction and final
samples for SK-I. Figure 4.22 shows the scatter plots of the NHITAendcap and the NHITAside forthe same samples. In criteria (3) for SK-I and SK-II, the number of p.e.s in the ID hit cluster
located behind the OD hit cluster is used to reject the stopping muons. Figure 4.23 shows thescatter plots of the NCLSTA2 and the PE200 for the data, atmospheric neutrino Monte Carloevents after the PC 1st reduction and final samples for SK-I. Figure 4.24 shows the scatter plotsof the NHITAC2 and the PE200 for the data, atmospheric neutrino Monte Carlo events after thePC 1st reduction and final samples for SK-II.
The event rate after the PC 2nd reduction is 2000 (5500) events/day for SK-I (SK-II).

4.3.3

Third Reduction

In the third reduction flasher events and cosmic ray stopping muons are rejected. 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 cut. Events which satisfy the following criteria arerejected as flasher events for both SK-I and SK-II :78

Number of events

10

8

10

6

10

4

10

2

10

5

10

4

10

3

10

2

(a) data

rejected

Figure 4.21: NHITACmin distributions for(a) the data, (b) partially contained atmospheric neutrino Monte Carlo events after
the PC 1st reduction and (c) final sampleof the data and the Monte Carlo events for
SK-I. The selection criteria are shown by arrows.

(b) MC

10
1
10

4

10

3

10

2

(c) final sample
data
MC

10
1
0

10

20

30

40

50

60

0

20

40

60

80

20

40

60

80

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

(a) data

100

120

140

rejected

0

NHITAendcap

200
180
160
140
120
100
80
60
40
20
0

rejected

NHITAendcap

NHITAendcap

200
180
160
140
120
100
80
60
40
20
0

NHITAendcap

NHITACmin

160

180

200

(b) MC

100

120

140

160

180

200

NHITAside

rejected

0

20

40

60

80

(a) final sample (data)

100

120

rejected

0

20

40

60

80

140

160

180

200

(b) final sample (MC)

100

120

140

160

180

200

NHITAside

Figure 4.22: Scatter plots of NHITAendcap and NHITAside for (a) the data and (b) partiallycontained atmospheric neutrino Monte Carlo events after the PC 1st reduction and (c) final
samples of the data and (d) the Monte Carlo events for SK-II. The size of the boxes shows thenumber of the events in each bin. The cut criteria are shown by the lines and arrows.

79

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

1

0

NCLSTA2

≥2

1

0
0

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

Figure 4.23: Scatter plots of NCLSTA2 and PE200 for (a) the data and (b) partially contained
atmospheric neutrino Monte Carlo events after the PC 1st reduction and (c) final samples ofthe data and (d) the Monte Carlo events for SK-I. The size of the boxes shows the number ofthe events in each bin. The cut criteria are shown by the dashed lines and arrows.

140(a) data
NHITAC2NHITAC2

80
60

80
60
40

20

20
250

500

750

140

1000

1250

1500

1750

0

2000

(b) MCNHITAC2

80
60

20
750

1000

1250

1500

1750

1000

1250

2000

PE200 (p.e.s)

1500

1750

2000

rejected(b) final sample
(MC)

60
40

500

750

80

20
250

500

100

40

0

250

120

100

0

0

140

rejected

120(a) final sample(data)

100

40

0rejected

120

100

0

NHITAC2140

rejected

120

0

0

250

500

750

1000

1250

1500

1750

2000

PE200 (p.e.s)

Figure 4.24: Scatter plots of NHITAC2 and PE200 for (a) the data and (b) partially contained
atmospheric neutrino Monte Carlo events after the PC 1st reduction and (c) final samples ofthe data and (d) the Monte Carlo events for SK-II. The size of the boxes shows the number ofthe events in each bin. The cut criteria are shown by the lines and arrows.

80

10

8

10

6

10

4

10

2

(a) data

rejected

Number of events

1
5

Figure 4.25: NHITAin distributions for(a) the data and (b) atmospheric neutrinoMonte Carlo events after the PC 2nd reduction and (c) final samples. The number ofthe Monte Carlo events in (c) is normalized
to that of the data for SK-I. The selection
criteria are shown by arrows.

(b) MC

10
4
10
3
10
2
10
10
1
10

4

10

3

(c) final sample

10

2

data
MC

10
1
0

10

20

30

40

50

60

70

NHITAin(1)

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

The stopping muon events are removed by the number of the hit OD PMTs near the entrance
position. Simple vertex and direction fitter called point fit is applied. The vertex position isdetermined by point fit as the point where the timing residual distribution of hit PMTs has the
sharpest peak assuming the photons are emitted from a point source (see Section 5.2.1). The
direction of the ring is determined by summing up the charge weighted vector for all PMTs, andthe entrance position is estimated by a backward extrapolation from the reconstructed vertex.
Events satisfying the following criteria are rejected as stopping muons for both 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.

The NHITAin distributions for the data and atmospheric neutrino Monte Carlo events for SK-I
after the PC 2nd reduction, and final samples are shown in Figure 4.25.
The event rate after the PC 3rd reduction is 100 (200) events/day for SK-I (SK-II).

4.3.4

Fourth Reduction

The PC 4th reduction rejects further cosmic ray muons, which pass through the PC 3rd
reduction due to the relatively small OD activities. The two types of event reconstruction tools
are used in this reduction stage, that is a point fit and a though going muon fit. The through81

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 the ID, and the goodness tends to be worse comparedwith 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 raymuons. The selection criteria in the PC 4th reduction are as follows for SK-I and SK-II :(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
corner clipping muons. The distribution for SK-I of the d~pfit · d~PMT and the DCORN are shownin Figure 4.26 and Figure 4.27, respectively. Through-going muons which have long track length
are rejected by the third cut. Figure 4.28 shows the scatter plots of the TLMU and the goodnessof the through-going muon fit for the data and atmospheric neutrino Monte Carlo events after
the PC 3rd reduction, and final samples for SK-I. The events in the upper-right box region are
rejected as through-going muons.
The event rate after the PC 4th reduction is 20 events/day for both SK-I and SK-II.

4.3.5

Fifth Reduction

The aim of the 5th reduction stage is the final rejection of the remaining background events
for the PC sample. This is done by some elaborate criteria specialized for each backgroundsource. The event rate after the PC 5th reduction is 1 events/day for both 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.

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.
82

Number of events

10

6

10

5

10

4

10

3

10

2

10

4

rejected(a) data

Figure 4.26: d~pfit · d~PMT distributions for (a)the data and (b) partially contained atmospheric neutrino Monte Carlo events after
the PC 3rd reduction and (c) final samples.
The number of the Monte Carlo events in (c)is normalized to that of the data for SK-I.
The selection criteria are shown by arrows.

(b) MC
10

3

10

2

10

3

10

2

(c) final sample
data
MC

10
1

Number of events

-1

10

5

10

4

10

3

10

2

10

3

10

2

10

2

-0.8

-0.6

-0.4 -0.2

0

0.2

→
→
dpfit • dPMT

0.4

0.6

0.8

1

(a) data

rejected

Figure 4.27: DCORN distributions for for(a) the data and (b) partially contained atmospheric neutrino Monte Carlo events after the PC 3rd reduction and (c) final samples. The number of the Monte Carlo eventsin (c) is normalized to that of the data for
SK-I. The selection criteria are shown by arrows.

(b) MC(c) final sample

10

0

200

400

600

800

data
MC
1000 1200 1400 1600

DCORN (cm)83

goodness(a) data

goodnessgoodnessgoodness

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

(b) MC0500 1000 1500 2000 2500 3000 3500 4000 4500 5000

muon track length (cm)

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

(c) final sample (data)(d) final sample (MC)0500 1000 1500 2000 2500 3000 3500 4000 4500 5000

muon track length (cm)Figure 4.28: Scatter plots of muon track length and goodness of through-going muon fit for (a)the data and (b) partially contained atmospheric neutrino Monte Carlo events after the PC 3rd
reduction, and (c) final samples of the data and (d) the Monte Carlo for SK-I. The events inthe upper-right box region are rejected as through-going muons.

Through-going muon cut
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 :(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.
84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.

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 bya precise fit (MS-fit) using the Cherenkov ring pattern (see Section 5.5). The entrance and theexit points on the detector wall are estimated by an extrapolation. The cut criteria for both
SK-I and SK-II are as 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
Three types of stopping muon cuts are applied. In the first cut, the number of hit OD PMTs
near the entrance position is counted. The entrance position of the stopping muons are estimatedby reconstructing the vertex position and direction using MS-fit and back extrapolating to thewall. 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.
85

In the second stopping muon cut, the opening angles between the OD hit cluster and thering estimated by two different fitters, TDC-fit and MS-fit are compared. In case of the stopping
muon event, the opening angle is expected to be large, while it is small for th PC event. The
cut criterion is the following :(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).
In the third stopping muon cut, the charge inside a 42◦ cone in the ID is used. The vertex
and direction are determined by stopping muon fit, which estimate the entrance as the positionof the earliest hit cluster in the ID. The entrance position can not be reconstructed correctly
for the PC events since the vertex of the PC events are not on the wall. Events satisfying thefollowing criteria are rejected as stopping muons :(1)

goodness of stopping muon fit > 0
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.

These cut criteria are common for SK-I and SK-II.
Cable hole muon cut
The veto scintillation counters are placed over the four cable holes on top of the detector.
Unlike the FC event selection, simple condition that only requiring veto counter hit is inapplicable for the PC event selection. Cut criteria rejecting for cable hole muons are as follows andcommon for SK-I and SK-II :(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.

86

10000(a) data
EVIS/2 (cm)

8000
6000
4000
2000

Figure 4.29: Scatter plots of Evis /2 andTRACK for (a) the data and (b) atmospheric
neutrino Monte Carlo events after the PC 4th
reduction for SK-II. The selection criteria areshown by arrows.

rejected
0

0

1000

2000

3000

4000

5000

10000

(b) MC
EVIS/2 (cm)

8000
6000
4000
2000

rejected
0

0

1000

2000

3000

4000

5000

TRACK (cm)Corner clipping muon cut
The corner clipping muon events are rejected in the PC 2nd and 3rd reduction steps. In thisreduction step, the remaining corner clipping muon events. The corner clipping muon events
have a small hit cluter in the ID, then the vertices are occasionally mis-reconstructed inside
the ID. As a result, the track length from the vertex to the exit point reconstructed by MS-fit
is large, while the track length which is estimated by the visible energy using the energy lossof muons ∼2 MeV/cm is small. This cut is applied in SK-II and the cut criterion for rejecting
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.

Figure 4.29 shows the scatter plots of Evis /2 and TRACK.

4.3.6

PC Reduction Summary

Finally, the fully contained neutrino events are selected by applying the PC event cuts :• Vertex of neutrino interactions should be inside the fiducial volume (2 m from the ID PMT
surface).
• The number of hit PMTs in the highest charge OD cluster (NHITAC) should be larger
than 9 (15 for SK-II).
• Visible energy (Evis ) should be greater than 350 MeV.
The detection efficiency in each reduction step is estimated by the atmospheric neutrino Monte
Carlo events as shown in Table 4.3. The systematic uncertainty of PC reduction is estimated tobe 2.6 % and 4.0 % for SK-I and SK-II, respectively.

87

Reduction step

SK-I (%)

SK-II (%)

1st reduction
2nd reduction
3rd reduction
4th reduction
5th reduction

99.0
94.2
93.1
87.9
85.8

98.7
94.2
93.1
85.8
83.2

Table 4.3: The detection efficiencies in the each reduction step for SK-I and SK-II for events
whose true vertices are in the fiducial volume, NHITAC larger than 9 (15 for SK-II) and Evislarger than 350 MeV. In the last line, the fitted vertex is used.

The remaining background for PC sample is cosmic ray muons. After all reduction process,the remaining events are scanned by physicists and the contamination of background is estimatedto be 0.2 % and 0.7 % for SK-I and SK-II, respectively. The event rate of PC events for real datais shown in Figure 4.34 and is 0.61±0.02 (0.53±0.03) events/day for SK-I (SK-II) as shown in
Figure 4.34.

4.4

Reduction for Upward-Going Muon Sample

The upward going muon events are classified into two categories : (i) upward stopping muon
events with only an entrance point in the OD and (ii) upward through-going muon events withan entrance point and exit point in the OD. In order to separate the upward stopping muon
events and the upward through-going muon events, the number of hit OD PMTs within 8 mfrom the exit position (NHITEX) is used. Figure 4.30 shows the NHITEX distributions for
upward stopping and upward through-going muon final samples. The events with NHITEX lessthan 10 (16 for SK-II) are regarded as stopping muons and the rest of events are categorized tothrough-going muons. In SK-II, the cut criterion is shifted larger due to the change for the OD
responses. The systematic uncertainty for the stopping/through-going separation is estimated tobe 0.4 % and 0.3 % for SK-I and SK-II, respectively, by comparing the NHITEX distributions forthe observed data and the Monte Carlo events. Low energy events and downward-going muon
events are rejected by the automated reduction process and the remaining background after
automated process, mainly mis-reconstructed horizontal-going cosmic ray muons are rejected byphysicists with eye-scanning.
Although the oscillation analysis uses upward-going muons, the solar term effect, which is the
topic of this thesis, is only relevant to Sub-GeV neutrinos. Hence, the reduction for upward-going
muons will be described only briefly.

4.4.1

Charge cut

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

PEtot should be > 8000 p.e.s (3000 for SK-II)

and
88

10

4

10

10

3

10

2

stopping

Number of events

Number of events

stopping

4

through-going

10

1

10

3

10

2

through-going

10

1
0

10

20

30

40

50

60

70

NHITEX

0

10

20

30

40

50

60

70

NHITEX

Figure 4.30: NHITEX distributions for upward stopping and through-going muon final samples
for SK-I (left) and SK-II (right). The points show the observed data, and histograms show theatmospheric 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).
(2)

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

PEtot =8000 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 and the 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.

4.4.2

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 [118].
89

All events from the output of the upward-going muon reduction are passed through theprecise fitter which is described in 5.10. The direction reconstructed by the precise fitter is used
to select upward-going events by the criterion for cosΘ <0.

4.4.3

Scanning

In order to eliminate the background events such as horizontal-going, corner clipping or
bremsstrahlung cosmic ray muons and noise events, the selected events as upward are scannedby physicists using a visual display and upward-going muon events are selected one by one. All
events are checked by two independent scanners not to miss neutrino events. The possibility to
miss a upward-going event is at most 1 % by a scanner. Therefore, the efficiency of scanningis estimated to be almost 100 %. About a half of the events remaining after all the automatedreduction steps are rejected by this final scan. We note that the scanners only check thatthe reconstructed vertex and direction is not largely wrong. The reconstruction results in theprevious stage are used in the final physics analysis.

4.4.4

Efficiency and Background for Upward-Going Muon Sample

The final samples of the upward going muons are required the following conditions :• The reconstructed particle direction is upward.
• The reconstructed track length should be greater than 7 m.
The second criterion is to keep the quality of the event reconstruction. The systematic error forthe track length cut is estimated to be 1.1 % for both SK-I and SK-II.
The reduction efficiency of the upward stopping and through-going muon events are estimated by the upward-going muon Monte Carlo events to be 102 % (101 %) for stopping muons
and 96 % (94 %) for through-going muon events for SK-I (SK-II). Efficiency higher than 100 % for
upward stopping muons is due to a slight bias in the separation of stopping and through-going
muons, causing a small fraction of the more numerous through-going muons to be misidentified
as stopping muons. The systematic uncertainty in the data reduction process for upward-goingsample is estimated by comparing the distribution of each cut variable of the data with that of
the Monte Carlo events. The estimated errors are 1.3 % (1.9 %) for upward stopping muons and
0.5 % (0.9 %) for upward through-going muons for SK-I (SK-II).
The final samples of upward-going muons contain cosmic ray muon background in the mosthorizontal bin (−0.1 < cos Θ < 0). Some of the down-going cosmic ray muons are fitted to be
upward-going because of the fitter resolution and multiple Coulomb scattering of muons in the
neaby rock. Figure 4.31 shows the zenith versus azimuth directions for the upward going muon
sample. Clusters of cosmic ray downward muons are seen in the regions zenith angle cosΘ > 0(downward) and azimuth angle around φ = 120◦ , 180◦ and 270◦ because the rock of the mountaincovered over the detector is thin for these directions. In order to estimate the contaminationof the background events, the upward going muon samples are divided into two azimuth angle
regions. Figure 4.32 shows the azimuth angle distributions for upward going muon samples, inwhich region (2) shows the thinner mountain direction, which is defined as 60◦ < φ < 310◦ forstopping muons and 60◦ < φ < 240◦ for through-going muons and region (1) shows the thicker
mountain direction.

90

Azimuth-Zenith Angle Distribution of Observed Through Going Muons

cosθ

cosθ

Azimuth-Zenith Angle Distribution of Observed Stopping Muons

0

0

-0.25

-0.25

-0.5

-0.5

-0.75

-0.75

-1

0

100

200

-1

300

φ [degree]

0

100

200

300

φ [degree]

Figure 4.31: Scatter plots of the zenith angle and the azimuth angle of the muon directions for
stopping (left) and through-going muon data (right) for SK-I. Vertical axis shows the cosine ofzenith angle and horizontal axis shows the azimuth angle of the muon direction.

Figure 4.33 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 upward direction in the region (2) with (exponential + constant)function. The numbers of background events in the range of −0.1 < cos Θ < 0 are estimatedto be 40.3±13.7 for upward stopping muon and 14.4±7.2 for upward through-going muon for
SK-I, respectively. For SK-II, 19.4±6.3 events for upward stopping muons and 9.0±5.2 events
for through-going muons are estimated. The errors represent the statistical uncertainties of thefitting. The event rate of upward going muon events before background subtraction is shownin Figure 4.34. The event rate after background subtractio is 0.25±0.01 (0.25±0.02) events/day
for stopping muon events and 1.12±0.03 (1.06±0.04) events/day for through-going muon eventsfor SK-I (SK-II).

91

3

10

2
(1)

(2)

Number of Events

Number of Events

10

(1)

10

4

10

3

10

2

(1)

(2)

(1)

10

1
0

100

200

300

0

φ [degree]

100

200

300

φ [degree]

10

10Number of Events

Normalized Number of Events / φ=360°

Figure 4.32: Azimuth angle distributions for stopping (left) and through-going muons (right) for
SK-I. Region (2) shows the thinner-mountain direction and region (1) shows the thicker-mountaindirection. White histograms show the distributions of downward-going muons (0 < cos Θ < 0.1)
and the hatched histograms show the upward-going muons (cos Θ < 0). Peaks due to thecontamination from cosmic ray muons are seen in region (2) for downward-going muons.

3
Region (1) φ<60°,φ>310°
Region (2) 60°<φ<310°

10

2

4
Region (1) φ<60°,φ>240°
Region (2) 60°<φ<240°

10

3

10

2

10

10
1
1

-1

-0.1

-0.05

0

0.05

10

0.1

cosΘ

-0.1

-0.05

0

0.05

cosΘ

Figure 4.33: Zenith angle distributions for stopping muons (left) and through-going muons(right) for SK-I. Reverse triangles show those for region (1) (thicker-mountain direction) andopen circles show those for region (2) (thinner-mountain region).

92

12

SK1

SK2

Event rate (event/day)

10

8

6

FC
PC
Upmu
4

2

0
1996

1998

2000

2002

2004

2006

Year

Figure 4.34: The event rate for FC, PC and upward-going muon events as a function of theelapse days from 1996. The dashed line shows the averaged event rate for each sample.

93

Chapter 5

Event Reconstruction
5.1

Overview

Event reconstruction process is applied to the atmospheric neutrino events which passthrough the data reduction. The same programs are used for both the observed data and theatmospheric neutrino Monte Carlo events. The event reconstruction process is fully automated.
The detailed explanations of the procedures are described in this chapter.
The outline of the reconstruction process for FC and PC events is the following :(1) Vertex Fitting
The vertex position is determined as the point at which the timing residual distribution of
hit PMTs has the sharpest peak. The direction and the outer edge of the dominant ringis estimated.
(2) Ring Counting
The other possible rings are searched using the vertex position and the direction of thedominant ring. The ring candidates are tested whether the ring is true or not by a likelihood method, and the number of rings is determined.
(3) Particle Identification
Each ring is classified into two types, a showering type (e± , γ) and non-showering type(µ± , π ± ), by the ring pattern and the opening angle. The showering type and nonshowering type are commonly called e-like and µ-like, respectively.
(4) Precise Vertex Fitting (only for single-ring event)The precise vertex position is obtained using the Cherenkov ring pattern assuming theparticle type, a non-showering type or a showering type. This precise vertex fitter is called
MS-fit.
(5) Momentum Determination
The momentum of each ring is determined from the charge detected inside a Cherenkov
cone. The conversion from the charge to the momentum is determined based on a Monte
Carlo simulation and the detector calibration.
94

(6) Ring Number Correction
Rings which have the low momentum and overlapped with the other energetic rings are
discarded as fitting mistakes.
(7) Decay Electron finding
Decay electrons following the primary events are searched.
(8) π 0 Fitting
The other ring candidate of π 0 events is searched among the events which are recognizedas single-ring e-like by the above reconstruction processes.

5.2

Vertex Fitting

The event reconstruction process starts from the vertex fitting. The vertex position is reconstructed using the timing information of hit PMTs in three steps.

5.2.1

Point-Fit

Vertex position is roughly estimated in the first step assuming all of the photons are emitted
at the same time from a point source. The time of flight of photon is subtracted from the hit
timing for each PMT. The vertex is estimated as the point where the timing residual distribution
is peaked. This vertex reconstruction algorithm is called point-fit, and the goodness of the fit isdefined as follows :(ti − t0 )2
1 X
exp −
(5.1)
Gp =
N
2 (1.5 × σ)2i

where N is the number of hit PMTs, ti is the time residual of the i-th PMT, t0 is a freeparameter chosen to maximize Gp and takes a value near the mean value of ti , and σ is the
PMT timing resolution taken to be 2.5 nsec. The numerical factor 1.5 is chosen to optimize thefitter performance. The fitter searches for the vertex point which gives the maximum value ofGp .
The particle direction is also roughly estimated by summing up the charge weighted vector
over all PMTs :X~0
P~i − O
(5.2)
d~0 =
qi ×
~ 0|
|P~i − Oi~ 0 is the vertex position obtained by point fit, P~i is thewhere d~0 is the particle direction, O
position and qi is the detected charge of the i-th PMT.

95

5.2.2

Ring Edge Search

The direction and the outer edge of the dominant ring is measured in the second step. To
determine the direction of the ring, the estimator Q(θedge ) is defined as follows :Z θedge!2PE(θ)dθ(θedge − θexp )2dPE(θ)0(5.3)
Q(θedge ) =×× exp −sin θedgedθ
2σθ2θ=θedgewhere θexp and σθ are the Cherenkov opening angle expected from the charge within the cone
and its resolution, respectively. PE(θ) is the angular distribution of the observed charge as afunction of the opening angle θ from the particle direction. The observed charge is corrected
for the effect from a water transparency and a PMT acceptance. The ring edge position θedge isdetermined to satisfy the following two conditions :(1) θedge > θpeak(2)

d2 PE(θ)
=0
d2 θ
θedge

where θpeak is the opening angle where PE(θ) takes maximum. PE(θ) distribution and thesecond derivative of PE(θ) for a typical event are shown in Figure 5.1.
The estimator Q(θedge ) is calculated changing the particle direction around d~o which isestimated by point fit. The direction of the ring and the ring edge position are determined tomaximize the estimator Q(θedge ).

5.2.3

TDC-FitFinally the vertex position is more precisely determined by TDC-fit. Instead of assuming
a point-like, the track length of the charged particle and the scattered Cherenkov photons aretaken into account in the TDC-fit.
The time residual of the i-th PMT is calculated by :1
~ i − O|~ − n × |P~i − X~ i | for PMTs inside the Cherenkov ring t0i − × |X
c
c
(5.4)
ti =

 t0 − n × |P~i − O|
~
for
PMTs
outsidetheCherenkov
ringic~ is the vertex position, X~ i is the position at which Cherenkov photons are emitted
where O
toward the i-th PMT, n is the refractive index of water, P~i is the position and t0i is the hit
timing of the i-th PMT.
The estimator of the TDC-fit consists of three parts which are calculated by the different
way based on the position of the PMT. For the hit PMTs inside the Cherenkov ring, the fitting
estimator GI is defined by :X 1(ti − t0 )2
exp −
GI =
(5.5)
2 (1.5 × σ)2σi2iwhere σi is the timing resolution of the i-th PMT depending on the detected p.e.s, σ is thetiming resolution for the average p.e.s of all hit PMTs and ti is the time residual of the i-th
PMT calculated by Equation (5.4). The definition for t0 is same as that in Equation (5.1).
96

number of p.e.

150
100
50

second derivarive

0

0

20

40

60

80

0

20

40

60

80

10

0

-10

degreeFigure 5.1: The typical PE(θ) distribution (upper) and its second derivative (bottom) as afunction of Cherenkov opening angle.

For the hit PMTs outside the Cherenkov ring, the definition of the estimator changes withthe time residual. If the hit timing is later than t0 , the contribution from the scattered photons
are considered. The definitions of the estimator for hit PMTs outside the Cherenkov ring are as
follows :



X 1 
(ti − t0 )2
exp −
×2−1
( for ti ≤ t0 )
(5.6)
GO1 =
2 (1.5 × σ)2
σi2
i





X 1 
(ti − t0 )2, Gscatt (ti , t0 ) × 2 − 1
max exp −GO2 =2 (1.5 × σ)2σi2i( for ti > t0 )

(5.7)

where

 



Rq
(ti − t0 )2Rq
ti − t0
Gscatt (ti , t0 ) =× exp −
+ 1−
× exp −
1.52
2 (1.5 × σ)2
1.5260nsec(5.8)

Rq is the fraction of the charge detected inside the Cherenkov ring. The numerical factors in
the equations are chosen to optimize the fitter performance.
Then total estimator of TDC fit is obtained by :GT =GI + (GO1 or GO2 )
X 1
σi 2
i

97

(5.9)

42 deg. ringhit PMT(possible center)

centerCherenkov ring

(most probable)

Figure 5.2: 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.

The vertex position and the ring direction are determined to maximize the GT .
The resolutions of the TDC-fit are estimated to be 57 cm (55 cm) and 60 cm (60 cm) for FC
single-ring Sub-GeV e-like and µ-like events, 103 cm (87 cm) and 76 cm (83 cm) for FC singlering Multi-GeV e-like and µ-like events, 62 cm (85 cm) for FC multi-ring µ-like events and 90 cm(109 cm) for PC events for SK-I (SK-II). For single-ring events, the vertex positions are more
precisely estimated later by MS-fit using the ring pattern.

5.3

Ring Counting

Once the event vertex and the first Cherenkov ring is found, a ring counting algorithm isapplied to search for any other Cherenkov rings in the event.

5.3.1

Ring Candidate Search

Cherenkov ring candidates are searched by an algorithm using a Hough transformation [119].
Figure 5.2 illustrates the basic concept to find other possible rings, in which a spherical coordinate centered on the vertex position is considered. The shaded circle in this figure represents the
Cherenkov ring image projected to a plane perpendicular to the ring direction. Hit PMTs are
picked up and virtual circles (dashed line) centered on the position of the hit PMT are drawn
with 42 ◦ half angle. As a result of this procedure, the direction of a Cherenkov ring is identified
as a intersection point of these circles. In practice, instead of drawing virtual circles, expectedcharge distribution function f (θ) with the weight of the observed charge is mapped on a (Θ, Φ)
plane for each hit PMT. As a result of this Hough transformation method, ring center candidates
are visible as the peaks of charge on the map. A typical charge map is shown in Figure 5.3. The
two peaks correspond to the directions of Cherenkov rings.

98

600
400
200
0
150

Θ(

de 100
gr
ee 50
s)

300

200
100
00

)

grees

Φ (de

Figure 5.3: A charge map from Hough transformation algorithm for a typical two ring events.
The peaks are the direction of the Cherenkov rings.

5.3.2

Ring Candidate Test

The ring candidates are tested by a likelihood method. When N rings are already found in
the event, the test whether the (N +1)-th ring candidate is true or not by the likelihood function.
The likelihood function for the assumption of N +1 rings is defined as :



X
P +1
exp
(5.10)
LN +1 =
log prob qiobs , N
α
·
q
n
n=1
i,n
i

where hit PMTs inside N +1 Cherenkov rings is summed up. qiobs is the observed p.e.s in theexp
i-th PMT and αn · qi,nis the expected p.e.s in the i-th PMT from the n-th ring. The LN +1
is maximized by changing the scale factors αn (n=1 , · · · , N +1) with a constraint of lower
momentum limit. The probability function in Equation (5.10) is defined by :
(qiobs − qiexp )2
1


 √
exp −
(for qiexp > 20 p.e.)

2

2
σ
2π σ




(5.11)
prob (qiobs , qiexp ) =Probability obtained by the probability density distribution





function based on the convolution of a single p.e. distribution and a Poisson distribution (for q exp < 20 p.e.)iwhere σ is the resolution for qexp . If no candidate satisfies LN +1 ≥ LN , the number of rings
is determined to be N and the ring counting procedure is finalized. For the ring candidates
satisfying LN +1 ≥ LN , following four evaluation functions are calculated :99

F1: The difference L(N + 1) − L(N ) corrected for the total p.e.s. When the
difference is larger, the candidate is more likely to be true.

F2: The average of the expected p.e.s near the edge of the Cherenkov ring fromthe (N + 1)-th ring, Qedge . The charge from the other rings is not included in
the calculation. When the Qedge is larger, the candidate is more likely to be
true.

F3

: The average of the expected p.e.s outside the (N +1)-th ring, Qout . The charge
from the other rings is not included in the calculation. When the Qedge -Qoutis larger, the candidate is more likely to be true.

F4: The residual p.e.s from the expectaion with N rings are calculated. Thenthe residual-charge weighted vector from all PMTs are calculated. When theabsolute value of the vector sum is large, the candidate is more likely to betrue.

For SK-I, the final evaluation function is defined to be a linear combination of four evaluation
functions with optimized weights (αi ) :FSK−I =

4X

[αi Fi ](5.12)

i=1

If FSK−I is positive, the N + 1-th ring is determined to be true. In SK-II, other two evaluation
functions are added :F5: The difference in p.e’s between the peak of a candidate ring and the average ofinside and outside the ring. The larger F5 is, the more probable the candidate
is to be a true ring.

F6: The azimuthal symmetry of a ring with respect to the direction of the ring. A
single-ring event is more symmetric than a multi-ring event. F6 is only used
to separated single and multi ring events.

In SK-II, the ring counting is modified to use a true log likelihood method. Using MC samples ofsingle-ring and multi-ring events, the probability density functions (PDFs) for each evaluation
function are determined, and the likelihood is calculated as :FSK−II ==6X

log [(Pi )](5.13){log [(Pi )N +1 ] − log [(Pi )N ]}(5.14)i6X
iwhere Pi is the probability density function for i-th evaluation function, and PN +1 and PN
are the probability for (N + 1)-ring and N -ring events. Figure 5.4 shows FSK−I and FSK−II
distributions.
The efficiency for identifying Sub-GeV CCQE νe (νµ ) events as single-ring is 93.2 % (96.0 %)for SK-I. The efficiency for identifying Multi-GeV CCQE νe (νµ ) events as single-ring is 84.4 %(93.3 %) for SK-I. The efficiency for identifying Sub-GeV CCQE νe (νµ ) events as single-ringis 93.2 % (96.6 %) for SK-II. The efficiency for identifying Multi-GeV CCQE νe (νµ ) events assingle-ring is 92.9 % (96.3 %) for SK-II.
100

120

Single-ringMulti-ring0

10 20 30 40 50
Multi-ring100

Number of events

Number of events

400
Single-ring
350
300
250
200
150
100
50
0
-50 -40 -30 -20 -10

80

900
800
700
600
500
400
300
200
100
0
-15

Single-ring

250

Single-ring

-10

-5

Multi-ring

0

5

10

15

Multi-ring

200
150

60

100

40
20

50

0
-50 -40 -30 -20 -10 0 10 20 30 40 50
Ring counting likelihood

0
-15

-10

-5
0
5
Ring counting likelihood

10

15

Figure 5.4: The ring-counting likelihood distributions for FC Sub-GeV events (top) and MultiGeV events (bottom) of data (dot) and the Monte Carlo events (solid line) assuming 2-flavorνµ ↔ ντ oscillation assumed with (sin2 2θ, ∆m2 ) = (1.00, 2.5 × 10−3 eV2 ) for SK-I (left twopanels) and SK-II (right two panels). The hatched histograms show the CCQE interactions.

5.4

Particle Identification

A particle identification (PID) program is applied to classify the Cherenkov rings into two
types:showering (e-like) or non-showering (µ-like). Due to the electromagnetic shower and multiple scattering, electrons or gamma-rays produce e-like rings, which give diffused ring patterns.
Non-showering (µ-like) rings are produced by muons or charged pions and have sharper ringedges. Figure 5.5 shows the event display of electron and muon neutrino MC events with amomentum of 1 GeV. In addition, the Cherenkov rings from electrons and gamma-rays have theCherenkov opening angle of 42 ◦ , but the Cherenkov rings from muons or charged pions can havesmaller angles if they are not highly relativistic (β = v/c < 1) and when they lose energy. The
PID algorithm exploits these differences in the patterns and the opening angles of the Cherenkov
rings. The details of PID are described below.

5.4.1

Expected Charge Distributions

The first step of PID is to calculate the expected p.e.s in each PMT from an electron or
muon. The expected p.e.s in the i-th PMT from an electron is calculated as follows :qiexp (e) = αe × Qexp (pe , θi ) × 1.5
1
R× f (Θi ) + qiscatt×ri
exp(ri /L)

where
αe

: normalization factor
101(5.15)

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 5.5: Event display of single-ring electron (left) and single-ring muon (right) neutrino
MC event. Each small circle on the figure represents a hit ID PMT, and the size of a circle
corresponds to the number of photoelectrons detected. An electron event gives diffused ringpattern, and a muon event has a sharp ring edge.

ri: distance from the vertex to the i-th PMT

θi: opening angle between the i-th PMT direction and the ring direction

L: light attenuation length in waterf (Θi )

: correction for the PMT acceptance as a function of the photon incidence
angle ΘiR

: radius of the virtual sphere (16.9 m)

Qexp (pe , θi ) : expected p.e. distribution from an electron as a function of the opening
angle and the electron momentum
qiscatt: expected p.e.s for the i-th PMT from scattered photons

The expected p.e. Qexp (pe , θi ) as a function of the opening angle of the ith PMT and theelectron momentum is obtained by a Monte Carlo simulation. The distance dependence ofthe light intensity is corrected by a factor (R/li )1.5 . In addition to the direct photons, theexpectation from scattered photons are also considered.
In the case of muon, the expected p.e.s in i-th PMT is analytically calculated as follows :21
sin θxi
 + qiknock  ×× f (Θi ) + qiscatt (5.16)
qiexp (µ) = αµ × 
dθ
exp(r
/L)
i
r sin θ + r ·
i

xi

i

dx x=xi

where

102

r⋅d
θ
dx⋅
sin
θ
dθ

r
θ
dx

muon
Figure 5.6: A schematic view of Cherenkov radiation from a muon. Cherenkov photons are
emitted to the area of 2πr sin θ(dx sin θ + rdθ) while a muon travels the distance of dx.

αµ: normalization factorx: track length of the muonxi

: track length of the muon at which Cherenkov photons are emitted toward
the i-th PMTθ: Cherenkov opening angle of the muon traveling at x

θxi

: Cherenkov opening angle of the muon traveling at xi

qiknock: expected p.e.s. for the i-th PMT from knock-on electrons

sin2 θ comes from the Cherenkov angle dependence of the Cherenkov light intensity. The termr(sin θ+r(dθ/dx)) comes from the area where Cherenkov photons are emitted to. The Cherenkov
angle shows a decrease of dθ while a muon travels a distance of dx, and Cherenkov photons are
emitted to the area of 2πr sin θ(dx sin θ + rdθ) as shown in Figure 5.6. The contribution fromknock-on electrons qiknock is estimated by a Monte Carlo simulation.

5.4.2

Estimation of Particle Types

The definition of likelihood for the n-th ring is expressed by :


X
Y
exp
exp 
(e or µ) +Ln (e or µ) =prob qiobs , qi,nqi,n′(5.17)

n′ 6= nθi <(1.5×θc )where the product is made over the PMTs inside the n-th ring. qiobs is the observed p.e.s inexp
the i-th PMT, qi,n(e or µ) is the expected p.e.s from the n-th ring assuming an electron or aexp′
muon (Equation (5.15), (5.16)) and qi,n′ are the expected p.e.s from the n -th ring without any
assumption of the particle type. The function prob gives the probability to detect qiobs in theexp
exp
i-th PMT when qiexp is expected (Equation (5.11)). The qi,n(e) and qi,n(µ) is optimized by103changing the direction and the opening angle of the n-th ring to give the maximum likelihood
value.
In order to combine with another estimator which uses the Cherenkov opening angle, thelikelihood for the ring pattern is translated into a χ2 parameter :χ2n (e or µ) = −2 log Ln (e or µ) + constant(5.18)The probability from the ring pattern is given by :Pnpattern (e or µ) = exp − 2 !
χ2n (e or µ) − min χ2n (e) , χ2n (µ)
2 σχ2 2

(5.19)n√
The resolution of the χ2 distribution is approximated by σχ2n = 2N , where N is the numberof PMTs used in the calculation.
The probability from the Cherenkov opening angle is given by :
 
2 
obs
exp
θn − θn (e or µ) (5.20)
Pnangle (e or µ) = exp −2 (δθn )2

where θnobs and δθn are the reconstructed opening angle of the n-th ring and the fitting error,
respectively. θnexp (e or µ) is the expected opening angle of the n-th ring, which is estimatedfrom the reconstructed momentum assuming an electron or a muon.
The probability functions of the PID for single-ring events and multi-ring events are definedas :pattern
angle
Psingle (e, µ) = Psingle(e, µ) × Psingle(e, µ)(Single-ring event)(5.21)

Pmulti (e, µ) =(Multi-ring event)(5.22)

pattern
Pmulti(e, µ)For a single-ring event, the final probability is defined as the product of the probabilities
from the ring pattern and the opening angle. On the other hand, for multi-ring events, only
pattern
Pmulti(e or µ) is used since the opening angle and its error is not estimated precisely. Thisalgorithm was tested
test experiment at KEK [120]. The distributions of PID likep by a beam p
lihood, PP ID ≡ − log P (µ) − − log P (e), for single-ring events and multi-ring events areshown in Figure 5.7 and Figure 5.8. PP ID < 0 is e-like and PP ID ≥ 0 is µ-like. Also, Figure 5.8
shows the PID likelihood distributions for SK-II. The misidentification probabilities are estimated to be 0.8 % (0.7 %) and 0.7 % (1.0 %) for FC Sub-GeV single-ring νe and νµ using CCQE
events for SK-I (SK-II).

5.5

Precise Vertex Fitting

For single-ring events, the vertex resolution in the longitudinal directions fitted by vertex
fitting algorithm described in Section 5.2 is not optimized since it only uses the timing information. This can be improved by re-fitting the vertex position using the ring pattern. The
MS-fit modifies the vertex position and the particle direction by using the PID likelihood, andthe vertex position is adjusted parallel to the particle direction using the first vertex fitter with
104

300

µ-likee-like

250
200
100
50
0
-10
140
120
100
80
60
40
20
0
-30

-8

-6

-4

-2

0

2

-10

6

8

10

µ-likee-like

-20

4

0
10
PID likelihood

20

Number of events

Number of events

150

30

70
60
50
40
30
20
10
0
-10
200
175
150
125
100
75
50
25
0
-15

µ-likee-like

-8

-6

-4

-2

0

2

-5

6

8

10

µ-likee-like-1040
5PID likelihood

10

15

225
e-like200
175
single-ring
150
Sub-GeV
125
100
75
50
25
0
-10 -8
-6
-4
-2
120
100
80

e-likeµ-like

40
30

0

2

4

6

8

10

µ-likesingle-ringMulti-GeV

60
40
20
0
-30

-20

-10
0
10
PID likelihood (SK-II)

2030µ-likee-like50

Number of events

Number of events

Figure 5.7: The PID likelihood distributions for FC Sub-GeV single-ring (left top), MultiGeV single-ring (left bottom), Sub-GeV multi-ring (right top) and Multi-GeV multi-ring events(right bottom) of data (dot) and the Monte Carlo events (solid line) assuming 2-flavor νµ ↔ ντ
oscillation with (sin2 2θ, ∆m2 ) = (1.00, 2.5 × 10−3 eV2 ) for SK-I. The hatched histograms showthe νµ CC interactions.

multi-ring
Sub-GeV

20
10
0
-10

-8

-6

-4

-2

0

2

4

140
e-likeµ-like120
multi-ring
100
Multi-GeV
80
60
40
20
0
-15
-10
-5
0
5
PID likelihood (SK-II)

6

10

8

10

15

Figure 5.8: The PID likelihood distributions for FC Sub-GeV single-ring (left top), MultiGeV single-ring (left bottom), Sub-GeV multi-ring (right top) and Multi-GeV multi-ring events(right bottom) of data (dot) and the Monte Carlo events (solid line) assuming 2-flavor νµ ↔ ντ
oscillation with (sin2 2θ, ∆m2 ) = (1.00, 2.5 × 10−3 eV2 ) (solid lines) for SK-II. The hatchedhistograms show the νµ CC interactions.

105

3000
5000

FC Sub-GeV
1ring e-like

2500

4000

2000

σ = 28 cm

1500
1000
500

1000
0

50

100

150

σ = 24 cm

3000
2000

0

FC Sub-GeV
1ring µ-like

200

0

0

∆vertex (cm)

50

100

150

200

∆vertex (cm)400

FC Multi-GeV
1ring e-like

350
300

500

FC Multi-GeV
1ring µ-like

1000

400

σ = 49 cm

200
150

500

600

50

100

150

∆vertex (cm)200

σ = 65 cm

400
300

200

400

0

σ = 62 cm

300

σ = 24 cm

200

100
50

PC

600

800

250

0

700

FC Multi-ringµ-like

200

100

0

0

0

50

100

150

200

100
0

50

100

150

∆vertex (cm)∆vertex (cm)

200

0

0

50

100

150

200

∆vertex (cm)Figure 5.9: Distance between the true vertex and the reconstructed vertex for FC single-ring
events, FC multi-ring µ-like events and PC events in the atmospheric neutrino Monte Carlosample for SK-I.

the timing information. This process is iterated until the changes in the vertex position and theparticle direction is less than 5 cm and 0.5 degrees. The vertex resolution for single-ring events
is approximately 30 cm.
The performance of the fitting algorithm, TDC-fit or MS-fit, is examined by applying it to
the atmospheric neutrino Monte Carlo sample. Figure 5.9 and 5.10 show the distance between
the true vertex and the reconstructed vertex for FC single-ring events, FC multi-ring µ-likeevents and PC events for SK-I and SK-II. The vertex resolutions are estimated to be 28 cm(32 cm) and 24 cm (31 cm) for single-ring Sub-GeV e-like and µ-like events, 49 cm (47 cm) and24 cm (26 cm) for single-ring Multi-GeV e-like and µ-like events, 62 cm (85 cm) for multi-ringµ-like events and 65 cm (68 cm) for PC events for SK-I (SK-II). Figure 5.11 shows the angulardifferences between the true electron or muon direction and the reconstructed direction for CC
quasi-elastic scattering events in the atmospheric neutrino Monte Carlo sample. The angularresolutions are estimated to be 3.3◦ (3.5◦ ) and 2.0◦ (2.2◦ ) for Sub-GeV e-like events and µ-likeevents, 1.7◦ (1.6◦ ) and 0.8◦ (1.0◦ ) for Multi-GeV e-like events and µ-like events for SK-I (SK-II).
The resolution is defined as the width where 68 % of the events are included.

5.6

Momentum Determination

The momentum of each particle is estimated by the observed p.e.s inside a cone with half
opening angle of 70◦ . In order to determine the momentum for individual rings, the observed
p.e.s in hit PMTs are separated to the contribution from each ring. The separation of theobserved p.e.s is carried out based on the expected p.e. distribution from each ring as a functionof the opening angle θ and uniform in azimuthal angle φ. The observed p.e.s in the i-th PMT

106

1600

2500

FC Sub-GeV
1ring e-like

1400
1200
1000

σ = 32 cm

800

1500

σ = 31 cm

1000

600
400

500

200
0

FC Sub-GeV
1ring µ-like

2000

0

50

100

150

200

0

0

∆vertex (cm)

50

100

150

200

∆vertex (cm)250

FC Multi-GeV
1ring e-like

250

FC Multi-GeV
1ring µ-like

600
500

200

σ = 47 cm

150

200
150

σ = 26 cm

400
300

100

σ = 85 cm

100
50

100
0

50

100

150

200

0

0

∆vertex (cm)

50

100

150

0

200

0

50

100

150

PC

350
300
250
200
150

200

50
0

400

FC Multi-ringµ-like

200

100
50
0

σ = 68 cm

0∆vertex (cm)∆vertex (cm)

50

100

150

200

∆vertex (cm)Figure 5.10: Distance between the true vertex and the reconstructed vertex for FC single-ring
events, FC multi-ring µ-like events and PC events in the atmospheric neutrino Monte Carlosample for SK-II.

2000

800

FC Sub-GeV
1ring e-like

700
600
500

σ = 3.3°

400

1500
1250
750

200

500

100

250

0

0

2

4

6

8

σ = 2.0°

1000

300

0

FC Sub-GeV
1ring µ-like

1750

10

0

∆direction (degree)
225
200
175
150
125
100
75
50
25
0

2

4

6

8

10

450
400
350
300
250
200
150
100
50
0

FC Sub-GeV
1ring e-like800

σ = 3.5°

200
0

2

4

6

8

10

FC Multi-GeV
1ring e-like

600

120

0

2

4

6

8

∆direction (degree)

10

400

250

σ = 1.6°

80
60

150

40

100

100

20
2

4

6

8

10

∆direction (degree)

0

6

8

10

σ = 1.0°

200

200

0

4

FC Multi-GeV
1ring e-like

350

300

0

2

300

100

σ = 0.8°4000∆direction (degree)

FC Multi-GeV
1ring e-like

140

500

σ = 1.7°0∆direction (degree)
160

700

σ = 2.2°600
400∆direction (degree)

FC Multi-GeV
1ring e-like

FC Sub-GeV
1ring µ-like

1000

50
0

2

4

6

8

∆direction (degree)

10

0

0

2

4

6

8

10

∆direction (degree)

Figure 5.11: Angular difference between the true electron or muon direction and the reconstructed direction for CC quasi-elastic scattering events in the atmospheric neutrino Monte
Carlo events. The left four panels show for SK-I and the right four show for SK-II.

107

from the n-th ring are estimated as :
exp
qi,n
obs
obs
qi,n = qi × P exp
n′ qi,n′(5.23)

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 for the light attenuation in water and the acceptance of the PMT as follows : r  cos Θ
X
X

GMC 
i
i
obs
 α×
×
−
Si 
q
×
exp
RT OTn =i,n (5.24)
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: TOF 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 TOF subtracted hit timing distribution to reject the effect from muon decay electrons. For
the attenuation length in water L, the continuous measurement using cosmic ray through-going
muons (Section 2.6.4) is used to correct the time variation for the observed data. Figure 5.12
shows the momentum resolution for electrons and muons estimated from the Monte Carlo simulation as a function of the true momentum. The resolution is defined as 1 σ width of theGaussian fit.
The absolute energy scale is calibrated using the four independent calibration sources as
described in Section 2.6.5. The uncertainty of the energy scale is estimated to be less than 2.5 %for the momentum range from a few ten MeV/c to about 10 GeV/c.

108

Momentum resolution (%)

20

Momentum resolution (%)

6e-like

15
10
5
0

0

250

500

750 1000 1250 1500 1750 2000
Momentum (MeV/c)µ-like
4

2

0

0

250

500

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

Figure 5.12: Momentum resolutions for electron (top) and muon (bottom) estimated by theMonte Carlo simulation as a function of the true momentum. The closed circles show those for
SK-I and the open circles show those for SK-II.

5.7

Ring Number Correction

Next, if the number of rings are larger than 2, the number of rings is corrected using theenergy and angle information obtained for each ring, and mis-fit rings are removed. The criteria
for the correction to remove the i-th ring are :(1-1) pi < pj ; i 6= jpi is the momentum of i-th ring.
and(1-2) θij < 30◦
θij is opening angle between the i-th and j-th rings.
and(1-3) pi cos θij < 60 MeV/cpi cos θij is the momentum of i-th ring projected perpendicularly to j-th ring.
or(2-1) pi < 40 MeV/c
and

109

(2-2) pi /ptot < 0.05
ptot is the total momentum for all rings.
In SK-I, the momentum for the ring correction is determined without using the PID information, but in SK-II, the momentum is calculated using the PID information.

5.8

Decay Electron Finding

In this section, the detection method of electrons from muon decays following the primary
events is described. There are three types of observed decay electrons :• sub-event type
Decay electrons observed as a separate event (sub-event) as shown in Figure 5.13.
• primary-event type
Decay electrons observed in the same event as primary event.
• split type
Decay electrons occurred around the end of event timing window. The event is recorded
separately in the primary event and the sub-event.
For sub-event type events, the following criteria are required :(1)

The time interval from a primary event(∆t) is < 30 µsec.

and
(2)

The total number of hit PMTs is greater than 50 (25 for SK-II).

and
(3)

The goodness of vertex fit is greater than 0.5.

and
(4)

The number of hit PMTs in a 50 nsec time window is greater than 30 (15 for
SK-II).

and(5)

The total number of photoelectrons is less than 2000 (1000 for SK-II).

and
(6)

The number of hit PMTs in a 50 nsec time window (N50 ) is greater than 60(30 for SK-II).

For primary-event type decays, another peak after the primary event is searched with requirenment more than 20 hits in 30 nsec window above the background level. Additional conditions for decay electrons are as follows :110

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

Comnt;

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

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

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

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

Comnt;

Figure 5.13: An event display of a sub-event type decay electron. The left figure shows theprimary event (cosmic ray stopping muon) and the right figure shows the following decay electron
event. The time difference between two events is about 3.5 µsec.

(7)

The number of hit PMT in a 30 nsec time window is greater than 40 (20 for
SK-II). (for primary-event type and split type)

and
(8)

0.1 µsec < ∆t < 0.8 µsec or 1.2 µsec < ∆t < 30 µsec (for all types)

N50 =60 (30 for SK-II) corresponds to about 11 MeV of electron energy. Criterion (1) rejects
the gamma emission from µ− captured on 16 O nuclei. Criterion (8) rejects the decays in theinefficient time interval around 1 µsec. The efficiency of detecting these electrons are 80 % and
63 % for µ+ and µ− , respectively.

5.9

Event reconstruction for single-ring π 0 sample

Figure 5.14 shows the momentum distribution of FC Sub-GeV single-ring e-like events in
the Monte Carlo events with each interaction mode. The 9 % of NC events is mostly originatedfrom a NC single π 0 production (NC1π 0 ) and is a background to νe -induced CCQE events. A
π 0 immedeately decays into two γ’s and can be identified as a π 0 event by the reconstruction
algorism described in Section 2.6.5. However, one γ is sometimes missed by the standard reconstruction algorism and then such events are classified as single-ring e-like events. A γ ray fromπ 0 decay could not identified by the following reasons :1. The energies of two γ’s are highly asymmetric and the energy of the second γ (Eγ2 ) is too
small to be reconstructed as a ring.
2. The two rings of γ’s are overlapped and reconstructed as one ring when the opening angle
between two γ’s is small.
111

single-ring e-like

14000

νe CCQE
νe CC-nonQE

Number of events

12000
10000

71.3%
17.6%

NC

8.7%

νµ CC

2.4%

8000
6000
4000
2000
0

0

200

400

600

800

1000 1200

1400

Momentum (MeV/c)

Figure 5.14: The momentum distributions of FC Sub-GeV single-ring e-like events in the atmospheric neutrino events. The dashed line shows νe -induced CCQE events, the dotted line
shows νe -induced CC non-QE events (such as single- or multi- meson production), the dashedand dotted line shows NC events and the hatched region shows νµ -induced CCQE events. The
values are the fraction of each interaction mode in FC single-ring e-like events.

Figure 5.15(a) shows the scatter plot between the energy of the second γ and the opening
angle between two γ’s for NC1π 0 events in the atmospheric Monte Carlo events. To separate
these π 0 events from the CCQE events, π 0 fitter is introduced. Figure 5.15(b) shows the momentum distribution for NC1π 0 events in the atmospheric Monte Carlo events and the hatchedregion shows the events identified as single-ing e-like. About 33 % of NC1π 0 events are identified
as single-ring e-like in our standard reconstruction procedure.

5.9.1

Reconstruction algorism for π 0 fitter

The π 0 fitter reconstructs the second gamma-ray on the assumption of the existence of two
Cherenkov rings, comparing the observed charge distribution on the PMTs with the expectedcharge distribution of two gamma-rays. The two gamma-rays are supposed to be emitted at one
point and the direction of the more energetic gamma-ray is set to the direction derived from thevertex fitter (Section 5.5). Varying the direction of the second gamma-ray and the fraction ofenergy shared among the two gamma-rays, the combination of two gamma-rays which matches
best with the observed charge distribution is determined.
The expected charge distribution for the i-th PMT is calculated as follows :


1
r
i
×
× f (Θi ) + qiscatt(5.25)
qiexp = αe × Qexp (θi , pγ , ri ) × 1 − q
exp(r
/L)
2
2
i
r +R
i

P MT

where

αe

: normalization factor
112

3500

500
450

3000

400
350

Number of events

Momentum of second γ (MeV/c)

(b)

(a)

300
250
200
150

2500
2000
1500
1000

100

500
50
0

0

20

40

60

80

100

120

140

160

180

Angle between two γs (degree)

0

0

200

400

600

800

1000 1200

1400

True π Momentum (MeV/c)
0

Figure 5.15: (a) The scatter plot between the true momentum of the second gamma and thetrue opening angle between two gammas for NC1π 0 events in the atmospheric Monte Carloevents. (b) The distribution of the true π 0 momentum for NC1π 0 events in the atmosphericMonte Carlo events. The hatched region shows the NC1π 0 identified as single-ring e-like events.

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

θi: opening angle between the i-th PMT direction and the ring directionpγ: initial gamma-ray momentumri: distance from the vertex to the i-th PMT

RP M T: PMT radius of 25 cm

L: light attenuation length in waterf (Θi )

: correction for the PMT acceptance as a function of the photon incidence
angle Θiqiscatt: expected p.e.s for the i-th PMT from scattered photonsqThe factors of 1 − ri / ri2 + RP2 M T and exp(-ri /L) represent the corrections of solid angle
and light attenuation, respectively. Qexp is the expected photon flux which is estimated as afunction of the photon incident angle Θi , the initial gamma-ray momentum pγ and the distancefrom the vertex to the i-th PMT ri in the Monte Carlo simulation.

5.9.2

Best-fit Ring Configuration

To determine the best-fit configuration of two gamma-rays, a log likelihood method is used.
Using the same probability function as Equation (5.11), the likelihood is defined as :X
(5.26)
L=log prob qiobs , qiexp
i=1

113

1800

22.83
Constant
Mean
Sigma

1600
1400

/ 6
1132.
135.9
23.57

Constant
Mean
Sigma

1400

/ 4
1133.
133.7
20.76

21.19
0.35520.4124

1200

1200

1000

1000

800

800
600

600

400

400

200

200

0

1.177

1600

19.44
0.34430.3701

0

100

200

300

2

0

0

100

200

300

2

Minv (MeV/c )Minv (MeV/c )Figure 5.16: The invariant mass distributions for NC1π 0 events with the line of the Gaussian
fit for SK-I (left) and SK-II (right).

The qiexp is optimized by changing the direction of the second gamma and the momentum fraction
for the second gamma to give the minimum likelihood value, using the simplex method [121, 122].
In the procedure, any kinematical constraint of π 0 decay is not required. The π 0 events such as
two rings are almost overlapped have a narrow L distribution in the direction of the first ring.
To improve the fitting performance for such events, the second fit is performed with the finer
stepping size to look for the second gamma-ray near the first gamma-ray ring. The smaller L
between the first wide fit and the second narrow fit is chosen as a fitting result.
Figure 5.16 shows the distribution of the invariant mass (Minv ) reconstructed from two
gamma-rays of NC1π 0 events found by the π 0 fitter. Minv is defined as :q
(5.27)
Minv = 2Eγ1 Eγ2 (1 − cos θγγ ).

The mean value of the reconstructed invariant mass is 135.9 MeV/c2 and 133.7 MeV/c2 and theresolution of the invariant mass is 23.6 MeV/c2 and 20.8 MeV/c2 for SK-I and SK-II, respectively.
Figure 5.17 shows the invariant mass distributions for νe CCQE events and NC events among
the FC single-ring e-like events. The invariant mass for CCQE events is reconstructed smaller
than π 0 mass, while that for NC is reconstructed close to π 0 mass. Therefore 100 MeV/c2 cut
can separate between CCQE events and NC events among FC single-ring e-like events. For
νe CCQE events, there is no real second ring and the fitter could reconstruct the second fakering with hits around the first energetic ring. Since the electromagnetic shower defuses theedge of the first ring and hits around the first ring, the fitter recognizes these hits as a ring of
gamma-ray. Then, the opening angle between the first ring and the fake second ring is small andthe energies of the first ring and the fake second ring are expected to be small after sharing theinitial visible energy of electron as shown in Figure 5.18. The left plot on Figure 5.18 shows the
opening angle between two gammas found by π 0 fitter for νe CCQE events and NC events among
the FC single-ring e-like events and the right plot on Figure 5.18 shows the distribution of theenergy fraction of the second gamma E2 /(E1 + E2 ) for the same events. The separation of thesingle-ring π 0 events among the FC single-ring e-like events will be described in Section 6.1.1.

114

0.18
0.225

0.16

0.2

0.14

0.175

0.12

0.15

0.1

0.125

0.08

0.1

0.06

0.075

0.04

0.05

0.02

0.025

0

0

100

200

0

300

0

100

2

200

300

2

Minv (MeV/c )Minv (MeV/c )Figure 5.17: The invariant mass distribution for νe CCQE (solid line) and NC (dashed line)
among the FC single-ring e-like events for SK-I (left) and SK-II (right). Each distribution isnormalized to 1.

0.35
0.225

0.3

0.2

0.25

0.175
0.15

0.2

0.125

0.15

0.1

0.1

0.075
0.05

0.05
0

0.025

0

50

100

0

150

Opening angle (degree)

0

0.2

0.4

0.6

0.8

1

Energy fraction of 2nd ring

Figure 5.18: The reconstructed opening angle between two gammas (left) and the energy fractionof the second gamma E2 /(E1 + E2 ) (right) for νe CCQE (solid line) and NC (dashed line) among
the FC single-ring e-like events. Each distribution is normalized to 1.

115

0.7

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

0.6

Stopping

0.5
0.4
0.3
0.2
0.1
0
1

10

2

10
10
Eυ (GeV)

3

10

4

10

5

Figure 5.19: The energy spectrum of primary neutrino for upward stopping (dotted line), nonshoweing (dashed line) and showering (solid line) muon events.

5.10

Event Reconstruction for Upward-Going Muon Sample

Among the upward through-going muon events, some of them are accompanied with an electromagnetic shower. The typical energy of these muons are higher than that for non-showering
upward through-going muons. Figure 5.19 shows the primary neutrino energy distributions forthese events.
Therefore, the upward through going muons are separated to non-showering and showering
events. Details of the criteria to separate them can be found in [123]. The event reconstruction
algorithm for upward-going muons is different from that for FC and PC events. The same
program is used for both observed data and atmospheric neutrino Monte Carlo events whichpass through the data reduction process. The reconstruction is based on MS-fit (see Section 5.5)
with the assumption that the particle is a muon and the vertex is at the inner detector surface.
However, when the muon produces an electromagnetic shower, almost all the ID PMTs are hitor the Cherenkov ring cannot observed. For these events, the information of OD hit is used todetermine the particle direction. The angular resolution is 2.0◦ (2.3◦ ) for stopping muon eventsfor SK-I (SK-II), 1.3◦ (1.6◦ ) for non-showering muon events for SK-I (SK-II) and 1.5◦ (2.7◦ ) for
showering muon events for SK-I (SK-II) as shown in Figure 5.20.

116

2000
1750

UPmu stopping

1200

UPmu non-showering

1000

1500
1250
1000

800

σ = 2.0°

750

400

500
250

200

0

0

2

4

6

8

σ = 1.3°

600

10

0

0

2

4

6

8

10

200
175
150
125
100
75
50
25
0

UPmu showering
σ = 1.5°

0

2

4

6

8

10

∆direction (degree)∆direction (degree)∆direction (degree)

60
1000

UPmu stopping

40

400

σ = 2.3°600

σ = 1.6°

20

200

200

10

100
0

2

4

6

8

∆direction (degree)100

σ = 2.7°

30

300

400

UPmu showering

UPmu non-showering 50

500

800

0

600

0

2

4

6

8

∆direction (degree)

10

0

0

2

4

6

8

10

∆direction (degree)

Figure 5.20: Angular difference between the true direction and the reconstructed direction for
upmu stopping event, upmu non-showering events and upmu showering events in the atmospheric
neutrino Monte Carlo sample. The upper three panels are for SK-I and the bottom three arefor SK-II.

117

Chapter 6

Data setIn this analysis, FC, PC and UPMU data during the SK-I period (1489.2 days exposure for
FC and PC, 1645.9 days for UPMU) and the SK-II period (798.6 days for FC and PC, 827.7
days for UPMU) are used. Since the reconstruction of long path length muon is less sensitive tothe detector condition, the live time of the UPMU data is larger than that of FC and PC data.
The amount of simulated the atmospheric neutrino events are equivalent to an exposure of 100
years for SK-I and 60 years for SK-II.
Final samples of FC and PC events are selected by requiring the following criteria after data
reduction and event reconstruction processes :• FC samples(1) Number of hit PMTs in the OD hit cluster (NHITAC) < 10 (16 for SK-II)(2) Distance from event vertex to the nearest ID wall (Dwall ) > 200 cm(3) Visible energy assuming electrons (Evis ) > 30 MeV– FC single-ring µ-like samples(4) pµ > 200 MeV/c– FC single-ring e-like samples(5) pe > 100 MeV/c– FC multi-ring µ-like sample(6) The most energetic ring is identified as µ-like and the momentum pµ > 600 MeV/cand Evis > 600 MeV– FC multi-ring e-like sample(7) The most energetic ring is identified as e-like and Evis > 1330 MeV• PC samples(1) NHITAC ≥ 10 (16 for SK-II)(2) Dwall > 200 cm(3) Evis > 350 MeV– PC OD stopping sample
118

(4) 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, where PEexp
is the expected number of p.e.s in the OD from the potential track length in the
OD(5) The most energetic ring or the second one should be identified as µ-like– PC OD through-going sample(6) PEanti > PEexp /1.5
FC and PC samples are separated by the number of hit PMTs in the OD hit cluster (NHITAC, see Chapter 4). The fiducial volume for the FC and PC samples defined by Dwall >200 cm
corresponds to 22.5 kton. The visible energy (Evis ) is defined as the sum of the energy ofeach ring assuming all rings are produced by electrons. The criterion (3) for the FC sample,
Evis > 30 MeV, is required to reject remaining low energy background events. FC samples are
divided into “Sub-GeV sample” whose visible energy is below 1.33 GeV and “Multi-GeV sample” whose visible energy is above 1.33 GeV. The FC samples are classified into single-ring andmulti-ring samples according to the number of Cherenkov rings. The FC single-ring sample
is classified into µ-like and e-like samples by the results of particle identification. To increase
the purity of the interaction mode in each sample, the FC Sub-GeV single-ring events are separated into more finer samples as described in Section 6.1. For multi-ring sample, the mostenergetic ring is used to identify the particle type. For PC events, the requirement of Evis >350 MeV, which corresponds to muon momentum of 530 MeV/c, is safe for PC events because
the exiting muons must have at least momentum of 700 MeV/c to reach the OD. PC events are
separated into two categories, “OD stopping sample” and “OD through-going sample” using theobserved p.e.s in the OD and the expected charge derived from the track length. Muons in the“OD stopping sample” are assumed to stop in the OD, while more energetic muons in the “OD
through-going sample” are assumed to pass through the OD.
The upward stopping and through-going muon samples are selected by requiring the following
criteria after data reduction and event reconstruction processes :• Upward stopping muon sample(1) Number of hit OD PMTs within 8 m from the exit point (NHITEX) < 10 (16 for
SK-II)(2) Fitted direction is upward (zenith angle cosΘ ≤ 0)(3) pµ ≥ 1.6 GeV/c (equivalent to track length of 7 m for muon)• Upward through-going muon sample(1) NHITEX ≥ 10 (16 for SK-II)(2) Fitted direction is upward (zenith angle cosΘ ≤ 0)

(3) Distance from the entrance point to the exit point ≥ 7 m
Upward stopping and through going muon samples are separated by the number of hit OD
PMTs near the exit point (NHITEX, see Chapter 4). The third criterion is required to keep the
performance of the event reconstruction. The track length is determined by the distance between
the entrance and exit points for upward through-going muons. The upward through-going muon119sample is further separated into showering muon sample and non-showering muon sample. The
showering muon events consist of the high energy muons which lose energy through radiative
processes such as bremsstrahlung, e+ e− pair production and photo-nuclear interactions and theenergy of the parent neutrino is approximately 1 TeV. The momentum of upward stopping muon
is determined from the observed charge by the same way as that for FC and PC events (seeSection 5.6).
The number of events for the atmospheric neutrino data in each final sample is shown in
Table 6.1 together with the Monte Carlo predictions.

6.1

Event Samples for FC Sub-GeV

As mentioned in Section 1.3, the excess of νe in the low energy region is expected by the
sub-dominant oscillation effect due to the solar neutrino parameters. In order to enhance thepurity of CCQE interactions and to improve the sensitivity for the solar term effect, FC SubGeV events are further separated to sub-samples using the information such as the number ofdecay electrons.

6.1.1

FC Sub-GeV single-ring e-like events

The FC Sub-GeV single-ring e-like sample contains about 10 % of NC events which aremainly π 0 events from νN→ νNπ 0 interactions and can be background for the observationof the excess of νe events. Therefore these π 0 events are separated from νe -enriched samples
using the likelihood analysis with the information of π 0 fitter (Section 5.9.1). Figure 6.1 andFigure 6.2 show the invariant mass distributions from the π 0 fitter for νe + ν e charged current(CCQE) and neutral current (NC) events among FC Sub-GeV single-ring e-like Monte Carloevents in the each energy region. In the figure, the area of each distribution is normalized to
1. For e-like events with the electron momentum below 250 MeV/c, π 0 mass > 100 MeV/c2 cut
well separates between CCQE and NC. For e-like events with the electron momentum above
250 MeV/c, since the π 0 mass > 100 cut is not efficient enough to separate the two samples, thelikelihood selection using π 0 mass, energy fraction of the second ring and ∆-likelihood from π 0
fitter is applied. ∆-likelihood is defined as L(wide) − L(narrow) using Equation 5.26 with wide
fitting and narrow fitting. Figure 6.3 and Figure 6.4 show the distributions of these valuables
used in the likelihood. CCQE events and NC events are separately shown and normalized byarea. The likelihood distributions are shown in Figure 6.5 together with the criteria values shownby arrows. To keep the statistics of CCQE events, the different cut criteria in each energy region
are applied.
After the likelihood selection of π 0 -like sample, the rest of e-like events are separated to two
categories according to the number of decay electrons, 0µedecay sample which has no decay
electron or 1µedecay sample which has one or more decay electrons. Figure 6.6 show the distributions of number of decay electrons of FC Sub-GeV single-ring e-like events π 0 -like events
are subtracted for data and Monte Carlo events assuming no oscillation and 2-flavor νµ ↔ ντ
oscillation with (sin2 2θ23 , ∆m223 ) = (1.00, 2.5 × 10−3 eV2 ). The live time of the Monte Carloevents is normalized to that of the observed data. As shown in Table 6.2, the CCQE fraction of
FC Sub-GeV single-ring e-like 0µedecay sample compared with that of FC Sub-GeV single-ringe-like sample is improved to 80.4 % from 71.3 % and the fraction of NC is decreased to 4.9 %from 8.7 %. The momentum distributions of the separated three e-like samples are shown in
120

SK-I

SK-II

Data

Monte Carlo

Data

Monte Carlo

3353
2930
259
164
3227
1028
2049
150
506

2789.22434.8179.1
175.2
4132.71338.62644.4
149.7
379.3

1842
1607
147
91
1718
576
1059
83
266

1494.81302.1103.4
89.3
2155.3727.5
1347.9
79.8
203.2

746
651

667.7
881.3

417
379

415.7
496.2

647
911

1017.41143.8

349
427

504.6
588.4

Upward-going muon
417.7
stopping
non-showering 1544.8
showering
296.5

721.4
1429.1
251.0

208.6
771.2
105.4

338.3
741.9
89.9

FC Sub-GeV
single-ringe-like0µedecay
1µedecay
π 0 -likeµ-like0µedecay
1µedecay
2µedecay
2-ring π 0 -like
FC Multi-GeV
single-ringe-likeµ-likemulti-ringµ-likePC

Table 6.1: Summary of atmospheric neutrino events for data and Monte Carlo events of FC, PC
and upward-going muon samples for SK-I and SK-II. The live time of FC and PC is 1489.2 days
for SK-I and 798.6 days for SK-II and the live time of upward-going muon samples is 1645.9 days
for SK-I and 827.7 days for SK-II. The number of the Monte Carlo events is normalized by thelive time of the data.

121

(a)

(b)

0.2
0.1
0

0

100

0

0

100

0

100

0.1

(c)

0.1

0

200

200

0

200

(d)

0

100

200

(e)

0.05

0

0

100

200
2

invariant mass (MeV/c )Figure 6.1: The invariant mass distributions for νe +ν e charged current (solid line) and neutral current (dashed line) events for FC Sub-GeV single-ring e-like Monte Carol events in (a)Pe < 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, (e) 1000 MeV/c ≤ Pe for SK-I. Each distribution is normalized
to 1.

Figure 6.7 for data and Monte Carlo events. The distributions for data and the oscillated Monte
Carlo events are consistent, if the uncertainty in the absolute normalization of the Monte Carlo(which is about 20 %) is taken into account. The systematic errors in the event separation will
be described in Section 7.3.3.

6.1.2

FC Sub-GeV single-ring µ-like events

The FC Sub-GeV single-ring µ-like events are separated to three samples according to thenumber of decay electrons. They are 0µedecay, 1µedecay and 2µedecay samples which haveno decay electron, one decay electron and two or more decay electrons, respectively. Figure 6.8
shows the distribution of the number of decay electrons for FC Sub-GeV single-ring µ-like events.
The fraction of each interaction mode in three samples is summarized in Table 6.2. The numberof 0µedecay events is about 30 % of the total FC Sub-GeV single-ring µ-like events. The fractionof CCQE events is about 73 % in which decay electrons are not detected and the fraction of NC
events is about 11 %, most of which are νN→ νN′ π ± . The number of 1µedecay events is about
64 % of the total FC Sub-GeV single-ring µ-like events. The fraction of CCQE events is about
80 %, which is improved from 74.4 % in FC Sub-GeV single-ring µ-like events. The momentum
distributions for the separated three µ-like samples are shown in Figure 6.9 for data and Monte
122

(a)

(b)

0.2

0.2

0

0

100

0

200

0

100

(c)

(d)

0.1

0.1
0

200

0

100

200

0

0

100

200

(e)

0.1

0

0

100

200
2

invariant mass (MeV/c )Figure 6.2: The invariant mass distributions for νe +ν e charged current (solid line) and neutral current (dashed line) events for FC Sub-GeV single-ring e-like Monte Carol events in (a)Pe < 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, (e) 1000 MeV/c ≤ Pe for SK-II. Each distribution is normalized
to 1.

123

Energy fraction of second ring

π mass
0

(a)

0.15

0.3

(a)

0.3

(b)

0.2

0.2
0.1

0.2

0.05

0.1

0.1

0
100

(b)

150

200

0
100

0
150

0.1

(c)

200

0

0

0.5

0

0.5

0.3

(c)

(d)

(d)

0.2

0.075

0.1

0.1

0.2

0.05

0.1

0.05

0.1

0.025
0
100

150

200

0
100

150

0

200

0.15

0.15

0.1

0.1

0.05

0.05

0
-200

100

0
-200

400

0

0.5

0.2

(a)

0.2

0

∆-likelihood

0

0.5

(b)

100

400

0.2
(c)

(d)

0.15

0.15

0.1

0.1

0.05

0.05

0
-200

100

400

0
-200

100

400

Figure 6.3: The distributions of valuables used in the likelihood analysis of the π 0 -like sample
selection likelihood for CCQE (solid line) and NC (dashed line) among FC Sub-GeV singlering e-like Monte Carlo events in (a) 250 MeV/c ≤ Pe < 400 MeV/c, (b) 400 MeV/c ≤ Pe <630 MeV/c, (c) 630 MeV/c ≤ Pe < 1000 MeV/c, (d) 1000 MeV/c ≤ Pe for SK-I. Each distribution
is normalized to 1. The upper left four panels show π 0 mass distributions above 100 MeV/c2 ,the upper right four panels show the energy fraction of the second ring which is found by theπ 0 fitter and the bottom four panels show the ∆-likelihood.

124Energy fraction of second ring

π mass
0

0.2

(a)

0.2

(a)

0.3

(b)

0.2

0.15

0.2

0.1

0.1

(b)

0.1

0.1
0.05

0
100

150

200

0

0
100

150

200

0

0.15
(c)

0
100

150

0.2

0.05

0.1

0.1

0
100

150

200

0

0

(a)

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0

0

250

0.4

0

0.5

(b)

0

250

0.4
(c)

(d)

0.3

0.3

0.2

0.2

0.1

0.1

0

0

0.5

∆-likelihood

0.4

0.5

(d)

0.2

200

0

(c)

(d)

0.1
0.1
0.05

0

0.5

0

0

250

0

250

Figure 6.4: The distributions of valuables used in the likelihood analysis of the π 0 -like sample
selection likelihood for CCQE (solid line) and NC (dashed line) among FC Sub-GeV singlering e-like Monte Carlo events in (a) 250 MeV/c ≤ Pe < 400 MeV/c, (b) 400 MeV/c ≤ Pe <630 MeV/c, (c) 630 MeV/c ≤ Pe < 1000 MeV/c, (d) 1000 MeV/c ≤ Pe for SK-II. Each distribution is normalized to 1. The upper left four panels show π 0 mass distributions above 100 MeV/c2 ,the upper right four panels show the energy fraction of the second ring which is found by theπ 0 fitter and the bottom four panels show the ∆-likelihood.

125

0.2

(a)

(a)

0.2

(b)

0.15

0.15

0.1

0.1

0.1

0.05
0

0.2
0.1

0.05
-4

0

4

0

-4

0

0

4

(c)

-4

0

(d)

0.1

4

0

4

(d)

0.3

0.1

0.1

0

0

0.2

0.1

-4

-4

0.2

0.2

0

0

4

(c)

0.3

0.2

(b)

0.3

0.2

-4

0

0

4

-4

0

4

0

-4

0

4

Figure 6.5: The distributions of the likelihood of the π 0 -like selection for CC (dotted line) and NC(dashed line) events among FC Sub-GeV single-ring e-like events (solid line) in (a) 250 MeV/c≤ Pe < 400 MeV/c, (b) 400 MeV/c ≤ Pe < 630 MeV/c, (c) 630 MeV/c ≤ Pe < 1000 MeV/c, (d)1000 MeV/c ≤ Pe . The left four panels show for SK-I and the right four panels show for SK-II.
The selection criteria for single-ring π 0 -like events are shown with arrows.

4

3

10

2

Number of events

10

10

e-like

10

1

4

10

3

10

2

Number of events

10e-like

10

0

1

1

≥2

Number of µedecay

0

1

≥2

Number of µedecay

Figure 6.6: The distributions of number of decay electrons for FC Sub-GeV single-ring e-likeevents for data (dot) and Monte Carlo events assuming no oscillation (dashed line) and 2-flavorνµ ↔ ντ oscillation assumed with (sin2 2θ, ∆m2 ) = (1.00, 2.5 × 10−3 eV2 ) (solid line). The left
plot is SK-I and the right is SK-II. Single-ring π 0 -like events are subtracted.

126

60

e-like 0µedecay

600

400

e-like 1µedecay

Number of events

Number of events

800

40

400
20
200
0
0

500

1000

1500

Momentum (MeV/c)
80

0
0

500

1000

1500

e-like 1µedecay
40

300
200

20
100
0
0

Momentum (MeV/c)

500

1000

1500

Momentum (MeV/c)1ring π -like

0
0

500

1000

1500

Momentum (MeV/c)1ring π -like

0

0

30

60
40

20

20

10

0
0

e-like 0µedecay

500

1000

0
0

1500

Momentum (MeV/c)

500

1000

1500

Momentum (MeV/c)

Figure 6.7: The momentum distributions for FC Sub-GeV single-ring e-like events for data (dot)and Monte Carlo events assuming no oscillation (dashed line) and 2-flavor νµ ↔ ντ oscillation
with (sin2 2θ, ∆m2 ) = (1.00, 2.5 × 10−3 eV2 ) (solid line). The hatched regions show νe CCQE
events in the oscillated events. The left three plots are for SK-I and the right are for SK-II.

Carlo events. The distributions for data and the oscillated Monte Carlo events are consistent.
The systematic errors in the event separation will be described in Section 7.3.3.

6.1.3

FC Sub-GeV two-ring π 0 -like events

To constrain the contamination from the neutral current events, the two-ring π 0 -like sample
is categorized. The selection criteria are the same as Section 2.6.5. The momentum distributionsof FC Sub-GeV two-ring π 0 -like samples are shown in Figure 6.10 for data and Monte Carloevents.

6.2

Vertex Distributions

Figure 6.11 and Figure 6.12 show the reconstructed vertex distributions for the FC and PC
samples projected to R2 = (X 2 +Y 2 ) and Z axes. Points show the observed data and histograms
show the atmospheric neutrino Monte Carlo events assuming no oscillation and 2-flavor νµ ↔ ντ
oscillation with (sin2 2θ, ∆m2 ) = (1.00, 2.5×10−3 eV2 ). The live time of the Monte Carlo events
is normalized to that of the observed data. The vertex distributions of the data and the Monte
Carlo events with neutrino oscillation agree well in the fiducial volume, which are shown witharrows. For Multi-GeV single-ring µ-like events, the excess due to the cosmic ray muons at Z =1810 cm, the upper edge of the inner detector is seen in Z distribution for both SK-I and SK-II,
but they are rejected by the fiducial volume cut.

127

4

3

10

2

Number of events

10

10µ-like

10

1

4

10

3

10

2

Number of events

10µ-like

10

0

1

1

≥3

2

0

Number of µedecay

1

≥3

2

Number of µedecayµ-like 0µedecay

600

300

400

200

200

100
0
0

µ-like 1µedecay

Number of events

Number of events

Figure 6.8: The distributions of the number of decay electrons for FC Sub-GeV single-ring µ-likeevents for data (dot) and Monte Carlo events assuming no oscillation (dashed line) and 2-flavorνµ ↔ ντ oscillation assumed with (sin2 2θ, ∆m2 ) = (1.00, 2.5 × 10−3 eV2 ) (solid line). The left
plot is for SK-I and the right is for SK-II.

500

1000

1500

Momentum (MeV/c)

0
0

500

1000

1500

200

µ-like 0µedecayµ-like 1µedecay
200

150
100

100
50
0
0

Momentum (MeV/c)

500

1000

1500

Momentum (MeV/c)
30µ-like 2µedecay

0
0

500

1000

1500

Momentum (MeV/c)µ-like 2µedecay

30
20
20
10

10
0
0

500

1000

0
0

1500

Momentum (MeV/c)

500

1000

1500

Momentum (MeV/c)

Figure 6.9: The momentum distributions for FC Sub-GeV single-ring µ-like events for data (dot)and Monte Carlo events assuming no oscillation (dashed line) and 2-flavor νµ ↔ ντ oscillation
with (sin2 2θ, ∆m2 ) = (1.00, 2.5 × 10−3 eV2 ) (solid line). The hatched regions show νµ CCQE
events in the oscillated events. The left three plots are SK-I and the right are SK-II.

128

0

Number of Events

Number of Events

02-ring π -like

200
150
100
50
0
0

500

1000

2-ring π -like

100
75
50
25
0
0

1500

500

1000

1500

Reconstructed π0 Momentum (MeV/c) Reconstructed π0 Momentum (MeV/c)

Figure 6.10: The reconstructed π 0 momentum for data (dot) and Monte Carlo events assuming
no oscillation (dashed line) and 2-flavor νµ ↔ ντ oscillation with (sin2 2θ, ∆m2 ) = (1.00, 2.5 ×
10−3 eV2 ) (solid line) for Sub-GeV two-ring π 0 -like events. The hatched regions show NC eventsin the oscillated events. The left figure shows SK-I and the right shows SK-II.

FC Sub-GeV single-ring e-like
0µedecay 1µedecay
π 0 -likeall
Q.E.
single meson
multi πcoherent π

FC Sub-GeV
2-ring π 0 -like

80.4 %
12.2 %
1.5 %
0.1 %

2.2 %
56.1 %
14.6 %
0.7 %

17.2 %
9.1 %3.3 %0.07 %

71.3 %
14.9 %
2.5 %0.2 %0.07 %
3.0 %1.0 %0.01 %CC νµ + ν µ0.7 %17.7 %

10.0 %

2.4 %

16.7 %

NC

4.9 %

8.9 %

59.9 %

8.7 %

79.2 %

CC
νe + ν e

FC Sub-GeV single-ring µ-like
0µedecay 1µedecay 2µedecayall72.9 %
13.0 %
1.4 %
0.1 %79.3 %
15.9 %
2.1 %
0.2 %

2.9 %
71.5 %
21.1 %
0.7 %

74.4 %
17.0 %
2.6 %0.2 %CC νe + ν e

1.5 %0.07 %

0.05 %

0.5 %NC

10.9 %

2.3 %

2.2 %

5.1 %

CC
νµ + ν µQ.E.
single meson
multi πcoherent π

Table 6.2: Fraction of each neutrino interaction mode in FC Sub-GeV atmospheric neutrinoMonte Carlo events.

129

FC Multi-GeV
1-ring e-likeQ.E.
single meson
multi πcoherent πCCνe + ν e38.2 %
24.9 %
18.8 %0.8 %CC νµ + ν µ

7.1 %

NC

10.1 %

Table 6.3: Fraction of each neutrino interaction mode in FC Multi-GeV atmospheric neutrinoMonte Carlo events.

FCmulti-ring µ-like

FC Multi-GeV
1-ring µ-like

PC

UPµ
stop
through

5.7 %35.8 %
51.7 %
0.5 %

51.8 %
30.7 %
16.0 %
0.9 %

18.8 %
22.1 %
54.7 %
1.8 %

13.9 %
20.2 %
64.3 %
0.3 %

3.4 %
6.6 %
89.8 %
0.0 %CC νe + ν e

2.7 %

0.4 %

1.8 %

0.9 %

0.1 %

NC

3.7 %

0.2 %

0.9 %

0.4 %

0.1 %

CC
νµ + ν µQ.E.
single meson
multi πcoherent π

Table 6.4: Fraction of each neutrino interaction mode in FC Multi-GeV, PC and UPMU atmospheric neutrino Monte Carlo events.

130

FC Sub-GeV e-like

200
150

150

100

100

50

50
0

60
40

40

20

20

0

0
FC multi-ring µ-like

80

60

40

40

20

20
0

1000

2000

2

2

R (cm )

0

× 10

3

100

50

50
0
FC Multi-GeV e-like

40

30

20

20

10

10

0

0
FC multi-ring µ-like

60

0

1000

2000

2

2

R (cm )

PC

60

40

40

20

20

0

× 10

3

FC Multi-GeV µ-like

40

30

PC

80

60

100

0

60 FC Multi-GeV µ-like

FC Multi-GeV e-like

FC Sub-GeV µ-like

FC Sub-GeV e-like

Number of events

Number of events00

FC Sub-GeV µ-like

200

0

1000

0

2000

2

2

R (cm )

0

1000

× 10

3

2000

R2 (cm2)

× 10

3

Figure 6.11: The vertex distributions projected on R2 axis for data (dot) and Monte Carlo events
assuming no oscillation (dashed line) and 2-flavor νµ ↔ ντ oscillation with (sin2 2θ, ∆m2 ) =(1.00, 2.5 × 10−3 eV2 ) (solid line). The left six panels show those for SK-I and the right panels
show those for SK-II.

FC Sub-GeV µ-like

FC Sub-GeV e-like

150

150

150

100

100

100

100

50

50

50

50

40

0
FC Multi-GeV e-like

40

30

30

20

20

10

10

0

0

40

FC multi-ring µ-like0
FC Multi-GeV µ-like

Number of events0

Number of events

FC Sub-GeV µ-like

FC Sub-GeV e-like

150

PC

80

40

0
FC Multi-GeV e-like

40

30

30

20

20

10

10

0
40

0
FC multi-ring µ-like

60

30

60

20

40

20

40

10

20

10

20

0

0

0

0

Z (cm)

1000

-1000

0

1000

Z (cm)-1000

0

Z (cm)

1000

PC

80

30

-1000

FC Multi-GeV µ-like

0

-1000

0

1000

Z (cm)

Figure 6.12: The vertex distributions projected on Z axis for data (dot) and Monte Carlo events
assuming no oscillation (dashed line) and 2-flavor νµ ↔ ντ oscillation with (sin2 2θ, ∆m2 ) =(1.00, 2.5 × 10−3 eV2 ) (solid line). The left six panels show those for SK-I and the right panels
show those for SK-II.

131

Number of events

10

4

Sub-GeV

10

4Multi-GeV

10

3

10

3

10

2

10

2

10

1

2

3

4

≥5

10

1

Number of rings

Number of events

10

4

Sub-GeV

4

10

3

10

2

10

2

1

2

3

4

3

4

≥5

Multi-GeV

3≥5

Number of rings

6.3

10

10

10

2

Number of rings

10

1

2

3

4

Figure 6.13: The number of rings for data(dot) and Monte Carlo events assuming
no oscillation (dashed line) and 2-flavorνµ ↔ ντ oscillation with (sin2 2θ, ∆m2 ) =(1.00, 2.5 × 10−3 eV2 ) (solid line) for SubGeV events (left) and Multi-GeV events(right). The upper two panels show thosefor SK-I and the bottom panels show thosefor SK-II.

≥5

Number of rings

Number of Rings and Momentum Distributions

Figure 6.13 shows the distribution of the number of rings for data and Monte Carlo events.
Figure 6.14 shows the distribution of the momentum for data and Monte Carlo for FC MultiGeV single-ring events. The distributions for data and the oscillated Monte Carlo events are
consistent.

6.4

Flavor Ratio

The flavor ratio of the atmospheric neutrino, νµ + ν µ /νe + ν e is predicted with the 3 % of theuncertainty described in Section 3.2. As shown in Table 6.2-6.4, the results of the reconstructedparticle identification in single-ring events can be used to estimate the flavor of the parent
neutrino well, and the ratio of the number of e-like events and µ-like events can be used toestimate the flavor ratio of the atmospheric neutrino flux. The double ratio of the µ/e ratios isdefined by :R(µ/e) ≡ (Nµ /Ne )Data /(Nµ /Ne )M C
(6.1)
where Ne and Nµ are the number of e-like and µ-like events, respectively. If the ratio of dataagrees with that of the expected result, the double ratio should be 1.
The measured double ratios for Sub-GeV and Multi-GeV for SK-I are as follows :+0.016 0.650 −0.016 (stat) ± 0.035(syst) (Sub-GeV)(6.2)
R(µ/e) =+0.0320.695 −0.030 (stat) ± 0.100(syst) (Multi-GeV + PC)
132

Number of events

Multi-GeV single-ring e-like

Multi-GeV single-ring µ-like

300
200
200
100

100

0
1

Number of events

400

300

200

10

100

0
1

10

Momentum (GeV/c)

Momentum (GeV/c)

Multi-GeV single-ring e-like

Multi-GeV single-ring µ-like

100

200

150
100

100
50
0
1

10

Momentum (GeV/c)

100

0
1

10

Figure 6.14: The momentum distributions for data (dot) and Monte Carloevents assuming no oscillation (dashed line)
and 2-flavor νµ ↔ ντ oscillation with(sin2 2θ, ∆m2 ) = (1.00, 2.5 × 10−3 eV2 )(solid line) for Multi-GeV single-ring e-likeevents (left) and µ-like events (right). The
upper two panels show those for SK-I andthe bottom panels show those for SK-II.

100

Momentum (GeV/c)

and for SK-II are as follows :+0.022 0.647 −0.021 (stat) ± 0.033(syst) (Sub-GeV)
R(µ/e) =
0.741 +0.046−0.043 (stat) ± 0.055(syst) (Multi-GeV + PC)(6.3)

where PC events are treated as the Multi-GeV µ-like sample because of the high fraction ofνµ CC interactions. The systematic errors of the double ratio are estimated to be 5.3 % for SKI Sub-GeV and 14.4 % for SK-I Multi-GeV. For SK-II, since the systematic error on the ringnumber estimation is smaller, the systematic errors of the double ratio are smaller than thoseof SK-I and estimated to be 5.0 % for Sub-GeV and 7.4 % for Multi-GeV. The sources of thesesystematic of the double ratio are listed in Table 6.5. The explanation of each source is describedin Section 7.3. For Sub-GeV and Multi-GeV sample, the double ratios are significantly smaller
than 1 and the double ratios are consistent for SK-I and SK-II.

6.5

Zenith angle Distributions

Figure 6.15 shows the zenith angle distributions for data and the Monte Carlo expectations
for FC Sub-GeV single-ring samples, FC Multi-GeV samples, FC Multi-GeV µ-like + PC samples
and upward-going muon samples. Zenith angle cosΘ=-1(+1) indicates upward-going (downwardgoing) direction. FC Sub-GeV single-ring events are divided into two by the reconstructed lepton
momentum of 400 MeV/c. Since the angular correlation between the neutrino direction and thecharged lepton direction is weak below 400 MeV/c, the zenith angle distributions are almost flat
even though the prediction of the atmospheric neutrino flux has the zenith angle dependence
as shown in Figure 3.5. Figure 6.16 shows the angular resolution of the neutrino directions
as a function of the momentum. The angular resolution is defined as the angular difference
between the parent neutrinos and the reconstructed direction of outgoing leptons in which 68 %of the events are included. For Multi-GeV samples, the excess in the horizontal direction is
explained by the excess of the atmospheric neutrino flux due to the high decay probability of133

Prediction of νµ /νe ratio
Prediction of ν/ν ratio
K/π ratioEν spectral index
Sample-by-sample
ν interactionquasi-elastic scattering
single-meson production
deep-inelastic scatteringcoherent-pion production
NC/CC ratio
nuclear effects
Hadron simulation
FC reduction
PC reduction
Non-ν backgroundµ/e separation
Single-ring/multi-ring separation
Energy calibration
MC statistics

Sub-GeV(%)

Multi-GeV + PC(%)

2.7 (2.7)
1.6 (1.6)
0.6 (0.6)
0.6 (0.5)– (–)

2.2 (2.2)
0.8 (0.7)
1.9 (1.8)
2.3 (2.0)
2.9 (2.7)

1.4 (1.4)<0.1 (<0.1)
0.2 (0.3)
0.4 (0.4)
0.5 (0.6)
1.3 (1.2)
0.7 (0.7)
0.1 (<0.1)– (–)
0.5 (0.3)
1.3 (1.7)
3.2 (2.6)
0.6 (0.6)
0.5 (0.6)

1.0 (0.8)
0.3 (0.1)
0.5 (0.5)
0.2 (0.4)
2.0 (2.1)
0.8 (1.0)<0.1 (<0.1)
0.1 (0.1)
1.5 (2.2)
0.3 (1.1)
0.6 (0.3)
13.2 (4.4)
1.2 (1.4)
0.9 (1.1)

5.3 (5.0)

14.4 (7.4)

Total

Table 6.5: Sources of the systematic errors on double ratio R for the Sub-GeV and MultiGeV+PC samples for SK-I (SK-II).

134

high energy muon in the atmosphere. Comparing data with Monte Carlo expectations assuming
no oscillation, the clear deficit for higher energy µ-like upward-going events is observed. The
data are consistent with the Monte Carlo expectations assuming two-flavor νµ ↔ ντ oscillation
with (sin2 2θ, ∆m2 ) = (1.00, 2.5 × 10−3 eV2 ), while data and the no-oscillation Monte Carlo
expectations for e-like events are consistent.

135

400
300

400
Sub-GeV e-like (P<400MeV/c)

150

200
Sub-GeV e-like (P<400MeV/c)

150

200

100

100

100

100

50

50

0
0.5
cosΘ

1

400

0
0.5
cosΘ

Sub-GeV µ-like (P<400MeV/c)

300
200

100

100

0
-1

-0.5

0
0.5
cosΘ

1

150

-0.5

0
0.5
cosΘ

1

400
Multi-GeV e-like

300

100

200

50

100

0
-1

Sub-GeV µ-like (P<400MeV/c)

0
-1

0
-1

1

400

200

150

-0.5

-0.5

0
0.5
cosΘ

1

0
-1
300

Upmu stopping

100

200

50

100

0
-1 -0.8 -0.6 -0.4 -0.2
cosΘ
100
Upmu
showering
80
60
40
20
0
-1 -0.8 -0.6 -0.4 -0.2
cosΘ

0

0

Multi-GeV µ-like + PC

-0.5

0
0.5
cosΘ

1

0
-1 -0.8 -0.6 -0.4 -0.2
cosΘ

150

1

-0.5

0
0.5
cosΘ

1

200
Sub-GeV µ-like (P<400MeV/c)

150

50

50

0
-1

-0.5

0
0.5
cosΘ

1

Sub-GeV µ-like (P<400MeV/c)

0
-1

-0.5

0
0.5
cosΘ

1

200
Multi-GeV e-like

150

Multi-GeV µ-like + PC

100
50
-0.5

0
0.5
cosΘ

1

0
-1
150

Upmu stopping

40

100

20

50

0
-1 -0.8 -0.6 -0.4 -0.2
cosΘ
40
30

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

0
-1

100

60

0

0
0.5
cosΘ

100

80

Upmu non-showering

-0.5

200

100
80
60
40
20
0
-1

Number of Events

300

0
-1

Number of Events

-0.5

200

Number of Events

200
Sub-GeV e-like (P>400MeV/c)

200

0
-1
Number of Events

300

0

-0.5

0
0.5
cosΘ

1

Upmu non-showering0-1 -0.8 -0.6 -0.4 -0.2
cosΘ0Upmu showering20
100-1 -0.8 -0.6 -0.4 -0.2
cosΘ0Figure 6.15: Zenith angle distributions for data (dot) and Monte Carlo events assuming nooscillation (box) and two-flavor νµ ↔ ντ oscillation with (sin2 2θ, ∆m2 ) = (1.00, 2.5 × 10−3 eV2 )(dotted lines) for FC Sub-GeV single-ring e-like events, FC Sub-GeV single-ring µ-like events, FC
Multi-GeV e-like events, FC Multi-GeV µ-like + PC events and upward-going muon events. FC
Sub-GeV single-ring events are divided by the reconstructed lepton momentum of 400 MeV/c.
The left nine figures show those for SK-I and the right figures show those for SK-II. Zenith anglecosΘ=-1(+1) indicates upward-going (downward-going) direction.

136

2

10

FC single-ring µ-likeFC multi-ring µ-likePC
Upward stopping µ
Upward through-going µ10Angular Resolution (degree)

Angular Resolution (degree)

10

1
10

2

10

2

FC single-ring µ-likeFC multi-ring µ-likePC
Upward stopping µ
Upward through-going µ

10

1
10

2

10
FC single-ring e-like

FC single-ring e-like

1

1
10

-1

1
Momentum (GeV/c)

10

10

-1

1

10

Momentum (GeV/c)

Figure 6.16: Angular resolution of the neutrino direction as a function of the outgoing charged
lepton momentum for SK-I (left) and SK-II (right).

137

Chapter 7

Oscillation analysis with solar terms
As described in Chapter 1, the atmospheric neutrino data are well explained by the νµ ↔ ντ
two-flavor mixing framework. The 2-3 neutrino oscillation parameters are constrained to be
1.9 × 10−3 < ∆m223 < 3.1 × 10−3 eV 2 , sin2 2θ23 > 0.92,(7.1)

at 90 % C.L. [25]. On the other hand, the 1-2 oscillation parameters are precisely measured by
combining the solar neutrino data and the KamLAND reactor neutrino data as follows [13]:7.3 × 10−5 < ∆m212 < 8.1 × 10−5 eV 2 , 0.28 < sin2 θ12 < 0.37 (1σ).

(7.2)

If these LMA-MSW oscillation parameters are taken into consideration, the oscillation of atmospheric νe in the Sub-GeV energy region is expected to appear regardless of the θ13 value. The
effect of this sub-dominant oscillation depends on the size of the deviation of sin2 2θ23 from the
maximal mixing and on the octant of θ23 .
In this chapter, an analysis to observe the sub-dominant oscillation by the 1-2 parametersand determine the octant of θ23 is described.

7.1

Overview

As discussed in Section 1.3, the oscillation effect on the atmospheric νe flux driven by thesolar neutrino parameters can be written from Equation (1.25) as follows [28, 29]:
Feosc
− 1 = P2 (r cos2 θ23 − 1),
Fe0

(7.3)

where Feosc and Fe0 are the atmospheric νe flux with and without oscillations and r is the ratio of
the original muon and electron neutrino fluxes Fµ0 /Fe0 . P2 is the neutrino transition probability(νe → νx , x = µ or τ ) in matter driven by the 1-2 parameters as shown in Figure 1.2.
The factor in parentheses in Equation (7.3) is called the ‘screening’ factor. Since the flavor
ratio r is approximately 2 in the Sub-GeV neutrino energy region as described in Section 3.2,
the screening factor is ∼ 0 in the case of the maximal 2-3 mixing (i.e. θ23 = 45◦ ). According to
the screening factor, the appearance of the sub-dominant oscillation effect depends largely on
the deviation of θ23 from the maximal mixing. If θ23 < 45◦ , the screening factor is positive andan excess of the Sub-GeV e-like sample is expected. If θ23 > 45◦ , the screening factor is negative
138

1.1
Sub-GeV e-like

1.08

Neosc / Ne2flavor full-mixing

Neosc / Ne2flavor full-mixing

1.1

sin θ23=0.4
2
sin θ23=0.5
2
sin θ23=0.6
2

1.06
1.04

Sub-GeV e-like 0decay

1.08

sin θ23=0.4
2
sin θ23=0.5
2
sin θ23=0.6
2

1.06
1.04

1.02

1.02

1

0.98

1

0.98

0.96

0.96
-1

-0.5

0

0.5

1

cosθ

-1

-0.5

0

0.5

1

cosθ

Figure 7.1: Zenith angle dependence of the ratio of the number of FC Sub-GeV single-ring e-likeevents (left) and e-like 0µedecay events (right) including sub-dominant oscillations due to the 1-2
oscillations (Neosc ) and with pure νµ ↔ ντ two-flavor full-mixing oscillations (Ne2f lavorf ull−mixing ).
For calculation of Neosc , ∆m212 = 7.9 × 10−5 eV 2 , sin2 θ12 = 0.31, ∆m223 = 2.5 × 10−3 eV 2 andthree different values of sin2 θ23 , sin2 θ23 = 0.4 (dashed line), 0.5 (solid line) and 0.6 (dotted line),
are assumed. For calculation of Ne2f lavorf ull−mixing , ∆m223 = 2.5 × 10−3 eV 2 and sin2 θ23 = 0.5
are assumed.

and the Sub-GeV e-like sample is expected to be reduced. Figure 7.1 shows the zenith angle
dependence of relative effect due to 1-2 oscillations on the number of FC Sub-GeV single-ringe-like events (left) and e-like 0µedecay events (right) for different values of sin2 θ23 . The 1-2 oscillation parameters are assumed as ∆m212 = 7.9 × 10−5 eV 2 and sin2 θ12 = 0.31. In the figures, thenumber of e-like events with sub-dominant 1-2 oscillations (Neosc ) is normalized by the numberof e-like events with pure νµ ↔ ντ two-flavor full-mixing oscillations (Ne2f lavorf ull−mixing ). Although νe does not oscillate at all in the pure νµ ↔ ντ two-flavor full-mixing oscillation scenario,
Ne2f lavorf ull−mixing differs from Ne0 (the number of e-like events without oscillations) because
the FC Sub-GeV single-ring e-like sample contains ∼ 1 % contamination of νµ CC interactions.
The sub-dominant oscillation effect is negligible for sin2 θ23 =0.5. For sin2 θ23 =0.4 and 0.6, thesub-dominant oscillation effect appears as a ∼2 % excess and deficit of Sub-GeV e-like events,
respectively. The size of the sub-dominant oscillation effect for e-like 0µedecay events is larger
than that of e-like events by ∼0.5 % due to the enrichment of CCQE interactions in the 0µedecay
sample. When the right plot of Figure 7.1 is divided by the reconstructed electron momentum
as shown in Figure 7.2, the sub-dominant oscillation effect is expected larger in the lower energy
region. In the higher energy region, the size of the effect is smaller but its zenith angle dependence is stronger. This is because the direction of a higher momentum lepton has a stronger
correlation to the parent neutrino direction (see Figure 6.16).
The sub-dominant oscillation effect due to the solar neutrino parameters also appears on
the Sub-GeV single-ring µ-like events as shown in Figure 7.3. The relative effect is foundto be very small for sin2 θ23 =0.5. Since the number of FC Sub-GeV single-ring µ-like events
is modified largely by the dominant oscillation channel νµ ↔ ντ , the relative effect of subdominant oscillations for sin2 θ23 =0.4 and sin2 θ23 =0.6 can be extracted by comparing with the
139

1.1
Sub-GeV e-like 0decay P<400(MeV/c)

1.08

sin θ23=0.4
2
sin θ23=0.5
2
sin θ23=0.6
2

1.06
1.04

Neosc / Ne2flavor full-mixing

Neosc / Ne2flavor full-mixing

1.1

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

1.08

sin θ23=0.4
2
sin θ23=0.5
2
sin θ23=0.6
2

1.06
1.04

1.02

1.02

1

0.98

1

0.98

0.96

0.96
-1

-0.5

0

0.5

1

cosθ

-1

-0.5

0

0.5

1

cosθ

Figure 7.2: Zenith angle dependence of Neosc /Ne2f lavorf ull−mixing with electron momentum below
400 MeV/c and above 400 MeV/c. The line definitions are same as Figure 7.1.

case of pure νµ ↔ ντ two-flavor oscillations with the corresponding sin2 2θ23 =0.96. In Figure 7.3,
the effect of sub-dominant oscillation on the FC Sub-GeV single-ring µlike sample appears on
the opposite side of that on the e-like sample. The size of the sub-dominant oscillation effect forµ-like 1µedecay events is larger than that of µ-like events by ∼0.1 % due to the enrichment of
CCQE interactions in the sample. The discrimination between sin2 θ23 =0.4 and sin2 θ23 =0.6 isexpected to be larger by taking the µ/e ratio. Figure 7.4 shows the zenith angle dependence ofthe sub-dominant oscillation effect on the ratio between the number of FC Sub-GeV single-ringµ-like events and the number of FC Sub-GeV single-ring e-like events, and the ratio between
the number of FC Sub-GeV single-ring µ-like 1µedecay events and the number of FC Sub-GeV
single-ring e-like 0µedecay events. For upward-going events, the difference on the µ/e ratio
between the sin2 θ23 =0.4 and sin2 θ23 =0.6 cases reaches about 5 %, which is comparable to thesize of the estimated systematic error on the µ/e ratio measurement and it is enhanced by ∼1 %for the sub-samples separated by the number of decay electrons compared with that for the
Sub-GeV single-ring samples without decay-e cuts.
Figure 7.5 compares the sensitivity of the analysis for the determination of sin2 θ23 based on
the event samples without decay-e cuts and with decay-e cuts. It is found that the event samples
with decay-e cuts improve the sensitivity and give a more stringent arrowed region for sin2 θ23 .
The oscillation analysis of the Sub-GeV samples taking into account the sub-dominant 1-2
oscillation has the possibility to determine the octant of θ23 for the non-maximal sin2 2θ23 . Since
the value of sin2 2θ23 affects the distribution for the other samples, a combined analysis of all
the samples is needed.

7.2

Analysis Method

In the analysis, the FC, PC and UPMU samples of the SK-I and SK-II atmospheric neutrino
data are used. The SK-I samples are divided into 400 bins and the SK-II samples are divided
into 350 bins by the reconstructed event types, momenta and zenith angles : 310 bins for the
SK-I FC samples, 60 bins for the SK-I PC samples, 30 bins for the SK-I upward-going muon
140

1.1
Sub-GeV µ-like

1.08

Nµosc / Nµ2flavor full-mixing

Nµosc / Nµ2flavor full-mixing

1.1

sin θ23=0.4
2
sin θ23=0.5
2
sin θ23=0.6
2

1.06
1.04

Sub-GeV µ-like 1decay

1.08

sin θ23=0.4
2
sin θ23=0.5
2
sin θ23=0.6
2

1.06
1.04

1.02

1.02

1

0.98

1

0.98

2

0.96
-1

-0.5

0

2

0.96

2flavor (sin 2θ23=0.96)

0.5

12flavor (sin 2θ23=0.96)

-1

-0.5

0

cosθ

0.5

1

cosθ

1.1
Sub-GeV

1.08

2

1.04

1.1
Sub-GeV µedecay

1.08

sin θ23=0.4
2
sin θ23=0.5
2
sin θ23=0.6

1.06(Nµ/Ne)osc / (Nµ/Ne)2flavor full-mixing(Nµ/Ne)osc / (Nµ/Ne)2flavor full-mixing

Figure 7.3: Zenith angle dependence of the ratio of the number of FC Sub-GeV single-ring µ-likeevents (left) and µ-like 1µedecay events (right) including sub-dominant oscillations due to the 1-2
oscillations (Nµosc ) and the number of FC Sub-GeV single-ring µ-like 1µedecay events with pure
νµ ↔ ντ two-flavor full-mixing oscillations (Nµ2f lavorf ull−mixing ). For the dash-dotted line, thenumerator is the number of µ-like events in the case of pure νµ ↔ ντ two-flavor oscillations withsin2 2θ23 = 0.96, which corresponds to sin2 θ23 = 0.4 or sin2 θ23 = 0.6. The other line definitions
are same as Figure 7.1.

2

sin θ23=0.4
2
sin θ23=0.5
2
sin θ23=0.6

1.06
1.04

1.02

1.02

1

0.98

1

0.98

2

0.96
-1

-0.5

0

2

0.96

2flavor (sin 2θ23=0.96)

0.5

1

cosθ

2flavor (sin 2θ23=0.96)

-1

-0.5

0

0.5

1

cosθ

Figure 7.4: Zenith angle dependence of the Nµ /Ne ratio including sub-dominant oscillations
due to the 1-2 oscillations normalized by the Nµ /Ne with pure νµ ↔ ντ two-flavor full-mixing
oscillations. The line definitions are same as Figure 7.3. The left figure shows the Nµ /Ne ratio of
FC Sub-GeV single-ring events and the right figure shows the Nµ /Ne ratio between FC Sub-GeV
single-ring µ-like 1µedecay events and e-like 0µedecay events.

141

5

4

4
χ2 - χ2min

χ2 - χ2min

5

3
2
1

3
2
1

0

0
0.4

0.5
sin2θ23

0.6

0.4

0.5
sin2θ23

0.6

Figure 7.5: χ2 -χ2min distribution as a function of sin2 θ23 using the oscillated Monte Carlo eventsof a 6 year exposure with the true oscillation parameters of θ13 = 0, sin2 θ12 = 0.32, ∆m212 =7.8×10−5 eV2 , ∆m223 = 2.5×10−3 eV2 and sin2 θ23 = 0.4 (left), 0.6 (right). Details of thedefinition of χ2 will be described in the next section. The solid line shows the sensitivity basedon the standard event samples without decay-e cuts and the dotted line shows on the event
samples with decay-e cuts.

samples, 280 bins for the SK-II FC samples, 40 bins for the SK-II PC samples and 30 bins for
the SK-II upward-going muon samples as shown in Figure 7.6 and Figure 7.7. The number ofbins are different between SK-I and SK-II to avoid the binning effect on fitting results. The
small number of expected events (typically less than 5 events) in a bin gives a large reduced-χ2statistically. Since the expected number of events for SK-II is smaller than that for SK-I, thebinning is different especially in higher energy regions which have less statistics.
The FC samples are divided into ’single-ring e-like’, ’single-ring µ-like’, ’multi-ring e-like’ and’multi-ring µ-like’ events by the number of Cherenkov rings and the particle type. The single-ring
events are separated into ’Sub-GeV’ and ’Multi-GeV’ events at 1.33 GeV of the visible energy.
Additionally the FC Sub-GeV single-ring samples are separated according to the number ofµedecays. ’Single-ring π 0 -like’ events and ’2-ring π 0 -like’ events are selected as described in
Chapter 6. PC events are separated into ’PC stop’ and ’PC through’ events according to theamount of charge deposit in the outer detector. The FC and PC samples except for the FC
single-ring e-like 1µedecay sample, the FC single-ring µ-like 2µedecay sample and the π 0 -likesamples are divided into 10 bins equally spaced between cos Θ = −1 and cos Θ = +1, where cos Θis cosine of zenith angle of particle direction. The FC single-ring e-like 1µedecay sample, the FC
single-ring µ-like 2µedecay sample and π 0 -like samples are divided only by the momentum, notby the zenith angle because of small statistics and poor angular correlation for NC events. The
SK-I and SK-II data are separately considered due to the difference of the detector response.
A total of 750 bins are considered in the calculation of χ2 with their associated sets of theobservations and the Monte Carlo predictions.

142

Upmu shower(1 x 10 bins)

FC Multi-GeV
1-ring e-like

FC Multi-GeV
mulit-ring e-like

FC Mulit-GeV
1-ring µ-like

FC Multi-ringµ-likePC stop

PC through(4 x 10 bins)(4 x 10 bins)(2 x 10 bins)(4 x 10 bins)(2 x 10 bins)(4 x 10 bins)

Upmu non-shower(1 x 10 bins)

log10P (MeV/c)

5.0

Upmu stop

4.0
3.7

3.7

3.4

3.4

3.0

3.0

FC Sub-GeV
1-ring e-like

0 µedecay

1 µedecay(5 x 10 bins)(5 x 1 bins)

FC Sub-GeV 1-ring µ-like

FC 1-ring0
π-like(5 x 1 bins)(1 x 10 bins)

0 µedecay 1 µedecay 2 µedecay(5 x 10 bins)

FC Sub-GeV
02-ring π-like(5 x 10 bins) (5 x 1 bins)(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 7.6: The definition of the binning for the SK-I data for the neutrino oscillation analysis :310 bins for the FC samples, 60 for the PC samples, 30 for the upward-going muon samples.
The momentum bins shown in white boxes are divided into 10 bins equally spaced betweencos Θ = −1 and cos Θ = +1. The FC single-ring e-like 1µedecay sample, the µ-like 0µedecay
sample and the π 0 samples are not divided by the zenith angle bins.
Upmu shower(1 x 10 bins)

FC Multi-GeV
1-ring e-like

FC Multi-GeV
mulit-ring e-like

FC Mulit-GeV
1-ring µ-like

FC Multi-ringµ-likePC stop

PC through(4 x 10 bins)(4 x 10 bins)(2 x 10 bins)(4 x 10 bins)(2 x 10 bins)(4 x 10 bins)

Upmu non-shower(1 x 10 bins)

log10P (MeV/c)

5.0

Upmu stop
3.7

3.7

3.4

3.4

3.0

3.0

FC Sub-GeV
1-ring e-like

0 µedecay

1 µedecay(5 x 10 bins)(5 x 1 bins)

FC Sub-GeV 1-ring µ-like

FC 1-ring0
π-like(5 x 1 bins)

0 µedecay 1 µedecay 2 µedecay(5 x 10 bins)(1 x 10 bins)

3.4

FC Sub-GeV
02-ring π-like(5 x 10 bins) (5 x 1 bins)(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 7.7: The definition of the binning for the SK-II data for the neutrino oscillation analysis :280 bins for the FC samples, 40 for the PC samples and 30 for the upward-going muon samples.
143A χ2 value is defined according to the Poisson probability distribution as follows [131]:

 
!
80
750
obs
X
X
N
i
2 N exp (1 +fji · ǫj ) − Niobs  + 2Niobs ln
χ2 ≡
P
i
exp
80
i·ǫ )
(1
+
f
N
j
j=1 j
i
j=1
i=1
!
2
77
X
ǫj
+
+ ∆χ2solar ,
(7.4)
σjsys
j=1

where Niobs is the number of observed events in the i-th bin and Niexp is the expected numberof events in the i-th bin from the Monte Carlo simulation. During the fit, the values of Niexp
is recalculated to account for neutrino oscillations and 80 systematic errors which come fromuncertainties in the neutrino flux model, neutrino cross-section model and detector response. The
details of the systematic errors are described in the next section. The systematic uncertainties
are represented by 80 parameters ǫj . These 80 ǫj are varied to minimize χ2 for each choice ofoscillation parameters. The second sum in the χ2 definition represents the contributions fromthe systematic errors. Among 80 errors, only 77 errors contribute to the χ2 since the threeerrors related to the sample normalization are allowed to be free (the systematic uncertainties
on the absolute normalization and sample normalization of multi-ring e-like events for SK-I and
SK-II ). σjsys is an estimated 1-sigma value of the j-th systematic uncertainty. The factor fji
represents the fractional change in the predicted event rate in the i-th bin due to a variation of
the parameter ǫj .
At the minimum χ2 location for each choice of oscillation parameters, ∂χ2 /∂ǫj = 0 is realized
for each of the parameters ǫj . As a result, the minimization of χ2 in Equation (7.4) is equivalent
to solving the following k = 1, 80 equations [130]:



!
750
80
750
80
X
X
X
X
1
1 + (
fji ǫj )2 − ... Niobs − Niexp  fki . (7.5)
Niobs fji fki ǫj =sys 2 δjk +
(σj )
j=1

i=1

i=1

j=1

Since these equations have nonlinear terms of ǫj , an approximate solution by an iteration method
is obtained.
A value of χ2 is calculated at each grid point in the four-dimensional oscillation parameter
space of ∆m212 , ∆m223 , sin2 θ12 and sin2 θ23 , because θ13 = 0 is assumed in this analysis. To
compare the solar term effect, two scenarios are examined. In one scenario with the solar neutrino
parameters turned on, the solar neutrino parameters are chosen around the allowed region
obtained by a combined analysis of the solar neutrino data and KamLAND data [13] as shownin Figure 7.8. To constrain the solar neutrino parameter, a ∆χ2 value by the solar+KamLAND
combined analysis is added to χ2 from the atmospheric neutrino data (∆χ2solar in Equation (7.4)).
The other scenario is an ordinary “one mass scale dominance” approximation with ∆m212 = 0,
that is, the pure νµ ↔ ντ two-flavor oscillation scenario.

7.3

Systematic Uncertainties

In this section, the 80 systematic uncertainties are described. Though the systematic errors
related to the event selection and the detector response are different between SK-I and SK-II,
other parameters are identical for SK-I and SK-II. Among 80 systematic errors, 14 are for the144

2∆m 10**

-4

-4.05

-4.1

-4.15

-4.2

-4.25
0.2

0.25

0.3

0.35

0.4

0.45
sin θ122

Figure 7.8: The arrowed 1-2 parameters obtained from the results of the solar neutrino and
KamLAND data [13]. The 20 points show the scanned points and the ∆χ2solar from the solar
plus KamLAND data at the each point is added to the χ2 from the atmospheric neutrino data.
Three contours correspond to the 68 % (dashed line), 90 % (solid line) and 99 % (dotted line).

neutrino flux model, 16 for the interaction, 25 for the SK-I detector response and 25 for the
SK-II detector response. All systematic errors are considered to be uncorrelated.

7.3.1

Systematic Uncertainties in Atmospheric Neutrino Flux

The details of the atmospheric neutrino flux are described in Section 3.2. Systematic uncertainties in the prediction of the atmospheric neutrino flux are estimated by comparing the
Honda flux [14] with the other calculations, the Fluka flux [41] and the Bartol flux [42]. The
systematic errors in the atmospheric neutrino flux are listed in Table 7.1 and 7.2.
• Absolute normalization
The absolute normalization uncertainty in the atmospheric neutrino flux calculation isestimated to be about 20 % for Eν < 100 GeV 30 % for Eν > 100 GeV, which comes fromthe uncertainty in the primary cosmic ray flux and the hadronic interaction models usedin the calculation. The absolute normalization parameter is treated as a free parameter inthis analysis because of the large uncertainty.
• Flavor ratio
The flavor ratio (νµ + ν µ )/(νe + ν e ) of the atmospheric neutrino flux is estimated withinapproximately 3 % by the different models. The systematic error is 3 % for Eν < 5 GeV.
For Eν > 5 GeV, the error increases almost linearly with logEν from 3 % (5 GeV) to 10 %(100 GeV). In the low energy range, the uncertainty mainly comes from the pion spectrum
in the primary hadronic interactions. In the high energy range, the uncertainty mainly
comes from the K/π production ratio. A positive number means the number of MC νµ +ν µ
events is increased.

145

uncertainty (%)

best-fit (%)

free
3.0
3.0a
5.0
5.0b
5.0
5.0c
0.4
1.1
0.7
2.1
1.7
1.9
1.5
0.8
0.4
0.8
0.7
0.4
0.3
0.3
1.1
1.4
2.8
1.9
1.4
1.5
1.3

25.6
-2.8
2.1
2.7
-0.2
-1.3
3.4
-0.03
-0.08
-0.05
-0.1
-0.1
-0.1
-0.10
-0.06
-0.03
-0.06
-0.05
0.05
0.03
0.03
0.1
0.2
0.3
0.2
0.2
0.2
0.1

(A1) Systematic uncertainties in neutrino flux
Absolute normalization(νµ + ν µ )/(νe + ν e ) Eν < 5 GeV
Eν > 5 GeV
νe /ν eEν < 10 GeV
Eν > 10 GeVνµ /ν µEν < 10 GeV
Eν > 10 GeV
Up/down< 400 MeV e-like 0µedecayµ-like 0µedecayµ-like 1µedecay> 400 MeV e-like 0µedecayµ-like 0µedecayµ-like 1µedecay
Multi-GeV e-likeµ-likePC
Sub-GeV multi-ring µ
Multi-GeV multi-ring µ
Horizontal/vertical < 400 MeV e-like 0µedecayµ-like 0µedecayµ-like 1µedecay> 400 MeV e-like 0µedecayµ-like 0µedecay
Multi-GeV e-likeµ-likePC
Sub-GeV multi-ring µ
Multi-GeV multi-ring µ

Table 7.1: Summary of systematic uncertainties in the prediction of the atmospheric neutrino
flux. Estimated uncertainty and the best-fit value is listed for each error.
aError linearly increases with log Eν from 3 % (5 GeV) to 10 %(100 GeV).
Error linearly increases with log Eν from 5 %(10 GeV) to 10 %(100 GeV).
c
Error linearly increases with log Eν from 5 %(10 GeV) to 25 %(100 GeV).

b

146

uncertainty (%)(A2) Systematic uncertainties in neutrino flux
K/π ratio
20.0
Lν (production height)
10.0
Energy spectrum
Ek < 100 GeV
0.03
Ek > 100 GeV
0.05
Sample-by-sample FC Multi-GeV
5.0
PC + upward stopping µ
5.0
Solar activity
SK-I
0.2
SK-II
0.5

best-fit (%)
-8.6
0.03
0.010
0.017
-4.5
-9.1
0.01
0.43

Table 7.2: Summary of systematic uncertainties in the prediction of the atmospheric neutrino
flux. Estimated uncertainty and the best-fit value is listed for each error.

• Anti-neutrino/neutrino ratio
The systematic uncertainty of anti-neutrino to neutrino ratio comes from π + /π − ratio in
hadronic interaction in the flux calculation. The uncertainties of ν e /νe and ν µ /νµ are
estimated to be 5 % for Eν < 10 GeV. For Eνe > 10ĠeV, the errors increase linearly withlogEν from 5 % (10 GeV) to 10 % (100 GeV). For Eνµ > 10ĠeV, the error increases linearlywith logEν from 5 % (10 GeV) to 25 % (100 GeV). A positive number means the numberof MC νe (νµ ) events is increased.
• Up/down ratio
As described in Section 3.2, the neutrino flux below a few GeV has up/down asymmetry
because of the rigidity cutoff by the geomagnetic field, while flux above a few GeV isup/down symmetric. The systematic uncertainty in the up/down ratio of atmospheric
neutrino flux is derived from the treatment of the geomagnetic field in the calculation.
Since the up/down asymmetry in low energy neutrino flux is largely canceled due to thepoor angular correlation in neutrino interactions, the uncertainty of flux calculation does
not directly affect the zenith angle distributions observed in Super-Kamiokande. The
systematic uncertainties of up/down ratio are estimated by comparing the predictions ofzenith angle distributions in Super-Kamiokande from independent flux calculations. The
magnitude of the uncertainty in the up/down ratio is a function of energy, but is assumed
to be fully correlated. A positive number means the number of MC upward events isincreased.
• Horizontal/vertical ratio
The systematic uncertainty of horizontal/vertical ratio is due to the difference in the
3-dimensional calculation method for each model for neutrino energy below 3 GeV. For
higher energy neutrinos, the uncertainties comes from the predicted K/π ratio in hadronicinteractions in the atmosphere. The magnitude of the uncertainty in the horizontal/verticalratio is a function of energy, but is assumed to be fully correlated. A positive number means
the number of MC horizontally-going events is increased.

147

path length (km)

10

4

10

3

10

2

10

1

-0.2

0

0.2

0.4

0.6

0.8

1

Zenith angle (cosΘ)

Figure 7.9: Neutrino flight length as a function of zenith angle. Solid line shows the flight
length used in our analysis and dashed line shows that for the compressed density structure of
atmosphere by 10 %.

• K/π ratio
The uncertainty of K/π production ratio in cosmic ray interactions in the atmosphere isestimated to be 20 %. A positive number means that the fraction of K is increased.
• Neutrino flight length
The uncertainty of neutrino flight length affects on the oscillation probabilities of neutrinos.
The systematic uncertainty is negligible for upward-going neutrinos which pass throughthe Earth because the flight length for those neutrino can be easily calculated from thegeometry. For downward-going and horizontally-going neutrinos, the production height of
the neutrinos must be considered to calculate the flight length. The uncertainty of theneutrino production height is related to the structure of the atmosphere. To estimate thesystematic uncertainty in the production height and neutrino flight length, the calculation
is carried out by changing the density structure of atmosphere by 10 %. Figure 7.9 shows
the comparison of the flight length used in the analysis with that for the case of different
structure of atmosphere (compressed by 10 %). The effect of this difference is consideredas the systematic error. A positive number means a more compressed atmospheric density
structure.
• Energy spectrum
The energy spectrum of primary cosmic rays is proportional to E γ with the spectral index
for the primary proton flux, γ = -2.74. 0.03 and 0.05 uncertannies in the spectral index of
the primary cosmic rays below and above 100 GeV. Spectral index unertanties below andabove 100 GeV are assumed to be correlated. A positive number means that the spectrum
is harder.
• Sample-by-sample normalization
Different flux calculations predict different energy dependence that cannot be explainedby a simple spectral index uncertainty as shown in Figure 7.10. From a comparison of the148Flux×Eν2(m-2sec-1sr-1GeV)

10

3

10

2

(a)

10

10

Flux ratio

Honda flux
Bartol flux
Fluka flux

1
-1

1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
10

(b)

Bartol/Honda
Fluka /Honda
-1

1

10
Eν(GeV)

10

2

10

3

Figure 7.10: (a)The direction averaged atmospheric neutrino energy spectrum for νµ + ν µ calculated by Honda [14], Bartol [42] and Fluka [41]. (b) The ratio of the calculated neutrino flux.
The fluxes calculated in Bartol (solid line) and Fluka (dashed line) are normalized by the fluxin Honda.

predicted number of events based on different flux models, 5 % is assigned as the relative
normalization uncertanty for these samples.
• Solar activity
The primary cosmic ray flux is affected by the solar activity changing in every 11 year
period and the atmospheric neutrino flux calculations considers the solar activity. ±1 yearof uncertainty is assigned to the modulation of solar activity, which corresponds to 20 %for 5 years in SK-I period and 50 % for 2 years in SK-II period.

7.3.2

Systematic Uncertainties in Neutrino Interaction

The neutrino cross-section and kinematics are calculated theoretically and measured by several experiments as described in Section 3.3. The systematic errors in the neutrino interactions
are listed in Table 7.3.
• Quasi-elastic scattering and single-meson production (MA )The cross-section and kinematics in quasi-interaction scattering and single-meson production have a dependence on the axial vector mass MA , which is set to be 1.11 in our
simulation based on experimental data. The uncertainty of MA , which affects the crosssection and the angular correlations between incident neutrinos and outgoing leptons, isestimated to be 10 % as described in Section 3.3.1.
• Quasi-elastic scattering for bound nucleon (Total cross-section)
The CCQE cross-section for the bound nucleon is based on the model of Smith and Monitz.
The systematic uncertainty in the CCQE cross-section is estimated from the difference between the model of Nieves et al . [125] and our MC and is set to be 1σ. In the Nieves’149

(B) Systematic uncertainties in neutrino interaction
Quasi-elastic scattering and single-meson production (MA )Quasi-elastic scattering for bound nucleon (total cross-section)
Quasi-elastic scattering for bound nucleon (ν/ν ratio)
Quasi-elastic scattering for bound nucleon (νµ /νe ratio)
Single-meson production (total cross-section)
1 π + production (model dependence)
1 π 0 production (model dependence)
1 π − production (model dependence)
Multi-pion production (a few GeV region)
Multi-pion production (total cross-section)
Coherent pion production (total cross-section)
NC/CC ratio
Nuclear Effect in 16 O nucleus
Nuclear Effect in pion spectrumCC ντ interaction cross-section
Hadron simulation

uncertainty (%)

best-fit (%)

10.0
1.0a
1.0b
1.0c
20.0
20.0
20.0
20.0
1.0d
5.0
100.0
20.0
30.0
1.0e
30.0
10.0

-0.8
0.7
0.9
0.5
3.2
25.5
4.4
-11.3
1.75
-0.7
-97.1
-8.6
-17.9
1.7
-19.2
2.0

Table 7.3: Summary of systematic uncertainties in neutrino interactions. Estimated uncertainty
and the best-fit value are listed for each error.
aDifference from the Nieves model is set to 1.0
Difference from the Nieves model is set to 1.0cDifference from the Nieves model is set to 1.0
dDifference from the Bodek model is set to 1.0eDifference between NEUT [62] and NUANCE [124] is set to 1.0b150

Ratio(Nieves/Smith-Moniz)

2
1.8
1.6

Figure 7.11: The ratio of the calculated
CCQE cross-sections for νe +ν e (solid line)and νµ + ν µ (dashed line) of Nieves’ calculation to the calculation of Smith andMonitz.

1.4
1.2
1
0.8
0.6
0.4
0.2
50

100

150

200

250

300

350

400

450
500
Eν (MeV)1.5

ν / ν double ratio(Nieves/Smith-Moniz)µ/e double ratio(Nieves/Smith-Moniz)

0

1.4
1.3
1.2
1.1
1
0.9
0.8
0.7

1.2
1
0.8
0.6
0.4
0.2

0.6
0.5
0

1.4

50

100

150

200

250

300

350

400

450
500
Eν (MeV)

0

50

100

150

200

250

300

350

400

450
500
Eν (MeV)Figure 7.12: The ratio of the calculated CCQE cross-section of Nieves’ calculation to the calculation of Smith and Monitz for ν/ν ratio (left) and νµ /νe ratio (right).

model the effect of in target nucleous in the target nucleon is taken into account. Figure 7.11 shows the calculated CCQE cross-section in Nieves’ model normalized by that in
the model of Smith and Monitz as a function of neutrino energy.
• Quasi-elastic scattering for bound nucleon (ν/ν ratio)
The uncertainty in the CCQE cross-section of ν/ν ratio is estimeted from the difference
between the model of Nieves et al . and our MC. The left plot on Figure 7.12 shows thecalculated CCQE cross-section of ν/ν ratio in Nieves’ model normalized by that in themodel of Smith and Monitz as a function of neutrino energy.
• Quasi-elastic scattering for bound nucleon (νµ /νe )The uncertainty in the CCQE cross-section of νµ /νe ratio is estimeted from the difference
between the model of Nieves et al . and our MC. The right plot on Figure 7.12 shows thecalculated CCQE cross-section of νµ /νe ratio in Nieves’ model normalized by that in themodel of Smith and Monitz as a function of neutrino energy.
• Single pion production (Total cross-section)
The uncertainty in the single-meson production cross-section is estimated to be 20 %.
Details of the single pion production are described in Section 3.3.2.
• Single pion production (Model dependence)
The uncertainty in the single-meson production cross-section for each π + , π 0 and π − production is estimated by taking the difference between the FKR model [77] and the model
calculation based on [78] to be 20 %.
151

(C) Systematic uncertainties in event selection for SK-I
Reduction for FC event
Reduction for PC event
FC/PC separation
Non-ν background
Sub-GeV
e-like 0µedecayµ-like 0µedecayµ-like 0µedecay
Multi-GeV
e-likeµ-likePC
Reduction for upward going muon stopping µthrough-going µ
Separation of upward stopping/through-going muon

uncertainty (%)

best-fit (%)

0.2
2.6
0.9
0.5
<0.1
<0.1
0.2
0.1
0.2
1.3
0.5
0.4

-0.003
0.4
-0.3
0.02
-0.001
0.0
0.03
-0.01
-0.03
-0.1
-0.04
-0.05

Table 7.4: Summary of systematic uncertainties in event selection for SK-I. Estimated uncertainty and the best-fit value are listed for each error.

• NC/CC ratio
The systematic uncertainty in the NC/CC ratio is estimated to be 20 %. A positive number
means more NC events in the Monte Carlo.
• Nuclear Effect in 16 O nucleus
The systematic uncertainty in the mean free path of hadrons produced via neutrino interactions in the 16 O nucleus is estimated to be 30 %. A positive number means stronger
nuclear effect in 16 O.
• Nuclear Effect in pion spectrum
The difference in the predicted pion energy spectrum by NEUT [62] and NUANCE [124]interaction models is taken to be a systematic error.
• Charged-current ντ interaction
The systematic uncertainty in the ντ cross-sections for CC interactions is estimated to be
30 % by comparing two models of NEUT [62] and Hagiwara et al . [128].

7.3.3

Systematic Uncertainties in Event Selection• FC and PC reduction
Details of FC and PC reduction are described in Section 4.2 and ??. The uncertainty of
the FC and PC reduction efficiencies are estimated by comparing the distributions of theeach cut valuables of data and MC. The uncertainty is estimated to be 0.2 % (0.2 %) and
2.6 % (4.0 %) for FC and PC reduction of SK-I (SK-II), respectively.

152

(D) Systematic uncertainties in event selection for SK-II
Reduction for FC event
Reduction for PC event
FC/PC separation
Non-ν background
Sub-GeV
e-like 0µedecayµ-like 0µedecayµ-like 0µedecay
Multi-GeV
e-likeµ-likePC
Reduction for upward going muon stopping µthrough-going µ
Separation of upward stopping/through-going muon

uncertainty (%)

best-fit (%)

0.2
4.0
0.5
0.3
0.1
<0.1
0.7
0.1
0.7
1.9
0.9
0.4

0.007
-3.8
0.07
0.006
-0.0010.0
0.01-0.001-0.006
-0.08
-0.04
0.003

Table 7.5: Summary of systematic uncertainties in event selection for SK-II. Estimated uncertainty and the best-fit value are listed for each error.

• FC/PC separation
The FC and PC events are separated by the number of hit PMTs in the OD hit PMT
cluster (NHITAC). The systematic uncertainty in the FC/PC separation is estimated tobe 0.9 % (SK-I) and 0.5 % (SK-II) from the difference between data and MC NHITAC
distributions. A positive number means that the number of MC FC events is increased.
• Non-ν background
The possible sources of background in FC and PC samples are cosmic ray muons for µ-likesamples and flasher PMTs and neutron from the rock for e-like samples. The contaminationof the background is estimated to be 0.5 % (0.3 %) for FC Sub-GeV e-like events, 0.2 %(0.7 %) for FC Multi-GeV e-like events, <0.1 % (<0.1 %) for FC Sub-GeV µ-like events,
0.1 % (0.1 %) for FC Multi-GeV µ-like event and 0.2 % (0.7 %) for PC events.
• Upward going muon reduction
The methods of the reduction for upward going muon events are described in Section 4.4.
The systematic uncertainty in the data reduction process for upward-going sample is estimated by comparing the distribution of each cut variable of the data with that of theMonte Carlo events. The estimated errors are 1.3 % (1.9 %) for upward stopping muonsand 0.5 % (0.9 %) for upward through-going muons for SK-I (SK-II). Uncertainties for
upward stopping µ and upward through-going µ are assumed to be correlated.
• Upward stopping/through-going muon separation
The upward stopping muon events and the upward through-going muon events are separated by the number of hit OD PMTs within 8 m from the exit position (NHITEX). The
systematic uncertainty for the stopping/through-going separation is estimated to be 0.4 %
153and 0.3 % for SK-I and SK-II, respectively, by comparing the NHITEX distributions forthe observed data and the Monte Carlo events. A positive number means that the numberof MC stopping muons is increased.

7.3.4

Systematic Uncertainties in Event Reconstruction• Ring separation
The ring-counting algorithm is described in Section 5.3. The systematic error in singlering and multi-ring separation is due to several uncertainties such as water transparency
and detector calibration. The uncertainties are estimated by comparing the ring counting
likelihood distributions for the observed data with that for the Monte Carlo events.
• Particle Identification
The systematic errors in the particle identification are estimated by taking the differencein the PID likelihood distributions between data and MC (see Section 5.4).
• Energy calibration
The absolute energy scale is calibrated using several sources as described in Section 2.6.5
and the systematic uncertainty is estimated to be 2.0 % for SK-I and 2.5 % for SK-II.
• Up/down symmetry of energy calibration
The difference in the energy scale for upward-going and downward-going events is measured
by using decay electrons from cosmic ray stopping muons (see Figure 2.33). The up/down
asymmetry of the energy scale is estimated to be 0.6 % for SK-I and SK-II.
• Single ring π 0 -like sample selection
The FC Sub-GeV single-ring π 0 -like event sample is separated from the FC Sub-GeV
single-ring e-like events as described in Section 6.1.1. The uncertainty of the single-ring
π 0 -like sample selection is derived from the difference between data and MC of the invariant
mass distribution and total likelihood distributions. Figure 7.13 shows the invariant mass
distributions for FC Sub-GeV single-ring e-like events from the π 0 fitter (see Section 5.9.1).
In the separation of π 0 -like sample, the π 0 mass is required to be larger than 100 MeV/c2 .
The uncertainty of the π 0 mass cut is estimated to be 9.6 % (5.1 %) by comparing theπ 0 mass distributions for the observed data and the Monte Carlo events for SK-I (SKII). The distributions of the total likelihood for 1-ring π 0 like sample selection of data
and MC are shown in Figure 7.14. The uncertainties in the selection efficiency from thelikelihood distributions are estimated to be 6.2, 21.3, 16.9, 8.7 % (6.1, 16.6, 8.5, 9.6 %)from the low to high momentum bins for SK-I (SK-II). Since the selection criteria haveenergy dependent, the uncertainty in the absolute enregy scale is also a source of the π 0selection uncertainty. The uncertainties from the energy scale uncertainty of 2.0 % (2.5 %)are 5.7, 1.3, 0.6, 0.6, 1.3 % (5.5, 4.0, 2.3, 4.0, 2.0 %) for SK-I (SK-II). Combining these
uncertainties, the systematic errors on the 1-ring π 0 -like sample selection are estimated tobe 11.2, 11.5, 23.4, 19.1, 13.0 % (7.5, 8.9, 17.5, 10.7, 11.1 %) in the each momentum binsfor
SK-I (SK-II). Exactly the same methods are sed to evaluate the systematic uncertainties
for SK-II, which are summarized in Table 7.7.
• Two ring π 0 -like sample selection
The FC Sub-GeV two-ring π 0 -like sample is separated as described in Section 6.1.3. The
154

uncertainty (%)(E) Systematic uncertainties in event reconstruction for SK-I
Single-ring/multi-ring separation < 400 MeV e-like
6.3µ-like2.4> 400 MeV e-like
3.4µ-like
1.3
Multi-GeV e-like
15.9µ-like6.2
Sub-GeVmulti-ring µ3.7
Multi-GeV multi-ring µ7.2
Particle identification
Sub-GeV
e-like-0.6µ-like0.6
Multi-GeV e-like-0.4µ-like0.4
Sub-GeVmulti-ring µ3.4
Multi-GeV multi-ring µ
4.7Energy calibration for FC event
2.0
Up/down symmetry of energy calibration
0.6
FC Sub-GeV single-ring100< Pe <250 MeV/c
11.20
π -like sample selection
250< Pe <400 MeV/c
11.5
400< Pe <630 MeV/c
23.4
630< Pe <1000 MeV/c
19.1
1000< Pe <1330 MeV/c
13.00
FC Sub-GeV two-ring π -like sample selection
8.4
Decay electron tagging
e-like 0µedecay
0.2∼0.5
for FC Sub-GeV single-ring sample
e-like 1µedecay-4.4∼-4.2µ-like 0µedecay
1.6∼1.7µ-like 1µedecay-1.4∼-1.1µ-like 2µedecay-6.4∼-5.9

best-fit (%)
2.6
1.0
1.4
0.5
6.7
2.6
-1.5
-3.0
0.06
-0.06
0.04
-0.04
-0.7
-0.9
-1.8
0.1
-10.2
-10.5
-21.3
-17.4
-11.9
1.9
-0.2∼-0.4
2.9∼3.0-1.1∼-1.2
0.7∼0.9
4.0∼4.3

Table 7.6: Summary of systematic uncertainties in event reconstruction for SK-I. Estimateduncertainty and the best-fit value are listed for each error.

155

uncertainty (%)(F) Systematic uncertainties in event reconstruction for SK-II
Single-ring/multi-ring separation < 400 MeV e-like
1.3µ-like0.7> 400 MeV e-like
2.8µ-like0.5Multi-GeV e-like
3.7µ-like1.5
Sub-GeVmulti-ring µ-0.8Multi-GeV multi-ring µ-2.2
Particle identification
Sub-GeV
e-like-0.8µ-like0.9Multi-GeV e-like-0.2µ-like0.2
Sub-GeVmulti-ring µ13.9Multi-GeV multi-ring µ
9.1Energy calibration for FC event
2.5
Up/down symmetry of energy calibration
0.6
FC Sub-GeV single-ring100< Pe <250 MeV/c
7.50
π -like sample selection
250< Pe <400 MeV/c
8.9400< Pe <630 MeV/c
17.5
630< Pe <1000 MeV/c
10.7
1000< Pe <1330 MeV/c
11.1
0
FC Sub-GeV two-ring π -like sample selection
6.9
Decay electron tagging
e-like 0µedecay
0.2∼0.5
for FC Sub-GeV single-ring sample
e-like 1µedecay-4.2∼-4.4µ-like 0µedecay
1.6∼1.7µ-like 1µedecay-1.4∼-1.1µ-like 2µedecay-6.4∼-5.9

best-fit (%)
0.01
0.07
0.03
0.05
0.03
0.01
-0.007
-0.02
0.1
-0.2
0.04
-0.04
4.1
2.7
-3.1
-0.3
-4.7
-5.6
-11.0
-6.7
-7.0
-0.7
-0.1∼-0.3
2.4∼2.5-0.9∼-1.0
0.6∼0.8
3.4∼3.6

Table 7.7: Summary of systematic uncertainties in event reconstruction for SK-II. Estimateduncertainty and the best-fit value are listed for each error.

156

600
400
200
0

0

50

100

150

200

0

50
100
150
Invariant mass (MeV/c2)

200

Figure 7.13: The π 0 mass distributions
from the π 0 fitter for data (dot) and Monte
Carlo events (solid line) of FC Sub-GeV
single-ring e-like for SK-I (top) and SK-II(bottom) together with the shifted distribution (dotted line) by the fitted results.

300
200
100
0

0

Selection likelihood for single-ring π sample

35
30
25
20
15
10
5
0

(a)

-4

-2

0

2

60

4

(c)

50
40
30
20
10
0

-4

-2

0

2

4

40
35
30
25
20
15
10
5
0

70
60
50
40
30
20
10
0

0

Selection likelihood for single-ring π sample(b) 22.5

-4

-2

0

2

20
17.5
15
12.5
10
7.5
5
2.5
0
4

(d)

-4

-2

0

2

4

40
35
30
25
20
15
10
5
0

(a)

-4

-2

0

2

4

(c)

-4

-2

0

2

4

40
35
30
25
20
15
10
5
0

40
35
30
25
20
15
10
5
0

(b)

-4

-2

0

2

4

(d)

-4

-2

0

2

4

Figure 7.14: The π 0 -like sample selection likelihood distributions for data (dot) and Monte Carloevents (solid line) of FC Sub-GeV single-ring e-like for SK-I (left) and SK-II (right) together withthe shifted distribution (dotted line) by the fitted results in (a)Pe < 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, (e)1000 MeV/c ≤ Pe .

157

150

µ-likee-like

100
50
0

-200

0

Number of events

Number of events

200

100

50
25
0

200

-100

0

100

PID likelihood (first ring)Number of events

Number of events

PID likelihood (first ring)

300

400
300

µ-likee-like75µ-likee-likeµ-likee-like

200

200

100

100
0

-200

0

0

200

PID likelihood (second ring)

-100

0

100

PID likelihood (second ring)

Figure 7.15: The PID likelihood distributions of data (dot) and Monte Carlo events (solid line)
for FC Sub-GeV two-ring events with the shifted distribution (dotted line) by the fitted results.
The distributions shown by green histgrams are from the Monte Carlo π 0 events. The left twopanels show for SK-I and the right two show for SK-II.

uncertainty in two-ring π 0 -like sample selection comes from the ring-counting and PID.
Figure 7.15 shows the PID parameter distributions of FC Sub-GeV two-ring events for thedata and MC. In order to estimate the uncertainty due to PID, the Monte Carlo likelihood
distributions are renormalized (vertical) and scaled (horizontal) to get the best-fit to thedata. The differences in the number of events which identified to be e-like before and after
shifted are 0.6 % and 0.2 % for the first ring and the second ring, respectively. Figure 7.16
shows the ring-counting distributions of FC Sub-GeV events for data and MC with themost energetic ring being identified to be e-like. By fitting data and MC by the same as
method for PID, the differences of the number of the events which are identified to be tworing before and after the shift are 6.7 % and 5.0 % for the separation from a single ring tomore rings and the separation from two rings to more rings, respectively. Combined theuncertainties from ring-counting and PID, the uncertainty of the two ring π 0 -like sample
selection is calculated to be 8.4 % for SK-I. For SK-II, the same methods are used toevaluate the systematic uncertainties. The uncertainties due to PID are estimated to be
0.4 % and 0.2 % for the first ring and the second ring, respectively. The uncertainties due
to ring-counting are estimated to be 0.04 % and 6.9 % for the separation from a single ring
to more rings and the separation from two rings to more rings, respectively. Combined theuncertainties from ring-counting and PID, the uncertainty of the two ring π 0 -like sample
selection is calculated to be 6.9 % for SK-II.
• Decay electron tagging
The uncertainty in the decay electron tagging comes from that of muon decay and also π +
158

500

700
600

400

500
400

300

300

200

200

100

100
0
-50
160
140
120
100
80
60
40
20
0
-50

-40

-40

-30 -20 -10
010 20 30
Ring-counting likelihood (first ring)40-30 -20 -10
010 20 30 40
Ring-counting likelihood (second ring)

50

0
-15

80
70
60
50
40
30
20
10
0
50
-50

-40

-10
-5
0
5
10
Ring-counting likelihood (first ring)15-30 -20 -10
010 20 30 40
Ring-counting likelihood (second ring)

50

Figure 7.16: The ring-counting likelihood distributions of FC Sub-GeV events for data (dot)and Monte Carlo events (solid line) which the most energetic ring is identified to be e-like withthe shifted distribution (dotted line) by the fitted results. The distributions shown by green
histgrams are from the Monte Carlo π 0 events. The left two panels show for SK-I and the righttwo show for SK-II.

decay probability. The uncertainty in tagging efficiency of an electron from a muon decay
has been estimated to be 1.5 % using cosmic ray muons [129]. The uncertainty on π +interaction in water is mainly considered to come from the energy dependent uncertainties
on the cross-section of π + -16 O interaction. Figure 3.13 shows the cross-section for π + 16 O as a function of π + momentum. The uncertainties are estimated from the difference
between experimental data and the theoretical calculation used in our Monte Carlo to
be 20∼60 % depending on the pion energy. The uncertainty of π + decay probability is
estimated by the π + MC generated with increased or decreased mean free path by theuncertainties of π + -16 O interaction. Figure 7.17 shows the π + decay probabilities whenthe mean free path of π+ in water is increased and decreased. The differences of theprobability are 3∼13 % depending on the π + momentum and affect the uncertainty of
decay electron tagging by 0.3∼0.6 % for FC single-ring Sub-GeV e-like 0µedecay sample,
3.5∼4.2 % for FC single-ring Sub-GeV e-like 1µedecay sample, 0.5∼0.9 % for FC single-ring
Sub-GeV µ-like 0µedecay sample, 0.1∼0.3 % for FC single-ring Sub-GeV µ-like 1µedecay
sample and 5.0∼5.7 % for FC single-ring Sub-GeV µ-like 2µedecay sample. Combining theuncertainties from π + decay and tagging efficiency of an electron from a muon decay, thesystematic uncertainties on decay electron tagging are obtained as Table 7.6. Uncertainties
for 0µedecay and 1µedecay (2µedecay) are asuumed to be correlated. The unceratainty inthe decay electron tagging for SK-II is asuumed to be identical to SK-I.

159

1.2

Decay probability

1

Figure 7.17: The probabilities of π + decay
when the mean free path of π + in water
is increased (dashed line) and decreased(dotted line). The solid line shows the π +decay probability in our Monte Carlo simulation.

0.8

0.6

0.4

0.2

0

0

50

100 150 200 250 300 350 400 450 500
π momentum (MeV/c)+

7.4

Results

Since θ13 =0 is assumed in the analysis, χ2 is calculated in the four dimensional oscillation
parameter space of ∆m212 , ∆m223 , sin2 θ12 and sin2 θ23 . For the solar neutrino parameters, two
scenarios are examined. In one scenario with the solar neutrino parameters turned on, thescanning points of solar parameters are chosen around the allowed region obtained by a combined
analysis of the solar neutrino data and KamLAND data [13]. To take into account the constraint
on the solar neutrino parameters, the ∆χ2solar values from the combined analysis of the solar
neutrino and KamLAND data is added to χ2 from the atmospheric neutrino data for each(∆m212 , sin2 θ12 ) point as shown in Equation (7.4). The other scenario is the ordinary “one massscale dominance” approximation with ∆m212 =0, that is, the pure νµ ↔ ντ oscillation analysis
case.
Figure 7.18 shows the χ2 -χ2min distributions with and without the 1-2 parameters as a function of sin2 θ23 , where ∆m212 , ∆m223 and sin2 θ12 are chosen so that χ2 is minimized. The best-fit
point without the 1-2 parameters is located at sin2 θ23 =0.50, on the other hand, the best-fit pointwith the 1-2 parameters is 0.53. The minimum χ2 value is 828.1/743 d.o.f. in case with the 1-2
parameters at (sin2 θ23 , ∆m223 , sin2 θ12 , ∆m212 ) = (0.53, 2.5 × 10−3 eV2 , 0.32, 7.8 × 10−5 eV2 ).
There is no significant discrepancy of sin2 θ23 from 45◦ , but the asymmetric shape due to theeffect of the solar terms is seen in the χ2 distribution. The stringent constraint on sin2 θ23 hasbeen observed in the analysis with the solar terms. The 1σ arrowed region for sin2 θ23 , which
is defined to be ∆χ2 =1.0 is 0.43 < sin2 θ23 < 0.58, which is equivalent to sin2 2θ23 > 0.973 asshown in Figure 7.19. This is a more stringent constraint on θ23 compared with the publishedresult; sin2 2θ23 > 0.95 corresponding to 0.39 < sin2 θ23 < 0.61 [25].
The gray scale in Figure 7.20 shows the χ2 -χ2min distribution at each solar parameter point.
Since the difference of χ2 -χ2min in a 1-2 parameter plane is small (≤ 0.14), the global χ2 distribution has weak dependence on the details for the 1-2 parameters.
Constraint on ∆m223 is expected not to be affected by the solar terms as shown in Figure 7.21.
Compared with the published result [25] on χ2 as a function of ∆m223 , the χ2 distribution is lesssmooth. This is because the averaged oscillation probability technique can not be used in theanalysis involving the solar terms, and therefore the oscillated Monte Carlo prediction is lesssmooth.
Zenith angle dependence of the ratio between the number of FC Sub-GeV single-ring µ-like
160

5
4.5
4
3.5
∆χ2

3
2.5
2
1.5
1
0.5
0

0.4

0.5
sin2θ23

0.6

Figure 7.18: χ2 -χ2min distribution as a function of sin2 θ23 for oscillations without the 1-2 parameters (dotted line) and with the 1-2 parameters (solid line). For each sin2 θ23 point, ∆m223 ischosen so that χ2 is minimized. The green line corresponds to the 68 % confidence level which
is located at χ2min + 1.0.

5
4

∆χ2

3
2
1
0
0.9

0.925

0.95
sin22θ23

0.975

1Figure 7.19: χ2 -χ2min distribution as a function of sin2 2θ23 for oscillations without the 1-2
parameters (dashed line) and with the 1-2 parameters (solid line). There are 2 solid lines dueto the different χ2 curve below and above sin2 θ23 = 0.5 as seen in Figure 7.18.

161

0.14

2

∆m 10**

-4

0.12

-4.05

0.1
-4.1
0.08
-4.15

0.06
0.04

-4.2
0.02
-4.25
0.2

0.25

0.3

0.35

0.4

0.45

0

sin θ122

Figure 7.20: The gray scale shows χ2 -χ2min at each solar parameter point. The lines show thearrowed region from the result of the solar neutrino and KamLAND data [13], correspondng to
the 68 % (dashed line), 90 % (solid line) and 99 % (dotted line).

20
18
16
14
∆χ2

12
10
8
6
4
2
0 -3
10

10-2.810-2.6∆m232

10-2.410-2.2

Figure 7.21: χ2 -χ2min distribution as a function of ∆m223 for oscillations without the 1-2 parameters (dashed line) and with the 1-2 parameters (solid line).

162

1

Nµ/Ne

0.8

0.6

0.4

-1

-0.5

0
cosΘ

0.5

1

Figure 7.22: Zenith angle dependence of (Nµ /Ne ) for the data (dot), the best-fit results withsolar terms (solid line) and the best-fit results without solar terms (dotted line). Nµ (Ne ) is thenumber of events for FC Sub-GeV single-ring µ-like 1µedecay sample (e-like 0µedecay sample).

1µedecay events and FC Sub-GeV e-like 0µedecay events (Nµ /Ne ) for the data and the best-fit
results with and without the solar terms are shown in Figure 7.22. The two fitting results equally
explain the data and the effect of the solar terms on the (Nµ /Ne ) ratio has not been observed.
Figure 7.23, 7.24, 7.25 and 7.26 show the zenith angle distributions for FC Sub-GeV event
samples, FC Multi-GeV or multi-ring event samples, PC event samples and Upward going-muon
event samples for SK-I divided by each momentum bin. Figure 7.27, 7.28, 7.29 and 7.30 show
the zenith angle distributions for FC Sub-GeV event samples, FC Multi-GeV or multi-ring event
samples, PC event samples and Upward going-muon event samples for SK-II divided by each
momentum bin. The data are also consistent with the best-fit results with solar parameters forboth SK-I and SK-II.
The probability for χ2min = 828.1/743 d.o.f. or larger is calculated by “toy Monte Carlo”
events. Assuming the fitted number of events (Nf it ) in each bin is true, the expected numberof events (Nrnd ) is determined based on the Poisson distribution. χ2 for the toy Monte Carlo is
calculated as :


Nrnd
2
.
(7.6)
χ = 2(Nf it − Nrnd ) + 2Nrnd log
Nf it
The χ2 distribution shown in Figure 7.31 is obtained by 100000 calculations. The probability
for χ2min = 828.1/743 d.o.f. is estimated to be 11.7 %.

163

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

400-630630-1000

40

50

20

-1

0

1-1

0

1-1

0

1-1

0

200-250(MeV/c)1000 < Pl

100

0SK1 Sub-GeV 1-ring µ-like 0µedecay

60

1-1

0

1

0

-1

250-400

1-1

400-6301-1

630-1000

1-1

1-11000 < Pl1cosΘ
SK1 Sub-GeV 1-ring µ-like 1µedecaycosΘ

200-250(MeV/c)100

250-400

400-630630-1000

1000 < Pl

75
50
25
0

-1

1-1

1-1

1-1

1-1

1

cosΘ

Figure 7.23: Zenith angle distributions for all event bins for data (dot) and Monte Carlo events
assuming no oscillation (dotted line) and the best-fit results (solid line) for SK-I FC Sub-GeV
single-ring event samples. Each panel shows the distributions of each momentum bin.

SK1 Multi-GeV 1-ring e-like80

1 - 2.5(GeV/c)

2.5 - 55 - 10

60

40

40

20

20

-1

1-1

1-1

1-1

1

cosΘ
SK1 Multi-GeV multi-ring e-like50

1 - 2.5(GeV/c)

2.5 - 55 - 102.5 (GeV/c) < Pl

1 - 2.5 (GeV/c)10(GeV/c)< Pl

60

0

SK1 Multi-GeV 1-ring µ-like

80

0

-1

0

60

Pl< 1.0(GeV/c)10(GeV/c)< Pl

1-1

0

1

cosΘ
SK1 multi-ring µ-like
2.5 - 55 - 1010(GeV/c)< Pl

40
40

30
20

20

10
0

-1

1-1

1-1

1-1

1

0

cosΘ

-1

1-1

1-1

1-1

1

cosΘ

Figure 7.24: Zenith angle distributions for all event bins for data (dot) and Monte Carlo events
assuming no oscillation (dotted line) and the best-fit results (solid line) for SK-I FC Multi-GeVor multi-ring event samples. Each panel shows the distributions of each momentum bin.

164

SK1 PC stop30

SK1 PC through80

2.5 (GeV/c) < Pl

Pl < 2.5 (GeV/c)

Pl< 1.0(GeV/c)

2.5 - 55 - 1010(GeV/c)< Pl

60
20
40
10
20
0

-1

0

1-1

0

1

0

-1

cosΘ

1-1

1-1

1-1

1

cosΘ

Figure 7.25: Zenith angle distributions for all event bins for data (dot) and Monte Carlo events
assuming no oscillation (dotted line) and the best-fit results (solid line) for SK-I PC event
samples. Each panel shows the distributions of each momentum bin.

SK1 Upmu stopping

100

SK1 Upmu non-showering

400

80

SK1 Upmu showering

80

300

60

200

40

100

20

60
40
20
0

-1

-0.8

-0.6

-0.4

cosΘ

-0.2

0

0

0
-1

-0.8

-0.6

-0.4

cosΘ

-0.2

0

-1

-0.8

-0.6

-0.4

-0.2

0

cosΘ

Figure 7.26: Zenith angle distributions for all event bins for data (dot) and Monte Carlo events
assuming no oscillation (dotted line) and the best-fit results (solid line) for SK-I upward-going
muon event samples.

165

SK2 Sub-GeV 1-ring e-like 0µedecay

80100-250(MeV/c)250-400

400-630630-1000

60

30

40

20

20

10

0

-1

1-1

1-1

1-1

cosΘ

SK2 Sub-GeV 1-ring µ-like 0µedecay

40200-250(MeV/c)1000 < Pl

1-1

1

0

-1

250-400

1-1

400-6301-1

630-1000

1-11000 < Pl

1-1

1

cosΘ
SK2 Sub-GeV 1-ring µ-like 1µedecay

60

200-250(MeV/c)250-400

400-630630-1000

1000 < Pl

40

20

0

-1

1-1

1-1

1-1

1-1

1

cosΘ

Figure 7.27: Zenith angle distributions for all event bins for data (dot) and Monte Carlo events
assuming no oscillation (dotted line) and the best-fit results (solid line) for SK-II FC Sub-GeV
single-ring event samples. Each panel shows the distributions of each momentum bin.

SK2 Multi-GeV 1-ring e-like

40

2.5 - 5

1 - 2.5(GeV/c)

SK2 Multi-GeV 1-ring µ-like50

2.5 (GeV/c) < Pl

1 - 2.5 (GeV/c)

5 (GeV/c)< Pl
40

30

30
20
20
10
0

10

-1

1-1

1-1

1

cosΘ
SK2 Multi-GeV multi-ring e-like

20

2.5 - 5

1 - 2.5(GeV/c)

0

-1

0

1-1

0

1

cosΘ
SK2 multi-ring µ-like30

Pl< 1 (GeV/c)

5 (GeV/c)< Pl

1 - 2.5

2.5 (GeV/c)< Pl

15
20
10
10
5
0

-1

1-1

1-1

1

0

cosΘ

-1

1-1

1-1

1

cosΘ

Figure 7.28: Zenith angle distributions for all event bins for data (dot) and Monte Carlo events
assuming no oscillation (dotted line) and the best-fit results (solid line) for SK-II FC Multi-GeVor multi-ring event samples. Each panel shows the distributions of each momentum bin.

166

SK2 PC stop30

SK2 PC throughPl< 2.5 (GeV/c)40
20

2.5 - 55 (GeV/c)< Pl

30
20

10
10
0

-1

-0.5

0

0.5

1

0

-1

cosΘ

1-1

1-1

1

cosΘ

Figure 7.29: Zenith angle distributions for all event bins for data (dot) and Monte Carlo events
assuming no oscillation (dotted line) and the best-fit results (solid line) for SK-II PC event
samples. Each panel shows the distributions of each momentum bin.

SK2 Upmu stopping

60

SK2 Upmu non-showering

SK2 Upmu showering

30

150

40

20
100

20

0

10

50

-1

-0.8

-0.6

-0.4

-0.2

0

0

0
-1

-0.8

cosΘ

-0.6

-0.4

-0.2

0

-1

-0.8

-0.6

-0.4

-0.2

0

cosΘ

cosΘ

Figure 7.30: Zenith angle distributions for all event bins for data (dot) and Monte Carlo events
assuming no oscillation (dotted line) and the best-fit results (solid line) for SK-II upward-going
muon event samples.

Entries
Mean
RMS

4000

100000778.6
41.61

Number of Events

3500
3000
2500
2000

11.7%

1500
1000
500
0

700

800
χ2

900

1000

Figure 7.31: χ2 distribution of toy Monte Carlo events.

167

Chapter 8

Future Sensitivity on the θ23
Determination
The future sensitivity on the θ23 determination is studied by simulating the signal according
to the expectation from some specific choice of the true parameters. The oscillated Monte Carlosample renormalized to a 20 year exposure, which corresponds to about 3 times the whole SK-I
and SK-II data, is replaced to the observed data. As the true sin2 θ23 , three different values,
0.40, 0.50 and 0.60 are selected. For the other parameters, θ13 =0, sin2 θ12 =0.32, ∆m212 =7.8×10−5 eV2 and ∆m223 = 2.5×10−3 eV2 are assumed both for making fake signal and for the
expectation to calculate χ2 . Therefore, only sin2 θ23 is scanned for the χ2 minimum. The 80
systematic errors, which are related to the sample normalizations, neutrino flux calculations,
neutrino interaction models and detector responses, are considered in the oscillation analysis as
described in Section 7.3. In this study, in order to research which systematic errors should be
reduced for obtaining a better sensitivity, the size of specific systematic errors is reduced down
to 1/4 of the current estimation and then χ2 are compared.
Figure 8.1 shows the χ2 − χ2min as a function of sin2 θ23 for each true sin2 θ23 . The thinsolid line (black) and middle-thick solid line (green) show the case with all systematic errors
unchanged for a 6 year exposure and a 20 year exposure, respectively. While the best-fit point
is located at the true sin2 θ23 value, ∆χ2 distribution has the second local minimum around thefalse solution 1 − sin2 θ23 . It seems to be difficult to measure sin2 θ23 precisely in this statistics inthe case that the deviation of sin2 θ23 from the maximal is small. The other lines in Figure 8.1
show the cases of reducing the systematic errors in neutrino flux, neutrino interactions, detector
responses and all systematic errors by a factor of 0.25. It is found that reducing the systematic
uncertainties on neutrino interactions is most effective to distinguish the true sin2 θ23 from falseone.
To explore which systematic errors are dominant, χ2 distributions are compared by reducing
each systematic error one by one. As a result, the dominant systematic errors to be reduced forthe future determination of sin2 θ23 are as follows :• neutrino flux : (νµ + ν µ )/(νe + ν e ) for Eν < 5 GeV• neutrino flux : νe /ν e for Eν < 10 GeV• neutrino flux : νµ /ν µ for Eν < 10 GeV
168

6

5

5

4

4
χ2 - χ2min

χ2 - χ2min

6

3

3

2

2

1

1

0
0.3

0.4

0.5
sin2θ23
6

0.6

0.7

0
0.3

0.4

0.5
sin2θ23

0.6

0.7

5

χ2 - χ2min

4
3
2
1
0
0.3

0.4

0.5
sin2θ23

0.6

0.7

Figure 8.1: χ2 -χ2min distribution as a function of sin2 θ23 using the oscillated Monte Carlo eventsof a 20 year exposure with the true oscillation parameters of θ13 = 0, sin2 θ12 = 0.32, ∆m212 =7.8×10−5 eV2 , ∆m223 = 2.5×10−3 eV2 and sin2 θ23 = 0.4 (top left), 0.5 (bottom) and 0.6 (topright). The thin solid (black) line and middle-thick solid (green) line show the case with all
systematic errors unchanged for a 6 year exposure, which is equivalent to the period for SK-I
and SK-II, and a 20 year exposure, respectively. All the other lines are for 20 year exposure.
The dashed (blue) line shows for the case of reducing the systematic errors on neutrino flux by afactor of 0.25. The dotted (pink) line on neutrino interactions and the dashed-dotted (light-blue)line on the detector responses. The thick solid (red) line shows for the case of reducing all thesystematic errors.

169

6
5

χ2 - χ2min

4
3
2
1
0
0.3

0.4

0.5
sin2θ23

0.6

0.7

Figure 8.2: χ2 -χ2min distribution as a function of sin2 θ23 using the oscillated Monte Carlo eventsof a 20 year exposure with the true sin2 θ23 = 0.4. The size of the 7 dominant systematic errors
is assumed as 1/2 (dotted line) and 1/4 (dashed line).

• neutrino interaction : MA in quasi-elastic scattering and single-meson production• neutrino interaction : ν/ν ratio on CCQE cross-section• neutrino interaction : NC/CC cross-section ratio• detector response : decay electron tagging
Figure 8.2 shows the χ2 -χ2min distribution as a function of sin2 θ23 for the true sin2 θ23 = 0.4
with the 7 dominant systematic errors reduced down to 1/2 and 1/4 of the current estimates. It
is concluded that future higher statistics atmospheric neutrino experiments could improve ourknowledge on θ23 .

170

Chapter 9

Conclusion
The effect of the sub-dominant oscillation derived by the 1-2 parameters on the low energy
atmospheric neutrinos strongly depends on the deviation of sin2 θ23 from 0.5.
The oscillation analysis including the 1-2 oscillation parameters are performed using theatmospheric neutrino data during Super-Kamiokande-I (1489.2 days exposure for FC and PC,
1645.9 days for UPMU) period and Super-Kamiokande-II (798.6 days for FC and PC, 827.7 daysfor UPMU) period. This is the first experimental results that involves two ∆m2 s (∆m212 and∆m223 ).
In order to improve the sensitivity to observe the sub-dominant oscillation effect induced by
the 1-2 parameters, sub-GeV e-like events and sub-GeV µ-like events are separated to CCQEenriched sample and others.
The result of the oscillation analysis without the solar terms gives the best-fit value on sin2 θ23
at 0.50. On the other hand, the best-fit value on sin2 θ23 is 0.53 by the analysis with solar terms.
No significant deviation of θ23 from 45◦ is observed, but the asymmetric shape due to the effectof the solar terms is seen in the χ2 distribution. The stronger constraint on sin2 θ23 is observedin the analysis with the solar terms. The 1σ arrowed region for sin2 θ23 is 0.43 < sin2 θ23 < 0.58,
which is equivalent to sin2 2θ23 > 0.973. This is the more stringent constraint on θ23 comparedwith the published result.

171

Bibliography[1] S.L. Glashow, Nucl. Phys. 22, 579 (1961).
[2] S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967).
[3] A. Salam, In Proceedings of the 8th Nobel Symposium on Elementary particle theory,
relativistic groups and analyticity, edited by N. Svartholm (1969).
[4] Z. Maki, M. Nakagawa, and S. Sakata, Prog. Theo. Phys. 28, 870 (1962).
[5] R. Davis Jr., D.S. Harmer, and K.C. Hoffman, Phys. Rev. Lett. 20, 1205 (1968).
[6] K.S. Hirata et al., Phys. Rev. Lett. 63, 16 (1989);
K.S. Hirata et al., Phys. Rev. D 44, 2241 (1991), Erratum-ibid. D45, 2170 (1992).
[7] A.I. Abazov et al., Phys. Rev. Lett. 67, 3332 (1991);
J.N. Abdurashitov et al., JETP 95, 181 (2002).
[8] P. Anselmann et al., Phys. Lett. B 285, 376 (1992);
W. Hampel et al., Phys. Lett. B 447, 127 (1999).
[9] S. Fukuda et al., Phys. Lett. B 539, 179 (2002);
M.B. Smy et al., Phys. Rev. D 69, 011104 (2004).
[10] Q.R. Ahmad et al., Phys. Rev. Lett. 87, 071301 (2001);
Q.R. Ahmad et al., Phys. Rev. Lett. 89, 011301 (2002);
S.N. Ahmed et al., Phys. Rev. Lett. 92, 181301 (2002).
[11] J.N. Bahcall and M.H. Pinsonneault, Rev. Mod. Phys. 64, 885 (1992).
[12] J. Hosaka et al., Phys. Rev. D 73, 112001 (2006).
[13] M. Maltoni et al., hep-ph/0405172v6, (2007).
[14] M. Honda et al., Phys. Rev. D 70, 043008 (2004).
[15] K.S. Hirata et al., Phys. Lett. B 205, 416 (1988).
[16] K.S. Hirata et al., Phys. Lett. B 280, 146 (1992).
[17] D. Casper et al., Phys. Rev. Lett. 66, 2561 (1991).
[18] R. Becker-Szendy et al., Phys. Rev. D 46, 3720 (1992).
172

[19] M. C. Sanchez et al., Phys. Rev. D 68, 113004 (2003).
[20] Y. Fukuda et al., Phys. Lett. 81, 1562 (1998).
[21] The MACRO collaboration, Phys. Lett. B 434, 451 (1998).
[22] Y.Fukuda et al., Phys. Rev. Lett. 82 2644 (1999).
[23] E. Aliu et al., Phys. Rev. Lett. 94 081802 (2005).
[24] D. G. Michael et al., Phys. Rev. Lett. 97 191801 (2006).
[25] Y. Ashie et al., Phys. Rev. D 71, 112005 (2005).
[26] Y. Ashie et al., Phys. Rev. Lett. 93 101801 (2004).
[27] M. Apollonio et al., Eur. Phys. J. C 27 331 (2003).
[28] O. L. G. Peres and A. Y. Smirnov, Phys. Rev. Lett. B 456 204 (1999).
[29] O. L. G. Peres and A. Y. Smirnov, Nucl. Phys. B 680 479 (2004).
[30] I. Frank and I.Tamm, C. R. Acad. Sci. USSR 14, 109 (1937).
[31] H. Kume et al., Nucl. Inst. and Meth. 205, 443 (1983).
[32] A. Suzuki et al., Nucl. Inst. and Meth. A 329, 299 (1993).
[33] R. Claus et al., Nucl. Inst. and Meth. A 261, 540 (1987).
[34] S. Fukuda et al., Nucl. Inst. Meth. Phys. Res. Sect. A 501, 418 (2003).
[35] Y. Takeuchi et al., Phys. Lett. 452, 418 (1999).
[36] H. Ikeda et al., Nucl. Inst. and Meth. A 320, 310 (1992).
[37] T. Tanimori et al., IEEE Trans. Nucl. Sci. NS-36, 497 (1989).
[38] J. George, Ph.D. Thesis, University of Washington (1998).
[39] T. Shiabata, Master Thesis, Niigata University (2002).
[40] M. Yamada et al., Phys. Rev. D 44 617 (1991).
[41] G.Battistoni et al., Astropart.Phys. 19 269 (2003) [Erratum-ibid. 19 291 (2003)].
(http://www.mi.infn.it/ battist/neutrino.html)[42] G. Barr et al., Phys. Rev. D 70, 0423006 (2004).
[43] T. Sanuki et al., Astrophys. J. 545 1135 (2000).
[44] J.Alcarez et al., Phys. Lett. B 490 27 (2000).
[45] W.R. Webber, R.L. Golden and S.A. Stephens, In Proceedings of the 20th International
Cosmic Ray Conference (1 987).
173

[46] E.S. Seo et al., Astrophys. J. 378 763 (1991).
[47] P. Pappini et al., In Proceedings of the 23rd International Cosmic Ray Conference (1 993).
[48] M. Boezio et al., Astrophys. J. 518 457 (1999).
[49] W. Menn et al., Astrophys. J. 533 281 (2000).
[50] M.J. Ryan, J.F. Ormes and V.K. Balasubrahmanyan, Phys. Rev. Lett. 28 985 (1972).
[51] K. Asakamori et al., Astrophys. J. 502 985 (1998).
[52] I.P. Ivanenko et al., In Proceedings of the 23rd International Cosmic Ray Conference(1993).
[53] Y. Kawamura et al., Phys. Rev. D 40 729 (1989).
[54] A.V. Apanasenko et al., Astropart. Phys. 16 13 (2001).
[55] http://modelweb.gsfc.nasa.gov/atmos/usstandard.html.
[56] K. Hänssget and J. Ranft, Comput. Phys. 39 37 (1986).
[57] S. Roesler, R. Engel, and J. Ranft, Phys. Rev. D 57 2889 (1998).
[58] M.Motoki et al., Astropart.Phys. 19 113 (2003).
[59] P. Hansen et al., Phys.Rev. D 68 103001518 (2003).
[60] M.Honda et al., In Proceedings of the 27th International Cosmic Ray Conferences, H
amburg, Germany, (2001).
[61] G. Battistoni et al., Astopart. Phys. 12, 315 (2000).
[62] Y. Hayato, Nucl. Phys. Proc. Suppl. 112, 171 (2002).
[63] C. H. Llewellyn Smith, Phys. Rep. 3, 261 (1972).
[64] S. K. Singh and E. Oset, Phys. Rev. C 48, 1246 (1993).
[65] M. H. Ahn et al., Phys. Rev. Lett. 90, 041801 (2003).
[66] F.Cavanna et al., Nucl. Phys. Proc. Suppl. 139, 1-328 (2005)[67] R. A. Smith and E. J. Moniz, Nucl. Phys. B 43, 605 (1972). [Erratum-ibid. B 101, 547(1975).][68] C.H. Albright et al., Phys. Rev. D 14, 1780 (1976).
[69] K. Abe et al., Phys. Rev. Lett. 56, 1107 (1986).
[70] S. Barish et al., Phys. Rev. D 16, 3103 (1977).
[71] S. Bonetti et al., Nouvo Cimento 38, 260 (1977).

174

[72] M. Pohl et al., Nuovo Cimento 26, 332 (1979);
N. Arimenise et al., Nucl. Phys. B 152, 365 (1979).
[73] A.S. Vovenko et al., Yad. Fiz. 30, 1014 (1979).
[74] S. Belikov et al., Z. Phys. 320, 625 (1985).
[75] J.Brunner et al., Z. Phys. C 45, 551 (1990).
[76] D. Rein and L.M. Sehgal, Ann. of Phys. 133, 1780 (1981).
[77] R. Feynman et al., Phys. Rev. D 3, 2706 (1971).
[78] G.L. Fogli and G. Nardulli, Nucl. Phys. B 160, 116 (1979).
[79] S.K. Singh, M.J. Vicente-Vacas and E.Oset, Phys. Lett. B 416, 23 (1998).
[80] G. Radecky et al., Phys. Rev. D 25, 116 (1982).
[81] T. Kitagaki et al., Phys. Rev. D 34, 2554 (1986).
[82] P. Allen et al., Nucl. Phys. B 264, 221 (1986).
[83] P. Allen et al., Nucl. Phys. B 176, 269 (1980).
[84] W. Lerche et al., Phys. Lett. 4, 510 (1978).
[85] D. Allisia et al., Z. Phys. C 24, 119 (1984).
[86] S.J. Barish et al., Phys. Lett. B 91, 161 (1980).
[87] C.H. Albright and C. Jarlskog, Nucl Phys. B 84, 467 (1975).
[88] M. Glück, E. Reya and A. Vogt, Z. Phys. D 57, 433 (1995).
[89] P. Musset and J.P. Vialle, Phys. Rep. C 39, 1 (1978).
[90] J.E. Kim et al., Rev. Mod. Phys. 53, 211 (1981).
[91] S.J. Barish et al., Phys. Rev. D17, 1 (1978).
[92] S. Barlag et al., Z. Phys. C 11, 283 (1982).
[93] T. Sjöstrand et al., CERN-TH-7112-93 (1994).
[94] D. Rein and L.M. Sehgal, Nucl. Phys. B 233, 29 (1983).
[95] A. Kartavtsev, E. A. Paschos, and G. J. Gounaris, Phys. Rev. D 74, 054007 (2006).
[96] M. Hasegawa et al., Phys. Rev. Lett. 95, 252301 (2005).
[97] F. Bergsma et al., Phys. Lett. B 157, 469 (1985).
[98] J. L. Raaf, Ph.D. thesis, University of Cincinnati (2005).
[99] H. Faissner et al., Phys. Lett. B 125, 230 (1983).
175

[100] E. Isiksal, D. Rein, and J. G. Morfin, Phys. Rev. Lett. 52, 1096 (1984).
[101] R. Woods and D. Saxon, Phys. Rev. 95, 577 (1954).
[102] L. Salcedo et al., Nucl. Phys. A 484, 557 (1988).
[103] G. Rowe et al., Phys. Rev. C 18, 584 (1978).
[104] D. Ashery et al., Phys. Rev. C 23, 2173 (1981).
[105] C. Ingram et al., Phys. Rev. C 27, 1578 (1983).
[106] B. R. Martin and M. K. Pidcock, Nuc. Phys. B 126, 266 (1977).
[107] B. R. Martin and M. K. Pidcock, Nuc. Phys. B 126, 285 (1977).
[108] J. S. Hyslop et al., Phys. Rev. D 46, 961 (1992).
[109] D. A. Sparrow, Proc. of the Conf. on the intersection between partivle and nuclear physics,
1019 (1984).
[110] H. W. Bertini et al., Phys. Rev. C 6, 631 (1972).
[111] S. J. Lindenbaum et al., Phys. Rev. 105, 1874 (1957).
[112] L.V. Volkova, Sov. J. Nucl. Phys. 31, 784 (1980).
[113] GEANT, CERN Program Library Long Writeup W5013 (1994).
[114] C. Zeitnitz and T.A. Gabriel, Nucl. Inst. and Meth. A 349, 106 (1994).
[115] M. Nakahata et al., J. Phys. Soc. Jpn. 55, 3786 (1986).
[116] E. Bracci et al., CERN/HERA 72-1 (1972).
[117] A.S. Carrol et al., Phys. Rev. C 14, 635 (1976).
[118] S. Desai, Ph.D. Thesis, Boston University.
[119] E. R. Davies, Machine Vision: Theory, Algorithms, Practicalities, Academic Press, San
Diego (1997).
[120] S. Kasuga et al., Phys. Lett. B 374, 238 (1996).
[121] T. Barszczak, Ph.D. Thesis, University of California, Irvine (2005).
[122] J. A. Nelder and R. Mead, Computer Journal 7, 308 (1965).
[123] S. Desai et al., Astopart. Phys. 29, 42 (2008).
[124] D. Casper, Nucl. Phys. Proc. Suppl. 112, 161 (2002).
[125] J. Nieves et al., Phys. Rev. C 70, 055503 (2004).
[126] C.H. Albright and C. Jarlskog, Nucl. Phys. B 84, 467 (1975).
176

[127] A. Bodek and U. K. Yang, Nucl. Phys. Proc. Suppl. 112, 70 (2002)[128] K. Hagiwara et al., Nucl. Phys. B 668, 364 (2003).
[129] Y. Fukuda et al., Phys. Lett. B 433, 9 (1998).
[130] G.L. Fogli et al., Phys. Rev. D 66, 053010 (2002).
[131] S.Baker and R.D. Cousins, Nucl. Inst. and Meth. 221, 437 (1984).

177