Study of Non-Standard Neutrino Interactions with Atmospheric
Neutrino Data in Super-Kamiokande

Gaku Mitsuka
February 9, 2009

Abstract
The atmospheric neutrino data show the apparent deficit in upward-going ν µ events, which
was called as the atmospheric neutrino problem. They have been well explained by the 2-flavorνµ ↔ ντ oscillation scheme and also the ∆m223 and sin2 2θ23 parameters have been measured.
Neutrino oscillations in atmospheric neutrinos play a very important role in particle physics
because they present the clear evidence for physics beyond the standard model.
In addition to neutrino oscillations, a lot of attempts have been proposed to account for theatmospheric neutrino problem. Considering that atmospheric neutrino data are well fitted byneutrino oscillations, the contribution of the other models would be at most sub-dominant. In
this thesis, we focus on the additional neutrino interactions with matter, which are forbidden in
the standard model. The phenomena driven by such interactions in the Earth are assumed to
be dependent on the matter density and independent of the energy of neutrinos. We call themnon-standard neutrino interactions.
Our goals are, first, to measure the dominance of standard neutrino oscillations comparedto the possible presence of non-standard interactions, and next, to search for the signal of nonstandard interactions using large number of atmospheric neutrino data in Super-Kamiokande.
The analysis is performed with the two schemes: (1) Standard 2-flavor νµ ↔ ντ oscillation with
non-standard neutrino interactions in the νµ − ντ sector, and (2) Standard 2-flavor νµ ↔ ντ
oscillation with non-standard neutrino interactions in the νe − ντ sector. Scheme (1) assume
that the flavor transition is only occurred between νµ and ντ , whereas νe flux is kept as is. To
the contrary, scheme (2) allows the flavor transition with all flavored neutrinos. The effects
of non-standard neutrino interactions have never been evaluated simultaneously with neutrino
oscillations in the neutrino oscillation experiments. Then this analysis is the first experimental
approach to investigate their effects.
Our approach requires more accurate analysis than that with only neutrino oscillations, since
our interests would exist as sub-dominant effects. Not only improved atmospheric neutrino flux
model and neutrino interaction theories are employed in the simulation, but also the absoluteenergy scale is precisely examined using various calibration sources. Furthermore, the systematic
uncertainties are carefully inspected to investigate the possible signal of non-standard neutrino
interactions.
As a result of the analyses with the atmospheric neutrino data from the Super-Kamiokande-I(1996-2001) and Super-Kamiokande-II (2003-2005), no significant signal of non-standard neutrino interactions has been observed and the limits on parameters for non-standard neutrino
interactions have been obtained.

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. S.Moriyama and Prof. Y.Hayato. They were willingto give 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, Prof.
K.Kaneyuki, Prof. Y.Itow, Prof. M.Shiozawa, M.Miura, Y.Obayashi, K.Okumura, J.Kameda,
C.Saji, S.Nakayama, M.Ishitsuka, I.Higuchi, N.Tanimoto, Y.Shimizu, Y.Kaji, Y.Takenaga, H.Nishino,
C.Ishihara, S.Hazama, E.Kearns, H.W.Sobel, R.J.Wilkes, J.G.Learned, C.W.Walter, K.Scholberg,
S.Mine, J.L.Raaf, W.Wang, J.S.Jang, H.K.Seo, T.Kato, R.Terri, E.Thrane, R.Wendell, C.Regis,
F.Dufour, and M.Litos. Especially, this study could not achieved without the collaborationof the neutrino oscillation analysis group, E.Kearns, C.W.Walter, R.Wendell, F.Dufour, andC.Ishihara.
I would like to extend my gratitude to ICRR staffs and members, Prof. M.Nakahata, Prof.
Y.Takeuchi, Prof. M.Yamashita, Y.Koshio, K.Abe, A.Takeda, H.Sekiya, S.Yamada, H.Ogawa,
K.Kobayashi, H.Watanabe, T.Iida, M.Ikeda, K.Ueshima, K.Ueno, Y.Nakajima, Y.Kouzuma,
and T.Yokozawa.
I would also like to thank the SciBooNE collaboration, Prof. T.Nakaya, M.Yokoyama,
H.Tanaka, K.Hiraide, Y.Nakajima, Y.Kurimoto, and H.Takei, for their advice and friendship.
Many thanks to A.Minamino and M.Valverde.
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 Atmospheric Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Neutrino Oscillations in Atmospheric Neutrinos . . . . . . . . . . . . . . . . . . .
1.3 Non-Standard Interactions in Atmospheric Neutrinos . . . . . . . . . . . . . . . .
1.4 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
2
4
7

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10
10
12
12
14
14
18
18
21
21
22
22

3 Simulation of Atmospheric Neutrino Events
3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Atmospheric Neutrino Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Neutrino Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Elastic and Quasi-Elastic Scattering . . . . . . . . . . . . . . . . . . . . .
3.3.2 Single Meson Production . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Deep Inelastic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.4 Coherent Pion Production . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.5 Nuclear Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Detector Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Particle Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Cherenkov photon tracking in water . . . . . . . . . . . . . . . . . . . . .
3.4.3 Response of the PMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Upward-Going Muon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26
26
26
33
33
35
40
41
42
45
45
45
46
46

i

4 Calibration
4.1 Relative Gain Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Absolute Gain Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Relative Timing Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Water Transparency Measurement . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Light scattering measurement using a laser . . . . . . . . . . . . . . . . .
4.4.2 Measurement using cosmic ray muons . . . . . . . . . . . . . . . . . . . .
4.5 Absolute Energy Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.1 Decay electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.2 Neutrino induced π 0 events . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.3 Low energy stopping muons . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.4 High energy stopping muons . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.5 Time variation of energy scale . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.6 Uniformity of Energy Scale . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.7 Summary of the absolute energy calibration . . . . . . . . . . . . . . . . .

50
50
50
52
53
53
55
58
58
60
60
62
66
66
66

5 Data Reduction
and Event Reconstruction
5.1 Data Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Event Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70
70
72

6 Data set
6.1 Event Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Vertex Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Number of Ring and Momentum Distribution . . . . . . . . . . . . . . . . . . . .
6.4 Zenith angle Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76
76
79
79
79

7 Analysis Method and Systematic Uncertainties
93
7.1 Definition of χ2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7.2 Analysis Binning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.3 Systematic Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7.3.1 Atmospheric Neutrino Flux . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.3.2 Neutrino Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.3.3 Event Selections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.3.4 Event Reconstructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
8 Analysis with 2-Flavor Hybrid Model
103
8.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
8.2 Expected Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
8.3 Analysis Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
8.4 Result of 2-Flavor Non-Standard Interaction Analysis . . . . . . . . . . . . . . . 109
8.4.1 Limit on NSI parameters without relative phase . . . . . . . . . . . . . . 110
8.4.2 Limit on NSI parameters with relative phase . . . . . . . . . . . . . . . . 110
8.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113ii

9 Analysis with 3-Flavor Hybrid Model
125
9.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
9.2 Expected Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
9.3 Analysis Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
9.4 Result of 3-Flavor Non standard interaction analysis . . . . . . . . . . . . . . . . 130
9.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
9.6 Results with the uncertainty of matter density and sub-dominant effects . . . . . 135
9.6.1 Effect of the uncertainty of matter density profile . . . . . . . . . . . . . . 135
9.6.2 Effect of nonzero θ13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
9.6.3 Effect of nonzero ∆m212 and θ12 . . . . . . . . . . . . . . . . . . . . . . . . 139
10 Discussions
150
10.1 Deep inelastic scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
10.2 CHARM experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
10.3 NuTeV experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
10.4 Comparison with the existing results . . . . . . . . . . . . . . . . . . . . . . . . . 153
11 Conclusions

155

A Systematic Uncertainties
164
A.0.1 Atmospheric Neutrino Flux . . . . . . . . . . . . . . . . . . . . . . . . . . 164
A.0.2 Neutrino Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
A.0.3 Event Selections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
A.0.4 Event Reconstructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
B Atmospheric Neutrino Analysis
171
B.1 2-Flavor νµ ↔ ντ Oscillation Analysis . . . . . . . . . . . . . . . . . . . . . . . . 171
C Data Reduction
173
C.1 Reduction for Fully Contained Sample . . . . . . . . . . . . . . . . . . . . . . . . 173
C.1.1 First Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
C.1.2 Second Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
C.1.3 Third Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
C.1.4 Fourth Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
C.1.5 Fifth Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
C.1.6 FC Reduction Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
C.2 Reduction for Partially Contained Sample . . . . . . . . . . . . . . . . . . . . . . 190
C.2.1 First Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
C.2.2 Second Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
C.2.3 Third Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
C.2.4 Fourth Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
C.2.5 Fifth Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
C.2.6 PC Reduction Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
C.3 Reduction for Upward-Going Muon Sample . . . . . . . . . . . . . . . . . . . . . 202
C.3.1 Charge cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
C.3.2 Zenith angle cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
C.3.3 Scanning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203iii

C.3.4 Efficiency and Background for Upward-Going Muon Sample . . . . . . . . 203
D Event Reconstruction
207
D.1 Vertex Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
D.1.1 Point-Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
D.1.2 Ring Edge Finding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
D.1.3 TDC-Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
D.1.4 Performance of TDC-fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
D.2 Ring Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
D.2.1 Ring Candidate Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
D.2.2 Ring Candidate Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
D.3 Particle Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
D.3.1 Estimation of the Particle Type . . . . . . . . . . . . . . . . . . . . . . . . 214
D.3.2 The Expected p.e. Distribution for Electrons . . . . . . . . . . . . . . . . 216
D.3.3 The Expected p.e. Distribution for Muons . . . . . . . . . . . . . . . . . . 216
D.3.4 The Expected p.e. Distribution for Scattered Light . . . . . . . . . . . . . 217
D.3.5 Performance of Particle Identification . . . . . . . . . . . . . . . . . . . . 218
D.4 Precise Vertex Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
D.5 Decay Electron Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
D.6 Momentum Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
D.7 Ring Number Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
D.8 π 0 Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
D.9 Event Reconstruction for Upward-Going Muon Sample . . . . . . . . . . . . . . . 227

iv

Chapter 1

Introduction
This thesis aims to evaluate the dominance of the solution with pure neutrino oscillationsin the atmospheric neutrino data against the possible presence of non-standard neutrino interactions. Also we purpose to investigate the signal of non-standard neutrino interactions. First
we explain the feature of atmospheric neutrinos which work as the probe in this thesis. In
Chapter 1.2 neutrino oscillations in atmospheric neutrinos are discussed. Non-standard neutrino interactions are introduced in Chapter 1.3. At the end of this chapter, our motivations
and goals are presented.

1.1

Atmospheric Neutrinos

When primary cosmic rays interact with nuclei in the atmosphere of the Earth, secondary
particles, mostly pions and some kaons, are produced in hadronic showers. Atmospheric neutrinos are produced from the decay of those secondary particles, dominantly by the following
decay chains of pions :π + → µ+ + νµπ−→ e+ + νe + ν µ(1.1)→ e− + ν e + νµ

(1.2)

−

→ µ + νµ

Two muon neutrinos and one electron neutrino are generated by the decay of a charged
pion. Therefore, the flux ratio (νµ + ν µ )/(νe + ν e ) is expected to be two for relatively low energy
neutrinos. For high energy neutrinos above a few GeV, this flux ratio becomes greater than
two because high energy muons reach the ground without decay and the number of electron
neutrinos decreases. In Figure 1.1, the left-hand figure shows the flux ratio, (ν µ + ν µ )/(νe + ν e ),as a function of the neutrino energy, which is calculated by M. Honda et al. [1, 2, 3].
Since the primary cosmic rays isotropically pour on the Earth and the Earth has a sphericalform, the atmospheric neutrino flux is expected to be up-down symmetric. In Figure 1.1, two
figures on the right panel show the neutrino flux as a function of the zenith angle of neutrino
arrival direction where cos Θ = −1 (cos Θ = +1) indicates upward-going (downward-going)
directions. The flux of low energy neutrinos is affected by the rigidity (=momentum/charge)
1

10

(a)

3

3

5

-1

4

-1

6

–

νµ+νµ

(b-2)

-1

Flux×Eν(m sec sr GeV )

10

–

νe+νe

(b-1)

0.3-0.5 GeV

102

0.3-0.5 GeV 10 2

0.9-1.5 GeV

0.9-1.5 GeV

-2

–

–

Flux ratio (νµ+νµ)/(νe+νe)

10
9
8
7

3

10

2

10

-1

1

10

10

2

3.0-5.0 GeV

1Eν (GeV)

3.0-5.0 GeV

10

-1

-0.5

0

0.5

1

1

-1

-0.5

cosθ

0

0.5

1

cosθ

Figure 1.1: Flux ratio of νµ + ν µ to νe + ν e as a function of the neutrino energy (a), and thezenith angle dependence of atmospheric neutrino for νe + ν e (b-1) and νµ + ν µ (b-2). The flux
is calculated by M. Honda et al. [1, 2, 3].

cutoff of the primary cosmic rays due to the geomagnetic field. Since the geomagnetic fieldaround Super-Kamiokande is stronger than the average, the up-down symmetry is broken in low
energy neutrinos. For energies above a few GeV, the geomagnetic effect is negligible. The flight
length of atmospheric neutrinos ranges from ∼ 15 km to ∼ 13000 km depending on the zenith
angle of the arrival direction.
The detailed explanation about the expectation of the atmospheric neutrino flux is describedin Chapter 3.2.

1.2

Neutrino Oscillations in Atmospheric Neutrinos

The expected total flux of atmospheric neutrinos has ∼ 10 % uncertainty which mostly
comes from the uncertainties on cross sections of hadronic interactions and on fluxes of primarycosmic ray components [2]. However, the flux ratio (νµ + ν µ )/(νe + ν e ) can be predicted moreaccurately with only 3 % uncertainty thanks to the cancellation of each error. This ratio has
been measured by several underground experiments and reported in the style of a double ratioR ≡ (Nµ /Ne )Data /(Nµ /Ne )M C , where Nµ (Ne ) is the number of muon (electron) events producedby charged current interactions.
The double ratio R obtained by underground Cherenkov detectors, Kamiokande [4, 5, 6]
and IMB [7, 8] and an iron calorimeter detector Soudan 2 [9] was significantly lower than 1.
This was called as the atmospheric neutrino problem. Also it was known that the smallness
of R was caused by the observed zenith angle dependent deficit of νµ and ν µ . MACRO [10],Kamiokande [12] and Super-Kamiokande [11] have studied the upward-going muons produced
by energetic νµ charged current neutrino interactions in the rock surrounding the detector andtheir data are consistent with the 2-flavor νµ ↔ ντ oscillation.
2

Neutrino oscillation hypothesis is described as follows: In the standard model of elementary
particles, neutrinos are regarded as exactly massless particles [13, 14, 15]. However, there isno fundamental reason to forbid the finite masses of neutrinos. Neutrino oscillation has been
proposed by Maki, Nakagawa and Sakata in 1962 as a consequence of the finite masses ofneutrinos [16].
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,(1.3)

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 θ 12 , θ23 and θ13
for each flavor and a CP-violating phase δ as follows :




c12 s12 0c13
0 s13 e−iδ
1
0
0




(1.4)
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
(1.5)
i

XX

=

i

Uαi Uα∗0 i exp(−iEi t)|να0 (t = 0)i,

(1.6)

α0

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

X

(1.7)
2

∗
Uβi Uαiexp(−iEi t)(1.8)|Uαi Uβi |2 +

(1.9)

i

=

X
i

X
i6=j

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

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 θU=.
(1.10)− sin θ cos θ
The transition probability for να → νβ (eq.(1.9)) is expressed in a simpler form :22 (Ei − Ej )tP( να → νβ ) = sin 2θ sin.
2
3(1.11)

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

∆m2ij
,
2E

(1.12)

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 :!
∆m2ij Lν2
2
P (να → να ) = 1 − sin 2θ sin(1.13)
4Eν
!
1.27∆m2ij (eV2 )Lν (km)22
= 1 − sin 2θ sin(1.14)
Eν (GeV)Lν22= 1 − sin 2θ sin π,(1.15)lwhere l ≡ 4πEν /∆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 ].
Survival probability as a function of flight length of neutrino is demonstrated in Figure 1.2,
where sin2 2θ = 1.0 and ∆m2 = 2.2 × 10−3 eV−2 are assumed.
In 1998, Super-Kamiokande reported that the zenith angle distributions of muon neutrino
events were asymmetric and concluded that their measurement results gave an evidence forneutrino oscillation [17]. Super-Kamiokande also presented the observation of an oscillation signature with L/E dependence [18], which is well described by the hypothesis that νµ oscillates toντ with a probability of the nearly maximal mixing. Since typical atmospheric neutrino experiments have the neutrino energies above several hundred MeV, the observation of atmospheric
neutrinos is sensitive to ∆m2 down to 10−4 eV2 according to the relation between Lν ,Eν and∆m2 . Figure 1.3 summarize the recent measurements of the neutrino oscillation parameters,
sin2 2θ and ∆m223 , by the atmospheric and accelerator neutrino experiments.

1.3

Non-Standard Interactions in Atmospheric Neutrinos

In addition to neutrino oscillations, a lot of attempts have been proposed to explain theatmospheric neutrino problem [22]. Among many alternative models, we focus on the additional
neutrino interactions where the neutrinos take part in the flavor changing and non universal
processes in matter. Flavor changing neutral current (FCNC) represents the neutrino interactions with fermion f in matter, να + f → νβ + f . Figure 1.4 shows a possible diagram for
FCNC off d-quark. Lepton non universality (NU) is defined as the difference in the amplitudeof the neutral current scattering between να and νβ , which are exactly same under the standard
model. These interactions are forbidden in the standard model, however the theories concerning4

∆ m223 [10−3eV2]

Figure 1.2: Survival probability of να → να as a function of flight length. Horizontal axis is the
propagation length log10 (Lν (km)). (Black curve) Eν =1GeV. (Red curve) Eν =10GeV. (Green
curve) Eν =100GeV. sin2 2θ = 1.0 and ∆m2 = 2.2 × 10−3 eV2 are assumed.

4

3

2

0.5

0.6

0.7

0.8

0.9

1

sin22θ23

Figure 1.3: 90 % confidence level allowed regions of oscillation parameters for ν µ ↔ ντ oscillation.
The horizontal axis shows sin2 2θ23 and the vertical axis shows ∆m223 . The results are taken fromSuper-Kamiokande[19] (solid curve), SK L/E [19] (dashed curve), K2K experiment [20] (dottedcurve) and MINOS experiment[21] (dash-dotted curve).

5

Figure 1.4:

about neutrino masses predict them, for example the case in which neutrino masses follow fromthe admixture of isosinglet neutral heavy leptons [23]. For simplicity, we call these interactions
non-standard neutrino interactions (NSI) in the following text. Although NSI is predicted bythese theories, the expected phenomena driven by NSI itself do not depend on any model except
for the assumption that NSI is dependent on the density of matter, while is independent of theneutrino energy.
In the standard model, the weak interaction Hamiltonian is described as
GF 
HW = √ Jµ† Jµ + ρJµN C JµN C(1.16)
2
where Jµ and JµN C are the weak current. The part of eq.(1.16) relevant to να + f scattering isGF  ¯√ f γµ (1 − γ5 )να [ν̄α γ µ (1 − γ5 )f ]
2


√


1
2GF ρ ν̄α γµ (1 − γ5 )να f¯γ µ (gV − gA γ5 )f
+
2

eff
HW
=

(1.17)

where gV = gL + gR = −1/2 + 2 sin2 θW , gA = gL − gR = −1/2 and ρ = 1. L, R are theleft-handed and right-handed fermion, respectively. θW is the Weinberg angle.
After the Fierz transformation of the first term,

where



GF
eff
HW
= √ [ν̄α γµ (1 − γ5 )να ] f¯γ µ (cV − cA γ5 )f2α=fcV = g V + 1
cA = g A + 1(1.18)

α 6= fgV

gA(1.19)

When neutrinos propagate through matter, they are refracted due to the interaction withmatter. In it, neutrinos receive the extra energy V . Using eq.(1.18), V for the fermion f is givenas√
VSM = 2GF cV Nf(1.20)
with the fermion density in matter Nf . Suppose the interaction in α 6= f ,√√√VSM = 2GF cV Nf = 2GF gV Nf = 2GF (gL (f ) + gR (f ))Nf
6

(1.21)

Now, we extend eq.(1.21) to the case in NSI. In this case, V is replaced with√fR
VN SI = 2GF (εfαβL + εαβ)Nf(1.22)

where εfαβP (P = L, R) is the parameter to describe the strength of NSI. Note that NSI allow
the flavor transition through the neutral current interaction such as ν α + f → νβ + f . As seenin eq.(1.22), NSI coupling parameter εfαβP corresponds to the neutral current coupling gP (f ) of
the standard model. We cannot distinguish NSI to the left-handed and right-handed fermion inatmospheric neutrinos, so εfαβP is replaced with εfαβ ≡ εfαβL + εfαβR . Also, since we assume fermion
f as d-quark in matter, hereafter we remove the superscript f (= d) and use ε αβ for simplicity.
Thus the effective potential induced by 2-flavor NSI in the να − νβ sector can be written as
H(r) = H0 + Veff
!
√
p 0
=
+ 2GF Nf (~r)
0 p(1.23)
εαα ε∗αβεαβ εββ!

(1.24)

where p is the neutrino momentum, GF is the Fermi coupling constant, εββ − εαα represents
the difference in the amplitude of NC interaction between να and νβ , and εαβ represents theamplitude of FCNC. Nf (~r) is the fermion (d-quark) density as a function of the position ~r along
the neutrino path. The solution of the Schrödinger equation is written as: Z L
H(r)dr |ψ(0)i(1.25)|ψ(t)i = exp −i0where L is the flight length of neutrino. The integration is carried out along the neutrino path.
The survival probability of να is written as
P (να → να ) = 1 −ε2αβεαβ 2 + (εββ − εαα )2 /4

sin

2



q
√
2GF Xf εαβ 2 + (εββ − εαα )2 /4

(1.26)

where Xf is the column density of the fermions f along the neutrino path
Xf =

Z L

Nf (~r)dr(1.27)

0

We obtain the oscillating form of neutrino survival probability, however the survival probability is a function of the column density of the fermions, and has no dependence on the neutrinoenergy. Survival probability να → να as a function of propagation length is presented in Figure 1.5, where the matter density profile in the Earth is taken from the PREM model [24].

1.4

Motivations

NSI in the matter of the Earth gives the flavor transitions which possibly account for theatmospheric neutrino problem. However, the solution with pure NSI was strongly ruled out
because of the inconsistency between data and expectation in the high energy region [25]. Thatis, the observed high energy data well explained by neutrino oscillations show the small flavor
7

Figure 1.5: Survival probability of να → να as a function of the propagation length in the matterof the Earth. Horizontal axis is the propagation length log10 (Lν (km)). (Black curve) εαβ =1.0
and εββ − εαα =0.01. (Red curve) εαβ =1.0 and εββ − εαα =1.0. (Green curve) εαβ =10.0 and
εββ − εαα =0.1. Nf (~r) is replaced with the average matter density along the path of neutrinohNf (~r)i to simplify the calculation.
transition due the condition ∆m2 L/4Eν  1, whereas NSI can give rise to the sizable transition
wherever the fermion column density is sufficient.
Nevertheless, this result only states that the atmospheric neutrino problem cannot be explained by the pure NSI and NSI is most unlikely the dominant component of the characteristicsof the problem. In other words, NSI can coexist with neutrino oscillations although its effect is
at most sub-dominant.
Presence of NSI simultaneously with neutrino oscillations can be the test how neutrino
oscillations are robust to the atmospheric neutrino data by comparing the allowed neutrino
oscillation parameters between two cases, pure neutrino oscillations and the mixed with NSI (wecall it the hybrid model). Furthermore, suppose neutrino oscillations are stable enough or thetrue solution of the problem, neutrino oscillations capably act as the probe of NSI by measuring
the deviation of the hybrid model from pure neutrino oscillations. The situation where NSI andneutrino oscillations coexist simultaneously has never been tested by the neutrino oscillation
experiments so far, thus this thesis will give the first experimental results.
Propagation of three flavored neutrinos ν(t) in the hybrid model are derived from the solutionof the Schrödinger equation,d(1.28)i |ν(t)i = H|ν(0)idt
with the effective hamiltonian H for neutrinos (+) and anti-neutrinos (−)
εee ε∗eµ ε∗eτ
0
0
0
√
1




H=
U 0 ∆m221
0  U † + VMSW ± 2GF Nf (~r) εeµ εµµ ε∗µτ 2Eν
εeτ εµτ ετ τ
0
0
∆m231
8

(1.29)

where the first and second term are given by neutrino oscillations and the third term by NSI.
For convenience, the first and second term are called as the vacuum term and matter term,
respectively. Under the standard model, only νe receives an extra contribution from the electron
in matter. This effect is called as the MSW mechanism [26, 27, 28] and denoted as V MSW .
As an example, we briefly review the expected phenomena of the most simple case in which
2-flavor neutrino oscillation and NSI in the νµ − ντ sector are taken into account simultaneously.
In this case, the effective hamiltonian can be reduced to
!
!
∗
√
ε
ε
0
0
1
µµ
µτ
U
U † ± 2GF Nf (~r)
H=
(1.30)2Eν
0 ∆m232εµτ ετ τ
where VMSW is not appeared since the reduced H decouples to νe .
In the matter of the Earth, a contribution to the neutrino propagation can be known by thesize of their eigenvalues derived from the diagonalization of eq.(1.30), where the eigenvalues vary
according to the neutrino energy. Assuming all of εαβ have a equivalent size to the standard
model (εαβ ∼ 1), the vacuum term dominates below 1 GeV, while the matter term plays theleading role above a few GeV. In the energy range where the matter term has the dominant contribution, deficit of νµ events is enhanced or decreased and hence the zenith angle distributions
are changed due to nonzero εαβ .
Considering these expected phenomena, the features of atmospheric neutrinos advantageously work in the tests: (i) They have the wide energy range from a few hundred MeV to ahundred TeV, (ii) neutrinos incoming from the upward direction propagate in the dense matterof the Earth, and (iii) almost full mixing between νµ and ντ . The advantage of (i) is understoodby recalling that the inconsistency between NSI and neutrino oscillations is appeared in the highenergy events while NSI shows better agreement with the low energy events. (ii) enhances theamplitude of NSI compared to neutrino oscillations, and derive the conspicuousnes of NSI. (iii)
helps us to obtain the natural ντ beam oscillated from the atmospheric νµ flux, where we notethat NSI couplings to ντ , i.e. εατ , have been poorly constrained due to the lack of abundant ντ
beam.
Finally, let us summarize the motivation of this study:(1) Test of the robustness of neutrino oscillations to the atmospheric neutrino data.
(2) Investigation of the possible presence of NSI. If we would not able to obtain the signal of
NSI, we derive the constraints to them.
This study is, in a sense, equivalent to measure the deviation between atmospheric neutrino data and the solution with pure neutrino oscillations, then a highly accurate analysis isrequired. In order to achieve the study, not only improved atmospheric neutrino flux modeland neutrino interaction theories are employed in the simulation, the absolute energy scale isadjusted accurately, but also the systematic uncertainties are carefully inspected. We emphasize that the analysis assuming the existence of NSI is the first attempt among the neutrino
oscillation experiments so far.

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 K2K experiment and the forthcoming T2K experiment [29]. The
Super-Kamiokande experiment started taking data in April, 1996 and continued the observation
for five years within the running period referred to SK-I till the detector 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 Super-Kamiokande detector was
rebuilt after the accident with the half of the original PMT density in the inner detector andresumed 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 thereconstruction 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

10

(2.1)

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.

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 determinedby the refractive index of the medium and the mass of the particle. Since the refractive index
of water is about 1.34, the momentum thresholds of Cherenkov radiation for electrons, muonsand charged pions are 0.57, 118 and 156 MeV/c, respectively. As for proton, because of its large
mass the momentum thresholds is about 1.07GeV/c.
Cherenkov light is emitted on a cone with a characteristic half opening angle θ C along thedirection of the particle. The opening angle, called Cherenkov angle, is determined as follows :cos θC =

1
,
nβ

(2.2)

where β = v/c. For the particle with β ' 1 in water, the Cherenkov angle is about 42 ◦ .
The number of photons emitted by Cherenkov radiation is given as a function of the wavelength λ as follows :


d2 N
2πα
1
= 2 1− 2 2 ,
(2.3)
dxdλ
λ
n β
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/m 3 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/m 3 .
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(0.3 m)AIR DRIERCARBONCOLUMNHEATEXCHANGER

AIR FILTER(0.01 m)AIR FILTER(0.1 m)

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

START/STOP

QAC-A
TAC-B

QAC-A

TAC-B

to ADCCLEAR
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

interface

Super Memory Partner

VME
VME

interrupt reg.
TRG

20-inch PMT
SCH
TRIGGERATM

TRIGGER

x 20
x 240

PROCESSORHIT INFORMATION
ATM
GONG

Analog Timing ModuleTKO

PMT x 11200

ATM x ~1000

SMP x 48

online CPU(slave) x 9

Figure 2.14: 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.

24Analog signal
from PMT
ATM threshold

ADC gate
TDC startand stop
HITSUM

400 nsec width

stop

start

200 nsec width

Global triggerATM
Sum of HITSUM
Master threshold
Global trigger
CentralHut

Figure 2.16: An overview of the ID trigger scheme.

25Chapter 3

Simulation of Atmospheric Neutrino
Events3.1

Overview

Atmospheric neutrino events in Super-Kamiokande are simulated by a Monte Carlo method,in which all of the important factors are accurately simulated. Monte Carlo simulation consistsof three constituents: neutrino flux, neutrino interactions and particle tracking in the detector.
First, we expect number of events occured in the detector during a certain period by means of
the products of neutrino flux and cross sections. The kinematics of the neutrino interactions are
simulated following the atmospheric neutrino energy spectrum, and then the generated particles
are tracked in water with the detector simulation program.
Neutrino oscillations and the other physical features of atmospheric neutrinos are studied bycomparing the observed data with the prediction by the Monte Carlo simulation.

3.2

Atmospheric Neutrino Flux

There have been several models for atmospheric neutrino flux, which are calculated by M.
Honda et al. [1, 2, 3] (Honda flux), G. Battistoni et al. [39] (Fluka flux) and G. Barr et al. [40](Bartol flux). In our Monte Carlo simulation program, the Honda flux is adopted as a default
model and other two flux calculations are used to estimate the systematic uncertainties in theflux calculation.The way of estimating systematic uncertainlies are discussed in Chapter 7.3.1.
The energy range supported by the Honda flux is up to 10 TeV, so the Volkova flux [41] is adopted
to simulate the event above 10 TeV. In order to connect the Volkova flux with the Honda flux
smoothly at 10 TeV, we adjust the absolute normalization of the Volkova flux.
The primary cosmic ray flux model as an input of the neutrino flux calculation is determinedbased on the experimental measurements. Current status of the measurements of cosmic ray
proton flux is shown in Figure 3.1 with the model used in the Honda flux calculation. The
primary cosmic ray spectrum has been precisely measured by BESS and AMS experiments up
to 100 GeV [42, 43]. The cosmic ray flux changes depending on the turbulence of the solar
wind, which is higher when the solar activities are high (solar maximum) than when the solar
activities are low (solar minimum). The difference of the cosmic ray flux at solar maximum andsolar minimum is more than a factor of two for 1 GeV cosmic rays, while it decrease to ∼ 10 %26

for 10 GeV. The effect of geomagnetic field is represented as the rigidity (= momentum/charge)
cutoff. The geomagnetic field works as a shield, by which cosmic ray with lower momentum thanthe cutoff cannot arrive at the Earth. Cosmic ray flux above 100 GeV, which are responsible for≥ 10 GeV neutrinos, is not affected by the solar activity nor the geomagnetic field.

Figure 3.1: Measurements of primary cosmic ray proton flux and the model used in Honda
flux calculation. The data are taken from Webber [44] (crosses), LEAP [45] (upward triangles),MASS1 [46] (open circles), CAPRICE [47] (vertical diamonds), IMAX [48] (downward triangles),
BESS98 [42] (circles), AMS [43] (squares), Ryan [49] (horizontal diamonds), JACEE [50] (downward open triangles), Ivanenko [51] (upward open triangles), Kawamura [52] (open squares) andRunjob [53] (open diamonds).
In the atmosphere, primary cosmic ray protons and nuclei interact with air nuclei, andsecondary particles, mostly pions and some kaons, are consequently generated. In the Honda flux
the US Standard Atmosphere model [54] is employed for the density structure of the atmosphereand the structure gives the zenith angle dependence of the atmospheric neutrinos. Geomagnetic
filed effects are calculated based on the IGRF2005 model [55].
In the hadronic interactions of the cosmic rays with air nuclei, the two theoretical models:
NUCRIN [56] simulation program is used for the primary cosmic ray energies < 5 GeV andDPMJET-III [57] is used for > 5 GeV. Mesons, mostly pions and some kaons, generated throughthe hadronic interactions create atmospheric neutrinos when they decay in the atmosphere.
Likewise, secondary cosmic ray muons are generated through the decay of mesons. The
flux of the secondary cosmic ray muons have been measured by several experiments such as
BESS [58] or L3+C [59]. These observations capably work for the calibration of the hadronicinteraction model by comparing the momentum spectrum with the expectation. DPMJET-III
employed in the Honda flux is modified using the measurements by BESS and L3+C as follows:
first, tunable parameters are assigned to a valence quark and the average energy of produced
meson, for example π + , is expressed as
hEπ+ i = (1 + cu )hEπ0+ i(3.1)

where Eπ+ is the modified energy of π + , Eπ0+ is the original energy, and cu is the tunable27

parameter for u-quark. Produced mesons, π ± , π 0 , K + , etc, are connected through such tunable
parameters. Second, determine the best-fit tunable parameters by adjusting the predicted muon
spectrum to be consistent with the observations. After that, nucleon spectrum is derived tokeep the energy conservation. Figure 3.2 shows the ratio of cosmic ray muon spectrum after themodification, and results in that the hadronic interaction model agrees with the measurement
within .10% in the 1∼100 GeV muon energy range.
1.5

Data/Calculation

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

0

10

10

1

10

2

10

3

10

4

pµ (GeV/c)

Figure 3.2: Secondary cosmic ray muon fluxes normalized by the calculation [60]. Data areobserved at Tsukuba(open box) and Norikura(open triangle) with the BESS detector [61, 62],
and at CERN(open circle) with the L3 detector [59]. For reference, DEIS [63] and MUTRON [64]are plotted. Dashed lines represent the sum of errors in data and calculation. Figure is takenfrom [60].
In the calculation of neutrino flux, interactions and propagation of particles are treated ina 3-dimensional way, including the curvature of charged particles in geomagnetic field. An advantage of the 3-dimensional way compared to the 1-dimension is discussed. The prominent
features of the 3-dimensional calculation in comparison with the 1-dimensional one are (1) an
enhancement of neutrino flux for near-horizontal direction and (2) lower production height ofneutrinos in the atmosphere which is also prominent for near-horizontal direction. The first
feature is explained as a difference of the effective area for primary cosmic rays which generate horizontally incoming neutrinos, as illustrated in Figure 3.3. This effect is important for
low energy neutrinos (< 1 GeV), in which transverse momentum of the secondary particles inhadronic interactions and the bending of muons in the geomagnetic field are considerable. Figure 3.4 shows the zenith angle distributions of the Honda and Bartol 3-dimensional atmospheric
neutrino flux and the Honda 1-dimensional neutrino flux, using 1-dimensional approximation
with older primary flux and hadronic interaction model. Since the geomagnetic field above
Super-Kamiokande is stronger than the average, neutrino flux below a few GeV is predicted to
be up-down asymmetry. This up-down asymmetry for low energy neutrinos is smeared due tothe small angular correlation in neutrino interactions and cannot be observed in charged leptons. The reason of the second feature is explained as follows. The primary cosmic rays fromhorizontal direction travel a longer distance in atmosphere than the vertically incoming cosmic
28

rays to reach the same altitude. Therefore, interaction vertex of the vertically incoming cosmic
rays have a lower altitude compared to the horizontal one. Unlike a 1-dimensional calculation,
the incidence angle of primary cosmic rays which produce horizontally going neutrinos is not restricted to be horizontal direction in a 3-dimensional calculation. As a result, production heightof neutrinos in the 3-dimensional calculation is lower than that in the 1-dimensional calculation
for near-horizontal direction. This effect is also prominent for low energy neutrinos (< 1 GeV).
The simulations for neutrino production height from the Honda flux and Honda-1D flux are
shown in Figure 3.5.

Figure 3.3: A schematic view of the effective areas of primary cosmic rays interacting with air
nuclei for 1-dimensional and 3-dimensional calculations. Arrows written by solid lines show theprimary cosmic rays and dotted lines show the neutrinos. The 3-dimensional calculation gives
larger areas for near-horizontal direction.
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 3.6, 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, however it becomes larger than two as the neutrino energy increases because
more cosmic ray muons reach the ground before decaying. The flavor ratio highly depends onthe zenith angle of incoming neutrinos as seen in Figure 3.7. The zenith angle dependence ofthe flavor ratio plays the key role in 3-flavor neutrino oscillation analysis.

29

300

4000

Honda flux
Fluka flux
Honda-1D
flux

-2

-1

-1

Flux(m sec sr )0.1-0.3 GeV

1.0-3.2 GeV

–

νµ+νµ

200

2000
100
–

νe+νe
0
-1.0

0

1.0

0
-1.0

0

1.0

0

1.0

1200

>3.2 GeV

0.3-1.0 GeV-1-1

Flux(m sec sr )

40

800

-2

20
400

0
-1.0

0

1.0

cosΘ

0
-1.0

cosΘ

Figure 3.4: Zenith angle dependence of the atmospheric neutrino flux. Boxes show the Honda
flux, dots show the Bartol flux and solid lines show the Honda-1D flux. The higher flux is forνµ + ν µ and the lower flux for νe + ν e in each plot. An enhancement of the neutrino flux is seen
for near-horizontal directions at energies below 1 GeV in 3-dimensional calculations, the Honda
flux and the Fluka flux. On the other hand, no enhancement is seen in the Honda-1D flux.

30

20

34

(a) vertical

νµ

16
14
12

νe

10
8

30
28
26
24
22
20
18

6
10

(b) horizontal

32
Production height (km)Production height (km)

18

16
-1

1
Eν (GeV)10-11Eν (GeV)( ν µ+ ν µ)/( ν e + ν e )νµ + νµ102νe + νe10
Honda flux
Fluka flux
Bartol flux2−2

Flux*Eν (m sec −1sr −1GeV)

Figure 3.5: 50 % accumulation probability of neutrino production height for (a) nearvertical (cosθ > 0.95) and (b) near-horizontal (|cosθ| < 0.05) directions. Thick and thin solid
curves are for νµ and νe by the 3-dimensional calculation, and thick and thin dashed curves arefor νµ and νe by the 1-dimensional calculation, respectively.

Honda flux

10

Fluka flux
Bartol flux

1 −1
10

1

10

10
Eν (GeV)

1 −1
10

2

1

10

10
Eν (GeV)2

Figure 3.6: Predictions of the direction averaged atmospheric neutrino flux (Left) and the flavor
ratio (Right). Solid curves show the Honda flux, dashed curves show the Fluka flux and dottedcurves show the Bartol flux.

31Flux flavor ratio

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

3

0.5
co
s

1

1 -1
θ 10

10

10

2

10 GeV)E ν(

Figure 3.7: Flavor ratio (νµ + ν µ )/(νe + ν e ) as a function of neutrino energy and zenith angle.
Calculation is based on the Honda flux.

32

3.3

Neutrino Interaction

Atmospheric neutrinos interact with nucleons and electrons in water. The neutrino interactions are treated using the NEUT program library [66, 67] in our Monte Carlo simulation. Thislibrary 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
experiment with a lot of modifications. In the NEUT library, the following charged current(CC) and neutral current (NC) interactions are simulated :CC/NC (quasi-)elastic scatteringν + N → l + N0

CC/NC single meson production

ν + N → l + N 0 + meson

CC/NC single gamma production

ν + N → l + N0 + γ

CC/NC deep inelastic interaction

ν + N → l + N 0 + hadrons

CC/NC coherent pion production

ν + 16 O → l + 16 O + π

where N and N 0 are nucleons (proton or neutron) and l is a lepton.
Since the cross section of neutrino-electron elastic scattering is about 10 3 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 amplitude of the CC quasi-elastic scattering (CCQE) for free protons (i.e. hydrogen
atom in water) is described by the product of the hadron and lepton weak currents:
GF
T = √ ul (k2 )γ µ (1 − γ5 )uν (k1 )hN 0 (p2 )|Jµhadron |N (p1 )i
2

(3.2)

where GF is the Fermi coupling constant, p1 (p2 ) is a initial(final) nucleon 4–momentum, andk1 (k2 ) is a initial(final) lepton 4–momentum, respectively. The hadronic weak current J µhadronis expressed as:iσµν q ν ξFV2 (q 2 )2
1 20
hadron0+ γµ γ5 FA (q )(3.3)
hN (p2 )|Jµ
|N (p1 )i = cos θc uN (p2 ) γµ FV (q ) +M
where M is the mass of the target nucleon, θc is the Cabbibo angle, q is the 4–momentum
transfered of the lepton. We use dipole-type form factors both for vector form factors, F V1 (q 2 )and FV2 (q 2 ), and axial vector form factor, FA (q 2 ), which are given as :

−1 

q2
q2
1 2
2
2
FV (q ) =1−
GE (q ) −GM (q )
(3.4)
4M 2
4M 2
−1



q2
2
2
2 2
G
(q
)
−
G
(q
)
(3.5)
ξFV (q ) =
1−
E
M
4M 2

−2
q2
2
FA (q ) = −gA 1 − 2
(3.6)
MA
−2

q2
2
−1
2
GE (q ) = (1 + ξ) GM (q ) = 1 − 2
(3.7)
MV
33

GE and GM are the electric and magnetic form factor, MV and MA are the vector and axial
vector mass.
Hence, the differential cross section is written as [68] :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

(3.8)

where Eν is the neutrino energy, and s and u are Mandelstam variables [68]. The factors A, B
and C are given as :m2 − q 2
q2
q2
2
4
−
|F
|
−
4
+
|FV1 |2
A
4M 2
M2
M2


4q 2 FV1 ξFV2
q2
q2
2 2
− 2 |ξFV | 1 +
−
M
4M 2
M2


m2
− 2 (FV1 + ξFV2 )2 + |FA |2
M
2

q
1
2
F
(F
+
ξF
)
B(q 2 ) =
A
V
V
M2

q2
1
2
1 2
2 2
2
|FA | + |FV | −|ξFV |C(q ) =
4
4M 2
A(q 2 ) =(3.9)
(3.10)(3.11)

where m is the lepton mass, ξ is defined by means of anomalous magnetic moment µ ≡ µ p −µn =3.71. In our simulation the vector mass MV is set to be 0.84 GeV and the axial vector mass
MA is set to be 1.21 GeV according to the experimental results [69, 70, 71]. Common M A value
is also used for single meson productions. gA is the axial vector coupling constant which ismeasured in (polarized) nucleon beta-decay [72] and set to be 1.232 in the NEUT library. Worldaverage gA value is 1.2673 ± 0.0035 [73], although the change of cross section caused by thedifferent gA value is less than 5%. In addition to the dipole form factor, several types of form
factor have been proposed [74, 75, 76]. The difference of the cross section between their models
and dipole form factor is at most 10%.
For larger MA values, interactions with higher Q2 values (and therefore larger scattering
angles) are enhanced. MA value which we use is decided to be consistent with the data fromthe other experiment, K2K and MiniBooNE. The uncertainty of the value is estimated to be
10 % following the total uncertaintiy addopted to the K2K and MiniBooNE results. The resultsof various measurements and the weighted average except for K2K and MiniBooNE results areshown in Figure 3.8.
For the scattering off bound nucleons in 16 O, nuclear effects such as the Fermi motion of thenucleons or the Pauli exclusion principle must be considered. We follow the Smith and Moniz [78]model to treat nuclear effects. In this model the Pauli exclusion principle is explained as follows:
Since nucleons are fermions, the outgoing momentum of the nucleons in the interactions isrequired to be larger than the Fermi surface momentum to allow quasi-elastic scattering tooccur. In the NEUT library, the Fermi surface momentum is set to be 225 MeV/c which isdetermined to reproduce the quasi-elastic peak in electron scattering, while the corresponding
value in association with the Spectral function [79], more sophisticated way to treat the Pauli
exclusion principle, is 209 MeV/c.

34

Figure 3.8: Axial vector mass MA extractions from (quasi-)elastic neutrino and anti-neutrino
scattering experiments. The weighted average except for the K2K and MiniBooNE results is
MA =1.026±0.021GeV. Figure is taken from [77].

Various theoretical models have been proposed to account for nuclear effects in the neutrino interactions at intermediate energies besides the Smith and Moniz model, for example
Nieves et al . [80], and Nakamura et al . [81] and so on. These models are used to estimate thesystematic uncerteinties of the neutrino interaction and nuclear effects.
The cross sections for neutral current elastic scatterings are estimated from the following
relations [82, 83] :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)(3.12)(3.13)(3.14)(3.15)

Figure 3.9 shows the cross section of the quasi-elastic scattering for the experimental data andthe calculation by the NEUT library, where solid and dashed curve indicate the scattering off
free and bound nucleon, respectively.

3.3.2

Single Meson Production

The single meson productions via baryon resonances are the dominant hadron production
processes in the region where the invariant mass of the hadron system (W ) is less than about
2.0 GeV/c2 . Our simulation for the process is based on the model of Rein and Sehgal [90]. Thismodel was originally developed for single pion productions, however in the NEUT library their
method is extended with some modification so as to simulate single η and K productions.
This method assumes an intermediate baryon resonances :ν + N → l + N∗
N ∗ → N 0 + meson35(3.16)

-

ANL
BNL
SKAT

Serpukhov
GGM (1977)
GGM (1979)

-38

2

cm )

(a) νµn→µ p

σ (10

2

cm )

-38

σ (10

2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

0

2

4

6
8
10
Eν (GeV)

12

14

2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

–

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

0

2

4

6
8
10
Eν (GeV)12

14

Figure 3.9: cross sections of (a) νµ and (b) ν µ with the experimental data from ANL [84],Gargamelle [85, 86], BNL [87], Serpukhov [88] and SKAT [89].

where N and N 0 are nucleons and N ∗ is a baryon resonance. The amplitude of a baryon
excitation is described as:
GF
T (νN → lN ? ) = √ ul γ µ (1 − γ5 )uν hN ? |Jµhadron |N i2

(3.17)

Where hN ∗ |Jµhadron |N i is the weak hadron current for this process. The matrix elements of thehadron current are calculated using FKR (Feynman-Kislinger-Ravndal) baryon model [91] inwhich the baryon is described by a relativistic three-body system. The differential cross sectionof single meson production is a product of the amplitude of each resonance production and theprobability of the baryon resonance decay to this meson. For negligible decay width of a baryon
resonance (N ∗ ), the differential cross section is :
d2 σ
1 X
1
2
·
=
T (νN → lNj∗ ) δ(W 2 − Mj2 )22
dq dEν
32πM Eν 2(3.18)

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), and M j is the mass ofthe baryon resonance (Nj∗ ). Intermediate baryon resonances with invariant mass W less than
2 GeV/c2 , totally 28 resonances, and 3 background terms are considered in the simulation aslisted in Table 3.1. For W larger than 2 GeV, the interactions are simulated as deep inelastic
scattering as described in Section 3.3.3.
Lepton mass effects from the non-conservation of lepton current and the pion-pole term in
the hadronic axial vector current dictated by PCAC are adopted to the CC interactions [92, 93],
which induce the supression of the cross section in lower Q2 .
The differential cross section for the resonance with finite decay width Γ can be derived byreplacing the δ-function with a Breit-Wignar factor :δ(W 2 − Mj2 ) →Γ
1·
2π (W − Mj )2 + Γ2 /4(3.19)For a single meson production, the axial vector mass MA is also set to be 1.21 GeV due to thesame reason as (quasi-)elastic scattering.
36

−38

cm2)

νµ p → µ− p π+σ(ν µ )(10

cm2)
−38

σ(ν µ )(10

1.2
1
0.8

1
0.9

νµ n → µ− n π+

0.8
0.7
0.6
0.5

0.6

0.4

0.4

0.3
0.2

0.2
0.1

0

0

1

10

Eν (GeV)σ(ν µ )(10

−38

cm2)

(a) νµ p → µ− pπ +
1
0.9

1

10

Eν (GeV)(b) νµ n → µ− nπ +νµ n → µ− p π0

ANL Radecky, Phys.Rev.D 25, 1161 (1982)

0.8

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

0.7

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

0.6

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

0.5

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

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

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

0.2

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

0.1
0

1

10

Eν (GeV)(d) The list of the experiments in this figure(c) νµ n → µ− pπ 0

Figure 3.10: Cross sections for charged current single pion productions of ν µ . Solid curvesindicate our calculations. Experimental data are summarized in the panel (d).
To determine the angular distribution of a pion in the final state, Rein’s method [94] is used
for the P33 (1232) resonance. For other resonances, the directional distribution of the generated
pion is set to be isotropic in the resonance rest frame. The angular distribution of π + has beenmeasured for νµ p → µ− p π + [95] and the results agree well with the NEUT’s prediction.
The Pauli exclusion principle in the decay of the baryon resonance is considered for theinteraction off bound nucleon by requiring the momentum of the nucleon to be greater than the
Fermi surface momentum.
Is is known that a baryon resonance in a nucleus can dissapear without meson emission via
the following interaction:
N ∗ + N → N 0 + N 00
(3.20)

where N ∗ is a baryon resonance and N, N 0 , N 00 are nucleons. The rate of the interactions isestimated from the theoreticl calculation [96]. In our simulation 20% of resonances in 16 O is
assumed to dissapear without meson emission.
Figures 3.10, 3.11 and 3.12 show the cross sections of charged current and neutral current
single meson productions with our calculations and the experimental data.
37

1

cm2)
−38

νµ p → µ+ p π−

0.9σ(ν µ )(10

cm2)
−38

σ(ν µ )(10

1
0.9
0.8
0.7

0.8
0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

1

10

0
Eν (GeV)σ(ν µ )(10

−38

cm2)

(a) ν µ p → µ+ pπ −10.9

νµ n → µ+ n π−

1

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

0.8
0.7

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

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

0.5
0.4

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

0.3
0.2

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

0.1
0

1

10

+

(c) ν µ p → µ nπEν (GeV)(d) The list of the experiments in this figure0

Figure 3.11: Cross sections for charged current single pion productions of ν µ . Solid curves show
our calculations.

38

cm2)

0.4

−38

0.35

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

−38

0.4

0.3

0.3

0.25

0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05

0

1

2

3

4

5

6

7

8

0

9
10
Eν (GeV)

cm2)

0.4

−38

νµ(νµ) p → νµ(νµ) p π0

0.3

0.35

0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05
1

2

3

4

5

6

7

8

2

3

4

5

6

7

8

9
10
Eν (GeV)910Eν (GeV)νµ(νµ) n → νµ(νµ) n π0

0.3

0.25

0

1

(b) Neutral current π − production

σ(ν µ )(10cm2)σ(ν µ )(10

0.4

−38

(a) Neutral current π + production

0.35

νµ(νµ) p → νµ(νµ) p π−

0

1

2

3

4

5

6

7

8

9
10
Eν (GeV)(d) Neutral current π + production(c) Neutral current π 0 production

Figure 3.12: Cross sections for neutral current single pion productions of ν µ (solid curve) andν µ (dashed curve). Experimental data are taken from [97] (dot) and [98] (box).

39

3.3.3

Deep Inelastic Scattering

The cross section of CC deep inelastic scattering is calculated by integrating the following
equation in the range of the invariant mass W > 1.3 GeV/c [99] :
G2F MN Eν
yd2 σ ν,ν̄y222=+ 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
2ml
C2 = −(3.21)
4MN Eν x
where x and y, Bjorken parameters, are defined as x = −q 2 /(2M (Eν −El )) and y = (Eν −El )/Eν ,MN is the nucleon 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 the Parton distribution function(PDF) GRV98 [100], where
the correction function given by A. Bodek and U. K. Yang [101] are adopted to improve theimplementation of the PDF in lower Q2 region.
In our simulation, the cross section of deep inelastic scattering induced by the neutral current
interactions are assumed to have the following relations which are estimated from experimental
results [102, 103] :0.26( Eν < 3 GeV )σ(νN → νX)(3.22)=
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)=
(3.23)
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 [104] :hnπ i = 0.09 + 1.83 ln(W 2 )(3.24)The number of pions in each event is determined by using the 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 [105] :0.35 + 0.41 ln(W 2 )nFπ=nB
0.5 + 0.09 ln(W 2 )π(3.25)In the region of W > 2.0 GeV/c2 , the kinematics of the hadronic system are calculated by
PHYTIA/JETSET package [106]. This package treats not only π but also K, η, ρ and so on.
Cross section of the CC νµ and ν µ interactions are plotted in Figure 3.13.
40

2

-1

cm GeV )-38

σ/Eν (10

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

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

CCFR 90
CDHSW 87
GGM-PS 79

0

10

CHARM 88
BNL 80
CRS 8020

30 50

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

CCFRR 84
SKAT

100

200

150

250

Eν (GeV)Figure 3.13: Cross sections for charged current νµ and ν µ interactions. Upper(lower) curves areνµ (ν µ ).

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 [107] is adopted to simulate the interactions, and the differential cross section is givenby :G2F M 2 2
1
fπ A Eν (1 − y)(σ πN )222π
16π total

2
MA2
2
×(1 + r )e−b|t| Fabs
MA2 + Q2
r = Re(fπN (0))/Im(fπN (0))

d3 σ
dQ2 dydt= β×(3.26)(3.27)

where β is the axial vector couping constant and is 1 (2) for NC (CC) interactions, G F is
the weak coupling constant, M is the nucleon mass, fπ is pion decay constant and is 0.93 mπ ,A is the atomic number ( = 16 for oxygen), Eν is the neutrino energy, y is the lepton fractional
πN is the averaged pion-nucleon cross section, b is in the order of the nucleus
energy loss, σtotaltransverse dimensions and is 80 GeV−2 , MA is the axial-vector mass, Q2 is the square of the
4–momentum transfer of the lepton, t is the square of the 4-momentum transfer to the nucleus,
and Fabs is a t-independent factor to account for the absorption of pions in the nucleus. f πN (0)in r is the πN scattering amplitude.
The measurement by the K2K-SciBar detector sets the upper limit on the cross section of CC
coherent pion production [108]. The upper limit was significantly lower than the predicted crosssection by Rein and Sehgal. Therefore some modification is necessary to obtain the reasonable
expectation. In NEUT library, the modification by Rein and Sehgal [109] is introduced to account41

σNC(10−40cm2/Carbon nucleus)σCC(10−40cm2/Carbon nucleus)

80
70
60
50
40
30
20

50
45
40
35
30
25
20
15
10

10
0
0

5
0.5

1

1.5

2

2.5

3

Eν (GeV)

0
0

1

2

3

4

5

6

Eν (GeV)Figure 3.14: The cross sections of coherent pion production off the carbon nucleus. (Left) The
charged current interaction. Solid and dashed curve indicate Rein and Sehgal with and without lepton mass effects, respectively. Dotted and dashed-dotted curve represent Kartavtsev etal. [110] and Alvarez-Russo et al. [111], respectively. The arrow indicates the experimental upper
limit by K2K [108] (circle) and SciBooNE [112] (boxes). The experimental limits are converted
from the cross section ratio σcoherent /σCCtotal with the NEUT prediction. (Right) Solid curve indicate Rein and Sehgal. Dashed curve indicate Kartavtsev et al.. Experimental data are takenfrom MiniBooNE [113] (open circle), Aachen-Padova [114] (open box) and Gargamelle [115](open cross).

for the non-vanishing lepton masses in CC interactions which suppress the cross section, in caseof νµ , about 25% at 1.3GeV due to the interference of the axial vector and pseudoscalar(pionexchange) amplitudes.

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 applied to π, K and η by using a cascade model in our simulation. The interactionsof pions are especially important since the cross section of neutrino interactions accompaniedwith pion production are large for Eν > 1 GeV, and the pion-nucleon interaction cross sectionis 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, is determined by the Wood-Saxon density
distribution [116] :Z
1ρ(r) = ρ0(3.28)r−cA
1 + expawhere 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 [117]. A type of pion interaction is determined from the calculated
mean free path of each interaction, which is modeled by L. Salcedo et al. [118]. The mean freepath of pions depends on their momentum and positions in the nucleus. In the interactions, the
42

+ 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 3.15: 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 [121].

Fermi motion of the nucleus and the Pauli exclusion principle are considered, and the outgoing
nucleon must have the energy above the Fermi surface momentum defined by :pF (r) =

3 2
π ρ(r)
2

1

3.

(3.29)

The angular and momentum distributions of the outgoing pions are determined by using theresults of a phase shift analysis from π - N scattering experiments [119]. 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) [120, 121] as shownin Figure 3.15. For kaons, the elastic scattering and charge exchange interactions are consideredusing the results from the cross sections measured by the K ± -N scattering experiments [122,123, 124]. For η mesons, the absorption (ηN → N ∗ → π(π)N ) is considered [125]. These pions
are tracked as described above.
Nucleons generated in nucleus often cause secondary interactions with nucleons bound innucleus. The nucleon-nucleon elastic scattering cross section is based on the measurements by
Bertini [126], which is used in GCALOR. The pion production caused by the decay of produced
deltas is also taken into account, according to the isobar production model by Lindenbaum etal. [127].
Probability of nucleon-nucleon interaction in 16 O is shown in Fig. 3.16. The productionof nucleons with nucleon momentum below 225MeV/c is suppressed by the Pauli exclusion
principle. In the momentum range above 300MeV/c, a half of nucleons interact mainly byelastic scattering, therefore more than one nucleon escapes from Oxygen.
Mean deflection angle is shown in Fig. 3.17, where the nucleon momentum is from 500MeV/c
to 1000MeV/c. The deflection angle has a peak around 40 degrees.
43

Probability of interaction(%)

100
90
80
70
60
50
40
30
20
10
0
0

500

1000

1500

2000

2500

3000

Figure 3.16: Probability of nucleon-nucleon inteactions as a function of incoming nucleon momentum(MeV/c). No interaction(solid curve), elastic scattering(dashed curve), single π production(dotted curve), and two π production(dashed-dotted curve) are plotted.

Arbitrary unit

ntum(MeV)
400
350
300
250
200
150
100
50
0

0

20

40

60

80 100 120 140 160 180

Proton scattering angle(deg.)

Figure 3.17: Deflection angle of proton by secondary interactions in 16 O nucleus. Proton modifang
mentum is distributed from 500MeV/c to 1000MeV/c.

44

The concept of formation zone(length) is also considered for all hadrons produced in nucleus.
This is a distance or time from the neutrino interaction point to the hadron production point.
The intermediate states are assumed to be non-bound quark states. The formation length ofeach hadron is expressed as follows:
L = p/µ2(3.30)
where p is the momentum of tha hadron and µ2 =0.08GeV2 is a fitted constant from the SKAT
experiment [128]. We can see the nucleon-nucleon intearction in nucleus has large effect andresults in the significant distortion of the geometry of particles observed in the detector.

3.4

Detector Simulation

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

3.4.1

Particle Tracking

The detector simulation has been developed based on the GEANT package [129]. Table 3.2
lists various processes which are considered in our simulation program. The hadronic interactions in water are simulated using the CALOR package [130]. This package is known toreproduce the pion interactions well including low energy region (∼ 1 GeV/c). For still lower energy region (pπ ≤ 500 MeV/c), a custom program [131] based on experimental data from π− 16 O
scattering [132] and π − p scattering [133] is used in our simulation code.
The uncertainty in the hadron simulation is estimated by comparing the model used inour detector simulation, CALOR [130], and the FLUKA model. The uncertainty affects thecontamination of NC interactions, especially for the νµ -enriched samples. The difference fromthe FLUKA model is assumed to be a systematic error.
The number of generated Cherenkov photons, the direction of each photons and its wavelength are calculated using Eq.(2.2) and Eq.(2.3). The Cherenkov photons are only generated
between 300nm and 700nm because PMTs are only sensitive to this region.

3.4.2

Cherenkov photon tracking in water

We take into account the dispersion of the refractive index. The group velocitycvg =n(λ) − λ ∂n(λ)∂λ(3.31)

where c is light velocity in vacuum, λ is light wavelength, is used for the light velocity in thewater. The generated Cherenkov light can be scattered and absorbed in water. The watertransparency as a function of wavelength is determined as follows:
Short wavelength
The relationC ∝ λ−4(3.32)

is employed, where C is the scattering coefficient and λ is wavelength. If the size of particles
with which a Cherenkov photon interacts is small compared to the wavelength, the scattering
length is propotional to the fourth power of light wavelength. This is Rayleigh scattering.
45

Long wavelength
We use the data taken from [134].
The absolute water transparency is determined to be consistent with the direct measurementof pure water. When the Cherenkov light arrives at the surface of a PMT or black sheet, reflection
on the surface can occur. The calculated reflection value, including the polarization, is used forthe simulation. The reflection by black sheet is calculated and measured in water, which areconsistent each other.

3.4.3

Response of the PMT

The average value of the collection efficiency of PMT is treated as a tunable parameter. Thisis used for the final adjustment of the entire scale.
Charge and timing resolution of PMTs are considered. The charge value of each hit PMTs
is simulated by the random number distributed to follow the measured one photon distribution
shown in Figure 2.7. The timing distribution dependent on the charge is measured using a laser.
In the simulation, the timing of each PMT is a Gaussian random variable with the sigma shownin Figure 4.5.

3.5

Upward-Going Muon

Upward-going muon (UPMU) is the event which muon produced by the charged current
interactions of the upward-going muon neutrinos enters the inner detector. For such events, there
are two targets in the neutrino interactions: water in the outer detector and rock around theentire Super-Kamiokande detector. About 85% of UPMU events are induced by the interaction
in rock, and the others are in water. Hereafter, we mention only the case in rock, since theinteraction in water is identical to the simulation we saw above.
As for the neutrino interaction taking place in the rock, the follwing rock profiles are assumed
in the simulation, ρ = 2.65g/cm3 , Z = 11, and A = 22. These profiles are based on the“standard rock” [135]. The propagation of daughter muons through rock is simulated by theroutines developed by [136].
Each UPMU has an associated volume through which it may traverse
V µ (Eµ ) =

4π 3
R
3 eff

(3.33)

where Reff (Ei , Emin ) is the effective muon range, Ei is the initial UPMU energy, and Emin =1.6GeV is the energy threshold for UPMU event. The effective detection volume V eff , consistingof the rock around the detector which is much bigger than the detector volume, is given by the
product of the effective detector area Aeff (θ) as a function of zenith angle θ and the muon range
Reff (Ei , Emin ).
Veff (E0 , θ, Emin ) = Aeff (θ)Reff (Ei , Emin )(3.34)
Effective area is approximately 1200 m2 for all θ.
We are thus able to constrain the volume around the detector for which events can begenerated by requiring that for each event the ratio of Veff to V µ (Eµ ) exceeds a random number
on [0,1] (denoted as ζ).
Veff (E0 , θ, Emin )/V µ (Eµ ) > ζ(3.35)46

We can visualize this requirement by embedding the volume associated with Veff inside V µ (Eµ ).
We pick random point inside V µ (Eµ ) and accept it if and only if it also lies within Veff .

47

Resonance/BackgroundMsss(GeV/c)

Width(GeV/c)

P33 (1234)p, n
D13 (1525)p
D13 (1525)n
S11 (1540)p
S11 (1540)nS31 (1620)p, n
S11 (1640)p
S11 (1640)n
D13 (1670)p
D13 (1670)n
D15 (1680)p
D15 (1680)n
D33 (1730)p, n
P11 (1450)p
P11 (1450)nP33 (1640)p, n
F15 (1680)p
F15 (1680)nP11 (1710)p
P11 (1710)n
P13 (1740)p
P13 (1740)n
P31 (1920)p, n
F35 (1920)p, n
F37 (1950)p, n
P33 (1960)p, n
F17 (1970)p
F17 (1970)nBackground Proton
Background Neutron
Background J=3/2

1.232
1.520
1.520
1.535
1.535
1.620
1.650
1.650
1.700
1.700
1.675
1.675
1.700
1.440
1.440
1.600
1.680
1.680
1.710
1.710
1.720
1.720
1.910
1.905
1.950
1.920
1.990
1.990
0.940
0.940
0.940

0.115
0.125
0.125
0.150
0.150
0.140
0.150
0.150
0.100
0.100
0.155
0.155
0.250
0.200
0.200
0.370
0.125
0.125
0.110
0.110
0.200
0.200
0.220
0.300
0.240
0.250
0.325
0.325
3.000
3.000
3.000

Table 3.1: List of resonances and background terms taken into account48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 3.2: List of the processes considered in simulator.

49

Chapter 4

Calibration
4.1

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 4.1 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
Gi =
· li · exp(4.1)
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 4.2 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.

4.2

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 γ-ray generated from neutron capture on Nickel nucleus is used as a calibration source to measure the single photoelectron distributions. The schematic view of the Nickel
50

Number of PMTs in Each Bin

Figure 4.1: 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 4.2: The relative gain distribution of all the ID PMTs, measured in 1996.

51

mean

2.055pC

5103Ni wire+water

200mm
Cf

2

1

0

190mm

0

pC

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

calibration source is shown in Figure 4.3. Neutrons produced by spontaneous fission of 252 Cf
are captured on surrounding Ni wires in polyethylene vessel, and low energy (6 ∼ 9 MeV) γ-rays
are generated simultaneously. The number of hit PMT is about 50 ∼ 80 in total, so that the
number of p.e. detected by each PMT is at most one. The charge distribution of a typical PMT
is also shown in Figure 4.3. The sharp peak near zero is caused by electrons that are emitted
from the photocathode but miss the first dynode, and the peak around 2 pC corresponds to thatof single photoelectrons. The mean value 2.055 pC is used as a constant to convert the PMT
charge from pico Coulomb to the number of p.e.s .

4.3

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 4.4 shows the schematic view of the relative timing calibration system. N 2 laser generator 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 4.4. The laser light injected to the diffuser ball isfirst diffused by a TiO2 diffuser tip at the center and then further diffused by the surrounding52

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

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 4.5. 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.

4.4

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
disturbing the normal data taking.

4.4.1

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 [137]. Figure 4.6 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 4.6. The PMT hits clustered at the bottom of the tank are due to the direct(unscattered) photons.
53

980
970

Linear Scale

Log Scale960

T (nsec)

950
940
930
920
910
900
890
880
1

5

10

100

Q (p.e.)Figure 4.5: TQ-map, a measured two-dimensional plot of timing vs. charge distribution.

LASER
Optical Fiber

Top

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

Scattering

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

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

54

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

In the analysis, the detector is separated into 6 regions, top and 5 in barrel, as shownin Figure 4.6. The total charge of bottom PMTs is used for the normalization of the lightintensity. Figure 4.7 shows the PMT hit timing distributions in each region for data and Monte
Carlo simulation. These PMT hits on the top and barrel wall are due to the photons scatteredin water, or the photons reflected by surfaces of bottom PMTs or black sheets. The first peaks
and slopes are 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 4.8 with astar symbol. The lines shows a model used in the Monte Carlo simulation, which are determinedby fitting the measurements from this method.

4.4.2

Measurement using cosmic ray muons

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

55

attenuation coefficient(1/m)

Light scattering measurement

-1

10

-2

10

absorption

Rayliegh scattering-3

10
Mie scattering

300

400

500

600

700

wavelength(nm)Figure 4.8: Wavelength dependence of attenuation coefficients (L−1atten ) obtained by the lightscattering measurement (points) together with a model in the detector simulation (lines, see
Chapter 3.4).

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

(4.2)

where Q0 is a constant, f (θ) is the PMT acceptance, l is the photon path length and L isthe attenuation length. The left panel on Figure 4.9 show the schematic view of the measurement. The right panel on Figure 4.9 shows log (Q·l/f (θ)) as a function l in a typical run. The
attenuation length is estimated to be 95 m for this data set.
Since the cosmic ray muons are measured during the normal data taking, continuous checkof the water transparency is possible by this method. Figure 4.10 shows the time variation of
the attenuation length. This time variation is corrected in event reconstruction processes.
This measurement also works for the tuning of the charge scale of the Monte Carlo simulation.
Figure 4.11 shows the intersection of the effective observed charge at l → 0 in the right panel on
Figure 4.9 as a function of elapse days. The variation of the attenuation length is negligible at thesmall path length, so the effective observed charge at l ∼ 0 is stable during the data taking. The
charge scale (intersection) of the Monte Carlo simulation is adjusted to be consistent with themeasured value by changing the number of photons generated by the traversing charged particle
and the acceptance of PMT, where the acceptance is defined to be independent of the photon
incident angle. The charge scale is adjusted within 0.4% between the Monte Carlo simulationand data. The residual of the adjustment is smaller than the statistical error of data.

56

20-inch
PMT

42o
Θi

li

log(Ql/f( θ)): Cherenkov
Photons

Inner Tank

500

RUN 3106

P1
P2

5.987
-0.9487E-04

1000

2000

3000

0.4609E-03
0.4444E-06

400
300
200

Cosmic Muon

0i’th PMT

4000

5000
l (cm)

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

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 4.10: Time variation of water attenuation length.

57

Figure 4.11: Effeftive charge log(Ql/f (θ)) at l → 0 as a function of elapse days. Black circle isthe observed data and black box is the average of the data with RMS. White circle is the Monte
Carlo simulation and white box is its average with RMS. This figure is the case for SK-II.

4.5

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 tens of MeV/c to about 10 GeV/c byusing independent calibration sources: decay electrons from stopping muons, π 0 events, low andhigh energy stopping muons.
The residuals between the data and Monte Carlo simulation are not corrected in the analysis,
but are combined with the time variation of the energy scale, and considered as the systematic
uncertainty in the absolute energy scale calibration.

4.5.1

Decay electrons

Many electron events produced by the decay of cosmic ray muons are observed in SuperKamiokande. The energy and angular spectrum of the electron is expressed as



1
Γ
2
2
2
= 4x 3x(1 − x) + ρ(4x − 3) − Pµ ξ cos θ 1 − x + δ(4x − 3)
(4.3)
Γ dxd cos θ33
where Pµ is the polarization of muon, x = (2Ee /mµ ), and θ is the angle between the spin ofmuon and the electron momentum. ρ, ξ and δ are called as the Michel parameters. In the V − A
theory, the Michel parameters are3
ρ=δ= ,
4

ξ=1

(4.4)

x is restricted to be less than 1, so the electon produced by the muon decay is distributed up
to ∼ mµ /2. ∼ 53 MeV, which is so called the Michel spectrum. The validity of this formula has
58

Events

Events

225
200

500

400

175
150

300

125
100

200
75
50

100

25
0

0

10

20

30

40

50
60
70
80
MOMENTUM (MeV/c)

0

0

10

20

30

40

50
60
70
80
MOMENTUM (MeV/c)

Figure 4.12: The momentum distribution of decay electrons of the data (dot) and the Monte
Carlo (line) for SK-I (left) and SK-II (right). Monte Carlo events are normalized by the numberof observed data.

been confirmed by the various high precision experiments using the polarized muon beam, forexample [138]. Thus the well known energy spectrum are used to check the absolute energy scaleby comparing the energy spectrum between the observed data and 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.
Figure 4.12 shows the momentum spectra of decay electrons compared with the prediction from
the Monte Carlo simulation. The momentum distribution is distorted from the Michel spectrum,for example as clearly seen in the tail of the spectrum extending up to ∼70 MeV. This is causedby the case of µ− in which almost all µ− is trapped by a oxygen atom in K-shell orbit and the
decay electron is influenced by electric potential of the oxygen nuclei and orbital motion of theparent µ− [139]. Criterion (2) rejects ∼6 MeV γ-rays from µ− capture on the nucleon. In thesimulation, measured µ+ /µ− ratio of 1.37 [140] and the effect of µ− capture by oxygen nuclei
are considered.
We use the vertex fitter which has been devised and developed for the low energy neutrino
observations such as solar and supernova neutrinos [141]. In this fitter, the vertex of electron isreconstructed using the timing information of hit PMTs, where the timing resolution of PMT
is taken into account. In the ideal case the goodness becomes 1.
The mean values of data agree with the Monte Carlo prediction within 0.6 %(1.6 %) for SK-I
(SK-II).

59

4.5.2

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(4.5)
Mπ0 = 2Pγ1 Pγ2 (1 − cos θ)
The π 0 events are selected from the atmospheric neutrino sample by the following criteria :1. Two Cherenkov rings are recognized and both of them are identified as electron-like.
2. Electrons from muon decay is not detected.
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. A typical π 0 event is displayed in
Figure 4.13. Figure 4.14 shows the invariant mass distribution of π 0 events, which is comparedwith the prediction from the Monte Carlo simulation. A peak of the mass distribution, forexample in SK-I they are 139.5MeV/c (data) and 138.7MeV/c (MC), is slightly shited from theπ 0 mass ∼ 135MeV/c toward a higher mass because of mainly two reasons: First, when a π 0
is produced in the oxygen nucleus, the remaining nucleus is often left in the excited state andthen emit the de-excitation γ-rays which add a small number of photons, i.e. energy, to an
event. Second, the two γ-rays as decay products of a π 0 do not convert into the electromagnetic
showers right after the decay. Thus the reconstructed vertex tends to be pulled toward theparticle direction. This causes the reconstructed opening angle of the two γ-rays to be slightly
larger than the true opening angle, resulting in the larger π 0 invariant mass. The de-excitationof oxygen nucleus is considered in the Monte Carlo simulation.
The peak position of the data is 0.6 %(1.6 %) higher than that of the Monte Carlo simulationfor SK-I (SK-II).

4.5.3

Low energy stopping muons

The Cherenkov angle of charged particles is expressed as a function of the momentum:s11m2
cos θC =
=
(4.6)
1+ 2
nβ
n
p
where 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 in the momentum determination by comparing it with the momentum derived fromthe Cherenkov angle (Pθ ). This method is not applicable to the electrons or the high energymuons because the Cherenkov angle approaches a limit when the momentum is high compared
to its mass. Low momentum stopping muon events are selected for this study by the following
criteria:
60

NUM
509
RUN
5346
SUBRUN
26
EVENT
1228025
DATE 97-Dec-10
TIME
14:55: 9
TOT PE:
2142.2
MAX PE:
15.4
NMHIT : 1115
ANT-PE:
12.5
ANT-MX:
1.9
NMHITA:17

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.:0.128E+06us
FEVSK :81002003
nOD YK/LW: 0/ 0
SUB EV : 0/ 0
Dec-e: 0( 0/ 0/ 0)*NO KEKGPS RED*

Comnt;

Figure 4.13: A typical π 0 event. Two Cherenkov rings from the γ-rays arefound.

160

70

140

60
120

50
100

40
80

30

60

20

40

10

20
0

0

50

100

150

200

250

300

0

0

50

100

150

200

250

300

Mγγ (MeV/c)Mγγ (MeV/c)

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

61

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.
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) selects low momentum muon events. Criterion (2) requires an entrance point of amuon in the OD, namely the stopping muons. Criteria (3) and (4) select the muon events which
are straight downgoing.
The upper left panel on Figure 4.15 shows the reconstructed momentum distribution for
the events passed through the above criteria. On the other hand, the upper right panel shows
the reconstructed opening angle distribution. In both panels, the Monte Carlo simulation isnormalized by the number of measured data. The correlation between P p.e. and Pθ is shown in
the lower panel on Figure 4.15. Since the expected momentum tends to show the smaller value
compared to the reconstructed momentum, these two momentums do not relate exactly one by
one in the low momentum region, although it does not affect the result of the calibration.
The energy scale of the data is compared with that of the Monte Carlo simulation by comparing the ratios of Pp.e. /Pθ . Figure 4.16 shows the averaged Pp.e. /Pθ for the data and theMonte Carlo simulation and the MC/data ratio as a function of the expected momentum P θ .
They agree within 0.7 % for SK-I and 1.3 % for SK-II.

4.5.4

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 is independent of the momentum reconstructedusing the observed charge, the estimated momentum from its range is used for checking theenergy 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).
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 defined as the distance between the entrance position of the stopping
muon and the vertex position of the subsequent decay electron. Figure 4.17 shows the averaged
value of the ratio 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 4.17.
Although the momentum dependence is seen, the deviations from unity are less than 0.7 % for
SK-I and 1.1 % for SK-II.

62

600

Number of events

Number of events

300
250
200
150
100

400
300
200
100

50
0

500

100

150

200

250

300

350

400

450

500

0

30

32

34

36

38

40

42

44

Opening angle (deg.)

Reconst. momentum (MeV/c)
600

reconst. mom (MeV/c)

550
500
450
400
350
300
250
200
150

150 200 250 300 350 400 450 500 550 600

expected mom. (MeV/c)

Figure 4.15: (Upper left) The reconstructed momentum distribution Pp.e. for the data (circle anderror) and the Monte Carlo simulation (histogram). (Upper right) The reconstructed opening
angle distribution. (Lower) The correlation between the two momentum P θ (horizontal axis)
and Pp.e. (vertical axis). Black dot represents the data and red dot for the simulation.

63

MC/DATA

P(p.e.)/P(θ)

1.4
1.3
1.2

1.05
1.04
1.03
1.02

1.1

1.01

1

1

0.9

0.99
0.98

0.8
0.97
0.7
0.6

0.96
0.95

200 225 250 275 300 325 350 375 400 425 450

200 225 250 275 300 325 350 375 400 425 450

expected momentum (MeV/c)

1.2

MC/DATAP(p.e.)/P(θ)

expected momentum (MeV/c)

1.15
1.1

1.05
1.04
1.03
1.02
1.01
1

1.05

0.99
1

0.98
0.97

0.95

0.96
0.9

200 225 250 275 300 325 350 375 400 425 450

expected momentum (MeV/c)

0.95

200 225 250 275 300 325 350 375 400 425 450

expected momentum (MeV/c)

Figure 4.16: Figure on left panel shows the averaged ratio of the momentum derived from thecharge to that from the opening angle (Pp.e. /Pθ ) as a function of the momentum Pθ for the data
(black points) and the Monte Carlo events (open boxes), and figure on right panel shows theratio 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.

64

1.05

M.C./DATA

Momentum(MeV/c)/Range(cm)

2.6

1.04

2.5

1.03
1.02

2.4

1.01
2.3

1
0.99

2.2

0.98
0.97

2.1

0.96
0

500

1000

1500

2000

2500

0.95

3000 3500
Range (cm)

2.6

0

500

1000

1500

2000

2500

3000 3500
range (cm)0500 1000 1500 2000 2500 3000 3500 4000
range (cm)1.05

M.C./DATA

Momentum(MeV/c)/Range(cm)

2

1.04

2.5

1.03
1.02

2.4

1.01
1

2.3

0.99
2.2

0.98
0.97

2.1

0.96
2

0

500

1000

1500

2000

2500

3000 3500
Range (cm)0.95

Figure 4.17: Figure on left panel shows the range v.s. averaged momentum/range of stopping
muon events for the data (black points) and the Monte Carlo events (open boxes), and figure
on right panel shows the ratio of the Monte Carlo events to the data for SK-I (top) and SK-II(bottom).

65

43
42
41
40
39

e momentum/track

Momentum(MeV/c)

44

1.05
1.04
1.03
1.02
1.01
1
0.99
0.98
0.97
0.96
0.95

0

250

500

0

250

500

45
44
43
42
41
40
39

750 1000 1250 1500 1750 2000
elapse dayse momentum/track

Momentum(MeV/c)

45

750 1000 1250 1500 1750 2000
elapse days

1.05
1.04
1.03
1.02
1.01
1
0.99
0.98
0.97
0.96
0.95

0

200

400

600

800
1000
elapse days

0

200

400

600

800
1000
elapse days

Figure 4.18: The time variation of the reconstructed momentum (top) and the averaged momentum/range (bottom) of the decay electrons as a function of elapse days from April, 1996(October,2002) for SK-I(SK-II). Left panel shows distributsions for SK-I and right panel for SK-II.

4.5.5

Time variation of energy scale

The stability of the energy scale is also monitored by the stopping muons and the decay
electrons. Figures 4.18 and 4.19 show the time variation of the averaged momentum/range andthe reconstructed momentum for the stopping muons and decay electrons as a function of elapse
days from April 1st, 1996. Energy scale variation is determined by the reconstructed momentumof the decay electrons (Upper two panels on Figure 4.18) and the averaged momentum/range ofthe stopping muons (Lower two panels on Figure 4.19). The RMS of the variation is given as
0.9 %(0.6 %) for SK-I (SK-II).

4.5.6

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 Θ e↔µ < 0.25,
where cos Θe↔µ is the opening angle between the electron and muon directions. Figure 4.20
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.

4.5.7

Summary of the absolute energy calibration

Figure 4.21 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 scale isestimated to be less than 0.7 % and 1.6%̇ for the momentum range from a few tens of MeV/c to
66

1.05
1.04
1.03
1.02
1.01
1
0.99
0.98
0.97
0.96
0.95

Momentum(MeV/c)
0

250

500

0

250

500

2.5
2.45
2.4
2.35
2.3
2.25
2.2

750 1000 1250 1500 1750 2000
elapse days
µ momentum/track

Momentum(MeV/c)µ momentum/track

2.6
2.55
2.5
2.45
2.4
2.35
2.3
2.25
2.2

750 1000 1250 1500 1750 2000
elapse days

1.05
1.04
1.03
1.02
1.01
1
0.99
0.98
0.97
0.96
0.95

0

200

400

600

800
1000
elapse days

0

200

400

600

800
1000
elapse days

Figure 4.19: The time variation of the reconstructed momentum (top) and the averaged momentum/range (bottom) of the stopping muons.

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 4.20: 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.

67

about 10 GeV/c for SK-I and SK-II, respectively. Combined with the RMS of the time variationof the energy scale, 1.1 % (1.7 %) is adopted to the systematic error for SK-I (SK-II).

68

SK-I
(MC - DATA) / DATA (%)

8
6
4
2
0
-2

µ rangeµ angle0π massdecay e

-4
-6
-8
10

2

3

10
10
10
momentum range (MeV/c)
SK-II

4

10

5

(MC - DATA) / DATA (%)

8
6
4
2
0
-2

µ rangeµ angle0π massdecay e

-4
-6
-8
10

2

3

10
10
10
momentum range (MeV/c)

4

10

5

Figure 4.21: 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.

69

Chapter 5

Data Reduction
and Event Reconstruction
5.1

Data Reduction

About 106 events are collected by the Super-Kamiokande detector per day, however mostof the events are cosmic ray muons and low energy backgrounds from radioactivities such asradon decay. The efficient event selection is necessary for selecting neutrino events from a hugequantity of data.
The atmospheric neutrino events observed in Super-Kamiokande consist of the following
four categories; fully contained (FC), partially contained (PC), upward stopping muons (UPMU
stopping) and upward through-going muons (UPMU through) as shown in Figure 5.1. For FC
and PC events, the vertices of neutrino interactions are required to be within a fiducial volume,
2 m inside from the ID wall. If the tracks of entire particles are contained inside the ID, theevent is classified into FC. If one of the particles exits the ID and deposits energy in the OD, theevent is classified into PC. The UPMU events are produced by the atmospheric muon neutrinos
via charged current interactions in the rock surrounding the detector. Since the downwardgoing neutrino-induced muons cannot be distinguished from the cosmic ray muons, only muons
traveling in the upward-going direction are selected. The UPMU stopping enter from outside
the detector and stop inside the ID, while the UPMU through enter the detector and exit theID.
Expected energy range for each event class are, ∼ 1 GeV for FC, ∼ 10 GeV for PC, ∼ 10 GeV
for UPMU stopping, and ∼ 100 GeV for UPMU through. Figure 5.2 shows the expected numberof neutrino events 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 Appendix C.2.2) is
employed, where hit cluster is defined as the spatial cluster of neighboring hit PMTs. Figure 5.3
shows the NHITAC distribution for both FC and PC final samples in the fiducial volume. FC
and PC 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 0.6 % for SK-I and 0.5 % for SK-II by comparing the
NHITAC distributions for the data and the Monte Carlo events.

70

Fully contained

Partially contained

Upward stoppingmuon

Upward through-going
muon

Figure 5.1: 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 5.2: The expected parent neutrino energy distribution for each event class.

71

10

5

10

4

10

3

10

2

10

PC
Number of events

Number of events

FC

10

1

0

4

10

20

30

40

50

60

70

80

90

10

3

10

2

NHITAC

PC

10

20

10

1

100

FC

0

30

40

50

60

70

80

90

100

NHITAC

Figure 5.3: 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, 2.1 × 10−3 eV2 ) (solid lines).
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.
As we mentioned, the UPMU events are classified into two categories : (i) upward stopping
muon events with only an entrance point in the OD (UPMU stopping) and (ii) upward throughgoing muon events with an entrance point and exit point in the OD (UPMU through). In order
to separate the UPMU stopping events and the UPMU through events, the number of hit OD
PMTs within 8 m from the exit position (NHITEX) is used. Figure 5.4 shows the NHITEX
distributions for UPMU stopping and UPMU through final samples. The events with NHITEX
less than 10 (16 for SK-II) are regarded as stopping muons and the rest of events are categorized
to through-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 estimatedto be 0.4 % both 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.
Detail of the data reduction processes for FC, PC, and UPMU are described in Appendix C.

5.2

Event Reconstruction

Event reconstruction processes are applied to the atmospheric neutrino events which passthrough the data reduction processes. The common programs are applied for both the observed
data and the atmospheric neutrino Monte Carlo events. The event reconstruction process isfully automated.
The outline of the reconstruction process for FC and PC events is described below and theflowchart of the processes is shown in Figure 5.5:
(1) Vertex Reconstruction

72

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

0

10

NHITEX

20

30

40

50

60

70

NHITEX

Figure 5.4: NHITEX distributions for UPMU stopping and UPMU through 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 ν µ ↔ ντ
2-flavor oscillation with (sin2 2θ, ∆m2 ) = (1.00, 2.1 × 10−3 eV2 ) (solid lines).
The vertex position is determined as the point where the timing residual distribution of
hit PMTs has the sharpest peak. The direction and the outer edge of the dominant ringare also reconstructed.
(2) Ring Counting
Other possible rings are searched using the vertex and direction information of the dominant ring determined by the vertex reconstruction. The ring candidates are tested whether
the ring is true or false by a likelihood method, and the number of rings is determined.
(3) Particle Identification
Candidate ring is classified into two types, a showering type (e± , γ) and non-showering type(µ± , π ± ), according to the ring pattern and the opening angle. For the sake of simplicity,
the showering type and the non-showering type is denoted as e-like and µ-like, respectively.
(4) Precise Vertex Reconstruction (only for single ring event)The precise vertex position is obtained using the Cherenkov ring pattern assuming theparticle type, a showering type(e-like) or a non-showering type(µ-like). This precise vertex
fitter is called MS-fit.
(5) Decay Electron finding
Primary event induced decay electrons are searched.
(6) Momentum Reconstruction
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.
73

(7) Ring Number Correction
Rings which have the low momentum and overlapped with other energetic rings are discarded as fitting mistakes.
(8) π 0 Reconstruction
Other ring candidate of π 0 events is searched among the single ring e-like events.
Detail of the event reconstruction processes are described in Appendix D.

74

Figure 5.5: Flowchart of the event reconstruction for FC and PC samples

75

Chapter 6

Data set
We summarize here our data sets for 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). Since the reconstruction of muon with long
path length is less sensitive to the detector condition, the live time of the UPMU data is larger
than that of FC and PC data. The statistics of the atmospheric neutrino Monte Carlo events
amounts to an exposure of 500 years for SK-I and SK-II.

6.1

Event Classification

Final samples of FC and PC events are selected and cateagorized by the following criteria
after event reconstruction processes :• FC samples(1-1) Number of hit PMTs in the OD hit cluster (NHITAC) < 10 (16 for SK-II)(1-2) Distance from vertex to the nearest ID wall (Dwall ) > 200 cm, i.e. Fiducial volume
cut(1-3) Visible energy assuming electrons (Evis ) > 30 MeV– FC Single-ring µ-like samples(1-4) pµ > 200 MeV/c– FC Single-ring e-like samples(1-5) pe > 100 MeV/c– FC Multi-ring µ-like sample(1-6) 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(1-7) The most energetic ring is identified as e-like and Evis > 1330 MeV• PC samples
76

(2-1) NHITAC ≥ 10 (16 for SK-II)(2-2) Dwall > 200 cm(2-3) Evis > 350 MeV– PC OD stopping sample(2-4) 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 theOD(2-5) The most energetic ring or the second one should be identified as µ-like– PC OD through-going sample(2-6) PEanti > PEexp /1.5
FC and PC samples are separated by the number of hit PMTs in the OD hit cluster (NHITAC). The fiducial volume for the FC and PC samples is defined by Dwall >200 cm whichcorresponds to 22.5 kton. The visible energy (Evis ) is defined as the sum of the energy of eachring assuming all rings are produced by electrons. The criterion (1-3) is introduced to rejectremaining low energy background events.
First FC events are divided into Single-ring and Multi-ring samples according to the numberof Cherenkov rings. Next the Single-ring sample is separated into “Sub-GeV sample” with
Evis < 1.33 GeV and “Multi-GeV sample” with Evis > 1.33 GeV. At last the Single-ring SubGeV sample is classified into e-like and µ-like samples according to the particle identification
result. As only for Single-ring Sub-GeV samples, more finer classifications are adopted to increase
the purity of a certain neutrino interaction mode in each sample according to number of decay
electrons and the π 0 reconstruction information(see Appendix D.5 and D.8). For Multi-ringsample the most energetic ring is used to identify the particle type.
For PC events, first the criterion (2-3) Evis > 350 MeV, which corresponds to muon momentum & 530 MeV/c, is adopted. This cut is sufficiently 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 the observed
p.e.s in the OD and the expected charge derived from the track length, where the criteria (2-4),(2-5) and (2-6) are applied. Muons in the “OD stopping sample” are assumed to be stopped
in the OD, while more energetic muons in the “OD through-going sample” are assumed to passthrough the OD.
The UPMU stopping and UPMU through samples are selected by requiring the following
criteria after data reduction and event reconstruction processes :• UPMU stopping sample(3-1) Number of hit OD PMTs within 8 m from the muon exit point (NHITEX) < 10 (16for SK-II)(3-2) Reconstructed ring direction is upward (zenith angle cosΘ ≤ 0)(3-3) pµ ≥ 1.6 GeV/c (corresponds to track length of 7 m for muon)• UPMU through sample(4-1) NHITEX ≥ 10 (16 for SK-II)
77

Figure 6.1: Classification of FC samples.

78(4-2) Reconstructed ring direction is upward (zenith angle cosΘ ≤ 0)(4-3) Distance from the muon entrance point to the exit point ≥ 7 m

UPMU stopping and through samples are separated by the number of hit OD PMTs near theexit point (NHITEX). The third criteria, (3-3) and (4-3), are adopted to keep the performance ofthe event reconstruction. The track length is determined by the distance between the entrance
and exit points for UPMU through samples. The UPMU through sample is further separated into
showering muon sample and non-showering muon sample. The showering muon events consistof the high energy muons which lose energy through radiative processes such as bremsstrahlung,e+ e− pair production and photo-nuclear interactions and the energy of the parent neutrino
is approximately 1 TeV. The momentum of UPMU stopping is determined from the observed
charge by the same way as that for FC and PC events (see Appendix D.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.

6.2

Vertex Distribution

Figures 6.2 and 6.3 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 ν µ ↔ ντ
oscillation with (sin2 2θ, ∆m2 ) = (1.0, 2.1 × 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 Sub-GeV samples observed data (points) show the little higer statistics than theMonte Carlo events taking account of neutrino oscillation (solid lines), which can be understoodas follows: the calculated neutrino flux corresponding to this energy range may be smaller
than the actual neutrino flux as indicated by the comparison between hadronic interaction
model and observed µ+ + µ− data (see Figure 3.2, where pµ . 3GeV corresponds to SubGeV neutrino events). These amounts of excesses are within the systematic uncertainty in theabsolute normalization. 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 forboth SK-I and SK-II, but they are rejected by the fiducial volume cut.

6.3

Number of Ring and Momentum Distribution

Figure 6.4 shows the distribution of the number of rings for data and Monte Carlo events.
Figures from 6.5 to 6.7 show the distribution of the momentum for data and Monte Carlo events.
The distributions for data and the oscillated Monte Carlo events are consistent.

6.4

Zenith angle Distribution

Figures from 6.8 to 6.12 show the zenith angle distributions. The data (dots and errors) arecompared with the Monte Carlo expectation without oscillations (dashed lines) and the bestfit expectation for 2-flavor νµ ↔ ντ oscillations (solid lines). The 2-flavor νµ ↔ ντ oscillation
hypothesis provides a consistent explanation to all data samples.
79

SK-I
FC Sub-GeV
Single-ringe-like0µedecay
1µedecay
π 0 -likeµ-like0µedecay
1µedecay
2µedecay
2-ring π 0 -like

SK-II

Data

Monte Carlo

Data

Monte Carlo

3426
2984
275
167
3219
1034
2035
150
528

3065.72663.1210.9
191.8
4321.91412.32745.3
164.3
615.0

1841
1605
155
81
1686
563
1043
80
284

1816.61456.6119.5
100.9
2343.0801.41457.1
84.5
322.1

FC Multi-GeV
Single-ringe-likeµ-likeMulti-ringe-likeµ-like
PC

829
694

894.1
1028.9

417
379

478.4
557.3

433
617
898

492.0
920.0
1185.4

260
361
430

278.8
484.7
589.6

UPMU
stopping
non-showering
showering

417.7
1544.8296.5

705.2
1482.8
289.3

208.6
771.2
105.4

359.2
763.0
105.9

Table 6.1: Summary of atmospheric neutrino events for data and Monte Carlo events of FC, PC
and UPMU 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 UPMU 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 the live time of the data.

80

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

77.7 %
12.5 %
1.0 %
1.3 %

3.8 %
50.7 %
10.1 %
8.5 %

10.6 %
7.2 %
2.0 %
0.5 %

68.4 %
14.8 %
1.7 %
1.7 %

2.7 %
3.7 %
0.6 %
0.7 %

CC νµ + ν µ

0.6 %15.2 %

7.0 %

2.0 %

2.5 %

NC

6.8 %

11.2 %

72.0 %

11.1 %

89.4 %

CC
νe + ν e

FC Sub-GeV Single-ring µ-like
0µedecay 1µedecay 2µedecayall71.3 %
12.9 %
1.1 %
0.8 %78.5 %
15.6 %
1.6 %
1.5 %

5.8 %
66.4 %
15.7 %
8.6 %

73.4 %
16.7 %
2.0 %1.6 %CC νe + ν e

1.8 %<0.1 %<0.1 %

0.6 %NC

11.8 %

2.6 %

3.3 %

5.7 %

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

Table 6.2: Fraction of each neutrino interaction mode in FC Sub-GeV Single-ring and Sub-GeV
π 0 -like atmospheric neutrino Monte Carlo events.

FC Multi-GeV
Single-ring e-like

FC Multi-GeV
Multi-ring e-like

33.6 %
23.9 %
13.7 %
2.0 %3.3 %22.9 %
39.8 %1.0 %CC νµ + ν µ

5.4 %10.6 %

NC

13.6 %

17.0 %

CC
νe + ν eQ.E.
single meson
multi πcoherent π

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

81

FC Multi-GeV
Single-ring µ-like

FC Multi-GeV
Multi-ring µ-like

51.2 %
32.5 %
12.3 %
2.7 %

5.1 %38.0 %
43.2 %
2.0 %CC νe + ν e

0.3 %

2.7 %

NC

0.3 %

7.4 %

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

Table 6.4: Fraction of each neutrino interaction mode in FC Multi-GeV µ-like atmospheric
neutrino Monte Carlo events.

PC
stop

UPMU
shower non-shower

20.8 %
25.3 %
49.4 %
1.7 %

16.1 %
22.8 %
60.0 %
0.3 %

1.3 %
2.9 %
95.8 %
0.0 %

4.0 %
7.7 %
88.3 %
0.0 %CC νe + ν e

2.0 %

0.6 %

0.0 %

0.0 %

NC

0.9 %

0.2 %

0.0 %

0.0 %

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

Table 6.5: Fraction of each neutrino interaction mode in PC and UPMU atmospheric neutrinoMonte Carlo events.

82

Number of events

FC Sub-GeV µ-like

200

150

150

100

100

50

50

0

0

60 FC Multi-GeV e-like

60 FC Multi-GeV µ-like

40

40

20

20

0

0
FC multi-ring µ-like

80

60

40

40

20
0

1000

2000

R2 (cm2)

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

R2 (cm2)

3

PC

60

40

40

20

20

0

× 10

0

1000

0

2000

R2 (cm2)

FC Multi-GeV µ-like

40

30

PC

20
0

100

0

80

60

FC Sub-GeV µ-like

FC Sub-GeV e-like

Number of events

FC Sub-GeV e-like

200

× 10

3

0

1000

2000

R2 (cm2)

× 10

3

Figure 6.2: 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 ) =(1.0, 2.1 × 10−3 eV2 ) (solid line). The left six panels show those for SK-I and the right panels
show those for SK-II. Excess of data compared to the Monte Carlo events (solid line) shown in
the entire Sub-GeV sample distribution is mentioned in the text.

Figure 6.13 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 areincluded.

83

FC Sub-GeV µ-like

FC Sub-GeV e-like

FC Sub-GeV µ-like

FC Sub-GeV e-like

150

150

150

150

100

100

100

100

50

50

50

50

0
FC Multi-GeV e-like40

FC Multi-GeV µ-like

40

30

30

20

20

10

10

0

0
FC multi-ring µ-like

40

0

Number of events

Number of events

0

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

1000

-1000

Z (cm)

01000

Z (cm)-1000

0

Z (cm)

1000

PC

80

30

-1000

FC Multi-GeV µ-like

0

-1000

0

1000

Z (cm)

Figure 6.3: 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 ) =(1.0, 2.1 × 10−3 eV2 ) (solid line). The left six panels show those for SK-I and the right panels
show those for SK-II. Excess of data compared to the Monte Carlo events (solid line) shown in
the entire Sub-GeV sample distribution is mentioned in the text.

Number of events

10

4

FC Sub-GeV

10

4

FC Multi-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

FC Sub-GeV

10

4

3

10

3

10

2

10

2

1

2

3

4

Number of rings

3

4

≥5

FC Multi-GeV

10

10

2

Number of rings

≥5

10

1

2

3

4

≥5

Number of rings

84

Figure 6.4: 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 ) =(1.0, 2.1×10−3 eV2 ) (solid line) for Sub-GeV
events (left) and Multi-GeV events (right).
The upper two panels show those for SK-I
and the bottom panels show those for SK-II.

Number of events

1000

1000

FC 1-ring Sub-GeV e-like

750

750

500

500

250

250

0
0

500

1000

1500

0
0

Number of events

Momentum (MeV/c)

400

200

200

1000

1500

0
0

Number of events

Momentum (MeV/c)

1500

500

1000

Figure 6.5: The momentum distributions
for FC Sub-GeV Single-ring e-like samples(left panel) and FC Sub-GeV Single-ring µlike samples (right panel). Upper panels
for SK-I and lower panels for SK-II. Dot
stands for data, dashed line for Monte Carloevents assuming no oscillation, and solidline for 2-flavor νµ ↔ ντ oscillation with(sin2 2θ, ∆m2 ) = (1.0, 2.1 × 10−3 eV2 ).

1500

Momentum (MeV/c)

FC 1-ring Multi-GeV e-like

400

FC 1-ring Multi-GeV µ-like

300
300
200

200

100
0
1
200

Number of events

1000

FC 1-ring Sub-GeV µ-like

FC 1-ring Sub-GeV e-like

500

500

Momentum (MeV/c)

400

0
0

FC 1-ring Sub-GeV µ-like

100
10

100

0
1

10

Momentum (GeV/c)

Momentum (GeV/c)

FC 1-ring Multi-GeV e-like

FC 1-ring Multi-GeV µ-like

150

100

200

100
100
50
0
1

10

Momentum (GeV/c)

100

0
1

10

100

Momentum (GeV/c)

85

Figure 6.6: The momentum distributions
for FC Multi-GeV Single-ring e-like samples (left panel) and FC Multi-GeV Singlering µ-like samples (right panel). Upper
panels for SK-I and lower panels for SK-II.
Dot stands for data, dashed line for Monte
Carlo events assuming no oscillation, andsolid line for 2-flavor νµ ↔ ντ oscillation
with (sin2 2θ, ∆m2 ) = (1.0, 2.1 × 10−3 eV2 ).

Number of events

200

FC Multi-ring µ-like

PC

300

150
200
100
100

50
0
0

1000 2000 3000 4000 5000

00Number of events

Momentum (MeV/c)
150

2000 4000 6000 8000 10000

Momentum (MeV/c)

FC Multi-ring µ-like

PC

100

100
50
50

0
0

1000 2000 3000 4000 5000

Momentum (MeV/c)002000 4000 6000 8000 10000

Momentum (MeV/c)

86

Figure 6.7: The momentum distributions
for FC Multi-ring µ-like samples (left panel)and PC samples (right panel). Upper panels for SK-I and lower panels for SK-II. Dot
stands for data, dashed line for Monte Carloevents assuming no oscillation, and solidline for 2-flavor νµ ↔ ντ oscillation with(sin2 2θ, ∆m2 ) = (1.0, 2.1 × 10−3 eV2 ).

SK−1+2 Sub−GeV 1−ring e−like 0mue
600

SK−1+2 Sub−GeV 1−ring e−like 1mue
140

500

120

400

100
80

300

60

200

40

100

20

0−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1
cosθ

02

(a) Sub-GeV Single-ring e-like 0 decay-electron(b) Sub-GeV Single-ring e-like 1 decay-electron

SK−1+2 Sub−GeV 1−ring mu−like 0mue

SK−1+2 Sub−GeV 1−ring mu−like 1mue

250

500

200

400

150

300

100

200

50

100

2.2

2.4

2.6 log momentum
2.8
3(MeV/c)3.2

0−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1cosθ

0−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1
cosθ(c) Sub-GeV Single-ring µ-like 0 decay-electron(d) Sub-GeV Single-ring µ-like 1 decayelectron

SK−1+2 Sub−GeV 1−ring mu−like 2mue
80
70
60
50
40
30
20
10
02

2.2

2.4

2.6 log momentum
2.8
3(MeV/c)3.2(e) Sub-GeV Single-ring µ-like 2 decay-electron

Figure 6.8: Zenith angle and momentum distributions of the observed data (points with statistical errors) and the Monte Carlo expectations (solid and dashed lines). Dashed line indicatesthe expectation assuming the best-fitted 2-flavor νµ ↔ ντ oscillations, while solid line assumes
the expectation with null oscillation.
87

SK−1+2 Sub−GeV 1−ring pi0−like
100SK−1+2 Sub−GeV 2−ring pi0−like
350
300

80

250

60

200

40

150
100

20
02

50
2.2

2.4

2.6 log momentum
2.8
3(MeV/c)3.2

02(a) Sub-GeV Single-ring π 0 -like

2.2

2.4

2.6 log momentum
2.8
3(MeV/c)3.2(b) Sub-GeV 2-ring π 0 -likeFigure 6.9: Momentum distributions of the observed data (points with statistical errors) andthe Monte Carlo expectations (solid and dashed lines). Dashed line indicates the expectation
assuming the best-fitted 2-flavor νµ ↔ ντ oscillations, while solid line assumes the expectation
with null oscillation.

88SK−1+2 Multi−Gev 1−ring e−like
SK−1+2 Multi−Gev 1−ring mu−like
220
200
200
180
180
160
160
140
140
120
120
100
100
80
80
60
60
40
40
20
20
0−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1 0−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1
cosθ
cosθ
(a) Multi-GeV Single-ring e-like

SK−1+2 Multi−Gev Multi−ring e−like140(b) Multi-GeV Single-ring µ-like

SK−1+2 Multi−Gev Multi−ring mu−like

180
160
140
100
120
80
100
80
60
60
40
40
20
20
0−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1 0−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1
cosθ
cosθ
120

(c) Multi-GeV Multi-ring e-like(d) Multi-ring e-likeFigure 6.10: Momentum distributions of the observed data (points with statistical errors) andthe Monte Carlo expectations (solid and dashed lines). Dashed line indicates the expectation
assuming the best-fitted 2-flavor νµ ↔ ντ oscillations, while solid line assumes the expectation
with null oscillation.

89

SK−1+2 PC stop

SK−1+2 PC through
250

50
200

40

150

30
20

100

10

50

0−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1cosθ

0−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1
cosθ(a) PC OD stopping(b) PC OD through going

Figure 6.11: Zenith angle distributions of the observed data (points with statistical errors) andthe Monte Carlo expectations (solid and dashed lines). Dashed line indicates the expectation
assuming the best-fitted 2-flavor νµ ↔ ντ oscillations, while solid line assumes the expectation
with null oscillation.

90

SK−1+2 Upmu stoppingSK−1+2 Upmu non−showering
180
500
160
400
140
120
300
100
80
200
60
40
100
20
0−1 −0.9−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.1 0 0−1 −0.9−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.1 0
cosθ
cosθ
(a) UPMU stopping(b) UPMU through non shower-like

SK−1+2 Upmu showering
100
80
60
40
20
0−1 −0.9−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.1 0cosθ(c) UPMU through shower-likeFigure 6.12: Zenith angle distributions of the observed data (points with statistical errors) andthe Monte Carlo expectations (solid and dashed lines). Dashed line indicates the expectation
assuming the best-fitted 2-flavor νµ ↔ ντ oscillations, while solid line assumes the expectation
with null oscillation.

91

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

92

Chapter 7

Analysis Method and Systematic
Uncertainties
We evaluate an agreement of the fitting by χ2 test, which is performed with the FC, PC and
UPMU samples of the SK-I and SK-II atmospheric neutrino data. These samples are divided into
400 bins for SK-I and 350 bins for SK-II according to the reconstructed event type, momentum
and zenith angle. The SK-I and SK-II data are individually handled due to the difference inthe detector response and the solar modulation effect in the atmospheric neutrino flux. First,
we define the formulation of χ2 , and next, analysis binning is presented in Chapter 7.2. Finallysystematic uncertainties are discussed.

7.1

Definition of χ2

Atmospheric neutrino data are divided into fine bins to have the high sensitivity dependingon the neutrino energy and zenith angle, i.e. propagation length. In order to treat the finerbinning, the likelihood is based on Poisson probabilities. Suppose we have totally n bins fordata indexed by i, the likelihood can be expressed as
N obs

L(N exp , N obs ) =

exp
i
n
Y
e−Ni N exp

i

Niobs !

i=1

(7.1)

where Niexp and Niobs is the number of expected and observed events in i-th bin, respectively.
The log likelihood ratio gives χ2 ,

n 
X
Niobs
L(N exp , N obs )
exp
obs
obs
=2
Ni − Ni + Ni ln exp
χ ≡ −2 ln
L(N obs , N obs )
Ni
2

(7.2)

i=1

The effects of independent systematic uncertainties are included into the bin following “pullmethod” [143]. The number of expected events in a bin varies according to the systematic
uncertainty, thus Niexp is replaced with
Niexp → Niexp (1 +
93

m
X
j

fji j )(7.3)

where j is a variation of j-th systematic uncertainty and fji is the fractional change of the event
rate in the i-th bin due to the j-th systematic uncertainty. m is the total number of systematic
uncertainties.
For each uncertainty, the parameter j is estimated during the fit to χ2 and the resulting
quantity j /σjsys follows Gaussian distribution, where σjsys is an estimated 1σ value of the j-thsystematic uncertainty. In order to constrain the range of j and hence the effect of systematic
uncertainty, additional term
!2
m
X
j
(7.4)
σjsys
j

is added to eq.(7.2). Thus the χ2 with totally n = 750 bins and m = 90 systematic uncertainties
is defined as


!2
90
750
90
obs
X
X
X

N
j
exp
2
i
obs
obs
i
+
N (1 +
χ =2
(7.5)
fj j ) − Ni + Ni ln exp
P
i
i )
σjsys
Ni (1 + 90
f
j
j=1 j
i=1

j=1

j=1

In this equation, 90’s j are varied to minimize χ2 for each choice of oscillation parameters.
The minimum χ2 is obtained when ∂χ2 /∂l = 0 is realized for every l with l = 1−90. ∂χ2 /∂l =0 can be reduced to
P obs i
X exp
X j
i
i N i flP i +
δjl = 0(7.6)
N i fl −(σjsys )2
1 + j fj  jijHere we add

XXj

Niobs fji fli j

(7.7)

i

to the both hands in eq.(7.6), and obtain the form
X X
j

Niobs fji fli +i

1(σjsys )2

δjl

!

j =

X
i



Niobs fli 

1+

1
P

i
j fj  j

+

X
j


exp
N
fji j − iobs NiThis means that the solution of ∂χ2 /∂l = 0 can be deduced from the matrix equationX
Mlj j = νl

(7.8)

(7.9)

j

where Mlk and νl are expressed by means of eq.(7.8)
X

Niobs fji fli +

i

X
i



Niobs fli 

1+

1
P

i
j fj  j

+

X
j

1

δjl → Mlj
(σjsys )2expNfji j − iobs  → νlNi

(7.10)(7.11)In the actual calculation, the first term of νl is computed by the series expansion upto O(5j ).
The matrix Mlk is symmetric and invertible. If the number of bins (n) is much larger than
the number of systematic uncertainties (m), this approach has the advantage rather than the94covariance matrix approach due to the difference between the inverse matrix m×m (pull-method)and n × n (covariance). Since νl contains j itself, eq.(7.9) is solved iteratively beginning with
j = 0 for every j until the value of χ2 stabilizes.
The effect of systematic uncertainties is taken into account through the coefficients f ji ineq.(7.5). These fji are calculated in advance using the Monte Carlo simulation. If we write thebin contents with only j-th uncertainty as (Niexp )j , and assume fji to be linear on the bin content
and taken to be the slope of the line between its content at ±σjsys , the coefficients fji are definedas(Niexp )j(+σ) − (Niexp )j(−σ)
ifj =(7.12)
2(Niexp )j(0)As a most simple case, we mention the uncertainty in the total cross section of single meson
production (j = 18). As is discussed later, this uncertainty is estimated to be 20%. It turns out
i is given as
the change of bin contents ±20%, then f18
if18=

1.2 − 0.8
= 0.2
2 × 1.0(7.13)

While for the energy dependent uncertainty, for example, fji of the absolute normalization uncertainty below 1 GeV (j = 1, see Figure 7.5) is expressed as
f1i = −0.2Eν (GeV) + 0.27

(7.14)

During the fit, the values of fji are taken from the table prepared in advance.

7.2

Analysis Binning

In order to obtain the high sensitivity the binning should be sufficiently fine. However, too
narrow binning causes the decrease of the number of expected events in a bin and then thecontribution to the χ2 grows up due to the increase of the error.
If the true number of expected events in a bin is assumed to be Ntrue with livetime L, thenumber of expected events NMC of a Monte Carlo simulation with livetime αL will be αNtrue .
Suppose we can observe Ñtrue with livetime L and ÑMC with livetime αL in a bin, they will not
be exactly Ntrue or αNtrue , but distribute following Poisson statistics about their true values.
We define the probability to observe Ñtrue and ÑMC events from a Poisson distribution of mean
Ntrue and NMC as P (Ñtrue |Ntrue ) and P (ÑMC |NMC ) = P (ÑMC |αNtrue ), respectively. By meansof the probabilities the average contribution to the reduced χ2 is expressed as an average overthe possible values of ÑMC and Ñtrue ,h(Ntrue ) =∞XÑtrue =0

P (Ñtrue |Ntrue )∞X

ÑMC =0

P (ÑMC |αNtrue )χ2 (Ñtrue , ÑMC /α)(7.15)

where χ2 is taken from eq.(7.2).
Figure 7.1 shows the average contribution to the reduced χ2 for varying amounts of uncertainty in the expected number of events. For less than three expected events, the contribution
to the χ2 is considerably greater than 1. Around five expected events the curve begins to flatten
and continues smoothly as increase in the number of expected events. Therefore the analysis
95

Reduced χ 2

1.4
1.2
1
0.8
0.6
0.4

2

4

6
8
10
Expected events

Figure 7.1: Average contribution to the reduced χ2 as a function of the expected number of
events in a bin. The factor of Monte Carlo simulation to the observation is 5 for blue curve, 20
for green curve, 100 for red curve, and no error for black curve.

binning has been chosen to avoid the feature at low expectation. The expected number of eventsin a bin in SK-II is smaller than that in SK-I, then the different binning are employed between
SK-I and SK-II to avoid the binning effect on the fitting results, especially in higher energy
regions which have less statistics. Also we can see the small factor of Monte Carlo simulation tothe observation yields the large reduced χ2 . In order to perform the precise analysis, we prepare
the Monte Carlo simulation corresponding to the 500 years of livetime for both SK-I and SK-II,in which the factors are approximately 120 for SK-I and 230 for SK-II. Therefore the statistics
of Monte Carlo simulation is sufficient.
Figures 7.2 and 7.3 show the definition of the binning for SK-I and SK-II, respectively. The
SK-I and SK-II data are individually handled due to the difference in the detector response andthe solar modulation effect in the atmospheric neutrino flux. A total of 750 bins are consideredin the calculation of χ2 with their associated sets of the observations and the Monte Carlopredictions. In this thesis, we employ the common binning to the other atmospheric neutrino
analysis in Super-Kamiokande to evaluate the results based on the same framework.
For the FC samples, the Sub-GeV Single-ring e-like 1µedecay, Sub-GeV Single-ring µ-like
2µedecay, and Sub-GeV π 0 -like samples are divided only by the momentum due to the small
statistics and poor angular correlation for NC events. Other FC samples are further divided
into 10 zenith angle bins equally spaced between cos Θ = −1 and cos Θ = +1, where cos Θ iscosine of zenith angle of Cherenkov ring direction. The PC samples are similarly divided into
10 bins equally spaced between cos Θ = −1 and cos Θ = +1.

7.3

Systematic Uncertainties

We describe the systematic uncertainties in this chapter. The systematic uncertainties ofthe event selection, detector response, and solar moduration effect in the atmospheric neutrino
96Upmu shower(1 x 10 bins)

FC Multi-GeV
1-ring e-like

FC Multi-GeV
mulit-ring e-like(4 x 10 bins)(4 x 10 bins)

FC Mulit-GeV
1-ring µ-like(2 x 10 bins)

FC Multi-ringµ-like(4 x 10 bins)

PC stop

PC through(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(5 x 10 bins)

1 µedecay(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 7.2: 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(4 x 10 bins)(4 x 10 bins)

FC Mulit-GeV
1-ring µ-like(2 x 10 bins)

FC Multi-ringµ-like(4 x 10 bins)

PC stop

PC through(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(5 x 10 bins)

1 µedecay(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 7.3: 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.
97flux are individually considered between SK-I and SK-II, while others are identical for SK-I and
SK-II. Among 90 systematic uncertainties, 19’s are for the neutrino flux, 15’s for the neutrino
interaction, 28’s for the SK-I detector response and 28’s for the SK-II detector response.
In this chapter, only uncertainties contributory to the analysis results are presented andothers are summarized in Appendix A. The effect of systematic uncertainty is taken into accountanalysis bin by bin, while all uncertainties are treated as uncorrelated each other.

7.3.1

Atmospheric Neutrino Flux

We discussed the atmospheric neutrino flux in Chapter 3.2. Systematic uncertainties in
the prediction of the atmospheric neutrino flux are estimated by the comparison of the Honda
flux [3] with other flux models, the FLUKA flux [39] and the Bartol flux [40]. The systematic
uncertainties in the atmospheric neutrino flux are listed in Table 8.1 and 8.2 in Chapter 8.
• Absolute normalization
The origin of the uncertainties in the atmospheric neutrino flux calculation is studied
in [3]. Figure 7.4 shows the uncertainty of each error source and their sum, in which δ πaccounts for an uncertainty due to pion production in the hadronic interaction model, δ K
for Kaon production uncertanty, δσ for hadronic interaction cross section uncertainty, andδair for atmosphere density profile uncertainty. Uncertainties due to pion production andKaon production are dominant sources above a few GeV of neutrino energy, while thecontribution of interaction cross section uncertainty grows up below 1GeV.
The sum of δπ and δσ is considered in this uncertainty term as shown in the Figure 7.5,
although the uncertainties of Kaon production and atmosphere density profile are included
in other term. Note that absolute normalization uncertainty is divided into two terms,
below 1 GeV and above 1 GeV, because the major error source is different below andabove 1 GeV.
• Flavor ratio
The flavor ratio (νµ + ν µ )/(νe + ν e ) of the atmospheric neutrino flux is estimated by thecomparison of the Honda flux with the FLUKA and the Bartol flux. According to thecomparison shown in the Figure 7.6, systematic uncertainty is estimated to be 2 % forEν <1 GeV, 3 % for 1 GeV< Eν <10 GeV, and 5 % for 10 GeV< Eν <30 GeV. Above
30 GeV, the uncertainty increases almost linearly with logEν from 5 % (30 GeV) to 30 %(1 TeV). In the low energy region, the uncertainty mainly comes from the pion spectrum
in the primary hadronic interactions, while in the high energy region, the uncertainty
originates in the K/π production ratio. Due to the difference of error source in eachenergy region, flavor ratio uncertainty is divided into three terms: Eν <1 GeV, 1 GeV<Eν <10 GeV, and Eν >10 GeV.
• K/π ratio
As seen in Figure 7.7, atmospheric neutrinos are mainly produced by π + /π − below 10 GeVof neutrino energy, while K’s contribution is sizable around a few tens of GeV, and dominant above 100 GeV. According to the SPY experiment [144], the measurement of theK/π ratio has been performed achieving an accuracy of 3%, where K and π momentum
ranges from 7 GeV/c to 135 GeV/c. According to the measurement of the SPY experimentand the correspondance between hadron momentum and neutrino energy, we estimate the98Relative Uncertainty

0.30

δ tot
δπ
δK
δσ
δ air

0.25
0.20
0.15
0.10
0.05
0

0

10

10

1

10

2

10
E ν (GeV)

3

Absolute normalization (%)

Figure 7.4: The uncertainty of each error source for atmospheric neutrino flux and their sum [3].
In this study, δπ is estimated from the comparison of the observation and the calculation of
the atmosoheric muon flux, however this method loses the validity below 1 GeV(shaded region).
Note the statistical and systematic uncertainties are not shown in this figure. Figure is takenfrom [3].
35
30
25
20
15
10
5
0
10−1

1

10

102

103
104
Eν (GeV)Figure 7.5: Absolute normalization uncertainty due only to δπ and δσ as a function of neutrinoenergy. δK and δair are not included in this figure. The uncertainty is indivisually consideredbelow and above 1 GeV due to the difference of their origin.

99

(νµ+νµ)/(νe+νe) double ratio

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

1

10

102

103
Eν(GeV)

Figure 7.6: The flavor ratio (νµ + ν µ )/(νe + ν e ) of the atmospheric neutrino flux. Vertical axis
stands for the double ratio, FLUKA flux/Honda flux(solid) and Bartol flux/Honda flux(dashed).
A kink in the FLUKA flux around 8 GeV due to the technical treatment in the original calculation is corrected in the application.

uncertainty of K/π ratio to be 5% below Eν =100GeV and 20% above Eν =1TeV. The
uncertainty is assumed to increase linearly from 5% to 20% between 100GeV and 1TeV.
• Sample-by-sample normalization
An energy dependence is inevitable in the comparison using several flux models and can
not be explained by a simple spectral index uncertainty as seen in Figure 7.8. From acomparison of the predicted number of events among three flux models, 5 % is assigned as
the relative normalization uncertanty for these samples.

7.3.2

Neutrino Interactions

Much work has been done to understand theoretically and experimentally the neutrinonucleus interactions. Estimation of systematic uncertainties are based on their studies. The
systematic uncertainties in the neutrino interactions are listed in Table 8.3.
• Nuclear Effect in pion spectrum
The systematic uncertainty of the predicted pion energy spectrum is estimated by the comparison of NEUT [66, 67], our simulation library, and other neutrino interaction simulator,
NUANCE [145] in which nuclear effects are treated based on [146].

7.3.3

Event Selections• PC stop/through separation
PC events are classified into two categories, PC stop and through events, according to theenergy deposit in OD region. The uncertainty of PC stop/through separation is estimatedas follows: P Eanti /P Eexp distribution of MC events is shifted so as the mean value of the
distribution is consistent with that of data, where P Eanti stands for the observed p.e.s
in the OD and P Eexp for the expected p.e.s from the potential track length in the OD.
The fractional change in number of events between nominal MC and shifted MC in eachcategory is considered as the systematic uncertainty of the separation. The uncertainty is
100

Particle Number per bin

π −Contribution

150
νe
100

K −Contribution

Vertical directions
100 GeV/c(x5)

1 GeV/c
0.1 GeV/c
10 GeV/c

50
0

νµ

400
200

10 GeV/c
1 GeV/c

100 GeV/c

0.1 GeV/c1000µ

10 GeV/c

100 GeV/c

1 GeV/c

500
0.1 GeV/c

0
−1
10

0

10

10

1

2

10
10
p π, K (GeV/c)

3

Flux ratio

Figure 7.7: The momentum distributions of π’s and K’s relevant to the atmospheric µ’s and ν’swith fixed momenta, 0.1, 1.0, 10, 100 GeV/c. Figure is taken from [60].

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

1

10

102

103
Eν(GeV)

Figure 7.8: The ratio of the calculated neutrino flux in which the fluxes calculated in Bartol(solid line) and FLUKA (dashed line) are normalized by the Honda flux. Also see ν µ + ν µspectrum shown in the left panel of Figure 3.6.

101

estimated in OD bottom, barrel, and top region separately because of the possible position
dependence of OD calibration.
• Background subtraction of Upward going muon events
The main background for upward going muon is cosmic ray muons. Most of thse events are
rejected in the UPMU reduction. After the reduction, the precise fitter gives more accurate
fitting informations and rejects downward going muon. The remaining backgrounds after
the selection from the precise fitter are eliminated by eye-scanning. Nevertheless, there may
still exist some background in near horizontal direction due to the finite fitter resolution
and multiple scattering of muons inthe nearby the rock. The cosmic ray contamination isestimated by the shape of zenith angle distribution above horizon which is extrapolated
by fitting the zenith angle distribution below horizon. The contamination is estimatedfor upward stopping muon, upward through-going shower-like, and upward through-goingnonshower-like events separately.

7.3.4

Event Reconstructions

Effects of systematic uncertainty related to the event reconstruction are small in this analysis.

102

Chapter 8

Analysis with 2-Flavor HybridModel
We test non standard neutrino interactions (NSI) with the whole atmospheric neutrino data
in SK-I and SK-II. NSI consist of either flavor changing neutral current (FCNC) and lepton
non universality (NU) in the matter of the Earth. FCNC represents the neutrino interactions
with matter as να + f → νβ + f which allow the flavor transition through the neutral current
interactions, and NU indicates the difference between the να + f and νβ + f neutral current
elastic forward scattering amplitudes, where f denote the fermion in matter.
The scheme we focus on this chapter is that of NSI in the νµ − ντ sector coexist with 2-flavorνµ ↔ ντ standard neutrino oscillations, here we call it the 2-flavor Hybrid model. Propagationof neutrinos from source to the detector assuming the 2-flavor hybrid model is expressed as in
Figure 8.1. In this scheme, the flavor transition is only occurred between ν µ and ντ , whereas νeflux is kept as is. The 3-flavor hybrid model with 2-flavor νµ ↔ ντ standard neutrino oscillations
and NSI in the νe − ντ sector is discussed in Chapter 9.
Our interests in this chapter are(1) Test of the robustness of neutrino oscillations in the atmospheric neutrino data.
(2) Investigation of the possible presence of NSI. If we would not able to obtain the signal of
NSI, we derive the constraints to them.
where we use the robustness of the implementation of 2-flavor νµ ↔ ντ neutrino oscillations to
the atmospheric neutrinos as a probe of NSI.
First, we explain the formalism of this model in Chapter 8.1. Expected phenomena are
discussed in Chapter 8.2, and results of this test are presented in Chapter 8.4.

8.1

Formalism

For the sake of the convenience for the latter discussion, we introduce the complete, i.e.
3-flavor, formula of Hamiltonian to govern the propagation of να to νβ :




0
0
0
εee ε∗eµ ε∗eτ
√
1




Uαj 0 ∆m221
Hαβ =0  (U † )kβ + VMSW + 2GF Nf (~r) εeµ εµµ ε∗µτ  (8.1)
2E
0
0
∆m231
εeτ εµτ ετ τ103

Figure 8.1: Propagation of neutrinos from source to the detector within the 2-flavor hybrid
model.

where U is the PMNS matrix and VMSW is the MSW potential in the flavor basis. The first andsecond term represent neutrino oscillations and the third term is the NSI contribution. N f (~r)
is the fermion number density of the medium along the neutrino trajectory ~r, here we assume
d-quark density in the matter of the Earth just for the convenience when we compare our results
to the bounds given by neutrino scattering experiments. Note that the first term is energy
dependent, while the second and third term is energy independent and matter dependent.
We reduce eq.(8.1) to that of the 2-flavor hybrid model we focus on this chapter. In this work
we follow the formalism by M.C. Gonzalez-Garcia and M. Maltoni [147], where the propagationof neutrinos(+) and anti-neutrinos(−) is governed by the following Hamiltonian:
!
!
√
−1 0
−1 0
∆m2
†
†
Uθ
Uθ ± 2GF Nf (~r)Uξ,±η
H≡
Uξ,±η
(8.2)
4E
0 1
0 1
The matrics Uθ and Uξ,±η are given by:Uθ =cos θ sin θ− sin θ cos θ

!

!
cos ξ
sin ξe±iη
Uξ,±η =− sin ξe∓iη
cos ξ


εµτ
1
ξ = arctan2(ετ τ − εµµ )/2

(8.3)
(8.4)
(8.5)

√
where a possible non-vanishing relative phase η(≡ δ(εµτ )) is considered.
2GF Nf (~r)εµτ isthe√ amplitude of the flavor changing neutral current(FCNC) process ν µ + f → ντ + f , while
2GF Nf (~r)(ετ τ − εµµ ) is the amplitude of lepton non universality(NU). These interactions aredefined in Chapter 1.3. Note that VMSW is not appeared in the νµ − ντ sector.
If the matter profile in the Earth is constant along the neutrino trajectory, P νµ →νµ is expressed as:22
2 ∆m LR(8.6)
Pνµ →νµ = 1 − Pνµ →ντ = 1 − sin 2Θ sin
4E
104

12
10

10

8
6
5
4

70
60
50
40
30
20

2
00

80

2GF NfSeismic velocity(km/s)

3

Density(g/cm )

15
14

10
2000

4000

6000
Depth(km)

0

0−1

−0.8

−0.6

−0.4

−0.2

0
cosθ

Figure 8.2: (Left) PREM predicted matter density (solid line) and seismic velocity (dashed line
for P-wave and dotted line for S-wave) as a function of the depth, where depth = 0 km is thesurface of the Earth. (Right) The average fermion number density as a function of zenith angle.

where the effective mixing angle Θ and NSI’s correction factor to the oscillation wavelength,
R, are given as1(sin2 2θ + R02 sin2 2ξ + 2R0 sin 2θ sin 2ξ cos η),2
Rq
R = 1 + R02 + 2R0 (cos 2θ cos 2ξ + sin 2θ sin 2ξ cos η).

sin2 2Θ =

(8.7)
(8.8)

R0 gives the ratio between standard oscillation and NSI to the oscillation wavelength
R0 = ±

λmatter 4E
2 ∆m2

(8.9)

NSI effect λmatter is given as:√
λmatter = 2 2GF Nf (~r)F
ρ(~r)Earth
F GeV≡ 4.58 × 10−22 (2 − Yp )3g/cm3r(ετ τ − εµµ )2
F = |εµτ |2 +4

(8.10)(8.11)

We employ the PREM model [24] to describe the matter density profile and chemical composition, as shown in Figure 8.2, where the proton/nucleus ratio in the mantle and core are set
to be Yp = 0.497 and 0.468, respectively. The fermion f in the above formulae are assumed to
be d-quark. This assumption is just for convenience when we compare our results to the bounds
given by neutrino scattering experiments. In the calculations we take the average matter density
along the path of neutrino.

8.2

Expected Phenomena

In this section we describe expected phenomena driven by NSI. As explicitly expressed ineq.(8.2), the first term (standard 2-flavor oscillations) depends on the neutrino energy, while the
105

1

cos θ

cos θ

0

0

1

-0.2

0.8 -0.2

0.8

-0.4

0.6 -0.4

0.6

-0.6

0.4 -0.6

0.4

-0.8

0.2 -0.8

0.2

-1
0

10

20

30
Eν (GeV)

0

-1
0

10

20

30
Eν (GeV)0

Figure 8.3: Survival probability of νµ → νµ for standard 2-flavor oscillation (left panel) andthe 2-flavor hybrid model (right panel). In both panels ∆m2 = 2.1 × 10−3 eV2 and θ = 45◦ areassumed. Only in the right panel εµτ (FCNC)=0.015 and ετ τ − εµµ (NU)=0.05, corresponding to
the present limits, are assumed. For simplicity, the matter density is assumed to be constant,ρ = 5.0g/cm3 .

second term (NSI) is energy independent and matter dependent, thus the relative effect of NSI in
the hybrid model is expected to vary according to the neutrino energy. First we demonstrate thesurvival probability of νµ → νµ for standard 2-flavor oscillation and the 2-flavor hybrid modelin Figures 8.3 and 8.4, where only the upward region is presented because NSI is effective only
in matter, so no difference is found in the angular region from downward to horizon.
We discuss how NSI affects the atmospheric neutrino oscillations in three energy ranges, (1)Eν <1GeV, (2) 1< Eν <30GeV, (3) Eν >30GeV.
(1) Eν <1GeV
As seen in Figure 8.5, an eigenvalueof the vacuum term in eq.(8.2), ∆m2 /2Eν , is much√larger than that of matter term, 2GF Nf ε, where ∆m2 = 2.1 × 10−3 eV2 , Nf ≡ Nd ∼ 3Ne ,and ε = 1.0 are assumed. This means that νµ → ντ transition is mostly governed by 2-flavor
standard oscillation below 1GeV and no significant contribution of NSI is expected. Note thatthe eigenvalue of the solar term(∆m221 and θ12 ) has a comparable size to the matter effects below
200MeV, which will be discussed later.
(2) 1< Eν <30GeV
In this energy range the matter term competes with the vacuum term. The ν µ → ντ transition is no longer dominated by the standard oscillation, but also sizably modified by the matterterm. Modified oscillation parameters, effective mixing angle (Θ) and eigenvalue (∆m 2eff ) in thematter, for several εµτ and ετ τ −εµµ are presented in Figure 8.6. As shown in Figure 8.6, nonzeroετ τ − εµµ gives the supression of νµ → ντ transition due to the condition Θ < θ23 ∼ 45◦ andthe magnitude of supression becomes larger as the neutrino energy increases, while nonzero ε µτ
affects the frequency of the oscillation, ∆m2eff > ∆m2 . As seen in the left panel on Figure 8.3,
maximum transition in the case of standard oscillation, i.e. P (νµ → νµ ) ∼ 0 , occurrs in the
106

1
0.9

1
0.9

1
0.9

0.8

0.8

0.8

0.7

0.7

0.7

0.6

0.6

0.6

0.5

0.5

0.5

0.4

0.4

0.4

0.3

0.3

0.3

0.2

0.2

0.2

0.1

0.1

0.1

0

0

-1

-0.8

-0.6

-0.4

-0.2

0

Zenith angle(cosθ)

-1

-0.8

-0.6

-0.4

-0.2

0

0

Zenith angle(cosθ)

-1

-0.8

-0.6

-0.4

-0.2

0

Zenith angle(cosθ)3−3

Eigenvalues(10 GeV)

Figure 8.4: Survival probability of νµ → νµ for standard 2-flavor oscillation and the 2-flavor
hybrid model as a function of the zenith angle of the neutrino direction. Solid curves (2-flavor
hybrid model) and dashed curves (standard 2-flavor oscillation) indicate the average probability
over neutrino energy: Eν <1.4 GeV (left), 1.4 < Eν < 5.0 GeV (center), and 5.0 < Eν <30.0 GeV (right). In all panels ∆m2 = 2.1 × 10−3 eV2 and θ = 45◦ are assumed. As for NSI
parameters εµτ =0.015 and ετ τ − εµµ =0.05 are considered. Constant density ρ = 5.0g/cm3 isassumed for simplicity.

2

1

0
−1
10

1

10

10

2

10
Eν (GeV)3

Figure 8.5: Eigenvalues of the vacuum term and matter term in eq.(8.2). Solid curve indicatesthe eigenvalue of the√vacuum term, ∆m2 /2Eν , and the hatched area represents the eigenvalueof the matter term 2GF Nf ε, where ∆m2 = 2.1 × 10−3 eV2 , Nf ≡ Nd ∼ 3Ne , and ε = 1.0 are
assumed. The hatched area ranges from the case with ρ = 2.7g/cm 3 (bottom edge) to the casewith ρ = 13g/cm3 (top edge). For reference, dashed curve indicates the eigenvalue of vacuum
term with ∆m221 = 7.6 × 10−5 eV2 , which are discussed in the 3-flavor hybrid model analysis.

107

2

6

-3

5

Effective ∆ m2 (10 eV )Effective mixing angle(deg.)

70
60
50
40
30
20
10
0
0

20

40

60

80
100
E ν (GeV)

4
3
2
1
0
0

20

40

60

80
100
E ν (GeV)

Figure 8.6: (Left) Effective mixing angle in matter. (Right) Effective mass squared differencein matter. In both panels, solid curves indicate the case with εµτ (FCNC)=0.015 and ετ τ −εµµ (NU)=0.05, dashed curves for εµτ =0.015, dotted curves for ετ τ − εµµ =0.015, dashed-dottedcurves for ετ τ − εµµ =0.05. As for vacuum parameters, θ23 = 45◦ and ∆m2 = 2.1 × 10−3 eV2 areassumed. ρ = 5.0g/cm3 is assumed.

nearly upward direction at about 20GeV. On the other hand, once nonzero ε µτ is considered
the maximum transition can be realized even at larger zenith angles. This is recognized in theright panel of Figure 8.3 and Figure 8.4. If we focus on the zenith angle distributions, since thisenergy range corresponds to FC Multi-GeV, PC, and UPMU stopping samples, we expect thatthe magnitude of µ-like deficit becomes smaller due to ετ τ − εµµ and the shape of zenith angle
distributions between near horizon and upward are modified because of ε µτ in higher energyνµ -rich samples.
(3) Eν >30GeV
As indicated in the solid curve in Figure 8.5, vacuum oscillation has less of a leading effectin the νµ → ντ transition in this energy range, while a large transition is realized with an energy
independent matter term when the neutrino propagates in the Earth sufficiently long. In otherwords, this is a different case from (2), εµτ plays the leading role in this energy range since
the modified frequency of the oscillation has a comparable size to the path length, therefore aνµ → ντ transition driven by εµτ is expected. In contrast, ετ τ − εµµ takes a sub-leading effectbecause this effect is subject to the standard oscillation induced νµ → ντ transition which is lessapparent in this energy range. Overall, these phenomena are expected to be visible in UPMU
through samples which range from a few tens of GeV up to extreme high energy ∼ 100TeV.
Finally we summarize this discussion: εµτ gives a higher frequency, thus leads the νµ → ντtransition for shorter path lengths in matter, while ετ τ − εµµ suppresses the transition even atrelevant energies.

108

8.3

Analysis Method

A value of χ2 is evaluated at each grid point in the four-dimensional parameter space ofsin2 2θ23 , ∆m2 , εµτ , and ετ τ − εµµ in case without relative complex phase η. If we consider thephase η, χ2 is calculated with five-dimensional parameter space. Note that εµτ is symmetricalbetween negative and positive value, then we only consider the positive value.

8.4

Result of 2-Flavor Non-Standard Interaction Analysis

First we mention the case when the relative phase η is not considered. As a result of the
global scan on the four-dimensional oscillation parameter space, best-fit values are derived atsin2 2θ = 1.00,εµτ = 1.0 × 10−3 ,∆m2 = 2.2 × 10−3 eV2 ,ετ τ − εµµ = −2.7 × 10−2

χ2min = 838.9 / 746.0 d.o.f.

(8.12)The best-fit values of the systematic uncertainty parameters j obtained at the global minimum are summarized in Tables from 8.1 to 8.7.
Figure 8.7 shows the allowed regions of the neutrino oscillation parameters (sin 2 2θ, ∆m2 ).
The three contour curves correspond to the 68%, 90% and 99% confidence level (C.L.) allowed
regions which are defined to be χ2 = χ2min +2.30, 4.61, and 9.21 respectively.
As for the case when the relative phase η is considered, a global scan on the five-dimensional
oscillation parameter space gives best-fit values,sin2 2θ = 1.00,εµτ = 6.9 × 10−3 ,∆m2 = 2.2 × 10−3 eV2 ,ετ τ − εµµ = −1.9 × 10−2 ,cos η = −1.0

χ2min = 837.5 / 745.0 d.o.f.

(8.13)

No significant difference is found in the allowed neutrino oscillation parameter region between
the two cases, with and without relative phase.
The minimum χ2 value derived from the 2-flavor hybrid model is larger than that with thestandard 2-flavor neutrino oscillation shown in Appendix B, even if two or three free parameters
are additionally considerd. This can be understood by the difference in the “averaging method”.
As seen, for example, in the left panel of Figure 8.4, the oscillation probability varies frequently
when ∆m2 L/4Eν  1, and these frequent variations bring notchy allowed regions. In order toavoid such a problem, an “averaging method” is introduced, which takes P (ν µ → ντ ) = sin2 2θ/2in the case that a oscillation probability frequently varies between 0 and 1. However it cannot beapplied to the 2-flavor hybrid model as keeping the original mechanism because the probability
equation is different between the two cases. Therefore a modified averaging method is introduced
and this method gives a little bit larger χ2 value. Considering the disparity of the averaging
method, the difference of the minimum χ2 value between the 2-flavor hybrid model and standard
2-flavor neutrino oscillation is less than 1, therefore the choice of averaging method can be
negligible in the analysis.
The two allowed neutrino oscillation parameter regions, that of the the hybrid model in Figure 8.7 and standard oscillation in Appendix B, are consistent each other and also the differenceof the minimum χ2 is very small. Hence we can conclude that 2-flavor neutrino oscillation is arobust hypothesis for the atmospheric neutrinos and no significant contribution of NSI is found.
109∆m223 (eV 2 )10

SK−I + SK−II−2

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

10

−3

0.7

0.8

0.9

2
1
sin 2 θ23

Figure 8.7: Allowed oscillation parameters assuming the 2-flavor hybrid model, where undisplayed parameters εµτ and ετ τ − εµµ are integrated out. The three contours correspond to the
68%, 90% and 99% C.L. allowed regions, respectively.

8.4.1

Limit on NSI parameters without relative phase

We present the allowed 2-flavor NSI paramters when relative phase η is not considered.
Figure 8.8 shows the allowed regions of the NSI parameters (εµτ , ετ τ − εµµ ), where undisplayedparameters sin2 2θ and ∆m2 are integrated out. Three contours correspond to the 68%, 90%
and 99% confidence level (C.L.) allowed regions which are defined to be χ2 = χ2min +2.30, 4.61,
and 9.21, respectively. Figures 8.9 and 8.10 show the χ2 − χ2min distributions as a function of εµτand ετ τ − εµµ , respectively, which are sliced at the best-fit position of other parameters. Limits
on the NSI parameters in the νµ − ντ sector at a 90 % C.L. are :|εµτ | < 1.1 × 10−2 ,−4.9 × 10−2 < ετ τ − εµµ < 4.9 × 10−2

(8.14)

Systematic uncertainties are summarized in Tables from 8.1 to 8.7. Best-fit values are withoutparentheses.

8.4.2

Limit on NSI parameters with relative phase

Next we take the more general case when the relative phase η is considered. The allowed
2-flavor NSI parameters region is presented in Figure 8.11, where the relative phase η is scanned
from 0 to 2π. Therefore a χ2 value is calculated at each grid point in the five-dimensionalparameter space including η.
Figure 8.11 shows the allowed regions of the NSI parameters (εµτ , ετ τ −εµµ ), where the undisplayed parameters sin2 2θ, ∆m2 , and cos η are integrated out. The three contours correspondto the 68%, 90% and 99% confidence levels (C.L.) allowed regions with the same definition as
110

10

−3

SK−I + SK−II 10 −2

ε (FCNC)

10 −1

ε ’(NU)

10 −2

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

10

−3

−3

−ε ’(NU)

10

10 −2

10 −1

10

−3

10

−2

ε (FCNC)

Figure 8.8: Allowed NSI parameters region assuming the 2-flavor hybrid model without therelative phase. The horizontal axis shows εµτ (≡ ε, FCNC) and the vertical axis shows ετ τ − εµµ(≡ ε0 , NU). The undisplayed parameters sin2 2θ and ∆m2 are integrated out. The three contours
correspond to the 68%, 90% and 99% C.L. allowed regions, respectively. Star represents thebest-fit point for the NSI parameters.

111

min

χ 2 −χ 2

14
12
10
8
6
4
2
0 −3
10

ε

−2

10

14

12

12

10

10

8

8

6

6

4

4

2

2

0

min

14

χ 2 −χ 2

χ 2 −χ 2

min

Figure 8.9: χ2 − χ2min distributions as a function of εµτ (≡ ε) in the negative side of ετ τ − εµµwithout the relative phase, where sin2 2θ and ∆m2 are integrated out. The three horizontal
lines correspond to the 68%, 90% and 99% C.L..

−ε ’

10−1

10−2

−3

−3

10

10

−2

10

−1

10

ε’

0

Figure 8.10: χ2 − χ2min distributions as a function of ετ τ − εµµ (≡ ε0 ) without the relative phase,where sin2 2θ and ∆m2 are integrated out. The three horizontal lines correspond to the 68%,
90% and 99% C.L..

112

Figure 8.8. Figures 8.12 and 8.13 show the χ2 − χ2min distributions as a function of εµτ andετ τ − εµµ . The limit on the NSI parameters in the νµ − ντ sector at a 90 % C.L. is :|εµτ | < 1.6 × 10−2 ,−5.3 × 10−2 < ετ τ − εµµ < 4.9 × 10−2

(8.15)

Systematic uncertainties are summarized in Tables from 8.1 to 8.7. Best-fit values are withinparentheses.

8.5

Discussion

At the end of this section, we discuss how NSI parameters are constranied by atmospheric
neutrinos. As mentioned in Chapter 8.2, the NSI effects are expected to be visible in ν µ -richsamples above a few GeV. In order to check it, we present the allowed NSI parameter region
derived by sets of sub-samples in Figure 8.14, where the solid curve indicates the allowed NSI
region given by UPMU through, the dashed curve for PC and UPMU stopping, and the dottedcurve for FC Single-ring Multi-GeV and Multi-ring. The plane with negative ετ τ − εµµ is
presented, since the minimum χ2 is located in negative ετ τ − εµµ space. strongest constraint
to εµτ is given by UPMU through, while ετ τ − εµµ is constrained by the other sub-samples:
FC Single-ring Multi-GeV, FC Multi-ring, PC and UPMU stopping. Note that FC Sub-GeV
samples are less effective to place a limit on NSI in the νµ − ντ sector.
The Constraints to εµτ (FCNC) and ετ τ − εµµ (NU) make sense as follows: nonzero εµτmodifies the frequency of oscillation and the modified frequency leads to a νµ → ντ transition
with a shorter path length in matter. Although, when neutrino energy is sufficiently small, thefirst octant of the transition can occur at a smaller zenith angle even without ε µτ modification.
Thus it is not possible to observe a clear inconsistency between standard oscillation and nonzero
εµτ below ∼30GeV. In the energy region corresponding to the UPMU through samples, thefirst octant of standard oscillation is no longer found at any zenith angle, on the other hand
the modified frequency by εµτ allows the νµ → ντ transition in the upward direction. Indeed,
nonzero εµτ yields a deficit of UPMU through events, and results in a apparent conflict with the
expected standard oscillation observation.
In Figure 8.15, colored boxes show the ratio of UPMU through events in three upward bins(cos Θ = −1 ∼ −0.7) to that in three horizon bins (cos Θ = −0.3 ∼ 0), we call it the Up/Horizon
ratio, and the three contours are also displayed for reference. The negative ε τ τ −εµµ plane is only
presented due to the same reason as in Figure [?]. We can see the Up/Horizon ratio becomes
smaller as εµτ increases due to the deficit of upward events, and results in a constraint on ε µτ .
In contrast, we know that the ετ τ − εµµ induced supression of the νµ → ντ transition is foundin any energy range, especially clear in the range 1< Eν <30GeV. As shown in Figure 8.14, theνµ -rich higher energy samples of PC, UPMU stopping, and also UPMU through contribute toput a constraint on ετ τ − εµµ . Moreover, in Figure 8.15 the Up/Horizon ratio grows larger as
the absolute value of ετ τ − εµµ increases, however it is inconsistent with the observations.
Finally, we test how our limits are changed when the systematic uncertainty of K/π ratio is
considered, which is effective to the fraction of UPMU stopping and through events. Figure 8.16
shows the allowed regions with two cases: (solid lines) with all of systematic uncertainties, and(dashed lines) K/π ratio uncertainty is neglected. εµτ is sensitive to the K/π ratio uncertainty
since εµτ is mostly constrained by UPMU through events as well as UPMU stopping, as seen in
Figure 8.14. On the other hand, the modification of ετ τ − εµµ is rather small. This makes sence
as we remember ετ τ − εµµ is also constrained by the other sub-samples.
113

10

−3

SK−I + SK−II 10 −2

ε (FCNC)

10 −1

ε ’(NU)

10 −2

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

10

−3

−3

−ε ’(NU)

10

10 −2

10 −1

10

−3

10

−2

ε (FCNC)

Figure 8.11: Allowed NSI parameters region assuming the 2-flavor hybrid model with the relative
phase η. The horizontal axis shows εµτ (≡ ε, FCNC) and the vertical axis shows ετ τ − εµµ(≡ ε0 , NU). The undisplayed parameters sin2 2θ, ∆m2 , and cos η are integrated out. The threecontours correspond to the 68%, 90% and 99% C.L. allowed regions, respectively. Star represents
the best-fit point for the NSI parameters.

114

min

χ 2 −χ 2

14
12
10
8
6
4
2
0 −3
10

ε

−2

10

14

12

12

10

10

8

8

6

6

4

4

2

2

0

min

14

χ 2 −χ 2

χ 2 −χ 2

min

Figure 8.12: χ2 − χ2min distributions as a function of εµτ (≡ ε) in the negative side of ετ τ − εµµwith the relative phase η, where sin2 2θ, ∆m2 , and cos η are integrated out. The three horizontal
lines correspond to the 68%, 90% and 99% C.L..

−ε ’

10−1

10−2

−3

−3

10

10

−2

10

−1

10

ε’

0

Figure 8.13: χ2 − χ2min distributions as a function of ετ τ − εµµ (≡ ε0 ) with the relative phase η,
where sin2 2θ, ∆m2 , and cos η are integrated out. The three horizontal lines correspond to the
68%, 90% and 99% C.L..

115

10

−3

−2

10

−1

−ε ’

10

SK−I + SK−II

10

−3

10

ε

−2

Figure 8.14: Allowed NSI parameters region derived by sets of sub-samples, where horizontal axis
is εµτ (≡ ε) and vertical axis is ετ τ − εµµ (≡ ε0 ). Solid curve indicates the allowed region given
by UPMU through, dashed curve by PC and UPMU stopping, dotted curve by FC Single-ringMulti-GeV and Multi-ring. Contour corresponds to χ2 = χ2 + 2.31.

10−3SK−I + SK−II

0.45

0.4

10

0.35

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

0.3

0.25

−1

0.2

−ε ’

10

−2

10

−3

10

−2

ε

Figure 8.15: Colored boxes show the UP/Horizon ratio of UPMU through events, where horizontal axis is εµτ (≡ ε) and vertical axis is ετ τ − εµµ (≡ ε0 ). Three contours, same as in Figure 8.8,
are displayed for reference.

116

-3

10

10-2

- ε’

10-1

-3

-2

10

10

ε

Figure 8.16: Allowed NSI parameters region, where horizontal axis is ε µτ (≡ ε) and vertical axis
is ετ τ − εµµ (≡ ε0 ). All of systematic uncertainties are considered in solid lines, while K/π ratio
uncertainty is neglected in dashed lines.

117

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

uncertainty (%)

best-fit (%)

25.0
15.0
2.0
3.0
5.0a
5.0
5.0
8.0b
2.0
6.0
6.0c
0.1
0.3
0.8
0.5
0.7
0.2
0.2
0.2
0.3
0.2
0.2
0.1
0.1
1.4
1.9
3.2
2.3
1.4
1.3
2.8
1.5
1.7

34.2 (33.1)
20.5 (17.5)-0.7 (-0.6)-1.3 (-1.3)
7.0 (6.9)
3.1 (3.0)
1.1 (1.1)-0.9 (-0.8)-0.3 (-0.3)-0.2 (-0.1)
1.2 (1.3)-0.05 (-0.05)-0.1 (-0.2)-0.04 (-0.04)-0.2 (-0.3)-0.3 (-0.4)-0.1 (-0.1)-0.1 (-0.1)-0.1 (-0.1)-0.1 (-0.2)-0.1 (-0.1)-0.1 (-0.1)
0.005 (0.005)
0.005 (0.005)
0.07 (0.08)
0.1 (0.1)
0.2 (0.2)
0.1 (0.1)
0.07 (0.08)
0.07 (0.07)
0.1 (0.2)
0.08 (0.08)
0.09 (0.09)

Table 8.1: Summary of systematic uncertainties in atmospheric neutrino flux.
within(without) parentheses are without(with) relative phase η.
aUncertainty linearly increases with log Eν from 5 % (10 GeV) to 30 %(1 TeV).
Uncertainty linearly increases with log Eν from 8 %(100 GeV) to 20 %(1 TeV).
c
Uncertainty linearly increases with log Eν from 6 %(50 GeV) to 40 %(1 TeV).

b

118

Valuesuncertainty (%)(A2) Systematic uncertainties in neutrino flux
K/π ratio
5.0aLν (production height)
10.0
Sample-by-sample FC Multi-GeV
5.0
PC + upward stopping µ
5.0
Solar activity
SK-I
20.0
SK-II
50.0

best-fit (%)-5.9 (-5.6)
1.2 (1.6)-5.6 (-5.5)-10.1 (-9.7)
4.0 (4.0)
30.3 (30.3)

Table 8.2: Summary of systematic uncertainties in atmospheric neutrino flux (continued). Values
within(without) parentheses are without(with) relative phase η.
aUncertainty is 20 % above Eν =1 TeV. The uncertainty increases linearly from 5% to 20% between 100GeV
and 1TeV.

119

(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)
Single meson production (ν/ν ratio)
Single meson production (π 0 /π ± ratio)
Deep inelastic scattering (Eν < 10GeV)
Deep inelastic scattering (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 (%)

1.0a
1.0 b
1.0c
1.0d
20.0
1.0e
40.0
1.0f
5.0
100.0(50.0) g
20.0
30.0
1.0h
30.0
10.0

0.9 (1.1)
4.3 (4.1)
10.1 (10.0)
4.3 (4.0)-2.1 (-2.0)-3.1 (-3.1)-37.1 (-37.9)
1.3 (1.6)
1.3 (1.2)
15.4 (18.6)-23.9 (-23.5)-24.2 (-23.0)
10.7 (10.5)-10.9 (-10.2)-3.7 (-3.7)

Table 8.3: Summary of systematic uncertainties in neutrino interactions. Values within(without)parentheses are without(with) relative phase η.
aDifference from the MA = 1.1GeV/c is set to be 1.0
Difference from the Nieves model is set to 1.0cDifference from the Nieves model is set to 1.0
dDifference from the Nieves model is set to 1.0eDifference from the Hernandez model is set to 1.0
fDifference from CKMT parametrization is set to 1.0g
100% for CC νµ . 50% for CC νe and NC.
hDifference between NEUT and NUANCE is set to 1.0b120

(C) Systematic uncertainties in event selection for SK-I
Reduction for FC event
Reduction for PC event
FC/PC separation
Non-ν background
Flasher for e-like
Sub-GeV
Multi-GeV
Cosmic ray for µ-like Sub-GeV
Multi-GeVPC
PC stop/through separation OD top region
OD barrel region
OD bottom region
Reduction for UPMU event UPMU stopping
UPMU through
UPMU stopping/through separation
UPMU stopping energy cut
UPMU through showering/non-showering separation

uncertainty (%)

best-fit (%)

0.2
2.4
0.6
0.5
0.2
0.1
0.1
0.2
15.0
7.4
11.3
1.8
0.3
0.4
0.8
2.8

0.03 (0.03)-0.8 (-0.7)-0.01 (-0.01)
0.06 (0.05)
0.02 (0.02)-0.01 (-0.02)-0.01 (-0.02)-0.03 (-0.04)
11.4 (11.5)-0.6 (-0.5)-7.3 (-7.3)
0.5 (-0.1)
0.09 (-0.02)-0.03 (-0.04)-0.08 (-0.08)-2.0 (-2.0)

Table 8.4: Summary of systematic uncertainties in event selection for SK-I. Values
within(without) parentheses are without(with) relative phase η.

121

(D) Systematic uncertainties in event selection for SK-II
Reduction for FC event
Reduction for PC event
FC/PC separation
Non-ν background
Flasher for e-like
Sub-GeV
Multi-GeV
Cosmic ray for µ-like Sub-GeV
Multi-GeVPC
PC stop/through separation OD top region
OD barrel region
OD bottom region
Reduction for UPMU event stopping µthrough-going µ
UPMU stopping/through separation
UPMU stopping energy cut
UPMU through showering/non-showering separation

uncertainty (%)

best-fit (%)

0.2
4.8
0.5
0.3
0.7
0.1
0.1
0.7
19.0
14.0
18.0
2.1
0.3
0.4
1.5
1.8

0.006 (0.007)-2.6 (-2.5)
0.06 (0.06)-0.03 (-0.03)-0.07 (-0.07)
0.02 (0.02)
0.02 (0.02)
0.1 (0.1)-18.0 (-18.0)-22.4 (-22.0)-19.5 (-19.4)
0.6 (-0.1)
0.09 (-0.02)-0.02 (-0.03)-0.04 (-0.04)-0.5 (-0.4)

Table 8.5: Summary of systematic uncertainties in event selection for SK-II. Values
within(without) parentheses are without(with) relative phase η.

122uncertainty (%)(E) Systematic uncertainties in event reconstruction for SK-I
Single-ring/Multi-ring separation
Single-ring< 400 MeV
e-like2.3µ-like0.7> 400 MeV
e-like0.4µ-like0.7
Multi-GeV
e-like3.7µ-like
1.7
Multi-ring
Sub-GeV
e-like3.5µ-like
4.5
Multi-GeV
e-like3.1µ-like
4.1
Particle identification
Single-ring
Sub-GeV
e-like0.1µ-like-0.1
Multi-GeV
e-like0.2µ-like-0.2
Multi-ring
Sub-GeV
e-like2.3µ-like-3.9
Multi-GeV
e-like1.7µ-like-2.9
Energy calibration for FC event
1.1
Up/down symmetry of energy calibration
0.6
0π -like sample selection
100< Pe <250 MeV/c
11.2
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
2.0+
Decay electron tagging (π decay)
e-like 0µedecay
1.5∼1.7
e-like 1µedecay-4.4∼-3.8µ-like 0µedecay
1.6∼1.8µ-like 1µedecay-1.5µ-like 2µedecay-6.4∼-5.9
Decay electron tagging (µ → e decay)
1.1

best-fit (%)

1.3 (1.3)
0.4 (0.4)
0.2 (0.2)
0.4 (0.4)
2.1 (2.1)
1.0 (1.0)-2.0 (-2.0)-2.6 (-2.6)-1.8 (-1.8)-2.4 (-2.4)
0.006 (0.006)-0.006 (-0.006)
0.01 (0.01)-0.01 (-0.01)
0.1 (0.1)-0.2 (-0.2)
0.1 (0.1)-0.2 (-0.2)
0.07 (0.07)
0.08 (0.08)-3.4 (-3.4)-3.5 (-3.5)-7.1 (-7.1)-5.8 (-5.8)-4.0 (-4.0)-0.5 (-0.5)
0.1 (0.1)-0.3 (-0.3)
0.1 (0.1)-0.1(-0.1)-0.5∼-0.4 (-0.5∼-0.4)
0.7 (0.7)

Table 8.6: Summary of systematic uncertainties in event reconstruction for SK-I.
123uncertainty (%)(F) Systematic uncertainties in event reconstruction for SK-II
Single-ring/Multi-ring separation
Single-ring< 400 MeV
e-like1.3µ-like2.3> 400 MeV
e-like1.7µ-like0.7
Multi-GeV
e-like2.6µ-like
1.7
Multi-ring
Sub-GeV
e-like3.8µ-like
8.2
Multi-GeV
e-like1.9µ-like0.8
Particle identification
Single-ring
Sub-GeV
e-like0.5µ-like-0.4
Multi-GeV
e-like0.1µ-like-0.1
Multi-ring
Sub-GeV
e-like1.2µ-like-2.2
Multi-GeV
e-like1.8µ-like-3.4
Energy calibration for FC event
1.7
Up/down symmetry of energy calibration
0.6
0π -like sample selection
100< Pe <250 MeV/c
7.5
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
2.0+
Decay electron tagging (π decay)
e-like 0µedecay
1.2∼1.7
e-like 1µedecay-4.2∼-3.8µ-like 0µedecay
1.6∼1.8µ-like 1µedecay-1.5µ-like 2µedecay-6.5∼-5.9
Decay electron tagging (µ → e decay)
1.1

best-fit (%)

1.2 (1.2)
2.1 (2.1)
1.6 (1.6)
2.4 (2.4)
2.4 (2.4)
1.6 (1.6)-3.5 (-3.5)-7.6 (-7.6)-1.9 (-1.9)-0.7 (-0.7)
0.08 (0.08)-0.07 (-0.07)
0.02 (0.02)-0.02 (-0.02)
0.2 (0.2)-0.4 (-0.4)
0.3 (0.3)-0.6 (-0.6)-1.7 (-1.7)-0.2 (-0.2)-4.0 (-4.0)-4.8 (-4.8)-9.4 (-9.4)-5.8 (-5.8)-6.0 (-6.0)-0.5 (-0.5)-0.1 (-0.1)
0.3 (0.3)-0.1 (-0.1)
0.1 (0.1)
0.4∼0.5 (0.4∼0.5)-0.7 (-0.7)

Table 8.7: Summary of systematic uncertainties in event reconstruction for SK-II.
124

Chapter 9

Analysis with 3-Flavor Hybrid
Model
Various direct limits on NSI with matter are available thanks to the neutrino scatteringexperiments. Among these experiments NuTeV gives a stringent limit on non-standard ν µinteractions, while the constrains to non-standard νe or ντ interactions are still poor.
In order to obtain the limits on NSI in the νe − ντ sector, we extend the framework of 2flavor hybrid model to the 3-flavor hybrid model in which NSI in the νe − ντ sector coexist with
2-flavor νµ ↔ ντ standard oscillation. The 3-flavor hybrid model allows the flavor transition
for all flavored neutrinos through the NSI coupling between νe and ντ . Propagation of flavoredneutrinos from their source to the detector is expressed in Figure 9.1.
The formalism we employ for this model is introduced in Chapter 9.1. Expected phenomena
and analysis method are mentioned in Chapters 9.2 and 9.3, respectively. Results of this testare presented in Chapter 9.4. In the analysis, we consider the three-dimensional parameter
space constructed by the NSI couplings, εee , εeτ , and ετ τ . Likewise, the effects of the other subdominant contributions to the size of such NSI paramters are studied assuming several possible
scenarios, which include the nonzero θ13 , ∆m212 and θ12 (solar term), and so on.

9.1

Formalism

An evolution matrix can be obtained by diagonalizing the hamiltonian eq.(8.1) in terms of
the leptonic mixing matrix in the matter U 0 , and the effective eigenvalues Ĥ = diag(E1 , E2 , E3 ).
In the case of constant matter density, the evolution matrix is represented as
Sβα (t, t0 ) =3
X
0 ∗ 0 −iEi L(Uαi) Uβi e,α, β = e, µ, τ

(9.1)

i=1

where L is defined as L ≡ t − t0 . Therefore the neutrino oscillation probability under the effectof NSI can be expressed asPαβ = |Sβα (t, t0 )|2(9.2)
However more realistically, as seen in Figure 8.2, the matter density in the Earth ranges fromabout 2.5 to 13. So as to solve it, we divide the neutrino propagation length by several steps
and derive the relevant evolution matrix. The oscillation probability can be calculated by the125Figure 9.1: Propagation of the 3-flavor hybrid model in which 2-flavor νµ ↔ ντ standard neutrino
oscillations coexist with NSI in the νe − ντ sector.
product of each evolution matrix and its eigenvalues. Note that our interests lie in the ν e − ντsector, so the NSI parameters other than εee , εeτ , and ετ τ are set to 0 in the calculation.

9.2

Expected Phenomena

In this chapter we describe expected phenomena driven by NSI. Similar to the case of the
2-flavor hybrid model, the relative effects of NSI in the hybrid model are expected to vary
according to the neutrino energy. Survival and oscillation probabilities are demonstrated in
Figure 9.2. Resonant peaks can be seen by the variation of the matter density as a function of
the depth.
We discuss how NSI affects the atmospheric neutrino distributions in three energy ranges:(1) Eν <1GeV, (2) 1< Eν <15GeV, and (3) Eν >15GeV.
(1) Eν <1GeV√
Because of the condition ∆m2 /4Eν  2GF Nf , νµ → ντ transition is mainly caused by
2-flavor standard oscillation thus we can disregard NSI effects.
(2) 1< Eν <15GeV
In this energy range, as we see in the 2-flavor hybrid model, the matter term has a sizableeffect to modify the νµ → ντ transition. Moreover the existing poor limits on NSI in the νe → ντsector allow the large modification compared to the 2-flavor hybrid model.
Fisrt we mention the NSI modification to the νµ → νe transition. This transition is unaffectedin the 2-flavor hybrid model, i.e. exactly 0, due to the lack of coupling between νe and νµ , howeverin the 3-flavor hybrid model, νµ → νe transition is allowed by a standard νµ → ντ transition
together with a non-standard ντ → νe transition due to εeτ :
θ

ε

eτ
23
→ νe
νµ −−
→ ντ −−(9.3)

The top right panel of Figure 9.2 shows the oscillation probability with NSI, which is exactly
0 unless NSI is considered. The survival probability of νe → νe is presented in the top left panel
126

0

-0.2

0.8

-0.2

-0.4

0.6

-0.6

0.4

-0.6

-0.8

0.2

-0.8

cos θ

1

cos θ

0

0.7
0.6
0.5

-0.4

0.4
0.3
0.2
0.1

cos θ

-1
0

5

10

15
Eν (GeV)

0

0

5

10

15
Eν (GeV)

0

1

-0.2

0.8

-0.4

0.6

-0.6

0.4

-0.8

0.2

-1
0

-1
0

5

10

15
Eν (GeV)0

Figure 9.2: Survival and oscillation probabilities for the 3-flavor hybrid model. (Top left) Survival probability of νe → νe . (Top right) Oscillation probability of νµ → νe . (Bottom) Survival
probability of νµ → νµ . In all panels, εee =0.5, εeτ =0.25 and ετ τ =0.05 are assumed, also standard
oscillation parameters are fixed as the best-fit parameters given by 2-flavor NSI analysis. For simplicity, the propagation length of the neutrino is defined as L ≡ −2REarth cos θ ∼ −13000 cos θ
(km).

127

Flavor ratio (νµ + νµ)/(νe + νe)

4
3.5
3
2.5
2
1.5

1

2

3

4

5

6

7

8

9 10
Eν (GeV)Figure 9.3: Flavor ratio (νµ + ν µ )/(νe + ν e ) as a function of neutrino energy, where calculation
is based on the Honda flux. Solid curve indicates the ratio at the horizon (cos θ = 0.). Dashedand dotted curves indicate the ratio at cos θ = −0.5 and cos θ = −1.0, respectively.
of Figure 9.2. These two probabilites in association with neutrino flux yield the expected e-like
distributions, namely, if we normalize by the νe flux (Φe ) and also disregard the νµ contaminations, e-like distributions can be expressed as P (νe → νe ) + (Φµ /Φe )P (νµ → νe ), where Φµ /Φeis the so called Flavor ratio shown in Figure 3.7. We notice that the flavor ratio grows withincreasing neutrino enegy, and also this ratio is highly zenith angle dependent, which is shownin Figure 9.3. Focusing the near horizontal direction, the flavor ratio is approximately 2. Thenif P (νe → νe ) + 2P (νµ → νe ) . 1, number of νe events decrease. This case is realized when εeτand ετ τ have a large and also comparable values each other. As seen in the chain of eq.(9.3),
νµ → νe transition is decreased since νµ → ντ transition is suppressed by large ετ τ . Also, at thesame time, νe → νµ or ντ is caused by εeτ . In contrast, focusing on upward direction, νe events
increase effectively due to the condition (Φµ /Φe )P (νµ → νe ) & 1 − P (νe → νe ). This condition
corresponds to large εeτ and εee , because ντ → νe transition is given by εeτ , while νe → νµ orντ is suppressed due to εee .
Therefore we expect that nonzero νe → νe probability leads the deficit of e-like events near
the horizon, while (Φµ /Φe )P (νµ → νe ) gives a sizable excess in upward e-like events. Indeed,
zenith angle distributions of higher energy νe -rich samples in Figure 9.4 indicate our expectations,where εeτ =0.4 is assumed to show these two phenomena simultaneously.
As for νµ → ντ transition, we mention the case with nonzero ετ τ , while other two parameters
εee and εeτ are assumed to be almost 0. This case can be reduced to the 2-flavor hybrid modelwith nonzero ε0 and negligible ε, hence the comparable constraint with ε0 is expected to beobtained for ετ τ . This case can be also realized when εee  εeτ , where νe → ντ transition ishighly suppressed, therefore phenomena in the νµ − ντ sector become close to that of the 2-flavor
hybrid model.
(3) Eν >15GeV
Above a few tens of GeV, the νe flux decreases so it allows us to neglect the νe → νµ orντ transition, corresponding to an e-like deficit, while an e-like excess due to the non-standard

128

SK−1+2 Multi−GeV 1−ring e−like

SK−1+2 Multi−GeV Multi−ring e−like

220
140
200
120
180
160
100
140
80
120
100
60
80
40
60
40
20
20
0−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1
0−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1
cosθ
cosθ

Figure 9.4: (Left) Zenith angle distribution of FC Single-ring Multi-GeV e-like sample. (Right)
Zenith angle distribution of FC Multi-ring Multi-GeV e-like sample. In both panels, solid line
indicates the 3-flavor hybrid model, dashed line 2-flavor standard oscillation, green hatched area
unoscillated νe events, and red hatched area appearance of oscillated νµ events. Note that nosystematic uncertainty is considered in the figure.

ντ → νe transition can be clearly recognized thanks to the small νe flux. As for the νµ − ντsector, if we replace ε0 with ετ τ , the case (3) of the 2-flavor hybrid model is applicable to thisrange.
Finally, we mention why atmospheric neutrinos are less sensitive for placing a limit on ε ee .
The reason is as follows: once εeτ and ετ τ are constrained to quite a small value, the sizable
element in the matter term is only in the (e, e) sector, i.e. εee . It turns out that the matter term
is disconnected from the standard oscillation and this condition allows εee to have any value.
Let us summarize the above discussions: εeτ can be constrained by higher energy νe -richsamples near the horizon and in the upward direction, while ετ τ can be constrained by the
deficit in νµ -rich samples.

9.3

Analysis Method

Analysis procedure for the 3-flavor hybrid model is essentially common to that used for the
2-flavor hybrid model, however we neglect a complex relative phase for flavor changing couplings
because other sub-dominant effects, for example nonzero θ13 , may have a larger effect on thesize of allowed NSI.
A value of χ2 is evaluated at each grid point in the three-dimensional parameter space of
εee , εeτ , and ετ τ . As we mentioned in Chapter 9.2, atmospheric neutrinos are less sensitive toconstrain εee , then we set the bound to the εee parameter space with the existing experimental
limit given by the CHARM experiment, −0.5 < εee < 0.5. Also εeτ is symmetrical between
negative and positive value when neither θ13 nor solar term (∆m212 and θ12 ) are considered,
therefore the evaluation is performed in the positive side of εeτ .
2-flavor standard oscillation parameters are fixed as (sin2 θ23 , ∆m2 )=(0.5, 2.1×10−3 eV2 ),i.e. best-fit parameters given by the 2-flavor standard oscillation analysis in Appendix B. Thiscondition is motivated by the result of the analysis with the 2-flavor hybrid model, in which129

allowed oscillation parameters region is similar to that of 2-flavor standard oscillation analysis.
Standard oscillation parameters in this hybrid model are assumed to take the fixed values. We
later verify this assumption by comparing the allowed NSI parameter region derived by several
different oscillation parameters.

9.4

Result of 3-Flavor Non standard interaction analysis

Our main results, allowed 3-flavor NSI paramters, are presented in Figure 9.5. In all panels,
undisplayed NSI parameter is integrated out, for example in top left panel ετ τ is integrated out.
Three contours correspond to the 68%, 90% and 99% confidence level (C.L.) allowed regions
which are defined to be χ2 = χ2min +2.30, 4.61, and 9.21, respectively. According to the bottom
panel of Figure 9.5, limit on NSI parameters in the νe − ντ sector at a 90 % C.L. is :|εeτ | < 0.16,−0.05 < ετ τ < 0.06

(9.4)

Best-fit values are
εee = −0.25,εeτ = 0.016,ετ τ = 0.024
2χmin = 829.9 / 747 d.o.f.

(9.5)
(9.6)

We only consider positive εeτ in this result, since εeτ is symmetrical between negative andpositive value, which is explained in Discussion again.
Systematic uncertainties are summarized in Tables from 9.2 to 9.8.
Allowed NSI paramters regions with four sets of standard oscillation parameters are presented
in Figure 9.6. Choise of the oscillation parameters is based on the standard 2-flavor oscillation
analysis. Two points are chosen along the ∆m2 direction within 90% C.L. with fixed valuesin2 θ23 =0.5, namely ∆m2 =1.7×10−3 eV2 and 2.7×10−3 eV2 . The other two points are chosen
along the sin2 θ23 direction within 90% C.L. with fixed value ∆m2 =2.1×10−3 eV2 , sin2 θ23 =0.39
and sin2 θ23 =0.61. We compare the allowed NSI regions at totally four points.
Allowed parameters regions are summarized in Table 9.1. Sizable changes of allowed NSI parameters region are not found except for (sin2 θ23 , ∆m2 )=(0.5, 2.7×10−3 eV2 ) indicated by dahsed
curve in Figure 9.6. Considering the difference of minimum χ2 values between (sin2 θ23 , ∆m2 )=(0.5,
2.1×10−3 eV2 ) and (0.5, 2.7×10−3 eV2 ), the latter allowed region shown by dashed curve could
be changed smaller. Therefore the assumption of the fixing standard oscillation parameters is
not significant in this analysis.

9.5

Discussion

We start the discussion with Figure 9.7, in which allowed NSI regions for five fixed ε ee =-0.5, -0.25, 0., 0.25, and 0.5, are presented. In this figure, parabola-like distributions of allowed
region as a function of εeτ are found except for εee = −0.25. Especially the allowed region
extends to negative ετ τ values when εee = −0.5, while they extend to the positive ones above
εee = 0. The parabola distributions can be explained as follows: For convenience, we introduce
the eigenvalues of matter term, which are derived by diagonalizaing the matter term. If we recall
eq.(8.1) and Nf ≡ Nd ∼ 3Ne , matter term including VMSW can be expressed as
130SK−I + SK−II

0.2

ε ττ

εeτ

0.4

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

0.35
0.3

0.15
0.1

0.25

0.05

0.2

0

0.15

−0.05

0.1

−0.1

0.05

−0.15

0

ε ττ

0.2

0.15
0.1

−0.4

−0.2

0

0.2

0.4

ε ee

−0.2

SK−I + SK−II
99% C.L.
90% C.L.
68% C.L.

−0.4

−0.2

0

0.2

0.4

ε ee

SK−I + SK−II
99% C.L.
90% C.L.
68% C.L.

0.05
0
−0.05
−0.1
−0.15
−0.2
0

0.1

0.2

0.3

ε e0.4τFigure 9.5: Allowed NSI parameters region. (Top left) εee vs. εeτ space. (Top right) εee vs. ετ τspace. (Bottom) εeτ vs. ετ τ space.

sin2 θ23∆m2 (eV2 )Allowed εeτAllowed ετ τ

Minimum χ2 (747 d.o.f.)

0.50

2.1×10−30.50|εeτ | < 0.16

−0.05 < ετ τ < 0.06

829.9

1.7×10−3

0.50

2.7×10−30.39|εeτ | < 0.17

−0.06 < ετ τ < 0.09

831.4

2.1×10−3

0.61

2.1×10−3−0.03 < ετ τ < 0.06

834.4

|εeτ | < 0.15
|εeτ | < 0.14
|εeτ | < 0.14

−0.04 < ετ τ < 0.04

836.1

−0.07 < ετ τ < 0.05

834.0

Table 9.1: Summary of NSI allowed parameters at the 90% C.L.. Our main result is in the first
line.

131

ε ττ

0.2

0.15
0.1
0.05
0
−0.05
−0.1
−0.15
−0.2
0

0.1

0.2

ε e0.4τ

0.3

Figure 9.6: Allowed region in εeτ vs. ετ τ space at the 90% C.L.. Solid curve is for the resultwith (sin2 θ23 , ∆m2 )=(0.5, 1.7×10−3 eV2 ), dashed curve for (0.5, 2.7×10−3 eV2 ), dotted curve for(0.39, 2.1×10−3 eV2 ), and dashed-dotted curve for (0.61, 2.1×10−3 eV2 ).

0.15
0.1

0.2

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

0.15
0.1

0.05

SK−I + SK−II

0.2

ε ττ

SK−I + SK−II

ε ττ

ε ττ

0.2

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

0.15
0.1

0.05

0

0

0

−0.05

−0.05

−0.1

−0.1

−0.1

−0.15

−0.15

−0.15

−0.2
0

−0.2
0

0.2

ε e0.4τ

0.3

0.1(a) εee = −0.5

0.15
0.1

0.2

−0.2
0

0.1

(b) εee = −0.25

SK−I + SK−II

0.2

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

0.15
0.1

0.05

0.2

0.3

ε e0.4τ(c) εee = 0.0

SK−I + SK−II
99% C.L.
90% C.L.
68% C.L.

0.05

0

0

−0.05

−0.05

−0.1

−0.1

−0.15

−0.15

−0.2
0

ε e0.4
τ

0.3

ε ττ

ε ττ

0.2

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

0.05

−0.05

0.1

SK−I + SK−II

0.1

0.2

0.3

ε e0.4τ−0.20(d) εee = 0.25

0.1

0.2

0.3

ε e0.4τ(e) εee = 0.5

Figure 9.7: Allowed NSI parameters region for fixed εee . ∆χ2 is defined as the deviation fromthe minimum χ2 value for each fixed εee .
132



1 + 3εee 0 3εeτ√Hmatter = 2GF Ne  0
0
0 
3εeτ
0 3ετ τ(9.7)

where we disregard complex relative phase for εeτ .
We know that a matrix A defined as
A=

α β
β γ

!

(9.8)

can be easily diagonalized in terms of eigenvectors P and rotation angle ω!
cos ω sin ω2βP =, tan 2ω =γ−α− sin ω cos ω

(9.9)

as
P † AP =

λ1 0
0 λ2

!

, where λ1 =γ cos2 ω − α sin2 ω
α cos2 ω − γ sin2 ω
,
λ
=
.
2
cos2 ω − sin2 ωcos2 ω − sin2 ω(9.10)

Thus the diagonalization of the Hmatter gives the eigenvalues Eν0 e ,νµ ,ντ and rotation angle,
i.e. mixing angle in matter, θ 0 as:


s
√
2
1
3 2GF Ne  1(9.11)+ 4ε2eτ + εee + ετ τ ++ εee − ετ τEν0 e =
2
3
3
Eν0 µ
Eν0 τ

= 0
=

tan 2θ 0 =(9.12)√
3 2GF Ne  1+ εee + ετ τ −23

2|εeτ |
1/3 + εee − ετ τs

1
+ εee − ετ τ
3

2



+ 4ε2eτ (9.13)(9.14)

Note that εeτ is included as |εeτ | in tan 2θ 0 . Therefore, the oscillation probability is symmetricalbetween negative and positive εeτ .
We first consider the case in which both Eν0 e and Eν0 τ are larger than the eigenvalue of vacuum
term, i.e. Eν0 e , Eν0 τ > ∆m2 /4Eν , which corresponds to 1/3+εee , ετ τ  εeτ . In this case, νµ → νetransition is quite small due to Eν0 e > ∆m2 /4Eν . However, at the same time, νµ → ντ transitionis also suppressed by large ετ τ . This is clearly inconsistent with the observation, therefore this
scenario is strongly excluded.
In addition to that case, we have another case where NSI parameters are allowed to have asizable value and also significant discrepancy between NSI and standard oscillation is not found.
This scenario is realised when Eν0 e > ∆m2 /4Eν > Eν0 τ . In this case, thanks to Eν0 e > ∆m2 /4Eν ,transition from νe to νµ or ντ is suppressed, thus it is approximately possible to regard the
3-flavor hybrid model as the 2-flavor hybrid model. If we assume Eν0 τ = 0 for simplicity, we
obtain the following relation:
3ε2eτ
ετ τ =(9.15)
1 + 3εee
133

ε ττ

0.2

0.15
0.1

SK−I + SK−II

0.5

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

0.4
0.3
0.2

0.05

0.1

0

0
−0.1

−0.05

−0.2
−0.1
−0.3
−0.15
−0.2
0

−0.4
0.1

0.2

0.3

ε e0.4τ−0.5

Figure 9.8: Allowed NSI parameters region, εee and εeτ at the 68%, 90% and 99% C.L. frominside. Colored boxes in each grid indicate the value of εee .

Suppose εee is fixed, this relation yields the parabolic χ2 distribution as seen in Figure 9.7.
In other words, atmospheric neutrinos are rather insensitive to constrain NSI parameters as longas one keeps the reration in eq.(9.15), because the frequency of the transition from ν µ to theother flavored neutrinos is unaffected. In fact, NSI parameters on the parabola distribution in
Figure 9.7 mostly satisfy eq.(9.15).
Besides the origin of parabolic χ2 distribution, the relation (9.15) yields the correlation of
the sign of allowed NSI parameters between εee and ετ τ . Namely if 1 + 3εee is negative, ετ τshould also be negative, since 3ε2eτ is always positive. The correlation is recognized in right panelof Figure 9.5, where allowed region ranges to negative ετ τ when εee is below about -0.3, while
the area changes from negative to positive ετ τ as εee increases. This can be more clearly seen in
Figure 9.8, in which negative ετ τ is constrained by negative εee , and this is also true for positiveεee and ετ τ .
If Eν0 e > ∆m2 /4Eν > Eν0 τ is satisfied, the frequency of νµ disapearance induced by the hybrid
model, i.e. νµ → νe or ντ transition, is same as that of 2-flavor νµ → ντ standard oscillation.
Nevertheless, effective mixing angle in matter is still under the effect of NSI through the NSI
mixing angle θ 0 . θ0 leads νµ → νe transition and suppresses νµ → ντ transition. Therefore
the inconsistency of higher energy e-like samples between NSI and observation contributes toconstrain θ 0 . Phenomena due to the nonzero θ 0 corresponds to the deficit of e-like events in near
the horizon and excess in the upward direction given by εeτ , as we discussed in chapter 9.2.
Our expectation that θ 0 can be constrained by the disapearance and appearance of νe events iscertainly consistent θ 0 ∼ 0, which is equivalent to tan 2θ 0 ∼ 0, in other words, εeτ ∼ 0.
To see more detail, we present the allowed NSI parameters regions derived by sets of subsamples in Figure 9.9, where FC Sub-GeV samples are not displayed because of their small
sensitivity. Higher energy νµ -rich samples indicated by solid curve (UPMU through) and dashedcurve (PC and UPMU stopping) show parabolic χ2 distributioins, which can be understood as
follows: Once the relation eq.(9.15) is satisfied, the frequency of νµ → νe or ντ is similar to thatof νµ → ντ in vacuum term. Futhermore, UPMU and PC samples can not distinguish whether
134

SK−I + SK−II

ε ττ

0.2

0.15
0.1
0.05
0
−0.05
−0.1
−0.15
−0.2
0

0.1

0.2

0.3

ε e0.4τFigure 9.9: Allowed NSI parameters derived by a set of sub-samples with εee = −0.25. Solid
curve indicates the allowed region given by UPMU through going, dashed curve by PC and
UPMU stopping, and dotted curve by FC Single-ring Multi-GeV and Multi-ring. Contour
corresponds to χ2 = χ2min + 3.53 (68% C.L.).

θ0 ∼ 0 or not because of no sensitivity to the e-like distributions. Figure 9.10 shows the allowed
NSI parameter region derived by UPMU through events (solid curve) and the UP/Horizon ratio(colored boxes) introduced in chapter 8.5. Indeed, the UP/Horizon ratios outside of the parabolaarea disagree with that of standard oscillation which is equivalent to ε eτ ∼ ετ τ ∼ 0.
In contrast, allowed region derived by FC Single-ring Multi-GeV and Multi-ring (dottedcurve) seems to constrain εeτ rather than other sub-samples. It means that higher energy νe rich samples constrain εeτ .

9.6

Results with the uncertainty of matter density and subdominant effects

There are several scenarios including sub-dominant effects possibly affecting the size of allowed NSI parameters. One is the systematic uncertainty of the matter density in the Earth,others are the sub-dominant oscillation parameters. In this section, we investigate how ourlimits can be modified when these parameters are taken into account.

9.6.1

Effect of the uncertainty of matter density profile

First, we consider the systematic uncertainty of the matter density in the Earth. It is known
that the variation of matter density along the neutrino propagation changes the oscillation
probabilities and it results in the systematic uncertainty of the limit on NSI. In order to estimate
the relaxations of the constraints, we derive limits on NSI assuming three cases: (1) matter
density is decrased by 10%, (2) original PREM prediction, i.e. results presented in the previous
chapters, and (3) matter density is increased by 10%. Systematic uncertainty of matter density,±10%, is motivated by the studies in geophysics [148, 149].
135

SK−I + SK−II

0.45

ε ττ

0.2

0.15

0.44

0.1

0.43

0.05

0.42

0

0.41

−0.05

0.4

−0.1

0.39

−0.15

0.38

−0.2
0

0.1

0.2

0.3

ε e0.4τFigure 9.10: Allowed NSI parameters with εee = −0.25. Solid curve indicates the allowed
region given by UPMU through going, where contour corresponds to χ 2 = χ2min + 3.53, i.e.
the 68% C.L.. Colored boxes show UP/Horizontal ratio defined as the ratio of upward 3 bins(cos Θ = −1 ∼ −0.7) to horizontal 3 bins (cos Θ = −0.3 ∼ 0).
Results of three cases are presented in Figure 9.11, where dashed curve indicates the resultwith 10% decreased density and dotted curve indicates the one with 10% increased density.
Solid curve is the original result, same as Figure 9.5. These curves show the allowed regions at
the 90% C.L.. According to the bottom panel of Figure 9.11, the limits on ε eτ or ετ τ are rather
insensitive to the variation of matter density, however we notice that the limits on ε eτ and ετ τare derived at the bound of parameter space, εee = 0.5. In contrast, if we see top two panels,
variations of allwed NSI parameters are maximum at εee ∼ 0.2. Therefore, according to thevariation at εee ∼ 0.2, we estimate the systematic uncertainties to be at most 0.03 for ε eτ and0.02 for ετ τ .

9.6.2

Effect of nonzero θ13

Next, we consider nonzero θ13 , which is expected to be quite small compared to θ23 orθ12 . Thanks to the smallness of θ13 , in the previous analysis, we neglected the effects of θ13 .
However θ13 has similar effects to εeτ in matter, then θ13 possibly interferes with εeτ . We test
the scenario with nonzero θ13 , assuming sin2 θ13 = 0.04, corresponding to the upper limit derived
by the CHOOZ experiment [150]. Note that nonzero θ13 breaks the symmetry between positiveεeτ and negative εeτ .
Results with two cases, normal hierarchy ∆m2 > 0 and inverted hierarchy ∆m2 < 0, arepresented in Figure 9.12. Allowed parameters at the 90% C.L. are−0.16 < εeτ < 0.10,−0.14 < εeτ < 0.13,−0.03 < ετ τ < 0.04,−0.03 < ετ τ < 0.04,(Normal hierarchy)(9.16)(Inverted hierarchy)(9.17)First we focus on normal hierarchy and compare it to the result with θ13 = 0 (see Figure 9.5).
Comparison of two results in the positive side of εeτ indicates that nonzero θ13 contributes
136

ε ττ

0.2

εeτ

0.4

0.35

0.15

0.3

0.1

0.25

0.05

0.2

0

0.15

−0.05

0.1

−0.1

0.05

−0.15

0

−0.4

−0.2

0

0.2

0.4

ε ee

−0.2

−0.4

−0.2

0

0.2

0.4

ε ee

ε ττ

0.2

0.15
0.1
0.05
0
−0.05
−0.1
−0.15
−0.2
0

0.1

0.2

0.3

ε e0.4τFigure 9.11: Allowed NSI parameters region with the uncertainty of matter density profile. (Topleft) εee vs. εeτ space. (Top right) εee vs. ετ τ space. (Bottom) εeτ vs. ετ τ space. In all panels,
solid curve, dashed curve, and dotted curve indicate the original PREM matter densty, 10%
decreased density, and 10% increased density, respectively. Contours are at the 90% C.L..

137

ε ττ

0.2

εeτ

0.4
0.3

0.15

0.2

0.1

0.1

0.05

0

0

−0.1

−0.05

−0.2

−0.1

−0.3

−0.15

−0.4

−0.4

−0.2

0

0.2

0.4

ε ee

−0.2
−0.4

−0.2

0

0.2

ε e0.4τFigure 9.12: Allowed NSI parameters region with nonzero θ13 . (Left) εee vs. εeτ space. (Right)εeτ vs. ετ τ space. In both panels, sin2 θ13 =0.04, upper limit by the CHOOZ experiment [150],
is considered. Solid curve and dashed curve indicate normal hierarchy and inverted hierarchy,
respectively. Contours are at the 90% C.L..

to constrain εeτ more tightly. This can be understood by the comparison between νµ → νetransition driven by θ13 and the effects of εeτ . Assuming standard 3-flavor oscillation withnormal hierarchy, the effective mixing angle in matter Θ23 is mostly equal to θ23 when θ12 = 0,
while Θ13 is modified by VMSW . Θ13 is approximately expressed asΘ13 ∼ θ13 + φ,tan 2φ ∼a sin 2θ13
2∆m31 − a cos 2θ13

(9.18)√
where a is the product of amplitude of matter term and neutrino energy, 2 2GF Ne Eν . Thus
Θ13 depends on neutrino energy as shown in Figure 9.13. Since sin2 Θ13 is not suppressed at
Eν ∼ 10GeV, νµ → νe transition is arised by θ23 together with Θ13 . As a result, this νµ → νetransition will increase number of higher energy νe events in upward. We have to notice thatthe excess is purely caused by standard oscillation with nonzero θ13 and VMSW , and also thisis similar with the effects of εeτ . In other words, εeτ is restricted to give upward excess, sincenonzero θ13 occupies a room of νe -rich samples once allowed for εeτ . It means that nonzero θ13
contributes to constrain εeτ .
In Figure 9.12, we also see that normal hierarchy (solid curve) has stronger limit on positiveεeτ rather than inverted hierarchy (dashed curve), while√ this is reversed for negative ε eτ . Thisdifference comes from the sign of the matter term, ± 2GF Ne , which is positive for neutrinos
and negative for anti-neutrinos.
For convenicence, we define the hierarchy of ∆m 2 and sign of√matter term as (∆m2 , 2GF Ne ) = (±, ±), for example (+, −) for anti-neutrino oscillation withnormal hierarchy. If we see Figure 9.13, (+, +) and (−, −) are represented by solid curves, while(+, −) and (−, +) are expressed by dashed curves. Namely, (+, +) and (−, −) give large upawrd
excess in νe -rich samples rather than (+, −) or (−, +), since solid curves have large mixing angle
at Eν = 5 − 10GeVFor simplicity, we discuss only in the positive side of εeτ . In case of normal hierarchy,138

Effective mixing angle Θ 13 (deg.)

90
80
70
60
50
40
30
20
10
0

0

10

20

30
E ν (GeV)

Figure 9.13: Effective mixing angle Θ13 for neutrino in the matter, corresponding to θ13 in vacuum. sin2 θ13 = 0.04 and constant matter density are assumed. Solid curves indicate the angle
with normal hierarchy and dashed curves indicate that with inverted hierarchy. For both curves,
thick curves are for ρ = 5.0g/cm3 and thin curves are for ρ = 3.0g/cm3 . These angles are also
applicable to anti-neutrinos by interchanging ν ↔ ν and normal hierarchy ↔ inverted hierarchy.
Note that no NSI is considered in this figure.

neutrino events have (+, +), then it gives an excess in upward νe -rich samples. To the contrary,for inverted hierarchy, the excess is caused by anti-neutrinos due to (−, −). In the energy rangeEν = 1 ∼ 15GeV, number of neutrino events is at least factor∼ 2 larger than anti-neutrino
events, because of the atmospheric neutrino flux (Φ(ν) > Φ(ν)) and also neutrino-nucleus corss
sections (σ(ν) > σ(ν)). As a result, neutrino events are expected to have larger contributions
to the excpected phenomena in atmospheric neutrinos rather than anti-neutrino events.
The excess in upward going events is more enhanced in normal hierarchy than inverted hierarchy, because, in normal hierarchy, large contribution is given by νµ , not by ν µ . In other words,
normal hierarchy occupies a room of νe -rich samples rather than inverted hierarchy. Figure 9.14
shows zenith angle distributions of FC Single-ring Multi-GeV e-like samples. Left panel andright panel show normal hierarchy and inverted hierarchy, respectively, where expectations of
2-flavor standard oscillation, 3-flavor standard oscillation, and the 3-flavor hybrid model are
displayed. If we normalize 3-flavor standard and hybrid model by 2-flavor oscillation, the magnitudes of the transition relative to 2-flavor oscillation is derived as in Figure 9.15. We can seethat the upward excess are caused by (+, +) and (−, −), while (+, −) and (−, +) have smaller
effects. Thus εeτ is constrained more tightly for normal hierarchy. This discussion is also true
for negative εeτ . In this case, sign of the matter term is reversed and an upawrd excess is given
by (+, −) and (−, +). Therefore the tighter constraint to εeτ is obtained with inverted hierarchy
rather than with normal hierarchy.

9.6.3

Effect of nonzero ∆m212 and θ12

The oscillation parameters ∆m212 and θ12 have been measured by solar neutrino experiments,
Super-Kamiokande [151] and SNO [152], and by KamLAND [153] using nuclear reactor antineutrinos, then we call ∆m212 and θ12 as Solar term in the following text. Results from theglobal analysis with their results, solar term is measured with high precision, and the combined139SK−1+2 Multi−GeV 1−ring e−like

220
200
180
160
140
120
100
80
60
40
20
0−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1cosθ

SK−1+2 Multi−GeV 1−ring e−like

220
200
180
160
140
120
100
80
60
40
20
0−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1cosθ

Events(/2−flavor OSC.)

Events(/2−flavor OSC.)

Figure 9.14: Zenith angle distribution for FC Single-ring Multi-GeV e-like samples. Left panel
shows normal hierarchy (∆m2 = 2.1 × 10−3 eV2 ), while right panel shows inverted hierarchy(∆m2 = −2.1 × 10−3 eV2 ). In both panels, crosses with error are for data, solid lines for 3-flavor
standard oscillation, and dashed lines for 2-flavor standard oscillation. Oscillation parameters are
assumed to be (sin2 θ23 , |∆m2 |) = (0.5, 2.1 × 10−3 eV2 ). sin2 θ13 = 0.04 is additionaly considerd
only in 3-flavor oscillation. Gray hatched area indicates the 3-flavor hybrid model with sin 2 θ13 =0.04. Especially the component of anti-neutrino is shown by red hatched area.

1.2
1
0.8
0.6
0.4
0.2
0−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1
cosθ

1.2
1
0.8
0.6
0.4
0.2
0−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1
cosθ

Figure 9.15: Number of FC Single-ring Multi-GeV e-like samples normalized by 2-flavor standard
oscillation as a function of zenith angle, where (sin2 θ23 , |∆m2 |) = (0.5, 2.1×10−3 eV2 ) is assumed.
Left panel shows normal hierarchy, while right panel shows inverted hierarchy. Blue solid line
indicates the ratio of 3-flavor standard oscillation and gray solid line indicates the 3-flavor hybrid
model. For the 3-flavor hybrid model, the contribution of neutrino is shown by dashed line andthat of anti-neutrino is shown by dotted line.

140Effective mixing angle Θ 12 (deg.)

90
80
70
60
50
40
30
20
10
0

0

0.5

1

1.5
2
E ν (GeV)

Figure 9.16: Effective mixing angle Θ12 in the matter, corresponding to θ12 in vacuum. sin2 θ12 =0.32 (tan2 θ12 = 0.47) and constant matter density are assumed. Solid curve indicates the angle
with neutrino and dashed curve indicates that with anti-neutrino. For both curves, thick curves
are for ρ = 5.0g/cm3 and thin curves are for ρ = 3.0g/cm3 . Note that no NSI is considered inthis figure.

result [153] is+0.21∆m221 = 7.59−0.21
× 10−5 eV2 ,+0.06
tan2 θ12 = 0.47−0.05(9.19)

We focus on the effects of solar term in atmospheric neutrinos observation. First if we neglect
VMSW , νe → νµ transition appears below Eν ∼ 2GeV. While in actual case, the probability ofνe → νµ transition gets smaller due to VMSW as increase of neutrino energy. It is explained asfollows: If we see Figure 8.5, vacuum term ∆m212 /4Eν indicated by dashed curve competes withmatter term even in low energy. Therefore the effective mixing angle Θ12 grows up rapidly due
to the matter effect, as shown in Figure 9.16, and transitions of νe to other flavored neutrinos
are suppressed. Effective mixing angle Θ12 is approximately expressed asΘ12 ∼ θ12 + ϕ,tan 2ϕ ∼a sin 2θ12
2
∆m21 − a cos 2θ12

(9.20)√
where a is defined as 2 2GF Ne Eν . νe → νµ transition is still visible in Sub-GeV samples, while
it fades out above Eν ∼400MeV. Hereafter, we investigate the effects of solar term to the sizeof allowed NSI.
With unoscillated neutrino fluxes Φ(νe ) and Φ(νµ ), e-like distribution is expressed as Φ(νe )P (νe →νe ) + Φ(νµ )P (νµ → νe ). If we normalize by Φ(νe ), relative e-like distribution to that of standard
2-flavor oscillation is expressed as P (νe → νe ) + Φ(νµ )/Φ(νe )P (νµ → νe ), where Φ(νµ )/Φ(νe ) is
approximately 2 in Sub-GeV region. Thus e-like distributions are mainly controled by ν µ → νetransition.
In Figure 9.17, we present νµ → νe oscillation probability with three cases: (1) standardνµ → νe oscillation with solar term, (2) the hybrid model without solar term, and (3) the hybrid
model with solar term. This figure indicates that the hybrid model without solar term givesνµ → νe transition due to nonzeo εeτ (top left panel), however, once solar term is considerd
as in bottom panel, this transition is governed by vacuum solar term below 100MeV, and issuppressed by the interference between vacuum and matter term in Eν = 100 ∼ 300MeV, while
141

0.5

cos θ

−0.2

0.4

−0.2

−0.4

0.3

−0.4

−0.6

0.2

−0.6

−0.8

0.1

−0.8

cos θ

0

0
0.7
0.6
0.5
0.4
0.3
0.2
0.1

cos θ

−1
0

0.5

1

1.5

2
Eν (GeV)

0

−1
0

0.5

1

1.5

2
Eν (GeV)

0

0
0.7

−0.2

0.6
0.5

−0.4

0.4

−0.6

0.3
0.2

−0.8
0.1

−1
0

0.5

1

1.5

2
Eν (GeV)0

Figure 9.17: Oscillation probabilities of νµ → νe . (Top left) Standard oscillation with solar
term, ∆m231 , ∆m221 , θ23 , and θ12 . (Top right) The hybrid model without solar term, ∆m231 , θ23 ,and εeτ . (Bottom) The hybrid model with solar term, ∆m231 , ∆m221 , θ23 , θ12 , εeτ . In this figure,∆m231 = 2.1 × 10−3 eV2 , ∆m221 = 7.6 × 10−5 eV2 , sin2 θ23 = 0.5, sin2 θ12 = 0.32 are assumed.
As for NSI, εeτ = 0.2 is assumed. For simplicity, we assume the matter density of constantρ = 5.0g/cm3 .

the transition is recovered again due to matter term εeτ above 300MeV. The importance what
we notice here is that we do not use e-like events with reconstructed momentum below 100MeVin the analysis to avoid electrons from muon decay. This means that, in the hybrid model,
solar term is expected to act as the suppression of νµ → νe transition in the energy range weare interested in. In other words, NSI parameters are allowed to have larger values when solar
term is considerd compared to without solar term. Indeed, analysis results in Figure 9.18 and
Figure 9.5 are consistent with our expectations. Limits on NSI parameters are slightly looser
than that without solar term. Allowed parameters at the 90% C.L. are−0.16 < εeτ < 0.18,−0.05 < ετ τ < 0.06

142

(9.21)

0.3
0.2

SK−I + SK−II

0.2

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

0.15
0.1

0.1

0.05

0

0

−0.1

−0.05

−0.2

−0.1

−0.3

−0.15

−0.4

−0.4

−0.2

SK−I + SK−II

ε ττ

εeτ

0.4

0

0.2

0.4

ε ee

−0.2
−0.4

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

−0.2

0

0.2

ε e0.4τFigure 9.18: Allowed NSI parameters region with solar term. (Left) ε ee vs. εeτ space. (Right)εeτ vs. ετ τ space. In both panels, sin2 θ12 =0.32 and ∆m221 =7.6×10−5 eV2 , best-fit parameters
given by the global analysis [154], are considered.

143

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

uncertainty (%)

best-fit (%)

25.0
15.0
2.0
3.0
5.0a
5.0
5.0
8.0b
2.0
6.0
6.0c
0.1
0.3
0.8
0.5
0.7
0.2
0.2
0.2
0.3
0.2
0.2
0.1
0.1
1.4
1.9
3.2
2.3
1.4
1.3
2.8
1.5
1.7

35.0
19.3
-0.6
-1.9
4.7
2.9
1.1
0.07
-0.2
-0.4
1.3
-0.05
-0.1
-0.04
-0.2
-0.3
-0.1
-0.1
-0.1
-0.1
-0.1
-0.1
0.005
0.005
0.08
0.1
0.2
0.1
0.08
0.07
0.2
0.08
0.09

Table 9.2: Summary of systematic uncertainties in atmospheric neutrino flux.
aUncertainty linearly increases with log Eν from 5 % (10 GeV) to 30 %(1 TeV).
Uncertainty linearly increases with log Eν from 8 %(100 GeV) to 20 %(1 TeV).
c
Uncertainty linearly increases with log Eν from 6 %(50 GeV) to 40 %(1 TeV).

b

144

uncertainty (%)(A2) Systematic uncertainties in neutrino flux
K/π ratio
5.0aLν (production height)
10.0
Sample-by-sample FC Multi-GeV
5.0
PC + upward stopping µ
5.0
Solar activity
SK-I
20.0
SK-II
50.0

best-fit (%)
-7.5
0.3
-5.1
-10.3
3.1
32.1

Table 9.3: Summary of systematic uncertainties in atmospheric neutrino flux (continued).
aUncertainty is 20 % above Eν =1 TeV. The uncertainty increases linearly from 5% to 20% between 100GeV
and 1TeV.

(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)
Single meson production (ν/ν ratio)
Single meson production (π 0 /π ± ratio)
Multi-pion production (Eν < 10GeV)
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 (%)

1.0a
1.0 b
1.0c
1.0d
20.0
1.0e
40.0
1.0f
5.0
100.0(50.0) g
20.0
30.0
1.0h
30.0
10.0

0.8
5.3
8.9
5.0
3.2
-3.4
-35.9
-0.6
1.9
24.2
-0.8
-22.9
9.3
-15.9
-2.2

Table 9.4: Summary of systematic uncertainties in neutrino interactions.
aDifference from the MA = 1.1GeV/c is set to be 1.0
Difference from the Nieves model is set to 1.0cDifference from the Nieves model is set to 1.0
dDifference from the Nieves model is set to 1.0eDifference from the Hernandez model is set to 1.0
fDifference from CKMT parametrization is set to 1.0g
100% for CC νµ . 50% for CC νe and NC.
hDifference between NEUT and NUANCE is set to 1.0b145(C) Systematic uncertainties in event selection for SK-I
Reduction for FC event
Reduction for PC event
FC/PC separation
Non-ν background
Flasher for e-like
Sub-GeV
Multi-GeV
Cosmic ray for µ-like Sub-GeV
Multi-GeVPC
PC stop/through separation OD top region
OD barrel region
OD bottom region
Reduction for UPMU event stopping µthrough-going µ
UPMU stopping/through separation
UPMU stopping energy cut
UPMU through showering/non-showering separation

uncertainty (%)

best-fit (%)

0.2
2.4
0.6
0.5
0.2
0.1
0.1
0.2
15.0
7.4
11.3
1.8
0.3
0.4
0.8
2.8

0.04
-0.8
-0.01
0.07
0.03
-0.01
-0.01
-0.03
11.3
-0.6
-7.1
0.1
0.02
-0.04
-0.08
-1.7

Table 9.5: Summary of systematic uncertainties in event selection for SK-I.

146

(D) Systematic uncertainties in event selection for SK-II
Reduction for FC event
Reduction for PC event
FC/PC separation
Non-ν background
Flasher for e-like
Sub-GeV
Multi-GeV
Cosmic ray for µ-like Sub-GeV
Multi-GeVPC
PC stop/through separation OD top region
OD barrel region
OD bottom region
Reduction for UPMU event stopping µthrough-going µ
UPMU stopping/through separation
UPMU stopping energy cut
UPMU through showering/non-showering separation

uncertainty (%)

best-fit (%)

0.2
4.8
0.5
0.3
0.7
0.1
0.1
0.7
19.0
14.0
18.0
2.1
0.3
0.4
1.5
1.8

0.007
-2.6
0.05
-0.01
-0.03
0.02
0.02
0.2
-18.0
-22.4
-19.2
-0.2
-0.02
-0.02
-0.04
-0.3

Table 9.6: Summary of systematic uncertainties in event selection for SK-II.

147

uncertainty (%)(E) Systematic uncertainties in event reconstruction for SK-I
Single-ring/Multi-ring separation
Single-ring< 400 MeV
e-like2.3µ-like0.7> 400 MeV
e-like0.4µ-like0.7
Multi-GeV
e-like3.7µ-like
1.7
Multi-ring
Sub-GeV
e-like3.5µ-like
4.5
Multi-GeV
e-like3.1µ-like
4.1
Particle identification
Single-ring
Sub-GeV
e-like0.1µ-like-0.1
Multi-GeV
e-like0.2µ-like-0.2
Multi-ring
Sub-GeV
e-like2.3µ-like-3.9
Multi-GeV
e-like1.7µ-like-2.9
Energy calibration for FC event
1.1
Up/down symmetry of energy calibration
0.6
0π -like sample selection
100< Pe <250 MeV/c
11.2
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
2.0+
Decay electron tagging (π decay)
e-like 0µedecay
1.5∼1.7
e-like 1µedecay-4.4∼-3.8µ-like 0µedecay
1.6∼1.8µ-like 1µedecay-1.5µ-like 2µedecay-6.4∼-5.9
Decay electron tagging (µ → e decay)
1.1

best-fit (%)

1.4
0.4
0.2
0.4
2.2
1.0
-2.1
-2.7
-1.9
-2.5
-0.01
0.01
-0.02
0.02
0.07
-0.1
0.05
-0.1
0.1
0.06
-3.4
-3.5
-7.1
-5.8
-4.0
-0.5
0.1
-0.3
0.1
-0.1
-0.5∼-0.4
0.7

Table 9.7: Summary of systematic uncertainties in event reconstruction for SK-I.
148

uncertainty (%)(F) Systematic uncertainties in event reconstruction for SK-II
Single-ring/Multi-ring separation
Single-ring< 400 MeV
e-like1.3µ-like2.3> 400 MeV
e-like1.7µ-like0.7
Multi-GeV
e-like2.6µ-like
1.7
Multi-ring
Sub-GeV
e-like3.8µ-like
8.2
Multi-GeV
e-like1.9µ-like0.8
Particle identification
Single-ring
Sub-GeV
e-like0.5µ-like-0.4
Multi-GeV
e-like0.1µ-like-0.1
Multi-ring
Sub-GeV
e-like1.2µ-like-2.2
Multi-GeV
e-like1.8µ-like-3.4
Energy calibration for FC event
1.7
Up/down symmetry of energy calibration
0.6
0π -like sample selection
100< Pe <250 MeV/c
7.5
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
2.0+
Decay electron tagging (π decay)
e-like 0µedecay
1.2∼1.7
e-like 1µedecay-4.2∼-3.8µ-like 0µedecay
1.6∼1.8µ-like 1µedecay-1.5µ-like 2µedecay-6.5∼-5.9
Decay electron tagging (µ → e decay)
1.1

best-fit (%)

1.3
2.3
1.7
0.7
2.7
1.8
-4.0
-8.5
-1.9
-0.8
0.08
-0.07
0.02
-0.02
0.2
-0.4
0.3
-0.6
-1.6
-0.2
-4.0
-4.8
-9.4
-5.8
-6.0
-0.3
-0.1
0.3
-0.1
0.1
0.4∼0.5-0.7

Table 9.8: Summary of systematic uncertainties in event reconstruction for SK-II.
149

Chapter 10

Discussions
As a result of the study using SK-I and SK-II atmospheric neutrino data, we obtained theconstraints on NSI parameters by means of the robustness of the implementation of 2-flavor ν µ ↔
ντ neutrino oscillations to the atmospheric neutrinos. We compare our results to the existing
results derived from the neutrino scattering experiments, the CHARM and NuTeV experiment,
where the cross section ratio for the deep inelastic scattering is measured. First, cross section
formulae are introduced in Chapter 10.1, and next, existing results from the CHARM and NuTeV
experiment are reviewed. Finally our results are compared to those results.

10.1

Deep inelastic scattering

If we discuss under the tree level, neutral current neutrino scattering off a quark is expressedasdσ(νq, ν̄ q̄) =dydσ(ν̄q, ν q̄) =
dy

G2F
2
s[gL2 + gR
(1 − y)2 ]πG2F2s[gL2 (1 − y)2 + gR]
π(10.1)

where s ' 2mN Eν , y is the Bjorken scaling, and gL , gR are the neutral current couplings assummarized in Table 10.1. mN is the nucleon mass.
Cross section for the neutrino scattering off nucleon is derived from eq.(10.1),dσ(νN ) =dxdydσ(ν̄N ) =
dxdy

G2F
2
2
s[gL2 xq(x) + gR
xq̄(x) + (gR
xq(x) + gL2 xq̄(x))(1 − y)2 ]
π
G2F
2
2
s[gR
xq(x) + gL2 xq̄(x) + (gL2 xq(x) + gR
xq̄(x))(1 − y)2 ]
π

u
d

gL
2
1
2
2 − 3 sin θW− 21 + 23 sin2 θW

gR2− 3 sin2 θW
1
2
3 sin θW

Table 10.1: Neutral current couplings. θW is the Weinberg angle.

150(10.2)

where x is the Bjorken scaling and q(x), q̄(x) are the parton distribution function. Integration
over x and y gives the total cross section,
 2


Z
Z
gL
G2F
g2
2
xq(x)dx +xq̄(x)dxσ(νN ) =s gL2 + R
+ gR
π
3
3


 2
Z
2
2 Z
GF
gR
gL
2
2
σ(ν̄N ) =xq̄(x)dx(10.3)
xq(x)dx + gL +
s
+ gR
π
3
3
Suppose a target of the scattering is isoscalar nucleus and parton disutibution functions for
u-quark and d-quark are denoted as u(x) and d(xa), respectively, neutral current couplings g L ,gR are relpaced with,
Z


1 2
U +D
2
gL xq(x)dx →gL (u)U + gL2 (d)D + (U ↔ D) ≡ g̃L2
2
2
Z



1
2
2
2 U +D
2gR(u)U + gR(d)D + (U ↔ D) ≡ g̃RgR
xq(x)dx →
2
2
Z
Z
U =
xu(x)dx, D = xd(x)dxg̃L2=

gL2 (u) + gL2 (d),

2
2
2
g̃R
= gR
(u) + gR(d)

The total cross sections eq.(10.3) are expressed as
 



G2
Q̄
Q
2
+ Q̄
+ g̃R
σN C (νN ) ≡ σ(νN ) = F s g̃L2 Q +
π
3
3

 


2
GF
Q
Q̄
2
2
σN C (ν̄N ) ≡ σ(ν̄N ) =s g̃L+ Q̄ + g̃R Q +
π
3
3
U +D
Q ≡
2
Using the same method, charge current total cross sections are given as


G2F
Q̄
σCC (νN ) =
s Q+
π
3


2
GF
Q
σCC (ν̄N ) =
s
+ Q̄
π
3

10.2

(10.4)(10.5)(10.6)CHARM experiment

The CHARM neutrino detector is a fine-grained calorimeter followed by an iron spectrometerwith a troidal magnetic field. The calorimeter is surrounded by a magnetized iron frame andhas a sampling step correnponding to one radiation length or 0.22 absorption length with otherdetecting elements, scintillators, propotional drift tubes and streamer tubes. The CHARM
detector was exposed to the neutrino beam in 1982. Neutrino beam was produced as follows:
The 400GeV proton beam was extracted from the CERN SPS and dumped onto thick copper
targets in the direction of the neutrino detector. Electron neutrinos with equal contributions ofneutrinos and anti-neutrinos were produced mainly from neutral kaon decay.

151

The CHARM collaboration measured the follwing combination of νe N and ν̄e N cross section [156]:σ(νe N → νX) + σ(ν̄e N → ν̄X)
2
Re == g̃L2 + g̃R= 0.406 ± 0.140(10.7)σ(νe N → eX) + σ(ν̄e N → ēX)

where the couplings g̃L and g̃R are taken from eq.(10.4). The eq.(10.7) does not depend on any
specific model beyond the standard model nor neutrino energy, thus we can compare it to our
results.
In order to investigate NSI in the CHARM result, we extend the couplings g̃ L and g̃R to thatwith NSI parameters,
X
X
2
2
dL 2
2
g̃L2 → (gL (u) + εuL
|εuL
|εdL
ee ) +
αe | + (gL (d) + εee ) +
αe |
2
2
g̃R
→ (gR (u) + εuRee ) +

α6=e

α6=e

X

X

α6=e

2
dR 2
|εuR
αe | + (gR (d) + εee ) +

α6=e

2
|εdR
αe |

(10.8)

According to this result, together with the corresponding couplings given by the standard
model, (g̃L )2SM = 0.3042 and (g̃R )2SM = 0.0301, the limits on flavor diagonal NSI and flavorchanging NSI off d-quark are−0.5 < εdLee < 0.5,

10.3

|εdP
τ e | < 0.5(P = L, R)(10.9)

NuTeV experiment

The NuTeV detector consisted of an 18m long, 690ton steel-scintillator target, followed byan iron-toroid spectrometer. The target calorimeter was composed of 168 (3m×3m×5.1cm)
steel plates interspersed with liquid scintillation counters (spaced every two plates) and drift
chambers (spaced every four plates). High-purity ν and ν̄ beams are provided by the Sign
Selected Quadrupole Train (SSQT) beam line at the Fermilab Tevatron. Neutrinos are produced
from the decay of pions and kaons resulting from interactions of 800GeV protons in a BeO target.
The NuTeV collaboration measures the cross section ratios of the neutrino-nucleon interactions. The ratios for an isoscalar target and at leading order are given byσ(νN → νX)
2= g̃L2 + rg̃R
σ(νN → µX)
σ(ν̄N → ν̄X)
1 2
R(ν̄) ≡= g̃L2 + g̃R
σ(ν̄N → µ̄X)rσ(ν̄N → µ̄X)r=σ(νN → µX)
R(ν) ≡(10.10)(10.11)(10.12)The effective couplings g̃L , g̃R are the same style in eq.(10.8), with changing e → µ in thecoefficients of the neutral current NSI.
The values of these couplings reported by NuTeV are [155]g̃L2 = 0.3005 ± 0.0014,

2g̃R= 0.0310 ± 0.0011(10.13)

Because of the same reason as we mentioned above, these results can be compared with our
results. Accompanied with the standard model prediction, (g̃L )2SM = 0.0301 and (g̃L )2SM =152

0.3042, the constraints at 90% C.L. to left-handed NSI are|εdLµµ | < 0.003(10.14)|εdL
τ µ | < 0.05

(10.15)For diagonal right-handed NSI at 90% C.L. allowed regions are−0.008 < εdRµµ < 0.015(10.16)

while for flavor changing intearctions,|εdR
τ µ | < 0.05

10.4

(10.17)Comparison with the existing results

As a result of the 2-flavor hybrid model with relative phase, more general case than thatwithout relative phase, the constraints to NSI in the νµ − ντ sector are written as|εµτ | < 1.6 × 10−2 ,−5.3 × 10−2 < ετ τ − εµµ < 4.9 × 10−2

(90% C.L.)(10.18)

As for 3-flavor hybrid model, results are given as|εeτ | < 0.16,−0.05 < ετ τ < 0.06

(90% C.L.)(10.19)If we combine the limit on ετ τ − εµµ with that on ετ τ , limit on εµµ can be extracted as−9.9 × 10−2 < εµµ < 0.11

(90% C.L.)(10.20)As explained in Chapter 1.3, the NSI coupling εfαβP (f = e, u, d P = L, R) corresponds to thestandard model NC coupling gP (f ), so εfαβP itself has no dependence on the matter density nor theneutrino energy. Therefore, we can compare the limits on εfαβP from the atmospheric neutrinos
with that from the neutrino scattering experiments. However, the measurement of NSI using
atmospheric neutrinos cannot distinguish the constraints between left-handed and right-handed
fermions, since NSI is not detected as the cross section, but appeared in neutrino propagation
through the matter of the Earth. Thus our detectable NSI parameter with d-quark is taken asdR
εαβ = εdL
αβ + εαβ . Considering the difference in the detections, the limits are compared in thedR
two-dimentional space of εdL
αβ and εαβ . Figure 10.1 shows the constraints on the NSI parameters.
Note that no constrant on εdPτ τ is available from the other experiments.

153

0.1

εdR
µτ

εdR
eτ

1

0.5

0.05

0

0

−0.5

−0.05

−1
−1

−0.5

0

0.5

1

−0.1
−0.1

−0.05

0

0.05

εdL
eτ
dR
(a) Constraints on εdLeτ and εeτdR(b) Constraints on εdLµτ and εµτ

0.1

εdR
ττ

εdR
µµ

0.1

0.05

0.05

0

0

−0.05

−0.05

−0.1
−0.1

0.1

εdL
µτ

−0.05

0

0.05

0.1

εdL
µµ

−0.1
−0.1

−0.05

0

0.05

0.1

εdL
ττ

dR
(c) Constraints on εdLµµ and εµµdR(d) Constraints on εdLτ τ and ετ τFigure 10.1: Constraints on the NSI parameters. Brue hatched areas indicate the allowed
parameter region obtained in this analysis. Red hatched areas indicate the existing results. For
both areas the hatched region corresponds to the 90% C.L.. Existing limit on the panel (a) is
derived from the CHARM experiment [156], while that on the panel (b) and (c) are derived
from the NuTeV experiment [155].

154

Chapter 11

Conclusions
We studied non-standard neutrino interactions (NSI) with the matter in the Earth, where
NSI consisted of either flavor changing neutral current (FCNC) and lepton non universality(NU). Considering that atmospheric neutrinos were well explained by neutrino oscillations, it
was assumed that neutrino oscillations had the dominant contribution and NSI had the subdominant contribution in the atmospheric neutrino data.
We analyzed atmospheric neutrino data to examine(1) How neutrino oscillations are robust to the atmospheric neutrino data.
(2) What amount of NSI is included in the atmospheric neutrino data. If we would not able
to obtain the signal of NSI, we derive the constraints to them.
where NSI was allowed to arise through the parameters εαβ (FCNC) and εαα (NU). The analysis
was performed with two schemes: (i) 2-flavor νµ ↔ ντ oscillation with NSI in the νµ − ντ sector,and (ii) 2-flavor νµ ↔ ντ oscillation with NSI in the νe − ντ sector.
Scheme (i) assumed that the flavor transition was only occurred between ν µ and ντ , whereas
νe flux was kept as is. In this scheme, in order to investigate (1) and (2), we set two oscillation parameters and two (or three if relative phase η was considerd) NSI parameters as
free parameters. As a result of the analysis without relative phase using SK-I and SK-II atmospheric neutrino data, minimum χ2 value χ2min = 838.9/746 d.o.f. at the best-fit position(sin2 2θ, ∆m2 , ε, ε0 )=(1.00, 2.2 × 10−3 eV2 , 1.0 × 10−3 , −2.7 × 10−2 ) was obtained. No possible
signal of NSI was found, while we constrained the NSI parameters,|εµτ | < 1.1 × 10−2 ,−4.9 × 10−2 < ετ τ − εµµ < 4.9 × 10−2

(90% C.L.)(11.1)

As for the case with non vanishing relative phase, χ2min = 837.5/745 d.o.f. at the best-fit position (sin2 2θ, ∆m2 , ε, ε0 , cos η)=(1.00, 2.2 × 10−3 eV2 , 6.9 × 10−3 , −1.9 × 10−2 , −1.0) was obtained.
Constraints to the NSI parameters were|εµτ | < 1.6 × 10−2 ,−5.3 × 10−2 < ετ τ − εµµ < 4.9 × 10−2

(90% C.L.)(11.2)

No significant difference was found in the allowed neutrino oscillation parameters region between scheme (i) and standard neutrino oscillation. Therefore neutrino oscillations were provento be robust to the atmospheric neutrino data even though NSI existed simultaneously.

155

On the other hand, scheme (2) allowed the flavor transition among all flavored neutrinos. In
this scheme we fixed the neutrino oscillation parameters as the best-fit values from the analysis
with only standard neutrino oscillation, then this scheme purposed to examine (2). As a result,no signal of NSI was given by the SK-I and SK-II data, while we obtained the constraints toNSI|εeτ | < 0.16,−0.05 < ετ τ < 0.06

(90% C.L.)(11.3)

Best-fit position was (sin2 2θ, ∆m2 , εee , εeτ , ετ τ )=(1.00, 2.1 × 10−3 eV2 , −0.25, 0.016, 0.024) andminimum χ2 value was χ2min = 829.9/747 d.o.f.
Limit on εµµ was extracted from the combination of the limit on ετ τ − εµµ and that on ετ τ ,−9.9 × 10−2 < εµµ < 0.11

(90% C.L.)(11.4)

Our constraints to the NSI parameters were compared with the existing limits derived fromthe neutrino scattering experiments, the CHARM and NuTeV experiment.
We concluded that we achieved the purposes, check the robustness of neutrino oscillations
and investigation of NSI, based on the first experimental approach assuming the existence of
NSI together with neutrino oscillations.

156

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163

Appendix A

Systematic Uncertainties
A.0.1

Atmospheric Neutrino Flux

• Anti-neutrino/neutrino ratio
The systematic uncertainty of the relativity between neutrino flux and anti-neutrino flux
comes from π + /π − and K + /K − ratio in hadronic interaction of the flux calculation. The
uncertainties of ν e /νe , shown in the left panel on Figure A.1, are estimated to be 5%
below 1 GeV and 5% from 1 GeV to 10 GeV. The uncertainties of ν µ /νµ are estimatedto be 2% below 1 GeV and 6% from 1 GeV to 10 GeV as shown in the right panel
on Figure A.1. The variations of both ratios are within 10% below 100 GeV, while it
increases almost linearly above 100 GeV. This can be explained as that π’s are the major
source of atmospheric neutrino below 100 GeV, however the contribution of K’s grows up
above 100 GeV, where the different K production model is employed in each flux model.
The anti-neutrino/neutrino ratio uncertainty is divided into three terms: E ν <1 GeV,
1 GeV< Eν <10 GeV, and Eν >10 GeV.
• Up/down ratio
As described in Section 3.2, the neutrino flux below a few GeV has up/down asymmetry
due to the rigidity cutoff by the geomagnetic field, while it becomes up/down symmetricabove a few GeV. The systematic uncertainty of up/down ratio is caused by the treatmentof the geomagnetic field in the flux calculation. However, the uncertainty does not directly
affect the zenith angle distributions, because the up/down asymmetry in low energy islargely washed out due to the poor angular correlation in neutrino interactions. The
systematic uncertainties of up/down ratio are estimated by the comparison of the zenith
angle distributions using flux calculations.
• Horizontal/vertical ratio
The systematic uncertainty of horizontal/vertical ratio is caused by the difference in the
3-dimensional calculation method in each flux model below 3 GeV of neutrino energy,
while, above 3 GeV, it arise due to the predicted K/π ratio in hadronic interactions in
the atmosphere. The magnitude of the uncertainty in the horizontal/vertical ratio is afunction of energy, however is assumed to be fully correlated.
• Neutrino flight length
An uncertainty of the altitude where atmospheric neutrinos are produced affects the neu164νµ/ νµ double ratioνe/ νe double ratio

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

1

10

102

103
Eν(GeV)

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

1

10

102

103
Eν(GeV)

Figure A.1: Anti-neutrino/neutrino double ratio for ν e /νe (left panel) and ν µ /νµ (right panel).
Double ratio is defined as (ν/ν)Fluka or Bartol /(ν/ν)Honda . Solid(dashed) line indicates the double
ratio with the Fluka(Bartol) flux.

trino oscillation probabilities. The uncertainty of the production altitude, distance fromthe surface of the Earth to the production point, is negligible for upward-going neutrinos passing through the Earth because the production altitude is much shorter than theneutrino propagation length in the matter of Earth. However, for downward-going andhorizontally-going neutrinos, the uncertainty of the production altitude has sizable effectto the calculation the oscillation probability. The systematic uncertainty is estimated by
the flux calculations which are carried out with the 10% enhanced and reduced density
structure of atmosphere, as shown in Figure A.2. The uncertainty of air density profile 10%
is given by the comparison between US-standard’76 and MSISE90 [157]. The variationof neutrino flux due to the increased or decreased density is considered as the systematic
uncertainty.
• Solar activity
The primary cosmic ray flux is affected by the solar activity changing in 11 year period
and the atmospheric neutrino flux calculations consider the solar activity. ±1 year of
uncertainty is assigned to the modulation of solar activity.

A.0.2

Neutrino Interactions• Axial vector mass(MA ) in quasi-elastic scattering and single meson production
Quasi-elastic scattering and single meson production have a dependence on the axial vector
mass MA which is set to be 1.21GeV/c as described in Section 3.3. MA is included withthe style of −1.23(1 − Q2 /MA2 )−1/2 , therefore the uncertainty should be considered as afunction of Q2 . Figure A.3 shows the ratio of Q2 spectrum with MA = 1.11GeV/c tothat with MA = 1.21GeV/c. Fitted curve of this ratio is considered as the systematic
uncertainty of MA .
• Quasi-elastic scattering for bound nucleon (Total cross section)
The CCQE cross section for bound nucleon is simulated based on the Smith and Monitz
model. The systematic uncertainty of the CCQE cross section is estimated by the comparison between the Smith and Monitz model and the Nieves model [80], and is set to165

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)

Ratio of MA=1.1 to MA=1.2

Figure A.2: Neutrino flight length as a function of zenith angle. Solid curve shows the flight
length used in our analysis and dashed curve shows that for the compressed density structureof atmosphere by 10%.

1.6

1.4

1.2

1

0.8

0.6

0.4

0

0.2 0.4 0.6 0.81

1.2 1.4 1.6 1.8 2
2
2
Q (GeV/c)

Figure A.3: The ratio of Q2 spectrum with MA = 1.11GeV/c to that of with MA = 1.21GeV/c.
Dot and error bar show the expectation of Monte Carlo events and the fitted curve is consideredas the systematic uncertainty.

166ν/ ν double ratio

Cross section ratio

2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2

1.2
1
0.8
0.6
0.4
0.2

0
νµ/ νe double ratio

1.4

1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
0

50 100 150 200 250 300 350 400 450 500
Eν(MeV)050 100 150 200 250 300 350 400 450 500
Eν(MeV)50 100 150 200 250 300 350 400 450 500
Eν(MeV)

Figure A.4: Top left: The ratio of the CCQE cross section, the Nieves model normalized by theSmith and Monitz model, for νe + ν e (solid curve) and for νµ + ν µ (dahsed curve). Top right:(ν e + ν µ )/(νe + νµ ) ratio. Bottom: (νµ + ν µ )/(νe + ν e ) ratio.
be 1σ. In the Nieves model the motion of target nucleon in nucleus, so called nuclear
effects, is considered using highly sophisticated way. Top left panel of Figure A.4 shows
the calculated CCQE cross section of the Nieves model normalized by Smith and Monitz
model.
• Quasi-elastic scattering for bound nucleon (ν/ν ratio)
The uncertainty of ν/ν ratio in CCQE cross section is estimeted using the same approach asthat for CCQE total cross section. The top right panel of Figure A.4 shows the calculated
CCQE cross section of ν/ν ratio of the Nieves model normalized by the Smith and Monitz
model.
• Quasi-elastic scattering for bound nucleon (νµ /νe )The uncertainty of νµ /νe ratio in CCQE cross section is estimeted by the comparison using
two models. The ratio of Nieves model to the Smith and Monitz model is shown in thebottom of Figure A.4.
• Single meson production (Total cross section)
Estimation of systematic uncertainties related to the single meson production is carriedout by using single π production because of the statistics. It means that the uncertainties
estimated using single π production are also applied to the other simgle meson productions,η or K. The uncertainty of the single meson production cross section is estimated to167

NC ν/ ν double ratio

CC ν/ ν double ratio

2.2
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0

0.5

1

1.5

2

2.5

3

3.5

4

4.5 5
Eν (GeV)

2.2
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0

0.5

1

1.5

2

2.5

3

3.5

4

4.5 5
Eν (GeV)Figure A.5: The ν/ν ratio of the Hernandez model normalized by the Smith and Monitz model.
Left panel shows charged current interactions and right panel shows neutral current interactions.
For the detail, see the text.

be 20 % according to the agreement between Monte Carlo simulation and experimental
results of the νµ p → µ− pπ + reaction, which is the dominant mode among the single meson
productions and various exmerimental results are available. The π 0 emitting interaction
has an additional uncertainly caused by the poor understanding of the π 0 interaction
compared to that with charged π.
• Single meson production (ν/ν ratio)
The uncertainty of ν/ν in single meson production cross section is estimeted by the comparison of the Rein and Sehgal model, used as default in our simulation, and the Hernandez model [158] which includes the recent progresses in the theoretical studies. Figure A.5
shows the ν/ν ratio of the Hernandez model normalized by the Rein and Sehgal model. Leftpanel shows CC interactions: νµ p → µ− pπ + (solid curve), νµ n → µ− pπ 0 (dashed curve),and νµ n → µ− nπ + (dotted curve). Right panel shows NC interactions: νµ p → νµ pπ 0 (solid
curve), νµ p → νµ nπ + (dashed curve), νµ n → νµ nπ 0 (dotted curve), and νµ n → νµ pπ −(dash-dot curve). Energy dependent difference between two models are adopted to thesystematic uncertainties.
• Single meson production (π 0 /π ± )Understanding of the interaction emitting π 0 is relatively poorer than that with charged
π due to the small statistics of the experimental results as seen in Figure 3.10, so thepredicted cross section has large variation among theoretical models. To estimate theuncertainty of the single π 0 production, the Rein and Sehgal model is compared with
the Hernandez model. The Figure A.6 shows the ratio of the CC single pion ν µ crosssection of the Hernandez model normalized by the Rein and Sehgal model, νµ p → µ− pπ +(solid curve), νµ n → µ− pπ 0 (dashed curve), and νµ n → µ− nπ + (dotted curve). According
to the comparison, 40% should be reasonably adopted to the uncertainty of π 0 emittinginteractions relative to the π ± emitting interactions.
• Deep inelastic scattering (Model dependence in lower energy region)
For neutrino energy above 50 GeV, the calculation for deep inelastic scattering agreewith the experimental measurement within 5 %. However, in the lower energy region the168CC νµ cross section ratio

2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

0.5

1

1.5

2

2.5

3

3.5

4

4.5 5
Eν (GeV)Figure A.6: The ratio of the predicted cross section by the Hernandez model to that of the Rein
and Sehgal model for CC νµ interactions. For the detail, see the text.

uncertainty gets larger. Estimation of the systematic uncertainty is carried out by taking
the difference between our default model and CKMT parametrization [159]. The differencein these two models depends on Q2 and is quite large in a few GeV energy region. The
difference of the these two models is assumed to be 1σ.
• Deep inelastic scattering (Total cross section)
The uncertainty in the deep inelastic scattering cross section at high energy (≥ a few tensof GeV) is estimated to be 5 % according to the agreement between measurements andexpectation.
• Coherent pion production
The systematic uncertainty of coherent pion production is estimated based on the comparison of the Rein and Sehgal model and experimental results. As for NC interaction,
several experimental results are available, and the difference between our simulation andthe experimental results is within about 50%. Thus the uncertainty for NC intearction isestimated to be 50%, and it is also applied to CC νe interaction because lepton mass, here
electron mass, is negligible in a few GeV region. However, it can not be capable to theCC νµ interaction due to no evidence for the existing of this mode derived from the K2K
result, so 100% uncertainty is considered for CC νµ interaction.
• NC/CC ratio
The systematic uncertainty in the NC/CC ratio is estimated to be 20 %, which is causedby the poor understanding of NC interactions.
• Nuclear effects in 16 O nucleus
Nuclear effects in 16 O nucleus affects the angular correlation between incident neutrino
and emitted mesons. Our predicted cross section and experimental results are shown in
Figure 3.15. The systematic uncertainty of nuclear effects in 16 O nucleus, the mean freepath of hadrons produced via neutrino interactions, is estimated to be 30 % according to
the agreement between experimental results and our prediction.
• Charged current ντ interaction
169

The systematic uncertainty in the ντ cross sections for CC interactions is estimated to be
30 % by comparing two models of NEUT and Hagiwara et al . [160].

A.0.3

Event Selections• FC and PC reduction
Details of FC and PC reduction are described in Chapter 5.1 and Appendix C. The uncertainty of the FC and PC reduction efficiencies are estimated by the comparison of the
distributions between data and Monte Carlo events as changing the cut variables. The
uncertainty is estimated to be 0.2 % (0.2 %) and 2.4 % (4.8 %) for FC and PC reduction of
SK-I (SK-II), respectively.
• Non-neutrino background
The possible sources of background in FC µ-like and PC samples are cosmic ray muons,
while in FC e-like samples, flasher PMTs and neutron from the rock are majour sources.
The estimated contamination of the background is summarized in Table 8.4 and 8.5.
• UPMU reduction
The methods of the reduction for UPMU events are described in Appendix C.3. The
systematic uncertainty is estimated by the comparison of cut variables between data andMonte Carlo events. The estimated uncertainties are 1.8 % (2.1 %) for UPMU stoppingand 0.3 % (0.3 %) for UPMU through for SK-I (SK-II). Uncertainties for UPMU stoppingand UPMU through are assumed to be correlated.
• UPMU stopping/through separation
The UPMU events are separated by the number of hit OD PMTs within 8 m from theexit position, we call it NHITEX. The systematic uncertainty for the stopping/throughseparation is estimated to be 0.4 % and 0.4 % for SK-I and SK-II, respectively.

A.0.4

Event Reconstructions• Ring counting
The ring counting process is described in Appendix D.2. The systematic uncertainty ofSingle-ring and Multi-ring separation is derived by several uncertainties such as watertransparency and detector calibration. The uncertainties are estimated by the comparisonof ring counting likelihood distributions between data and Monte Carlo events.
• Energy calibration
The absolute energy scale is calibrated with several calibration sources as described in
Section 4.5 and the systematic uncertainty is estimated to be 1.1 % for SK-I and 1.7 % 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 4.20). The up/down
asymmetry of the energy scale is estimated to be 0.6 % both for SK-I and SK-II.

170

Appendix B

Atmospheric Neutrino Analysis
B.1

2-Flavor νµ ↔ ντ Oscillation Analysis

νµ ↔ ντ oscillation analysis is performed SK-I and SK-II FC+PC+UPMU combined data
set. As the result of the global scan on the oscillation parameter grid, the minimum χ 2 value,χ2min = 832.8/748 d.o.f., is obtained at (sin2 2θ, ∆m2 ) = (1.02, 2.1 × 10−3 eV2 ), where θ isthe mixing angle and ∆m2 = m23 − m22 is the difference of the squared mass of ν3 and ν2 .
When only the grid in physical region is considerd, χ2min = 834.3/748 d.o.f. is obtained at(sin2 2θ, ∆m2 ) = (1.00, 2.1 × 10−3 eV2 . Assuming no oscillation, χ2 value takes 1395.6 for747 d.o.f. . The hypothesis of no oscillation is strongly rejected.
Figure B.1 shows the contour plots of the allowed regions of the neutrino oscillation parameters (sin2 2θ, ∆m2 ). Three contours correspond to the 68%, 90% and 99% confidence levels(C.L.) allowed regions which are defined to be χ2 = χ2min +3.34, 5.75, and 10.41 based on
the minimum in the physical region, respectively. These intervals are derived by using the 2dimensional Gaussian approximation from the minimum in the unphysical region [161]. The
90 % allowed parameter region is :1.7 × 10−3 eV2 < ∆m2 < 2.7 × 10−3 eV220.95 < sin 2θ

(B.1)
(B.2)

Figure B.2 shows the χ2 − χ2min distributions as a function of sin2 2θ and ∆m2 , which are sliced
at ∆m2 = 2.1 × 10−3 eV2 and sin2 2θ = 1.00 .
Zenith angle and momentum distributions are shown in Figures from 6.8 to 6.12 in Chapter 6.
The data (dots and errors) are compared with the Monte Carlo expectation without oscillations(dashed lines) and the best-fit expectation for 2-flavor νµ ↔ ντ oscillations (solid lines). The
2-flavor νµ ↔ ντ oscillation hypothesis provides a consistent explanation to all data samples.

171∆m223 (eV 2 )

10

−2

−3

10 0.70.8

0.9

1
sin 2 θ23
2

min

20

χ 2− χ 2

χ 2− χ 2

min

Figure B.1: Allowed oscillation parameters for 2-flavor νµ ↔ ντ oscillations. The horizontal axis
shows sin2 2θ and the vertical axis shows ∆m2 . Three contours correspond to the 68 %, 90 %,and 99 % C.L. allowed regions, respectively.

18
16

14
12
10
8

14
12
10
8

6

6

4
2

4
2
0 −3
10

20
18
16

10
∆m223 (eV 2 )

−2

0
0.7

0.8

0.9

1

1.1
1.2
sin 2 2 θ23

Figure B.2: χ2 − χ2min distributions as a function of ∆m2 (left) and sin2 2θ (right), in which
χ2min is the minimum value of χ2 in physical parameter region. sin2 2θ is set to be 1.00 in theleft-hand figure and ∆2m is set be 2.1 × 10−3 eV2 in the right-hand figure. Three horizontal lines
correspond to the 68 % (dotted line), 90 % (solid line) and 99 % (dashed line) C.L. intervals,
respectively.

172

Appendix C

Data Reduction
In this section the procedures of the data reduction for FC, PC and UPMU events aredescribed.

C.1

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.

C.1.1

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 C.1 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 thecosmic ray muon events. Figure C.2 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.
173

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10

10

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1
10

3

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10

data
MC

2

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10
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data
MC

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0

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100

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250

300

350

400

450

500

0

50

100

150

200

250

300

350

400

450

500

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

Figure C.1: 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.

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 10 6 events/day
to 3000 (2200) events/day for SK-I (SK-II).

C.1.2

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 C.2 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 C.3 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
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.

174

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(a) data

rejected

Number of events

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1010(b) MC101(a) data

rejected(b) MC

10
1
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10

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4

(c) final sample

104(c) final sample

10

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MC

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50

60

70

80

90

0

100

10

20

30

NHITA800

40

50

60

70

80

90

100

NHITA800

Figure C.2: NHITA800 distributions for (a) the raw data, (b) fully 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 arrows written
by dotted (solid) lines show the selection criteria in the FC 1st (2nd) reduction.

10

5

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1

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104(c) final sample

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0.1

0.2

0.3

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0.6

0.7

0.8

0.9

1

PEmax/PE300

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PEmax/PE300

Figure C.3: 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.

175C.1.3

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
(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

(C.1)

where ti and σi are the observed hit time of the i-th PMT and its resolution, and Ti is thehit time expected from the entering time of muon and its track. Figure C.4 and C.5 show
the number of 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 :(1)

NHITAin ≥ 10
or
NHITAin ≥ 5

if goodness of stopping muon fit > 0.5
176

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10(b-in) MC

10
10
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data
MC

10

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10
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(c-in) final sample

2

rejected(c-out) final sample
data
MC

10

1

1
0

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25

30

35

40

45

50

Nhitain

0

5

10

15

20

25

30

35

40

45

50

Nhitaout

Figure C.4: 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.

177

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2

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10

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Number of events

10(b-in) MC
10

2

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MC

10

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10

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2

rejected(c-out) final sample
data
MC

10

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0

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25

30

35

40

45

50

Nhitain

0

5

10

15

20

25

30

35

40

45

50

Nhitaout

Figure C.5: Number of hit OD PMTs within 8 m of the entrance point (left) of muons andwithin 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.

178

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rejected

4

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Number of events

Number of events

4

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10

10
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10

(a) data(b) MC

10

10

4

10

3

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2

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

data
MC

10

2

data
MC

10

1

1
0

5

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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 C.6: 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.

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 C.6 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 C.7). The rejection
criteria for cable hole muons are :(1)

One veto counter hit

and
(2)

Lveto < 4 m
Lveto is the distance from the cable hole to the the reconstructed vertex.
179

Cosmic ray µ

Veto Counter
Cable Bundle
from PMT
To
Electronics
Hut

Outer
Detector

Inner
Detector

Figure C.7: A schematic view of a cable-hole muon and a veto counter.

Figure C.8 shows the reconstructed vertex distribution for the FC 1-ring µ-like events before andafter 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 C.9 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 C.10 and C.11 shows NMIN100 distributions for the data after the 2nd reduction, theFC atmospheric neutrino Monte Carlo events and the FC final events.

180

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 C.8: 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 C.9: 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 .

181

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50

NMIN100 (NHITID > 800)

0

5

10

15

20

25

30

35

40

45

50

NMIN100 (NHITID < 800)

Figure C.10: 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.

182

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rejected(a) data

Number of events

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data
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NMIN100

Figure C.11: 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.
183

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 E vis >30 MeV.
After the FC third reduction, the event rate is 45 (21) events/day for SK-I (SK-II).

C.1.4

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 C.12 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 C.12.
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 :


B
A
B
1 X QAi − hQ i × Qi − hQ i
(C.2)
r=
N
σA × σ Biwhere 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,(C.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.

C.1.5

Fifth Reduction

The remaining background events are removed by several criteria specialized for each kindof background events.
184

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0.6

Maximum Correlation

Maximum Correlation

1.2

rejected

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0.2

(a)

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1.2
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0

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20

30

40

50

60

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(b)
0

10

20

30

40

50

60

70

Number of matched events

Number of matched events

Figure C.12: 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 ??). Figure C.13 shows NHITA in 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.
185

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3

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2

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1

(a) data

rejected10

5

Number of events

Number of events

1(b) MC

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MC

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data
MC

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25

30

35

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40

NHITAin

5

10

15

20

25

30

35

40

NhitainFigure C.13: 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 C.14 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
186

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rejected

rejected(a) data

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10

Number of events

Number of events

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MC

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data
MC

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20

25

30

35

40

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10

15

20

25

30

35

40

Nhitaearly

NHITACearly

Figure C.14: 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 C.15 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 ?? 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.
187

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2

5

(a) data

rejected

10
4
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(b) MC

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10

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1

2

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2

(a) data

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

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MC

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100

120

140

160

180

200

0

20

40

60

80

100

120

140

160

180

200

PElatePElateFigure C.15: 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.

C.1.6

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 C.1. The detection efficiency for final events is estimated to be 98.8 %and 99.5 % 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 C.30 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 C.2.

188

Reduction step

SK-I (%)

SK-II (%)

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

100.00100.00
99.92
99.33
99.30
98.84

100.00
99.94
99.76
99.45
99.37
99.51

Table C.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 E vis
larger 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 C.2: Summary of the estimated upper limit of the contamination of each background.

189

C.2

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.

C.2.1

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), PE tot
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 C.16 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).

C.2.2

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 C.17. 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.
190

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Number of events

Number of events

1010(b) MC

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10

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100

150

200

250

300

350

400

TWIDA (nsec)

0

50

100

150

200

250

300

350

400

TWIDA (nsec)

Figure C.16: 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 C.17: 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.

191

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 C.18 shows the NHITAC min distributions for the data, atmospheric neutrino Monte Carlo events after the PC 1st reduction and final
samples for SK-I. Figure C.19 shows the scatter plots of the NHITA endcap 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 C.20 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 C.21 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).

C.2.3

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 :192

Number of events

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rejected

Figure C.18: 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
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NHITACmin

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rejected

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0

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(b) final sample (MC)

100

120

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160

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200

NHITAside

Figure C.19: 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.

193

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 C.20: 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

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20

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750

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(MC)

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2000

PE200 (p.e.s)

0

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500

750

1000

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1500

1750

2000

PE200 (p.e.s)

Figure C.21: 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.

194

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rejected

Number of events

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10
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10

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MC

Figure C.22: 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.

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 ??). 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 C.22.
The event rate after the PC 3rd reduction is 100 (200) events/day for SK-I (SK-II).

C.2.4

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 through195

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 toreject corner clipping muons. The distribution for SK-I of the d~pfit · d~PMT and the DCORN
are shown in Figure C.23 and Figure C.24, respectively. Through-going muons which have longtrack length are rejected by the third cut. Figure C.25 shows the scatter plots of the TLMU andthe goodness of the through-going muon fit for the data and atmospheric neutrino Monte Carloevents 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.

C.2.5

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.
196

Number of events

10

6

10

5

10

4

10

3

10

2

10

4

rejected(a) data

Figure C.23: 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 C.24: 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)197

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 C.25: 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.
198

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 by aprecise fit (MS-fit) using the Cherenkov ring pattern (see Section D.4). 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.
199

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.

200

10000(a) data
EVIS/2 (cm)

8000
6000
4000
2000

Figure C.26: 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 C.26 shows the scatter plots of Evis /2 and TRACK.

C.2.6

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 C.3. The systematic uncertainty of PC reduction is estimatedto be 2.4 % and 4.8 % for SK-I and SK-II, respectively.

201

Reduction step

SK-I (%)

SK-II (%)

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

99.0
96.7
95.7
89.9
88.7

98.6
93.4
92.3
84.6
82.6

Table C.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 E vis
larger 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 C.30 and is 0.61±0.02 (0.53±0.03) events/day for SK-I (SK-II) as shown in
Figure C.30.

C.3

Reduction for Upward-Going Muon Sample

Finally, we mention the reduction processes for UPMU events.

C.3.1

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
(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.

C.3.2

Zenith angle cut

In order to reject the downward-going muon cosmic ray muons, seven different fitters specialized to fit stopping muons, through-going muon events and muon events with Bremsstrahlung
are used. The algorithm of the zenith angle cut is as following :(1) Apply a muon fitter.

202

(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 [162].
All events from the output of the upward-going muon reduction are passed through theprecise fitter which is described in D.9. The direction reconstructed by the precise fitter is used
to select upward-going events by the criterion for cosΘ <0.

C.3.3

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.

C.3.4

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 1.8 % and 2.1 % for SK-I and SK-II, respectively.
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 of203the Monte Carlo events. The estimated errors are 0.3 % (0.3 %) for upward stopping muons and
1.8 % (2.1 %) 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 C.27 shows the zenith versus azimuth directions for the upward going
muon sample. Clusters of cosmic ray downward muons are seen in the regions zenith anglecosΘ > 0 (downward) and azimuth angle around φ = 120◦ , 180◦ and 270◦ because the rock of
the mountain covered over the detector is thin for these directions. In order to estimate thecontamination of the background events, the upward going muon samples are divided into two
azimuth angle regions. Figure C.28 shows the azimuth angle distributions for upward going
muon samples, in which region (2) shows the thinner mountain direction, which is defined as
60◦ < φ < 310◦ for stopping muons and 60◦ < φ < 240◦ for through-going muons and region (1)shows the thicker mountain direction.
Figure C.29 shows the zenith angle distributions for upward-going muon samples, in which
region (1) and (2) are normalized by the coverage of the azimuth angle. While the zenith angle
distributions are almost flat for upward going events (cosΘ <0), the number of events in theregion (2) exponentially increase with the cosine of zenith angle for downward-going (cosΘ >0).
The contaminations of the cosmic ray muons into the upward-going direction are estimated by
an extrapolation from the 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 shown in
Figure C.30. 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).

204

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]

10

3

10

2
(1)

(2)

(1)

Number of Events

Number of Events

Figure C.27: 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.

10

4

10

3

10

2

(1)

(2)

(1)

10

1
0

100

200

300

0

φ [degree]

100

200

300

φ [degree]

Figure C.28: 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.

205

10

Number of Events

Normalized Number of Events / φ=360°

103
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

-0.1

-0.05

0

0.05

cosΘ

cosΘ

Figure C.29: 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).

12

SK1

SK2

Event rate (event/day)

10

8

6

FC
PC
Upmu
4

2

0
1996

1998

2000

2002

2004

2006

Year

Figure C.30: 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.

206

Appendix D

Event Reconstruction
Event reconstruction processes are applied to the atmospheric neutrino events which passthrough the data reduction processes. The common programs are applied for both the observed
data and the atmospheric neutrino Monte Carlo events. The event reconstruction process isfully automated.
The reconstruction process for FC, PC, and UPMU events are described in this chapter.

D.1

Vertex Reconstruction

First, the vertex is estimated using timing information of the PMTs. The procedure has
three steps.
(1) Point-fit: roughly find the vertex using the timing information(2) Ring edge search: find Cherenkov ring edge

(3) TDC-fit: fit precisely using (1) and (2) results

These steps are described below.

D.1.1

Point-Fit

The vertex fitter searches for the position where the timing residuals of the entire hit PMTs
are approximately equal. The time residual of the ith PMT, ti , is calculated by subtracting thetime of flight of the photons from the hit timing of the ith PMT, t0i :
→
− →
−
n
ti = t0i − × |Pi − O |(D.1)
cwhere n is the refractive index in water, O is the assumed vertex position, and P i is the positionof the ith PMT. To estimate the vertex, we use the estimator Gp for the goodness of the fitting:

 !
1 X
1 ti − t0 2
Gp =
exp −
(D.2)
N
2 1.5 × σiwhere N is the number of hit PMTs, σ is the typical timing resolution of a PMT (2.5 nsec), andt0 is the offset value of the time residual which is properly set in the fitting. The factor 1.5 inthe denominator is chosen to optimize the fitter performance. Gp takes a value between 0 and
1. The fitter searches for the vertex position with which Gp takes the maximum value.
207

number of p.e.

150
100
50

second derivarive

0

0

20

40

60

80

0

20

40

60

80

Figure D.1: Upper figure shows
the typical P E(θ) distribution andlower figure shows the second
derivative of P E(θ).

10

0

-10

degreeD.1.2

Ring Edge Finding

In this step, the edge of a Cherenkov ring is estimated. The information of the Cherenkov
ring is used in the precise vertex fitter in the next step. The procedures of this step are as
follows:
1. Make the angular distribution of the p.e.s, P E(θ), as a function of the opening angle
θ between the assumed ring direction and the Cherenkov photon direction. The PMT
acceptance and the transparency of water are taken into account in this calculation.
2. Obtain the Cherenkov opening angle θedge which satisfies the following criteria:•

d2 P E(θ)
=0
d2 θ
θedge

• θedge > θpeak where θpeak is the angle at which P E(θ) has a peak
If there are several θedge candidates, the θedge nearest to θpeak is selected.
3. Calculate the estimator, Q, which is defined as:
R θedgedθP E(θ)
Q= 0×
sin θedgedP E(θ)
dθ
θedge

!2

(θedge − θexp )2
× exp −
2σθ2




(D.3)

where θexp and σθ are the opening angle and its resolution of the Cherenkov cone expectedfrom the p.e.s in the assumed Cherenkov ring, respectively.
Fig. D.1 shows the typical P E(θ) distribution and its second derivative. Q is maximized bychanging the direction of the ring. These three steps are iterated until the maximum Q is found.

208

PMTs

Figure D.2: Schematic viewof Cherenkov radiation. The
Cherenkov photons to the ith
PMT were emitted at Xi . li
is the track length of thecharged particle from the vertex.

Opening
angle θ c

ΟXitrack length

li

D.1.3

Cherenkov
cone

TDC-Fit

The final step is a precise vertex fitting using the results of the previous steps. In thisprocedure, the track length of the charged particle and the scattered Cherenkov photons are
considered.
The time residual is calculated as:
(
−
→ →
−
→
− −
→
t0i − 1c × |Xi − O | − nc × |Pi − Xi | Inside the Cherenkov ring
ti =
(D.4)
→
− →
−
t0i − nc × |Pi − O |Outside the Cherenkov ring→−where Pi is the position of the ith PMT, Xi is the estimated point at which the photons were
emitted to the ith PMT, and n is the refractive index of water. Fig. D.2 shows the schematic
view of the Cherenkov radiation.
We define the estimators, GI and GO , for the inside and outside of the Cherenkov ring,
respectively. GI is defined as follows:

 !
X 1
ti − t0 2
1(D.5)
exp − ·GI =2
1.5 · σσi2iwhere σi is the timing resolution of the ith PMT as a function of qi (see Fig. ??), σ is the timing
resolution averaged over all hit PMTs, and ti and t0 are the time residual of the ith PMTs andits offset, respectively.
For the PMTs outside the Cherenkov ring, the effect of the scattered light is considered. We

209define the estimators, GO1 and GO2 according to the time residual of the PMTs:
#
!
"

 !
X 1
ti − t0 2
1, Gscatt (ti , t0) × 2 − 1
max exp − ·GO1 =2
1.5 · σσi2i( for ti > t0 PMTs) (D.6)

GO2

=

X 1
i

σi2

1
exp − ·
2ti − t0
1.5 · σ

2 !

×2−1

!

2 !



( for ti ≤ t0 PMTs) (D.7)

whereGscatt (ti , t0) =

Rq =

X



ti − t0
1.5 · σ

Rq
+ 1−
1.52





ti − t0exp −60nsec



(D.8)

qi

θ<θc +3.0

X

Rq1
× exp − ·
2
1.5
2

qi

: Fractional p.e.s detected within Cherenkov ring

(D.9)

θ<70◦

The numerical factors in Eqs.(D.5-D.9) are optimized by a Monte Carlo simulation study.
Finally, the estimator of the fitting, GT , is defined as:
GT =GI + G O1 + G O2
X 1
σi 2

(D.10)i

The vertex position which maximizes GT is defined to be the best fit vertex position by TDC-fit.

D.1.4

Performance of TDC-fit

Figures D.3 and D.4 show the resolution of TDC-fit estimated by atmospheric neutrinoMonte Carlo events. The resolution is defined as the distance where 68% of the total events arecovered. For SK-I, the resolutions are estimated to be 62 cm, 56 cm, 82 cm for FC single-ring,
FC multi-ring µ-like, and PC events, respectively. As for SK-II, the resolutions are estimatedto be 61 cm, 75 cm, 91 cm for FC single-ring, FC multi-ring µ-like, and PC events, respectively.
The event reconstruction process starts from the vertex fitting. The vertex position is reconstructed using the timing information of hit PMTs in three steps.

D.2

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.

D.2.1

Ring Candidate Search

Cherenkov ring candidates are searched by an algorithm using a Hough transformation [163].
Figure D.5 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 the21018000
16000
14000
12000
10000
8000
6000
4000
2000
0

2250
2000
1750
1500
σ = 62 cm 1250
1000
750
500
250
0
100
150
200
0

FC Single-ring

050∆pos (cm)FC Multi-ringµ-likeσ = 56 cm

50

100

150

200

∆pos (cm)

2500

PC
2000
1500

σ = 82 cm

1000
500
0

0

50

100

150

200

∆pos (cm)Figure D.3: The vertex resolution of TDC-fit for SK-I. (Left) FC single-ring samples (Center)
FC multi-ring µ-like samples. (Right) PC samples. Resolutions are estimated from atmospheric
neutrino Monte Carlo events. Hatched regions show the 68% of total events.
20000
15000
12500
10000
7500
5000
2500
0

1800
1600
1400
1200
σ = 61 cm 1000
800
600
400
200
0
100
150
200
0

FC Single-ring

17500

0

50

∆pos (cm)FC Multi-ringµ-likeσ = 75 cm

50

100

150

200

∆pos (cm)

PC

3500
3000
2500

σ = 91 cm

2000
1500
1000
500
0

0

50

100

150

200

∆pos (cm)Figure D.4: The vertex resolution of TDC-fit for SK-II. (Left) FC single-ring samples. (Center)
FC multi-ring µ-like samples. (Right) PC samples. Resolutions are estimated from atmospheric
neutrino Monte Carlo events. Hatched regions show the 68% of total events.

211

42 deg. ringhit PMT(possible center)

centerCherenkov ring

(most probable)

Figure D.5: 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.

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 D.6.
The two peaks correspond to the directions of Cherenkov rings.

D.2.2

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
(D.11)
LN +1 =
log prob qiobs , N
n=1 αn · qi,niwhere 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

212

600
400
200
0
150

Θ(

de 100
gr
ee 50
s)

300

200
100
00

Φ

)

ees
(degr

Figure D.6: A charge map from Hough transformation algorithm for a typical two ring events.
The peaks are the direction of the Cherenkov rings.

momentum limit. The probability function in Equation (D.11) is defined by :
(qiobs − qiexp )2
1


√

(for qiexp > 20 p.e.)
exp
−

2

2
σ
2π σ




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.)i(D.12)

where 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 :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.
213

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 be
true.

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.

The final evaluation function is determined by the probability density functions (PDFs) and log
likelihood method, and written as:
F

=
=

6
X

log [(Pi )](D.13){log [(Pi )N +1 ] − log [(Pi )N ]}(D.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 D.7 shows the distribution of the
final evaluation function for SK-I and SK-II.

D.3

Particle Identification

Particle identification(PID) procedure estimates the particle types of the reconstructed Cherenkov
rings using their patterns and the opening angles. The Cherenkov rings are categorized into 2
types: shower type which we call ‘e-like’, and non-shower type which we call ‘µ-like’.

D.3.1

Estimation of the Particle TypeWe define a likelihood function Ll for the electron assumption (l = e) and the muon assumption (l = µ). The definition of Ll for the nth ring is:


X
Y
exp,l
exp,l prob qiobs , qi,n+qi,n(D.15)Ll (n) =0n0 6=ni
θi,n <1.5θc,nexp,lwhere qiobs is the observed p.e.s for the ith PMT and qi,nis the expected p.e.s from the nth ringassuming the particle type is l (l = e or µ). The function prob(qiobs , qiexp,l ) is the same function
as used in the Ring-Counting procedure (defined by Eq.(??)). The product in Eq.(D.15) is madefor the PMTs whose opening angle from the nth ring direction is within 1.5 times the estimated
Cherenkov opening angle of the nth ring. The calculation of the expected p.e.s are described in

214

200
175
150
125
100
75
50
25
0
-20

-15

-10

-5

0

5

10

15

20

Number of events

Number of events

700
600
500
400
300
200
100
0
-20

350
Single-ring
300
250
200
150
100
50
0
-15
-10
-5
120

Single-ringMulti-ring
Sub-GeV

0

5

10

15

Multi-ring

100
Multi-GeV

80
60
40
20

-15

-10 -50
510Ring counting likelihood

15

20

0
-15

-10

-5
0
5
Ring counting likelihood

10

15

Figure D.7: 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.

the following subsections. Ll is maximized by changing the direction and the opening angle ofexp,lthe nth ring while qi,n(n0 6= n) are fixed.
0
In order to combine the information of Cherenkov ring pattern and Cherenkov opening angle,
the likelihood Ll is transformed into the χ2 function:
χ2l (n) = −2 ln Ll (n) − const.

(D.16)The probability from the Cherenkov pattern is written as:
Plpattern (n) = exp

1 χ2l (n) − χ2min
−
2
σχ2 2

2 !

(D.17)√
where χ2min = min[χ2e , χ2µ ], σχ2 is 2ND , and ND is the number of PMTs which were used in
the calculation of Ll .
The probability from the Cherenkov opening angle is written as:

2 

exp,l
θ
−
θ
n
 1 c,nPlangle (n) = exp −(D.18)22σθwhere θc,n is the reconstructed Cherenkov opening angle of the nth ring, θnexp,l is the expected opening angle for particle type l(electron or muon), σθ is the resolution of the estimated
Cherenkov angle.
215

The total probability is defined as the product of Ppattern (n) and Pangle (n):Pltotal (n) = Plpattern (n) × Plangle (n)(D.19)

where suffix l is e or µ.
If Petotal (n) > Pµtotal (n), the nth ring is determined as e-like and vise versa. For multi-ring
events, we adopt the ring pattern probability P pattern only, because the performance of thereconstruction of Cherenkov angle is relatively poor than in the case of single ring events.

D.3.2

The Expected p.e. Distribution for Electrons

The expected p.e. distributions for electrons are made using a Monte Carlo simulation. In
advance, we calculate the expected p.e. distribution, Qexpe (pe , θ), which will be detected by acircular area of 50 cm diameter(same size as the 20 inch diameter PMTs) on a hypotheticalspherical surface with the radius Rsph =16.9m(radius of the inner tank). Qexpe (pe , θ) is given as afunction of electron momentum pe (MeV/c) and the opening angle θ from the electron direction.
The expected p.e. for the ith PMT due to the nth ring is calculated as:eqi,n= αn,e × Qe (pe , θi,n ) ×



Rsph
ri

1.5

 r 
i
× exp −× f (Θi )λ(D.20)

where
e
qi,n
: expected p.e.s for the ith PMT due to the nth ring

αn,e : normalization factor
θi,n : opening angle between the nth ring direction and the
direction from the vertex to the ith PMT
ri : distance from the vertex to the ith PMT
Θi : angle of photon arriving direction relative tothe ith PMT’s facing directionf (Θi ) : correction function of the PMT acceptanceλ : attenuation length of light in water (see Section ??)The factor (Rsph /ri )1.5 takes into account the ri dependence of the intensity of the Cherenkov
light. The index 1.5 was determined by a Monte Carlo study.

D.3.3

The Expected p.e. Distribution for Muons

The expected p.e. distribution for a muon is analytically calculated by the following equation:
!
 r 
2
sin
θ
i,n
i
µ
knock

+
q
×
exp
−
= αn,µ ×qi,n× f (Θi )(D.21)idθ
λri sin θi,n + ri dx |x=xi

216

dxsinθ

r dθ

r
dθ

Cherenkov photons
trajectory

θ
muon
track

dx

Figure D.8: Schematic view of Cherenkov radiation from a muon. Cherenkov opening angle
θ changes as the muon momentum changes due to the energy loss in water. The Cherenkov
photons are emitted into the region dx · sin θ + r · dθ during the muon propagation dx.
whereµ: expected p.e.s for the ith PMT due to the nth ring
qi,n

αn,µ
knock
qi

: normalization factor: observed p.e.s for the ith PMT due to the knock-on electronsx : distance from the vertex along the muon trajectoryxi : distance from the vertex to the Cherenkov emission point
for the i th PMT
ri , θi,n , λ, Θi , f (Θ) : same quantities as in Eq.(D.20)
The numerator sin2 θ in Eq.(D.21) comes from the number of emitted Cherenkov photons todθ
the direction θi,n (see Eq.(??)). The denominator ri (sin θ + ri dx) comes from the area where
Cherenkov photons are emitted to. Fig. D.8 shows the schematic view of the Cherenkov photon
emission. The area changes due to the decreasing of the Cherenkov opening angle caused by theenergy loss of the muon.
qiknock is the contribution from the knock on electrons which is estimated by a Monte Carlosimulation.

D.3.4

The Expected p.e. Distribution for Scattered LightA PMT hit by the scattered light can be distinguished from a PMT hit by direct photons
using the timing information. The ’Off timing’ hit PMTs are chosen by the criteria:
tpeak − 30nsec < t0i < tpeak + 2σi + 5nsec
tpeak + 2σi + 5nsec < t0i: direct photons(D.22): scattering photons(D.23)

where t0i is the time residual for the ith PMT defined in Eq.(D.4), tpeak is the time at peak
position of the time residual distribution, and σi is the measured timing resolution as a function
217

300

µ-likee-like

250

60

200

40
30

100
50

160
140
120
100
80
60
40
20
0
-30

-8

-6

-4
e-like

-2

0

2

4

6

8

10

µ-likeNumber of events

Number of events

150

0
-10

µ-likee-like

50

20
10
0
-10

-8

-6

120

-4
e-like

-2

0

2

4

6

8

10

µ-like

100
80
60
40
20
-20

-10

0
10
PID likelihood

20

30

0
-15

-10

-5

0
5
PID likelihood

10

15

Figure D.9: The PID likelihood distributions of SK-I for 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). Dot with cross and solid line indicate the data and Monte Carlo events respectively. In the Monte Carlo events 2-flavor νµ ↔ ντ oscillation is considered with the parameter(sin2 2θ, ∆m2 ) = (1.00, 2.5 × 10−3 eV2 ). The hatched area indicates CC νµ interactions.
of observed p.e.s as shown in Fig. ??.
The number of p.e.s due to the scattered photons, qiscatt , is estimated from the ’Off timing’
PMTs and added to the expected p.e.s for the ith PMT:
qiexp,l = qil (direct) + qiscatt(D.24)

where the suffix l represents e or µ.

D.3.5

Performance of Particle Identification

The performanceof PIDp
is estimated by a Monte Carlo study. Figures D.9 and D.10 show the√PID parameter − log Pe − − log Pµ distributions for SK-I and SK-II final sample, respectively.
Sub-GeV sample is defined as the events with visible energy below 1.33 GeV, and Multi-GeV
sample is defined as the events with visible energy over 1.33 GeV. The peaks in e-like and µ-likeevents are clearly separated. Fig. D.11 shows the PID efficiency for CCQE events which are
identified as single ring, and a function of the charged lepton momentum. The efficiency is very
high for all energy region, and the misidentification probabilities are estimated to be 0.5% for
CCQE νe events and 1.2% for CCQE νµ events.

D.4

Precise Vertex Reconstruction

For single-ring events, the vertex resolution in the longitudinal directions fitted by vertex
fitting algorithm described in Section D.1 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
218

µ-likee-like50µ-likee-like

40
30

-8

-6

-4

-2

0

2

6

8

10

µ-likee-like120
100

4Number of events

Number of events

225
200
175
150
125
100
75
50
25
0
-10

20
10
0
-10
120

-8

-6

-4

-2

e-like

100

0

2

4

6

8

10

µ-like

80

80
60
40
20
0
-30

60
40
20
-20

-10

0
10
PID likelihood

20

30

0
-15

-10

-5

0
5
PID likelihood

10

15

Figure D.10: The PID likelihood distributions of SK-II for 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). Dot with cross and solid line indicate the data and Monte Carlo events respectively. In the Monte Carlo events 2-flavor νµ ↔ ντ oscillation is considered with the parameter(sin2 2θ, ∆m2 ) = (1.00, 2.5 × 10−3 eV2 ). The hatched area indicates CC νµ interactions.

PID efficiency

1
0.8
0.6

(a) νe Q.E.

0.4(b) νµ Q.E.

> 5 GeV/c> 5 GeV/c

0.2
0
10

2

10

3

10

4

10

Lepton Momentum (MeV/c)

2

10

3

10

4

Lepton Momentum (MeV/c)

Figure D.11: The PID efficiency for CC quasi-elastic single-ring event as a function of charged
lepton momentum for (a) νe and (b) νµ .

219

SK-I(cm)FC Sub-GeV
single-ringe-likeµ-likeFC Multi-GeV
single-ringe-likeµ-likemulti-ringµ-likePC

SK-II(cm)

27
26

32
31

49
24

47
27

56
56

75
64

Table D.1: The vertex resolutions for SK-I and SK-II. The resolution is defined as the width
where 68 % of the events are included. Note that the resolution for FC multi-ring samples are exactly same as the one after TDC-fit, since multi-ring events skip the precise vertex reconstruction
processes.

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 withthe 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 theatmospheric neutrino Monte Carlo sample. Figure D.12 and D.13 show the distance between thetrue vertex and the reconstructed vertex for FC single-ring events, FC multi-ring µ-like eventsand PC events for SK-I and SK-II. The vertex resolutions are summarized in Tables D.1. The
resolution is defined as the width where 68 % of the events are included.

D.5

Decay Electron Search

Improvements of the selection efficiency of µ-like event is achieved by requirng the detectionof the electron produced by the muon decay (decay electron). In this section the detection
method of the decay electrons followed by the primary events is described. Decay electron
search consists of several criteria so as to detect three types of decay electrons :• sub-event type
Decay electrons observed as a separate event (sub-event) as shown in Figure D.15.
• primary-event type
Decay electrons observed in the same event as primary event.
• split type
Decay electrons occurred around the end of timing window, therefore the event is recorded220

25000FC Sub-GeV
1ring e-like

14000
12000
10000

σ = 27 cm

8000

15000

σ = 26 cm

10000

6000
4000

5000

2000
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1ring µ-like

20000

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100

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200

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2000

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1ring e-like

1750
1500
1250

100

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200

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1ring µ-like

5000

σ = 49 cm

1000

50

∆vertex (cm)

750

4000

σ = 24 cm

3000
2000

500
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1000

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50

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2000
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PC

14000
12000

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10000

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8000
6000
4000
2000

0

50

100

150

200

0

0

∆vertex (cm)∆vertex (cm)

50

100

150

200

∆vertex (cm)Figure D.12: 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.

20000

FC Sub-GeV
1ring e-like

12000
10000

15000

σ = 32 cm

8000
6000

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σ = 31 cm

10000
7500

4000

5000

2000
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1ring µ-like

17500

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250
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200

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1ring e-like

FC Multi-GeV
1ring µ-like

5000
4000

σ = 47 cm

σ = 27 cm

3000
2000
1000

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150

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200

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1800
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σ = 75 cm

4000

σ = 64 cm

3000
2000
1000
0

50

100

150

∆vertex (cm)∆vertex (cm)

PC

6000

200

0

0

50

100

150

200

∆vertex (cm)Figure D.13: 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.

221

4000
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3000
2500
2000
1500
1000
500
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1ring e-like
σ = 3.1°

0

2

4

6

8

10000
8000

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1ring e-like

1000
800

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0

σ = 2.0°

2500
2000

4000

1500

2000

1000
500

10

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4

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8

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3500
3000

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2

4

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8

10

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1ring e-like

1000

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1ring µ-like

FC Multi-GeV
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3000
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2000

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600

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2000

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8000
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σ = 3.3° 5000
4000
3000
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1000
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1ring e-like

3500
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1500

400
200

4000

6000

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1200

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1ring µ-like

0

2

4

6

8

10

∆direction (degree)

0

0

2

4

6

8

10

∆direction (degree)

Figure D.14: 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.

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.

(2)

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

(3)

The goodness of vertex fit is greater than 0.5.

(4)

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

(5)

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

(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, another peak after the primary event is searched with the additional
coditions requring more than 20 hits in 30 nsec window above the background level. Additional
conditions are as follows :(7)

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

and
(8)

0.1 µsec < ∆t < 0.8 µsec or 1.2 µsec < ∆t < 30 µsec. This is addopted to alltypes.
222

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

NUM
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;

Comnt;

Figure D.15: 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.

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.

D.6

Momentum Reconstruction

The momentum estimation is based on the relation between the number of observed Cherenkov
photons inside a cone with half opening angle of 70◦ and the momentum of the particles. In orderto 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 the observed p.e.s is carried out based on
the expected p.e. distribution from each ring as a function of the opening angle θ and uniform
in azimuthal angle φ. The observed p.e.s in the i-th PMT from the n-th ring are estimated as :
exp
qi,n
obs
qi,n
= qiobs × P exp
n0 qi,n0(D.25)

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×
 (D.26)
×−
RT OTn =
S
q
×
exp
i
i,n

Gdata Lf (Θi )
θi,n <70◦−50nsec<ti <250nsec

θi,n <70◦223

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
direction

Θi: angle of photon arriving direction relative to the ith PMT facing
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 PMTf (Θ): correction function for the PMT acceptanceSi: 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 throughgoing muons (Section 4.4.2) is used to correct the time variation for the observed data. Fig. ??
shows the estimated momentum resolution for electrons and muons. The momentum resolution
is approximately described as:(
2.6% for electron
0.6 + √p(GeV/c)(D.27)
σp =0.7
1.7 + √p(GeV/c) % for muon
The resolution is defined as 1 σ width of the Gaussian fit.

D.7

Ring Number Correction

After the momentum reconstruction, the residual of a reconstruction is corrected by erasing
the mis-reconstructed rings with number of rings >2 among all of ring candidates. Ring number
correction process, suppose it aims to erase i-th ring, requires the criteria below :(1-1) pi < pj ; i 6= j,where pi is the momentum of i-th ring.
(1-2) θij < 30◦,where θij is opening angle between the i-th and j-th rings.
(1-3) pi cos θij < 60 MeV/c,where pi cos θij is the momentum of i-th ring projected perpendicularly toj-th ring.
224

3500

500

3000

400
350

Number of events

Momentum of second γ (MeV/c)

(b)

(a)

450

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

0True π Momentum (MeV/c)

Figure D.16: (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.

or(2-1) pi < 40 MeV/c(2-2) pi /ptot < 0.05
,where ptot is the total momentum for all rings.

D.8

π 0 Reconstruction

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 4.5.2. However some events are classified as single-ring e-like event
when one γ is missed by the standard reconstruction algorism. Mis-identification of γ ray fromπ 0 decay can be occurred by following reasons :1. The energies of two γ’s are highly asymmetric and the energy of the second γ (E γ2 ) 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.
Relation of the opening angle between two γ’s and the momentum of the second γ is shown in
Figure D.16, where plotted events are the expected NC single π 0 events. Figure D.16(b) shows
the momentum distribution of the expected NC single π 0 events. The hatched region indicatesthe mis-identified events as single-ring e-like which amounts to 33% of the total events.
The π 0 fitter reconstructs the second gamma-ray by comparing the observed charge distribution on PMTs with the expected charge distribution of two gamma-rays, where existence of
two Cherenkov rings is assumed. The two gamma-rays are supposed to be emitted at one vertex
and the direction of the more energetic gamma-ray is fixed on the direction derived from theprecise vertex reconstruction. The best-fitted combination of two gamma-rays is determined by225varying the direction of the second gamma-ray and the fraction of energy shared among the two
gamma-rays.
The expected charge distribution for the i-th PMT is calculated as follows :


ri
1
×
qiexp = αe × Qexp (θi , pγ , ri ) × 1 − q
× f (Θi ) + qiscatt(D.28)
exp(r
/L)
2
2
i
r +R
i

P MT

where

αe

: normalization factor

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: radius of PMT∼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) are introduced to correct the solid angle
and light attenuation, respectively. Qexp is the expected photon distribution as a function of
the photon incident angle Θi , the initial gamma-ray momentum pγ and the distance from thevertex to the i-th PMT ri .
A log likelihood method is adopted to determine the best-fitted configuration of two gammarays. The likelihood is defined as :



X
L=
log prob qiobs , qiexp(D.29)
i=1
exp
The qi is optimized to derive the minimum likelihood value by changing the direction of thesecond gamma and the momentum fraction for the second gamma. The reconstruction process
is iterated once again after the first process with finer stepping size to improve the fitting
performance for such events as two rings are almost overlapped. These events give a narrow L
distribution in the direction of the first ring and smallest L between the first wide fit and thesecond narrow fit is chosen as a fitting result.
Figure D.17 shows the reconstructed invariant mass (Minv ) distribution from two gamma
rays of NC single π 0 events. Minv is defined as :q
(D.30)
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.

226

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 D.17: The invariant mass distributions for NC1π 0 events with the line of the Gaussian
fit for SK-I (left) and SK-II (right).

D.9

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 D.18 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 [164]. 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 D.4)
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 D.19.

227

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

2

10

10
10
Eυ (GeV)

3

10

4

5

10

Figure D.18: The energy spectrum of primary neutrino for upward stopping (dotted line), nonshoweing (dashed line) and showering (solid line) muon events.

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 D.19: 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.

228