Precision solar neutrino measurementswith Super-Kamiokande-IV

Takaaki Yokozawa
December 18, 2012Abstract
The solar neutrino results from the fourth phase of the Super-Kamiokande (SK-IV) detector
are presented. The main physics goal of this thesis is the direct observation of the matter effectof neutrino oscillations, which is called the Mikheyev-Smirnov-Wolfenstein (MSW) effect, byobserving the day/night flux asymmetry and the energy spectrum distortion.
The SK detector is a ring imaging water Cherenkov detector which uses 50,000 tons of purewater as a target for neutrinos and is located 1000 m underground at Ikenoyama, Kamioka,
Japan. The front end electronics and data acquisition system were replaced in September of 2008
and observation was started as the SK-IV phase. With improvements of the water circulation
system, the detector simulation, and the analysis method, a lower background level was achievedin the central region of the detector and we were successful in setting the lower energy thresholddown from 4.5MeV to 4.0 MeV (in kinetic electron energy). The systematic uncertainty on
the total 8 B flux is reduced to ±1.7%, which is much improved compared with SK-I (+3.5%−3.2% )
8and SK-III (±2.2%). The B flux from 4.0 to 19.5 MeV(kinetic) of SK-IV is measured to be
2.34 ± 0.03(Stat.) ± 0.04(Syst.) [× 106 /cm2 /s], which is consistent with the SK-I,II,III results.
The neutrino oscillation analysis with combining all the SK phase gives the best fit oscillation
+1.80
−5
2
8
2
parameters of sin2 (θ12 ) = 0.339+0.028−0.024 and ∆m21 = 4.69−0.83 × 10 eV , where the total B flux
62
is constrained to (5.25±0.21)[×10 /cm /s], which is the result from the SNO NC measurement.
Combining further with other solar experiments and the KamLAND reactor result, the best fit
+0.20
−5
2
2
oscillation parameters are obtained as sin2 (θ12 ) = 0.304+0.013−0.013 and ∆m21 = 7.44−0.19 × 10 eV .
2The χ values of the spectrum fit of the SK data with MSW predictions for the two best
fit oscillation parameters described above, and for an assumption of a flat shape (constant
reduction factor independent of energy) are 76.54, 79.02 and 75.29 for 81 d.o.f, respectively. So
the flat shape is preferred at the 1.1-1.9σ level, compared with the best fit oscillation parameters
predictions.
By introducing a method to measure the day/night asymmetry with a constraint on the
expectation from matter effect and by improving systematic uncertainties, the SK combined
day/night asymmetry was obtained as ADN (=(day-night)/ 12 (day+night)) = -2.8 ± 1.1(stat.)±0.5(syst.) [%]. So, a non-zero day/night asymmetry is observed at the 2.3σ level.

Acknowledgments

I would like to express my gratitude to Prof. Masayuki Nakahata for introducing me to
the Super-Kamiokande experiment and for giving me many knowledge and suggestions for solar
neutrino analysis. This thesis would never exist without his close support and encouragement.
I would like to appreciate to Prof. Yoichiro Suzuki, the spokesperson of the Super-Kamiokande
experiment for giving me the oppotunity of studing in the Super-Kamiokande experiment.
I would like to thank grealty to members who worked with me on the solar neutrino analysis,
Dr. Y.Koshio, Dr. H.Sekiya, Prof. Y.Takeuchi, Prof. M.Smy and A.Renshaw.
I would like to thank to LOWE (low energy analysis group) members, Prof. Y.Kishimoto,
Dr. A.Takeda, Dr. L.Marti, Dr. S.Yamada, Dr. H.Watanabe Dr. T.Iida, Dr. M.Ikeda,
Dr. B.Yang, Dr. K.Ueno, Y.Nakano, Prof. M.Vagins, Prof. K.Martens, Dr. K.Bays, Dr.
G.Carminati, Prof. M.Sakuda, Prof. H.Ishino, Dr. A.Kibayashi, Dr. T.Yano, T.Mori, H.Toyota,
S.Mino, J.Takeuchi, R.Yamaguchi, Prof. Chen.S, Dr. H.Zhang, Y.Zhang, Prof. Y.Fukuda, Prof.
S.Tasaka and Prof. L.Labarga.
I would like to thank to all other ICRR staffs who worked with me, Prof. T.Kajita, Prof.
K.Kaneyuki, Prof. M.Shiozawa, Prof. Y.Hayato, Prof. S.Moriyama, Prof. M.Yamashita, Prof.
K.Okumura, Dr. M.Miura, Dr. Y.Obayashi, Dr. J.Kameda, Dr. K.Abe, Dr. R.Wendell, Dr.
H.Ogawa, Dr. K.Kobayashi, Dr. S.Nakayama, Dr. K.Hiraide, Dr. Y.Nishimura, Dr. T.Tomura,
Dr. H.Tanaka, Dr. N.Tanimoto, Dr. Y.Shimizu, Dr. H.Kaji, Dr. Y.Takenaga, Dr. N.Okazaki
and Dr. J.Liu.
I also would like to thank to my friends and stuffs who supported me all the time in Kamioka,
Dr. G.Mitsuka, Dr. H.Nishino, Dr. K.Ueshima, Dr. C.Isihara, Dr. T.Tanaka, T.F.McLachlan,
Y.Nakajima, S.Hazama, Dr. D.Motoki, Y.Yokosawa, Y.Kozuma, Maggie, K.Choi, P.de perio, S.Tobayama, Prof. S.Nakamura, I,Murayama, K.Fujii, H.Nishiie, A.Shinozaki, K.Iyogi,
H.Uchida, S.Hirano, Y.Nishitani, M.Miyake, K.Otsuka, K.Hieda, O.Takachio, D.Umemoto, Y.Nakano,
H.Takiya, K.Hosokawa, A.Murata, N.Oka, K.Nakagawa, I.Kametani, Y.Haga and S. Higashi.
Finally, I wish to express my deep gratitude to my friends and family.

Contents
1 Introduction

4

2 Solar Neutrinos
2.1 The Standard Solar Model and solar neutrinos . . . . . . . . . . . . . . . . . . .
2.2 Solar neutrino experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Radiochemical experiments . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 SNO experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Borexino experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Neutrino oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Oscillations in vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Oscillation in Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Possible solution of solar oscillation parameters . . . . . . . . . . . . . . .
2.4 Latest results of oscillation parameters . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 θ13 from short baseline reactor experiments . . . . . . . . . . . . . . . . .
2.4.2 KamLAND experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Physics of solar neutrino analysis in this thesis . . . . . . . . . . . . . . . . . . .

6
6
7
7
11
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13
14
15
16
17
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3 The Super-Kamiokande Detector
3.1 SK detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Detection principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 The Super-Kamiokande detector . . . . . . . . . . . . . . . . . . . . . . .
3.1.3 20 inch photomultiplier tubes . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.4 Phases of the SK detector . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Data acquisition system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Data acquisition system of SK-I,II,III . . . . . . . . . . . . . . . . . . . .
3.2.2 Hardware trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.3 Data acquisition system of SK-IV . . . . . . . . . . . . . . . . . . . . . . .
3.2.4 Software trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Water circulation system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Water purification system . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Water circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25
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34
37
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41

4 Event Reconstruction
4.1 Vertex reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Direction reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Energy reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43
43
45
45

1

5 Detector Calibration
5.1 Calibration sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Nickel source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.2 Xenon light source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.3 Laser light source for timing calibration . . . . . . . . . . . . . . . . . . .
5.1.4 Laser light source for water quality monitoring . . . . . . . . . . . . . . .
5.1.5 LINAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.6 DT generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 PMT calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 PMT gain calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2 PMT quantum efficiency measurement . . . . . . . . . . . . . . . . . . . .
5.2.3 PMT timing calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Water transparency measurement . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Measurement by decay-e . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Measurement by laser light injector . . . . . . . . . . . . . . . . . . . . . .
5.4 Input parameters to detector simulation . . . . . . . . . . . . . . . . . . . . . . .
5.5 Absolute energy scale calibration . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1 LINAC calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.2 DT calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.3 Systematic uncertainty of absolute energy scale . . . . . . . . . . . . . . .

51
51
51
52
52
54
55
56
57
57
59
59
64
64
64
68
74
74
76
79

6 Data Analysis
80
6.1 Run selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.2 Pre reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.3 Reduction steps for the solar analysis . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.3.1 Total number of hit PMT cut . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.3.2 Trigger cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.3.3 Flasher cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.3.4 Time difference cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.3.5 Spallation cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.3.6 gv2 -ga2 cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.3.7 Hit pattern cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.3.8 Gamma ray cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.3.9 16 N cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.3.10 Cluster hit event cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.3.11 Tight fiducal volume cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.4 Summary of reduction step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.5 Signal extraction method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.6 Multiple scattering goodness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7 Systematic Uncertainties
104
7.1 Energy scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
7.2 Energy resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
7.3 8 B spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
7.4 Trigger efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.5 Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
2

7.5.1 Fiducial volume cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.5.2 Gamma ray cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.5.3 gv2 − ga2 cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.5.4 Hit pattern cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.5.5 Cluster hit event cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.5.6 Spallation cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.6 Angular resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.7 Background shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7.8 Signal extraction method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7.9 Cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.10 Summary of systematic uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.11 Spectrum systematic uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.11.1 Energy-correlated systematic uncertainty . . . . . . . . . . . . . . . . . . 114
7.11.2 Energy-uncorrelated systematic uncertainty . . . . . . . . . . . . . . . . . 114
8 Results of the SK-IV solar analysis
117
8.1 8 B neutrino flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
8.2 Day/Night flux asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
8.3 Energy spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
9 SK-I,II,III,IV Combined Analysis
124
8
9.1 B neutrino total flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
9.2 Energy spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
9.3 Oscillation analysis method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
9.3.1 Neutrino oscillation predictions . . . . . . . . . . . . . . . . . . . . . . . . 127
9.3.2 Spectrum shape fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
9.3.3 Time variation fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
9.3.4 Flux constrained fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
9.4 Oscillation analysis results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
9.4.1 Oscillation parameter constraints from SK . . . . . . . . . . . . . . . . . . 131
9.4.2 Results of the energy spectrum fit . . . . . . . . . . . . . . . . . . . . . . 134
9.5 Day/Night asymmetry analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
9.5.1 Straight day/night asymmetry . . . . . . . . . . . . . . . . . . . . . . . . 135
9.5.2 Day/night amplitude fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
10 Discussion
141
10.1 Comparison/Combine with SNO result . . . . . . . . . . . . . . . . . . . . . . . . 141
10.2 Combined with other solar neutrino experiments . . . . . . . . . . . . . . . . . . 142
10.3 The future prospects of solar neutrino analysis . . . . . . . . . . . . . . . . . . . 145
10.3.1 Future Energy spectrum analysis . . . . . . . . . . . . . . . . . . . . . . . 145
10.3.2 Future day/night asymmetry analysis . . . . . . . . . . . . . . . . . . . . 145
11 Conclusion

149

3

Chapter 1

Introduction
The Sun continuously produces a huge amount of energy. The origin of this energy is now
understood to be nuclear fusion. The generated energy which is released as photons is slowly
diffused to the surface of the Sun, over the course of ∼105 years, due to high density of the Suninterior. On the other hand, neutrinos which are created in the solar core can provide direct
information of the center of the Sun, due to neutrinos having a very small reaction cross section,even given the high density region of the Sun. So, measurements of solar neutrinos give an
important test for the theories of stellar evolution and structure.
The neutrino was first introduced by Pauli in 1930 as ’verzweifeltes ausweg (desperate remedy)’, in order to preserve the energy consevation law in the beta decay process. It was discovered in 1956 by F.Reines and C.L.Cowan by observing electron anti-neutrinos from a nuclear
reactor[1]. In the Standard Model of the elementary particles, the neutrino is a lepton which
has a spin of 1/2 and no electric charge, has three flavors (νe , νµ and ντ ) and was originallyregarded as a massless particle.
The first solar neutrino observation was carried out by R.Davis and his colleagues, using 37 Cl
atoms[2]. This experiment started around 1968 and gave results which observed that the solar
neutrino flux is about 33% of the expected flux estimated by the Standard Solar Model (SSM).
This problem was dubbed the solar neutrino problem. In 1989, the solar neutrino problem wasconfirmed by the Kamiokande experiment[3], which was a 3,000 ton water Cherenkov detector.
The deficit of the solar neutrino flux was also observed in the Ga detectors (GALLEX andSAGE) in the early 1990’s[4][5].
In 2001, the solar neutrino problem was understood in terms of neutrino oscillations by comparing the 8 B flux results from Super-Kamiokande (SK)[6], via the electron-scattering method,
and the Sudbury Neutrino Observatory (SNO)[7], via the charged current interaction. The
theory of neutrino oscillations tells us that neutrinos have finite masses and that the mixing
happens between their mass eigenstates and interaction eigenstates, which are not the same.
Neutrino oscillations are purely quantum mechanical phenomena, which can be understoodin terms of a traveling wave function. If neutrinos travel through matter, a potential energy
induced by weak interactions with matter (called matter effect, hereafter) will affect the neutrino
oscillations. In order to understand current solar neutrino experimental results, the matter effectis necessary, but gives an indirect measurement of neutrino oscillations. This thesis focuses on
the matter effect introduced by S. P. Mikheyev, A. Yu. Smirnov and L. Wolfenstein in 1985,
often called the MSW effect[8][9]. Direct effects coming from the MSW theory are expected to beobserved by precise measurements of the solar neutrino energy spectrum and the flux difference4between daytime and nighttime. The former is due to the matter effect through the Sun, while
the latter is due to the matter contained within the Earth.
The structure of this thesis is as follows; the details of the standard solar model, neutrino
oscillations and the MSW effect and current results from solar neutrino experiments are describedin chapter 2. In chapter 3, the Super-Kamiokande experiment is described, and in chapters 4-7,
details of the solar neutrino analysis at Super-Kamiokande are described. The obtained solar
neutrino results from the 4th phase of SK are summarized in chapter 8, and SK combined results
are summarized in chapter 9. Finally, the discussion and conclusion are given in chapters 10
and 11.

5

Chapter 2

Solar Neutrinos
The Standard Solar Model (SSM) is the well-established theoretical model of the sun, whichtakes into account the properties of the Sun which have been measured or calculated independently. In this chapter, the basic ideas of the SSM, neutrinos which are generated in the Sun
and the solar neutrino experiments are described. Important neutrino phenomenon, such as
neutrino oscillations and its current parameters, are also described. Finally, the motivations ofcurrent solar neutrino analyses are described.

2.1

The Standard Solar Model and solar neutrinos

The Sun is considered to be a main sequence star. Because of its proximity to the Earth,studying about the Sun is the best way to understand stellar evolution. In the Standard Soalr
Model (SSM), the Sun is assumed to be spherical and to have evolved quasi-statically over a
period of about 4.6×109 years (age of Sun). The evolution of the Sun is manifested by the lossof photons from its surface, which is in turn balanced by the burning of protons into α particles
within the core of the Sun. It produces an intense flux of electron neutrinos as a consequenceof the nuclear fusion reaction, written as
4p →4 He + 2e+ + 2νe .

(2.1)

with releasing an energy of 26.7 MeV as a form of photons. The reaction given in (2.1) is in theform of a net reaction, which proceeds via two different branches. One is called the proton-proton
chain (pp chain), which is shown in Fig.2.1. In the pp chain reaction, five types of neutrinos are
generated by the following reactions, with unique or maximum neutrino energy;
p + p −→ 2 H + e+ + νe−(pp, Eν <0.42 MeV)

(2.2)

2

(pep, Eν =1.442 MeV)

(2.3)

7

7

( Be, Eν =0.861,0.383 MeV)

(2.4)

(8 B, Eν <14.06 MeV)

(2.5)

(hep, Eν <18.77 MeV).

(2.6)

p + e + p −→ H + νe7−

Be + e −→ Li + νe
83

B −→ 8 Be∗ + e+ + νe+

He + p −→ α + e + νe

The other fusion reaction branch is called the Carbon-Nitrogen-Oxygen cycle (CNO cycle) andis shown in Fig.2.2. There are three generated neutrinos by the following reactions;6Figure 2.1: Schematic view of the pp chain reactions.

13

N −→ 13 C + e+ + νe

15

17

O −→

F −→

(Eν <1.27 MeV)

(2.7)

15

+

N + e + νe(Eν <1.73 MeV)

(2.8)

17

+

(Eν <1.74 MeV).

(2.9)

O + e + νe

According to the SSM [10], about 98.4% of the total solar luminosity is predicted to be generated
by the pp chain, while the remaining 1.6% is thought to be generated by the CNO cycle.
Fig.2.3 shows the generated solar neutrino energy spectrum predited by the BP04 SSM [11],which was developed by J.N.Bahcall and M.H.Pinsonneault in 2004. The BP04 SSM predicted
a 8 B neutrino flux of 5.79 (1±0.23)[×106 /cm2 /s]. The upper figure in Fig.2.4 shows the electron
density in the Sun, predicted by the BP04 SSM. The lower figure in Fig.2.4 shows the matter
density of Earth, taken from the PREM model [12]. As described in section 2.3, these matter
density profiles are used as input to determine neutrino oscillations in matter.

2.2

Solar neutrino experiments

In this section, past and current solar neutrino experiments are introduced. These experiments have made great contributions to solar neutrino understanding, and are used in the global
oscillation analysis, which is described in section 10.

2.2.1

Radiochemical experiments

Chlorine experiment
The first observation of neutrinos coming from the sun was accomplished around 1968, by
Raymond Davis Jr., using an underground chlorine neutrino detector at the Homestake gold
7

Figure 2.2: Schematic view of the CNO cycle reactions.

Figure 2.3: The solar neutrino energy spectrum predicted by BP04 SSM. The value in percentage
written beside each neutrino spectrum correspond to total flux uncertainties.

8

Figure 2.4: Top figure: the electron density of the Sun as a function of distance to surface(r/r⊙ ) calculated by the BP04 SSM. Bottom figure: the mass density of the Earth as a fuctionof distance to the surface (left), the minimum (lower dashed line), the average (solid line) andthe maximum (upper dashed line) mass density as a function of cosine of zenith angle (cosΘz ).

9

Figure 2.5: The results of the production rate of 37 Ar with the Homestake chlorine detector.
The errors shown for individual measurements(except of right point) are statistical errors only,
while the error shown for the cumulative result is a combination of the statistical and systematic
errors in quadrature.

mine in the USA [2]. This detector was located 1778 m underground, and consisted of 615
tons of C2 Cl4 . This experiment started in 1967 and continued the observation until 1994. The
neutrino is detected by absorption on the chlorine, written asνe +37 Cl →37 Ar + e− .

(2.10)The energy threshold of this interaction is 0.814 MeV, so the Cl experiment was sensitive to
8 B, 7 Be, pep and CNO neutrinos. The generated 37 Ar atoms were extracted by bubbling He
gas through the detector’s tank, with a collection effeciency of 95%, and then counted by aproportional counter. This extraction was done once every 1-3 months. Fig.2.5 shows the resultsof the production rate of 37 Ar. The average rate was 2.56±0.16(Stat.)±0.16(Syst.) SNU1 , whichcorresponded to about one third from expectation.
Gallium experiments
The SAGE expriment in Russia [4] and the Gallex/GNO experiments [5][16] in Italy used
a gallium target for detecting neutrinos via neutrino absorption on gallium. The reaction is
written asνe +71 Ga →71 Ge + e− .
(2.11)The SAGE (Soviet-American Gallium Experiment) was located deep underground at the
Baksan Neutrino Observatory and began observation in 1990. It used 50 tons of liquid metal
Gallium target, in seven chemical reactors that are heated to 30◦ C, so that the gallium metal
remains molten. The resulting Ge atoms were counted by a proportional counter observing
Auger electrons and X-rays emitted by the 71 Ge electron capture.
1

A SNU (Solar Neutrino Unit) is the number of interaction per 1036 targets per second.

10

Figure 2.6: The combined SAGE results for each year. The shaded band is the combined best
fit and its uncertainty for all years. Error bars shown for individual measurements are statisticalerrors only.

The Gallex experiment began observation in 1991 and was replaced by GNO (Gallium Neutrino Observatory) in 1998. The Gallex/GNO detector was located in the Gran Sasso Underground Laboratory in Italy and used a 100 ton target of GaCl3 -HCl solution, which contains
30.3 tons of gallium. The extraction and the subsequent counting were done every 4-6 weeks.
The advantage of the gallium experiments is the low energy threshold, 0.233 MeV. This low
energy threshold allows for the observation of the pp neutrinos. Fig.2.6 shows the combined+2.6
SAGE results for each year, with the combined observed flux given as 65.4+3.1−3.0 (stat.)−2.8 (syst.)
SNU [4]. Fig.2.7 shows the combined Gallex/GNO result, with the joined flux of Gallex and
GNO found to be 69.3±4.1(stat.)±3.6(syst.) SNU [16]. The weighted average of the resultsof all three Ga experiments, SAGE, Gallex and GNO, is calculated as 66.1±3.1 SNU, whichcorresponded to about one second from expectation.

2.2.2

SNO experiment

The SNO (Sudbury Neutrino Observatory) experiment [7] was a Cherenkov detector whichused 1000 tons of heavy water (D2 O) and ran from 1999 to 2006. It was located 2092 m below
the surface in the Creighton mine in Sudbury, Canada. The SNO detector observed 8 B neutrinos
via three different reactions;νx + d → νx + p + n−

νe + d → e + p + p−−

νx + e → νx + e(Neutral Current(NC))(2.12)(Charged Current(CC))(2.13)(Elastic Scattering(ES)),(2.14)

where x=e,µ,τ . The characteristics of each reactions are;• Neutral Current (NC):11

Figure 2.7: The joined Gallex and GNO result. The error bars are statistical errors only.

Equally sensitive to all three active neutrino flavors. This reaction can determine thetotal 8 B neutrino flux (Φ8 B ) independently of any specific active neutrino flavor oscillation
hypothesis.
• Charged Current (CC):Only sensitive to νe . Comparing this rate to the NC reaction rate, it is possible to determine
the neutrino survival probability as a function of energy.

• Elastic Scattering (ES):Sensitive to all neutrino flavors, but the cross section for the νe is approximately six times
larger than that for the νµ,τ . This reaction is described in detail in section 3.1.1

The SNO detector operated in three phases which are distinguished by the method of theneutrino detection from the NC interactions (2.12).
• Phase1 : n + d → t + γ(6.25MeV)(D2 O)• Phase2 : n +35 Cl →36 Cl + γ(8.6MeV)(Salt)• Phase3 : n +3 He → p + t + 0.76MeV(NCD array)
The combined NC reaction rate, energy spectrum and day/night asymmetry results of allphases will be described in section 10.1.

12Figure 2.8: A Borexino Monte Carlo best fit result in the units of [counts/day/100 ton] and overthe energy region of 270 to 1600 keV.

2.2.3

Borexino experiment

The Borexino experiment [17] is a liquid scintillator experiment located in the Laboratori
Nazionali del Gran Sasso (LNGS). This detector is filled with ∼278 metric tons of organic liquid
scintillator and began observation in May of 2007. Borexino observed sub-MeV (low energy)
solar neutrinos by means of electron scattering.
Fig.2.8 shows a Monte Carlo best fit over the energy region of 270 to 1270 keV. The Borexino
experiment obtained the 7 Be, pep and CNO neutrino fluxes as
7

Be rate(862 keV line)
pep rate
CNO rate

2.3:

46.0 ± 1.5(Stat.)+1.5−1.6 (Syst.) [count/day/100 ton]

→ 2.78 ± 0.13[109 /cm2 /s](2.15)→ 1.6 ± 0.3[108 /cm2 /s](2.16)→ < 7.7[108 /cm2 /s](95%C.L.).

(2.17)::

3.1 ± 0.6(Stat.) ± 0.3(Syst.) [count/day/100 ton]

< 7.9count/day/100 ton

Neutrino oscillations

Neutrino oscillations were first proposed by Z. Maki, M. Nakagawa, S. Sakata, V. Gribov andB. Pontecorvo in 1969[14]. In this section, the theories of neutrino oscillations are described.

2.3.1

Oscillations in vacuum

The flavor eigenstates can be expanded in terms of the neutrino mass eigenstates (|νj i,
j=1,2,3) as13

|να i = U |νj i,(2.18)

where να (α=e,µ,τ ) are the flavor states and U is often called the Pontecorvo-Maki-NakagawaSakata (PMNS) or Maki-Nakagawa-Sakata (MNS) mixing matrix, and can be written as
C12 S12 0
C13
0 S13 e−iδ
1
0
0




U =  0 C23 S23  010  −S12 C12 0 
0
0 1−S13 eiδ 0
C13
0 −S23 C23
(2.19)

where Cij ≡cosθij and Sij ≡sinθij are the miximg angle between the ith and jth mass eigenstates.
δ is the CP violation phase in the neutrino sector.
For simplicity, neutrino oscillations between only two flavor eigenstates are described first.
The relationship between the flavor eigenstates (|νe i, |νx i) and the two mass eigenstates(|ν1 i,|ν2 i) can be written as!
!
!
!
cosθ sinθ
|ν1 i
|νe i
|ν1 i
=
.
(2.20)=U|ν2 i−sinθ cosθ
|ν2 i
|νx i
The time dependence of the mass eigenstates can be obtained by solving the time dependent
Schroedinger equation, which is described as|νj (t)i = |νj (0)iexp(−iEj t).

(2.21)

By solving this equation, the transition probability from νe to νx can be written as
1.27∆m221 [eV 2 ]L[m])E[M eV ]L)= sin2 2θ12 sin2 (π
LOSC
π
E
,
≡
1.27 ∆m221

P (νe → νx , L) = sin2 2θ12 sin2 (LOSC(2.22)

where L is the travel distance, E the neutrino energy and ∆m221 = m22 − m21 is a squared massdifference of mass eigenstates 2 and 1. Thus, the νe survival probability can be written as
1.27∆m221 L).
(2.23)
E
The electron neutrino survival probability can be described as a function of distance (L) after
traveling, neutrino energy (E), squared mass difference (∆m221 ) and mixing angle (θ12 ). The
values of ∆m221 and θ12 are called ’oscillation parameters’.
P (νe → νe , L) = 1 − sin2 2θ12 sin2 (2.3.2

Oscillation in Matter

When a neutrino travels through matter, the propagation is affected by the weak field ofelectrons, neutrons and protons via neutral current exchanging of Z 0 , while an electron neutrino
is only affected by the weak field of charged current interactions (i.e. via W± exchange) with14

electrons. This W ± weak field causes additional terms in the Hamiltonian, as seen in thefollowing equation. This effect is called the ’MSW2 effect’ [8][9]. The Hamiltonian H in the two
flavor eigenbasis can be expressed as
H = Hv + HM
=

∆m2
4E

(2.24)− cos 2θ sin 2θ
sin 2θ cos 2θ

!

+

V
0

0
0

!

,

(2.25)

where Hv is the Hamiltonian in vacuum, the HM is that while in matter and V is the potentialof the charged current interaction which can be described as√
V = 2 2GF Ne ,(2.26)

where GF is Fermi coupling constant and Ne is the electron density of the matter. By applying
this Hamiltonian to the Schroedinger equation, the eigenstates in matter can be written as
!
!
!
|νe i
cosθM −sinθM|ν1M i,(2.27)=|νx i
sinθM cosθM|ν2M iwhere θM is the effective mixing angle in matter and is written as
tan2θM =tan2θ.
E
1 − ∆mV2 cos2θ

(2.28)The survival probability in matter is then obtained by replacing θ in equation 2.23 with θm andreplacing LOSC with the oscillation length in matter Lm , which is written as
LM =

2.3.3

π
q
.
2
∆m
∆m2
2
2
1.27 (2V + E cos 2θ) + ( E sin 2θ)(2.29)

Possible solution of solar oscillation parameters

A simple example of solar neutrino oscillations is the transition of νe , generated within theSun, to νµ,τ during the trip to Earth, while undergoing vacuum neutrino oscillations. The
survival probability is calculated by equation (2.23), with L=150×106 km (distance from Sun
to Earth). This possible scenario is known as the vacuum (VAC) solution.
Other solutions are obtained by taking into account matter effects. Using the electron density
at the Sun and Earth (Fig.2.4), the νe survival probability, which takes into account the MSW
effect, can be calculated. Fig.2.9 shows how a generated neutrino can adiabatically change its
flavor while the electron density varies slowly. At very small electron densities, the electron
neutrino almost coincides with the lower mass eigenstate |ν1 i, for the small vacuum mixing
angle case. The second flavor eigenstate approximately coincides with the mass eigenstate |ν2 i.
The mass of the heavier mass eigenstate increases monotonically with ambient electron densityand becomes asymptotically proportional to the electron density. If m2 ≥ m1 , at large electron
densities the heavier mass eigenstate, |ν2 i, becomes close to |νe i, while the lower mass eigenstate,
2

S. P. Mikheev, A. Yu. Smirnov and L. Wolfenstein

15

Figure 2.9: The mass eigenstates in matter as a fuction of electron density.

|ν1 i, remains essentially unchanged as a function of electron density. An electron neutrino thatis created in the solar interior moves from high densities to lower densities. If the electron
density changes at a sufficiently slow rate, no transitions occur between the mass eigenstates.
The electron neutrino remains close to mass eigenstate |ν2 i and exits from the Sun as essentiallya different flavor eigenstate, |νx i. The regions within the plane of oscillation parameters allowedby current solar neutrino fluxes observed by the Cl, Kamiokande, and Gallium experiments
are separately distributed at the small mixing angle (SMA), the large mixinig angle (LMA)
and low mass difference (LOW) solutions. The location of these four solutions are shown in
Fig.2.10. Now, including measurements by SK and SNO (and with Borexino), the LMA solution
is uniquely selected at very high confidence level.
Because the MSW effect depends on the neutrino energy, the survival probability of generated
νe is energy dependent. This causes a distortion to the observed energy spectrum. As the MSW
effect converts νe to νx on its way out of the Sun, νx to νe conversion can happen when passing
through the Earth. This effect leads to the prediction that the observed electron neutrino flux
during the nighttime (when earth is between the detector and the sun) is larger than that ofdaytime (when the earth is not between the two). This flux difference is called the ’day/night
asymmetry’. As described in section 2.5, these direct observations of the MSW effect from theobserved energy spectrum distortion and from the day/night flux difference are the main topicsof this thesis.

2.4

Latest results of oscillation parameters

The mixing angle θ13 remained poorly known and only an upper limit was given before
2010. But recently, T2K and various short baseline reactor experiments observed a non-zeroθ13 value and is now precisely determined with an accuracy of 10% in sin2 (θ13 ). In this section,
the measurements and results of θ13 from the various experiments and the precise oscillation
parameter results from the KamLAND experiment are described.

16

Figure 2.10: The location of possible solutions of solar oscillation parameters (tan2 θ12 ,∆m2 in
eV2 ).

2.4.1

θ13 from short baseline reactor experiments

The short baseline reactor experiments give a precise value for θ13 . Since the value of θ13
affects the analysis of solar neutrino oscillations, some details of the short baseline reactor
experiments are described in this section. For experiments with L/E around 0.5 km/MeV,
which was adopted by the short baseline reactor experiments, the survival probability of ν̄e canbe writtern as
P (ν̄e → ν̄e ) = 1 − sin2 (2θ13 )sin2 (1.27∆m2atm L/E) + O(∆221 ),(2.30)

where ∆m2atm is the effective atmospheric mass squared difference, measured to be |∆m2atm |−3 eV2 [18]. This survival probability comes from a simple 2 flavor oscilla= (2.32+0.12−0.08 )×10
tion formula with no matter effects included, and shows that reactor experiments with a shortbase line (∼1km) can ignore the disappearance of ν̄e , driven by θ12 and ∆m221 . Thus, we canunambiguously determine the mixiing angle θ13 based on the survival probability of electron
antineutrinos. Reactor neutrinos are actually antineutrinos, but by assuming CPT invariance,
the value of θ13 obtained by antineutrino oscillation experiments can be adopted to the analysis
of solar neutrino experiments.
The reactor ν̄e neutrinos are detected via the inverse beta decay reaction, which is writtenasν̄e + p → e+ + n,

(2.31)

with a 1.8 MeV threshold. Using the coincidence of a prompt positron signal (e+ ) and a delayed
signal coming from the neutron capture by gadolinium or a proton, the inverse beta decay events
17

Figure 2.11: Energy spectrum obtained by the Double Chooz experiment. Top figure: Expectedenergy spectrum for the no-oscillation case (blue dashed line) and for the best fit oscillation
parameters (red solid line), along with measured spectrum (black). Middle figure: Difference
between data and the no-oscillation spectrum with the best fit expectation(red). Bottom figure:
Difference between data and the no-oscillation spectrum with the best fit expectation.

can be distinguished from backgrounds.
Double Chooz experiment
The Double Chooz detector [19] is located at the Chooz nuclear power station in the Ardennes
region of northern France. The far detector is placed ∼1000 m from the reactor cores, contains
a 10 m3 target of Gd loaded liquid scintillator and started observation in April 2011. The neardetector will be placed ∼400 m from the reactor cores, with same target as the far detector
and the plan is to start observation in 2013. Fig.2.11 shows the energy spectrum of the so far
observed data, with predictions of no-oscillation and best-fit oscillation parameters included.
Double Chooz indicated reactor ν̄e disappearance and gives sin2 (2θ13 ) as
sin2 (2θ13 ) = 0.109 ± 0.030(Stat.) ± 0.025(Syst.),(2.32)

with 227.9 days live time.
Daya BayThe Daya Bay detector [20] is located on the southern coast of China near 6 reactors located at the Daya Bay nuclear power plant (D1,D2) and the Ling Ao nuclear power plant(L1,L2,L3,L4), and is made up of 3 experimental halls (EH1,EH2,EH3) and 6 antineutrino detectors (AD1-AD6).

18

Figure 2.12: Energy spectrum obtained at the Daya Bay experiment. Top figure: The measured energy spectrum of the far hall (sum of three far ADs) compared with the no-oscillation
prediction based on the measurements of the two near halls. Bottom figure: The ratio between
measured and predicted spectra. The solid curve is the expected ratio with oscillations andassuming sin2 (θ13 )=0.089. The dashed line is the no-oscillation prediction.

The antineutrino detectors consist of ∼20 tons of gadolinium-doped liquid scintillator, with
0.1% Gd by weight in the innermost volume. Fig.2.12 shows the measured energy spectrum of
the far hall compared with the no-oscillation prediction based on the measurements of the twonear halls and also the ratio between those two. The Daya Bay experiment gives the result ofsin2 (2θ13 ) = 0.089 ± 0.010(Stat.) ± 0.005(Syst.),(2.33)

using data from the observation period of Dec. 2011 to May. 2012.
RENO experiment
The RENO experiment [21] is placed near the Yonggwang nuclear power plant and is locatedon the west coast of South Korea. There are six reactors, with the near and far detectors located
209 m and 1400 m from the center of the reactor array, respectively. The near detector is located
79 m underground from the ridge, while the far detector is located 260 m underground in a
mountain. The detectors hold 18.6 m3 of hydrocarbon liquid scintillator with gadolinium.
Fig.2.13 show the observed energy spectrum in the far detector compared with the nonoscillation prediction from the measurements in the near detector. A rate-only analysis obtains
the result ofsin2 (2θ13 ) = 0.113 ± 0.013(Stat.) ± 0.019(Syst.),with 228 live time days occurring between Aug. 2011 and Mar. 2012.
19(2.34)

Figure 2.13: Energy spectrum obtained at tha Reno experiment. Spectrum in the far detector
compared with the non-oscillation prediction from the measurements in the near detector.

20

Figure 2.14: Energy sepctrum obtained at the KamLAND experiment with the best fit oscillation
case.

2.4.2

KamLAND experiment

The KamLAND detector [22] is located in the Kamioka mine and consistsis of 1000 tonsof organic liquid scintillator. The KamLAND experiment started data taking in 2002. The
prompt scintillation light is emitted from the positron, while light from a 2.2MeV gamma ray isemitted when the neutron is captured by a proton. The KamLAND experiment had determineda precise value of the neutrino oscillation parameter ∆m221 and placed stringent constraints on
sin2 (θ12 ) by directly observing distortion of the reactor ν̄e energy spectrum. For L/E around 15
km/MeV (so-called solar L/E), the survival probability of ν̄e is writtern as
P (ν̄e → ν̄e ) = cos4 (θ13 )(1 − sin2 (2θ12 )sin2 (1.27∆m221 L/E)) + sin4 (θ13 ).

(2.35)The observed reactor energy spectrum from 2002 to 2007 is shown in Fig.2.14. The best fit
value of each oscillation parameter is given by−52∆m221 = 7.49+0.21−0.19 × 10 eV

sin2 (θ12 ) = 0.309+0.40−0.29 ,

(2.36)(2.37)

while using a fixed value of sin2 (θ13 )=0.025, obtained from the statistical average of the shortbaseline experiments.

2.5

Physics of solar neutrino analysis in this thesis

The main physics goal of this thesis is the direct observation of the MSW effect. As explaindin section 2.3.3, the key measurements for this goal are the observations of a day/night flux
21Figure 2.15: Day/night flux difference with statistical (blue) and systematic (red) uncertainties
for SK-I, II, III. The ’straight’ day/night value (see text) is used in this plot.

difference and a measurement of the energy spectrum distortion.
Fig.2.15 shows the observed ’straight’ day/night flux difference for each Super-Kamiokande(SK) phase. The definition of SK phase is described in Chapter.3. The straight day/night flux
difference is defined byΦday − Φnight,(2.38)ADN (straight) = 1
2 (Φday + Φnight )where Φday and Φnight are observed fluxes using the daytime and the nighttime data, respectively
The tendency is that the night flux is larger than the day flux at about the 2-5% level,
but the data is also still consistent with zero when taking into account statistical and systematic uncertainties. In order to get the combined value of all SK phases, a more sophisticatedmethod is necessary because of the different systematic uncertainties for each SK phase. The
main contribution of the systematic error comes from the directional dependence of the energy
scale systematic uncertainties (∼1.2-1.3%), which was made by the position dependence of water
quality in the SK detector. In the SK-III and SK-IV day/night analysis, this systematic uncertainty was greatly reduced, as shown in this thesis in the following chapters. Also, the combined
day/night result of all SK phases using the sophisticated analysis method is presented.
Fig.2.16 shows the solar neutrino νe survival probabilities obtained from solar experiments.
The flux of 7 Be (blue) and pep (red) neutrinos are used as the result for Borexino, 8 B neutrinos(green) are used as the result of Borexino+SNO-I,II (low-energy threshold analysis, LETA) [23]and 8 B neutrinos (black) are used as the result of SK+SNO. This figure also shows the MSWLMA predicted survival probability as a function of neutrino energy. As shown in Fig.2.16, the22

Figure 2.16: The global experimental constraints on the low energy solar neutrino νe survival
probability (Pee ). The gray band shows the MSW-LMA prediction Pee with the 1 σ range
estimated in [24].

survival probability is expected to distort in the few MeV energy region, in the MSW-LMA
model. The observed flux obtained by Borexino data (7 B and pep flux) validates the MSWLMA prediction, but the transition point in the spectrum is not measured yet and there aresome models which predict a shifting of the transition point. So, the spectrum measurement at
a few MeV (3-6 MeV) is important. Fig.2.17 shows the observed energy spectrum in SK-III,with predicted energy distortion. As shown in the figure, it was still difficult to decide whether
the observed spectrum was distorted or not due to the large statistical uncertainty for the lower
energy bins and the large energy correlated systematic uncertainties for the higher energy bins.
The large statistical uncertainty for the lower energy bins comes from the large amount ofbackground. So, it is important to get data with a lower background rate, lower the thresholdand lower the energy correlated systematic uncertainties, in order to observe the MSW effectfrom energy spectrum. SK has observed lower background data in the fourth phase (SK-IV)
and systematic uncertainty was also improved. In this thesis, details of the SK-IV analysis aredescribed. Also, by combining the energy spectrum of all SK phases, in which the correctionof systematic errors are taken into account, the energy spectrum distortion is discussed in thisthesis.

23

Figure 2.17: Observed energy spectrum/BP04 SSM, in SK-III phase. Horizontal axis shows theobserved number of solar neutrino over expected number of solar neutrino which is simulated
by solar neutrino MC for the flux of 5.79 [×106 /cm2 /s] (BP2004). The black points shows thefraction of data/SSM (Statistical uncertainty component in blue and energy-uncorrelated component in red) with energy-correlated systematic uncertainty(gray). The red line is a predictedenergy spectrum distortion for given best fit oscillation parameters.

24

Chapter 3

The Super-Kamiokande Detector
3.1

SK detector

The Super-Kamiokande(SK) detector is a ring imaging water Cherenkov detector which consists of 50,000 tons of pure water and an array of photosensors. It is located 1000 m undergroundat Ikenoyama, Kamioka, Japan. This chapter describes the detection principle and componentsof the detector, i.e. photosensors, data acquisition system and the water circulation system.

3.1.1

Detection principle

In SK, solar neutrinos are detected by the neutrino-electron elastic scattering reaction.
ν + e− → ν + e−

(3.1)

The advantages of the SK detector are as follows;• With large amount of pure water, a large number of neutrino events are detected.
• The recoil electrons are scattered in the forward direction in the case of higher energy
neutrinos, like 8 B solar neutrinos. The detection of the Cherenkov ring enables us toreconstruct the direction of the electrons, and so, the reconstructed events point back to
the Sun.
• Real time detection: Events are observed at the time of interaction. It is quite useful todiscuss time variation of the flux such as the day/night effect.
• The energy distribution of the recoil electrons are reflected by the energy spectrum of theincident neutrinos.
The cross section [29] of the reaction (3.1) is described as
G2 me
Te 2
me Te
dσ
[A0 + B0 (1 −= F
) + C0 2 ]
dTe
2π
Eν
Eν
G2F me= 4.31 × 10−45 (cm2 /MeV),
2π
25

(3.2)

where GF , me , Eν and Te are the Fermi coupling constant, the electron mass, the incident
neutrino enregy and the kinetic energy of the recoil electron, respectively. The parameters of
A0 , B0 and C0 are defined as
A0 = (gV + gA )2 ,2
− gV2 ),
B0 = (gV − gA )2 , C0 = (gA
1
1
for (νe ),gV = 2sin2 θW + , gA = +22
11
for (νµ ), (ντ ),gV = 2sin2 θW − , gA = −22

where θW is the Weinberg angle, and using the latest results, sin2 θw =0.2317 [15]. Fig.3.1 shows
the differential cross section of the reaction (3.1) for an incident neutrino energy of 10 MeV. The
horizontal axis shows the kinetic energy of the recoil electron (Te ). The radiative correctionsof the one-loop electroweak, QCD and QED processes are also considered. These corrections
reduce the relative probability of observing recoil electrons by about 4%, in our energy region.
The details of the radiative corrections are also described in [15].
The total cross section as a function of incident neutrino energy is obtained by integrating
equation (3.2) from 0 to Tmax . Tmax is limited by the incident neutrino energy and described as
Tmax =Eν
me .
1 + 2Eν(3.3)

Fig.3.2 shows the total cross section of νe and those of νµ , ντ . The cross section of νµ,τ -escattering is approximately six times lower than that of νe -e scattering, because the νµ and ντreaction occurs only through the neutral current, while νe interacts through both the charged
and neutral currents.
The recoil electron energy distributon resulting from 8 B solar neutrino interactions is described asZ Eν,maxdσ(3.4)
dEν dTe ,φ(Eν )F (Te )dTe =dTe0

where φ(Eν ) is the 8 B solar neutrino spectrum as shown in Fig.3.3. For the shape of the 8 B
solar neutrino spectrum, we adopted the calculation based on the α-spectrum measurement of
8 B →8 Be∗ + e+ + ν , 8 Be∗ → 2α by Winter et al. [25]. The recoil electron energy spectrum iseshown in Fig.3.4.
The angle between the directions of the recoil electron and the incident neutrino is given by
1 + me /Eν,cosθ = p
1 + 2me /Tewith the angular distribution of the recoil electrons given byZ Eν,maxdσ dTeφ(Eν )F (θ)dθ =dEν dθ.
dTe dθ
0

(3.5)

(3.6)

These energy and angular distributions of the recoil electrons, which are described in (3.4)
and (3.6), are used as the input to the Monte Carlo simulation of the 8 B and hep solar neutrinos.
They are described in Sec.8.1.
26

2

cm /mev100

dσ/dtex10

105

-46

110

95

90

85

80

75

70

65

60

0

2

4

6

8

10

electron kinetic energy(MeV)

Figure 3.1: Differential cross section of νe -e scattering in the case of 10 MeV incident neutrinoenergy.

Figure 3.2: The cross sections of νe and νµ , ντ -e scattering as a function of incident neutrinoenergy27

Figure 3.3: The upper figure shows the neutrino energy specturm measured by Winter [25]. The
lower plot shows the experimental uncertainty of the Winter spectrum (black band) and the ratio
between Winter spectrum and the Ortiz spectrum [26] and its ±1σ experimental uncertainty(dashed line).

eneel
Entries 19480
Mean
4.562
RMS
2.538

number of events/day/keV

10-1

10-2

10-3

10-4

10-5

10-6

2

4

6

8

10
12
14
16
electron total energy(MeV)

Figure 3.4: The total recoil electron spectrum observed with SK.

28

Figure 3.5: The SK detector

3.1.2

The Super-Kamiokande detector

SK is a cylindrical water Cherenkov detector with a 39.3 m diameter and a 41.4 m height.
Fig.3.5 shows the overview of the SK detector. The last three characters of Super-Kamiokande
stands for ’Neutrino Detection Experiment’ and ’Nucleon Decay Experiment’. To reduce thebackgrounds coming from cosmic ray muons, it is constructed at 1,000 m underground, which
can reduce the rate of cosmic ray muon to 1/100,000, compared with that at the surface. The
observed muon flux in the SK detector is about two Hz. The construction was started in 1991 andobservation in April, 1996. SK has four phases, whose differences are photo-coverage, existanceof PMT covers, front-end electronics and so on. The differencies is described in detail in section
3.1.4.
A PMT support structure, which is shown in Fig.3.6, divides the detector tank into two
optically isolated volumes, which are called the inner detector (ID) and the outer detector(OD). The ID has about 32,000 tons of pure water and about 11,000 PMTs (the exact number
is different for each SK phase). The wall of the ID tank, except for the PMT surfaces, is coveredby black polyethylene terephthalate sheets to block the light from leaking from the OD and to
reduce the effect of light reflections by the ID wall. The OD is an outer two meter layer fromdetector wall. The main purpose of the OD is to veto radioactive background, coming fromthe rock, and to tag cosmic ray muons. Totally 1885 eight inch PMTs are mounted at the OD
wall facing outward. In order to increase the photon detection efficiency, a wave length shifting
plate is mounted around each OD PMT. The inner surface of the OD region is covered by Tyvek
sheets, except for the PMT surfaces, to further increase the light collection efficiency.

3.1.3

20 inch photomultiplier tubes

The 20 inch photomultiplier tubes (PMT), which are used in the ID and shown in Figs.3.7
and 3.8, were produced by Hamamatsu Photonics. The photo-cathode uses Bialkali (Sb-K-Cs)
and has sensitivity between 300 and 600 nm wavelengths, which matches the Cherenkov light29

Figure 3.6: Schematic view of one unit of the PMT support structure. Thickness of the structureis about 55 cm. It holds twelve 20 inch PMTs for the inner detector and two eight inch PMTs
for the outer detector.

30

Figure 3.7: Picture of a 20 inch, eight inch and five inch PMT.

Figure 3.8: Schematic view of the 20 inch PMT.

after passage through the tank. The maximum quantum efficiency of the PMT is about 22% at
390 nm. Fig.3.9 shows the relationship of the Cherenkov light after traveling through 15 m ofwater and taking the quantum efficiency of the PMTs into account. The geo-magnetic field at
the PMTs must be reduced to less than 100 mG, in order to keep uniform response throughout
the dectecotr. A network of compensation coils surrounds the entire detector volume (the coilis attached on the tank wall). The geo-magnetic field is kept at less than 50 mG at each PMT
position.
The interior of the PMT is vacuum to avoid discharge and to transport photo-electrons. As
mentioned in Sec.3.1.4, all of the ID PMTs are enclosed in a protective case, consisting of anacrylic front and a Fiber Reinforce Plastic (FRP) back, which is shown in Fig.3.10 since thebeginning of the SK-II phase. So if a PMT is accidentaly broken, it causes a strong shock wave
which could in turn cause a chain reaction of exploding PMTs. The transparency of the front
acrylic cover is more than 96% for photon wavelengths above 350 nm.

3.1.4

Phases of the SK detector

From April 1996, the SK detector started observation as the SK-I phase, with 11,146 ID
PMTs and 1,885 OD PMTs. After 5 years of observation, in order to replace bad PMTs whichhave high dark noise, discarge in the dynodes, and so on, the SK-I observation phase wasstopped. After the replacement work was finished filling of the tank with pure water was started
in September 2001. When the water height became about ∼30 m in November 2011, one of
the bottom PMTs was broken and the released shock wave triggered a chain reaction which
destroyed 6777 ID PMTs and 1,100 OD PMTs.
31

Figure 3.9: Spectrum of Cherenkov light and quantum efficiency of the 20 inch PMTs.

Figure 3.10: PMT case to prevent a chain reaction in the case of an imploded PMT.

32

SK phase

SK-I

SK-II

SK-III

SK-IV

Start

Apr.1996

Oct.2002

Jul.2005

Sep.2008

FinishApr.2001

Oct.2005

Aug.2008

continued

livetime

1496days

791days548days(1069days)

Number of ID PMT

11146

5182

11129

11129

Number of OD PMT

1885

1885

1885

1885

Photo coverage

40%

19%

40 %

40%

Table 3.1: Summary of the features of each SK phase. Livetime of SK-IV phase is from Sep.
2008 to Mar. 2012, used in this thesis.

After this accident, the detector was reconstructed using the surviving ID PMTs. Broken
OD PMTs were all replaced by newly produced PMTs. After this reconstruction work in 2002,
the SK-II period was started with 5,182 ID PMTs and 1,885 OD PMTs, in October 2002. As
described before, all ID PMTs were each enclosed in a protective case to prevent another chain
reaction, even if one PMT is somehow destroyed.
In order to recover the original number of ID PMTs, about 6,000 20 inch PMTs were reproduced from 2002 to 2005. A work to mount those PMTs, called the full-reconstruction, wasconducted from October 2005 to July 2006. After this work, the SK-III phase was started with
11,129 ID PMTs.
The front end electronics and data acquisition system were replaced in September 2008,
marking the beginning of the SK-IV phase. The data acqusition system during the SK-I,II,III
phases and the new data acquisition system of SK-IV are described in the next section.

3.2

Data acquisition system

In this capter, the data acquisition systems are described. As described above, the dataacquisition system is different between SK-I,II,III and SK-IV. The PMT signal is converted todigital signals, which contain charge and timing information, by the electronics system installed
on top of the SK tank. For this purpose, there are 4 electronics huts and a central hut. In
the electronics huts, the front-end electronics and the high voltage systems which supply highvoltage to PMTs are contained. In the central hut, the trigger system and control electronics arepresent. A new type of trigger, special high energy trigger (SHE), are added in SK-IV for the
study of Supernova Relic Neutrino (SRN) study. If the event has a SHE trigger and doesn’t have
an OD trigger, an After Trigger (AFT) is issued to take 500µsec data of after SHE trigger todetect the prompt positron and delayed 2.2MeV gamma event from neutron capture by proton
which occurred from the inverse beta decay reaction. In September 2011, the SHE threshold
was lowered from 70 to 58 hits to study the lower energy SRN candidate event.

3.2.1

Data acquisition system of SK-I,II,III

The layout of the data acquisition system of SK-I,II,III is shown in Fig.3.11. The front-end
electronics (ATM, Analog Timing Module) were developed based on TKO (Tristan KEK Online)

33

standards. The integrated charge and timing information are digitized by an ADC (Analog to
Digital Converter) on board the ATM module. One ATM module processed signals from 12
PMTs, so about 1000 ATM modules were used in total.
Fig.3.12 shows the schematic view of the input components of the ATM. Input signals were
amplified 100 times and divided into four signals by the hybrid IC.
One of the divided signals would go through a discriminator, which had a threshold set to 100
mV, corresponding to about 0.32 photo-electrons. If this threshold was exceeded, two signals
were generated. One signal was a pulse which has 200 nsec width and -15 mV height. These
pulses of the input 12 PMT channels were then summed and output as a HITSUM signal. The
HITSUM signal was used for generating a global trigger which was used to determine whether
or not to save the event. This method of global trigger generation is described in the next
subsection. Another signal was a pulse of 900 nsec width and -15 mV height, and this pulse wascalled the selfgate signal. The selfgate signal was used as the trigger to start the QAC and TAC
integration, as described below.
The other two divided signals were sent to a QAC (charge to analog converter), which
integrated the charge for 400 nsec and to a TAC (time to analog converter), which generated
the hit timing. If a global trigger was generated within 1.3µ sec, the integrated QAC and TAC
information was digitized by an ADC.
The final divided signal was output as a PMTSUM signal. The PMTSUM is the sum of
the input PMT signals, for one ATM board, and was run into a Flash ADC that recorded the
waveform information.

3.2.2

Hardware trigger

The HITSUM signals from the ATMs were eventually summed at the centeral hut, and a
global trigger generated. The reason why the HITSUM signal had a 200 nsec width is that theevent duration time in the SK detector is up to 200 nsec, which is corresponded the time offlight would take to go from one edge of the detector to other edge. If the summed HITSUM
signal exceeded a threshold, a global trigger was generated. The overview of the global trigger
generation is shown in Fig.3.13.
There were, and still remain, three levels of the global trigger, called the High Energy trigger(HE), the Low Energy trigger (LE) and the Super Low Energy trigger (SLE). These triggers
were determined by the threshold of the summed HITSUM, with each trigger threshold, for each
SK phase, and are summarized in Table 3.2.
SK-I

SK-II

SK-III

SLE

186mV(*)

110mV

186mV

LE

320mV

152mV

302mV

HE

340mV

180mV

320mV

Table 3.2: Trigger threshold in each data taking period. The SLE threshold was lowered time
to time during SK-I. The final value is shown here.

34

Analog sum of 12PMT signals
20-inch PMT

:

:
Sparc

KPVGTHCEGSCH(NCUJ#&%ATM

8ch(12x8=96PMT)/board

online CPU(slave)

x 15
x 20
(NCUJ#&%

x 240

ATM

8/'

8ch x15 x8 =960ch

GONG
20-inch PMTSCHATMinterface
Sparc

x 20
x 240

SMP

ATM

online CPU(slave)

SMP

GONG

SMP
SMP
SMP

Fast Ethernet

SMP

x6

Super Memory Partner

VME
20-inch PMT
SCH

Sparc

ATM

online CPU(slave)

x 20
x 240

Sparc

ATM
GONG

online CPU(host)

Analog Timing Module

online CPU(slave)
TKO
Fast Ethernet

Sparc

Sparc

online CPU(slave)Sparc

online CPU(slave)

Fast Ethernet

Sparc

online CPU(slave)Sparc

online CPU(slave)

online CPU(slave)interface
Sparc
SMP

Sparc

online CPU(slave)

SMP
SMP
SMP
SMP
SMP

Super Memory Partnerinterface

VME
VME

interrupt reg.
TRG

20-inch PMT
SCH
TRIGGERATM

TRIGGERx 20

PROCESSORHIT INFORMATION

x 240(HITSUM)

ATM
GONG

Analog Timing ModuleTKO

PMT x 11129

ATM x ~1000

SMP x 48

online CPU(slave) x 17

FADC x 120

Figure 3.11: The layout of the data acquisition system of SK-I,II,III.

35

to PMTSUM
HITOUTDISCRI.

CURRENT SPLITTER&AMPLIFIER

PMT INPUT

W:200ns

to HITSUM

ONE-SHOT HIT
W:300ns

SELF GATESTART/STOP

TAC-A

TAC-A

CLEAR
GATE

QAC-A

QAC-A

to ADCSTART/STOP

TAC-B

TAC-B

CLEAR
GATE

THRESHOLD

QAC-B

QAC-B

DELAY

GATE
TRIGGERPED_START

Figure 3.12: The schematic view of the input components of the ATM.

36

Figure 3.13: The overview of the global trigger generation.

3.2.3

Data acquisition system of SK-IV

Fig.3.14 shows the block diagram of the data acquisition system of SK-IV.
To achieve stable observation for the next 10∼20 years, and to improve the sensitity of thedetector, new front-end electronics, called QBEEs (QTC-Based Electronics with Ethernet), were
installed in September 2008, and marked the start of the SK-IV phase. Fig.3.15 shows a pictureof one QBEE. The QBEEs are housed in TKO crates, re-used from the SK-I, II, III setup, while
the readout scheme is completely different, as described below. One QBEE module processes
signals from 24 PMTs. Fig.3.16 shows the block diagram of signal processing in the QBEEs.
The essential components on the QBEE for the analog signal processing and digitization are theQTC (Charge-to-Time Converter) ASIC [28] and the multi-hit TDC, respectively.
The QTC detects the PMT signals and drives output timing signals whose width represents
the integrated charge of the PMT signal. To satisfy the physics requirements of keeping goodcharge resolution at a few photo-electron level signal, and of setting a larger dynamic range
compared with the old front end electronics (ATM), the QTC has three input channels which
are called the small, medium and high gains. These gains are set to 1, 1/7 and 1/49. The
characteristics of each input channel are summarized in Table 3.3. The overall charge dynamic
rang is 0.2 pC to 2500 pC, which is about 5 times wider than that of the ATMs. The charge
integration gate is set to 400 nsec.
The output signal from the QTC is sent to a multi-hit TDC. The TDC measures the width of
the QTC output pulse by detecting the leading and trailing edges. A Field Programmable Gate
Array (FPGA) reads and processes the data stored in the TDC, with the width information dig37

Figure 3.14: The block diagram of the DAQ system of SK-IV.

Figure 3.15: Picture of the new QBEE electronics.

input channel

observable charge range

charge resolution

Small

0∼51[pC]

0.1[pC/count](∼0.05[p.e./count])

Medium0∼357[pC]

0.7[pC/count](∼0.35[p.e./count])Large0∼2500[pC](∼1250[p.e.])

4.9[pC/count](∼2.5[p.e./count])

Table 3.3: The characteristics of each input channel.

38

Figure 3.16: The block diagram of the signal processing in the QBEEs.

itized to charge infromation and the arrival timing inforamtion digitized to timing information.
The digitized signal is sent to the front end PCs via ethernet. The charge and timing resolutionof the QBEEs, for the single photo-electron level signal, are 10% and 0.3 nsec, respectively.
Both resolutions are better than the intrinsic resolutions of the 20 inch PMTs. Furthermore,
the QBEEs achieved high-seed signal processing by combining the pipelined components of the
QTC, TDC and FPGA.
Data from the QBEE boards is sent to the front-end PCs. The front end PCs sort thereceived data by time and then send it to the Merger PCs. In the Merger PCs, a software
trigger program, which is described in detail in the next section, is run to extract an event fromall the hit data, from all PMTs. The triggered event data are collected by the organizer PC andrecorded to disk.

3.2.4

Software trigger

Because of the achievement of high-speed signal digitization, all of every hit PMT’s information of all of hit PMTs is collected at the Merger PC. The software trigger searches for events
which have an Nhit,200ns (number of hits within a 200 nsec sliding time window) threshold. The
Nhit,200ns threshold for each trigger type is summarized in Table 3.4. The key feature of thesoftware trigger is that it is possible to select an event width dependent upon trigger type. In
SK-I,II,III, the width was always 1.3 µsec, because of the hardware trigger, while in SK-IV theevents with LE/HE trigger types have a width of 40 µsec. The wider event width is ideal for
detecting possible pre-activity (i.e. prompt γ rays) and post-activity (i.e. µ decay electrons).
The events with SLE trigger type have a width of 1.5 µsec, because of its high rate due toradioactive backgrounds. It is possible to tag external triggers for calibration events and now
simpler to program to Monte Carlo simulation. The systematic error of the trigger threshold of
SK-IV has been reduced compared with SK-I,II,III, and is described in chapter 7.

39trigger type

hit threshold[hits/200nsec]

event width[µsec]

SLE

34

1.5(-0.5∼+1.0)LE4740(-5∼+35)

HE

50

40(-5∼+35)

SHE

70(before Sep.2011)40(-5∼+35)

SHE

58(after Sep.2011)40(-5∼+35)

Table 3.4: Summary of software trigger types with SK-IV.

Figure 3.17: Schematic view of the water purification system.

3.3

Water circulation system

The water in the detector is continuously circulated through the water purification systemwith a flow rate of about 60 tons/hour. In this section, the water purification system and its
important improvements during the SK-III and IV phase are described.

3.3.1

Water purification system

The schematic view of the SK-IV water purification system is shown in Fig.3.17. The water
purification system consists of the following components;• 1 µm filter: Remove dust (∼1µm) and particles.
• Heat exchanger (HE): The water temperature rises due to the heat geneated by pumps,
PMTs, surrounding rock and so on. The temperature of the supply water is controlledwith an accuracy of 0.01◦ C, to suppress the convection inside the detector. The processedwater temperature is about 13◦ C, and is low enough to supress bacteria growth. Thiswater temperature control is only one of the improvements in SK-IV, and is described indetails in the next section.
40

• Ion exchanger: The resin for the high molecular, three dimensional network structure
within the ion exhange group. It is able to remove heavy ions such as N a+ , Cl− , Ca2+
and 218 P o.
• UV sterilizer: Kill bacteria by ultraviolet light.
• Radon(Rn)-free-air dissolving system: Dissolving the Rn free air into the water increases
the efficiency of the Rn removal by the vacuum degasifier.
• Reverse Osmosis filter (RO): Remove contaminants heavier than 1000 molecular weight.
The RO is newly added during the SK-III phase.
• Vacuum degasifier: Remove gas components dissolved in the water. The main purpose isto remove Rn. It is able to remove about 99% of oxygen and 96% of radon.
• Ultra Filter (UF): Removes small particles, down to the size of 10 nm.
• Membrance Degasifier: Remove radon with 83% efficiency.

3.3.2

Water circulation

Fig.3.18 shows the water circulation mode in the detector. The purified water from the water
purification system is supplied at the bottom of the ID detector and extracted from the tank at
the top of the ID and the edge of the OD. The detector has been running with this configuration
since August 2007 (during SK-III).
In addition, the new control system for the supply water temperature was installed in November 2009 (during SK-IV). This system can control the temperature within 0.01◦ C. Fig.3.19 shows
the time dependence of the supply water temperature. The rate of convection due to the different
supply water temperature is kept much lower. The reason the water circulation is designed toreduce convection is in order to reduce radon emanating from the PMT surface glass. Convection
transports such radon to the fiducial volume of the detector. The lowering of the backgroundlevel in the central region of the detector is discussed in chapter 6.

41

Figure 3.18: The water circulation mode.

Figure 3.19: Time dependence of the supply water temperature. In January 2010 the newtemperature control system was installed.

42

Chapter 4

Event Reconstruction
In this chapter, the reconstruction of solar neutrino events of SK-IV is described. SuperKamiokande observes solar neutrino events by detecting the Cherenkov light coming from therecoil electron of the ν-e elastic scattering. Because the observed recoil electron travels only
∼10 cm, it is treated as a point-like source. The observed light at each hit PMTs is close
to one photo-electron, and because the one photo-electron level charge information does nottell the absolute light deposit, only the timing information of the hit PMTs is used for event
reconstruction. In the event reconstruction, the vertex of the point-like source is reconstructedfirst. Next, the direction and energy are reconstructed. Because the direction and energy are
related to each other, the direction and energy reconstructions are iterative.

4.1

Vertex reconstruction

Low energy charged particles, such as recoil electrons coming from solar neutrino interactions,travel about 10 cm. Because of the large size of SK detector and its vertex resolution, thereconstructed vertex is treated as a point-like source. For the vertex reconstruction, a maximum
likelihood fit to the timing residuals of the Cherenkov signal is used. This vertex reconstructed
tool is called BONSAI (Branch Optimization Navigating Successive Annealing Iterations) [30].
The definision of the likelihood is as follows;
L(~x, t0 ) =

N
hit
X
i=1

log(P (t − tof − t0 )),(4.1)

where ~x is the testing vertex, t is the observed time for each PMT, tof is the time of flightfrom the test vertex (~x) to each hit PMT, t0 is the time offset of the vertex position, and thefunction P is a probability density function which describes the function of timing for a single
photoelectron signal. This probability density function is obtained from LINAC calibration
which is described in section 5.1.5, and is shown in Fig.4.1. Fig.4.2 shows the vertex resolution
as a function of recoil electron energy. The vertex resolution of SK-IV is slightly better thanthat of SK-III, because of the front end electronics replacement and better timing calibration
which is described in Section 5. The vertex resolution for 10 MeV electrons is ∼70 cm.

43

.KMGNKJQQF

661(6=PUGE?
Figure 4.1: Probability density function of the residual times used in the vertex reconstruction
program. Second and third peaks around the 30 nsec and 100 nsec are caused by after pulses of
the PMTs.

vres
Entries
160
Mean
8.622
3.251
RMS

Vertex Resolution
140
120
100
80
60
40
20
0
4

6

8

10

12

14

Figure 4.2: Vertex resolution as a function of energy. The vertex resolution is defined as thepoint which contains 68% of events when comparing the real vertex and the reconstructed vertex.
Red shows SK-IV, while black shows SK-III. Horizontal axis is the input recoil electron energy.

44

Figure 4.3: The angular likelihood function for 10 MeV electrons. The vertical axis shows arelative probability normalized to one at the maximum. The peak of this distribution is atθdir =∼42 degrees.

4.2

Direction reconstruction

For the direction reconstruction, a maximum pattern likelihood fit with the Cherenkov ringpattern is used. The definition of the likelihood is as follows;
~ ≡
L(d)

N20
X
i

log(fdir (cos θdir,i , E))i ×cos θi,a(θi )(4.2)

where d~ is a testing particle direction, N20 is number of hit PMTs within a 20 nsec timing
window around the residual timing(t − tof − t0 ), θdir is the expected opening angle distribution~ Because the magnitude of the multiple
between the vector from the vertex to each PMT and d.
coulomb scattering depends on particle momemtum, the probablity density function of opening
angle (fdir ) should be energy dependent. The energy dependence of fdir is generated using themono-energetic electron MC. As an example, Fig.4.3 shows the likelihood function for 10 MeV
electrons and Fig.4.4 shows energy dependence of the angular likelihood function.
θi
The term of cosa(θi ) is the correction for the PMT acceptance. θi is photon incident angle, anda(θi ) is the acceptance function which is shown in Fig.4.5. The angular resolution as a functionof energy is shown in Fig.4.6. For 10 MeV electrons, the angular resolution is about 25 degrees.

4.3

Energy reconstruction

The basic idea of the energy reconstruction is that the energy deposited by a charged particle
is approximately proportional to the number of hit PMTs. Before converting from the numberof hit PMTs to reconstructed energy, several corrections, which are described in equation (4.3),
are applied, and the result is called the number of effective hits, Nef f ;45

cos theta

1

1

0.8

0.9

0.6

0.8

0.4

0.7

0.2

0.6

0

0.5

-0.2

0.4

-0.4

0.3

-0.6

0.2

-0.8

0.1

-1

4

6

8

10

12

14

16

kinetic electron energy(MeV)
Figure 4.4: The energy dependence of the angular likelihood.

Figure 4.5: The incident angle dependence of the PMT acceptance.

46

ares
Entries
160
Mean
8.869
3.207
RMS

Angular Resolution
50
45
40
35
30
25
20
15
10
5
0
4

6

8

10

12

14

Figure 4.6: The angular resolution as a function of energy.

Nef f =

N50 
X
i


1
ri
Nall
×
× exp()/QEi .
(Xi + ǫtail − ǫdark ) ×Nalive S(θi , φi )λ(run)(4.3)

Items in Nef f are as follows;• N50 : Number of hit PMTs
N50 is the maximum number of hits within a 50 nsec timing window, after subtracting thetime of flight.
• Xi : Multi photo-electron hit correction
Although the majority of hit PMTs observe a single photo-electron, in the case of event
occurring near the wall or towards the wall, there is the possibility that the hit PMT
will observe multiple photo-electrons because there is not enough distance between theinteraction vertex and the PMTs to spread the Cherenkov light. Xi is a factor to correct
for the multiple photo-electron effects. It is defined as

Xi =


1
 log 1−xixi

 3.0,,

xi < 1xi = 1(4.4)

where xi is the fraction of hit PMTs of the surrounding PMTs around the i-th hit PMT.
If the i-th hit PMT detects multiple photons, many PMTs surrounding the i-th PMT are
also hit. In the case of xi =1, for all surronding PMT hits, Xi are assigned to 3, and in thecase of xi < 1, using Poisson statistics Xi is defined as equation (4.4).
• ǫtail : late hits correction

47

Figure 4.7: The definitions of (θi , φi ).

Some Cherenkov lights are observed outside of the 50 nsec timing window, due to scatteringin water or reflection on the inner surface of the detector. The correction of ǫtail is definedasN100 − N50 − Nalive × Rdark × 50nsec,
N50ǫtail =

(4.5)

where N100 is the maximum number of 100 nsec timing window. As described in the next
item, dark rate effects are subtracted out of the 50 nsec timing window.
• ǫdark : dark noise hits correction

The correction of ǫdark is defined asǫdark =

Nalive × Rdark × 50nsec
,
N50

(4.6)

where Nalive is the number of all live PMTs. For each run, the average dark noise rate
(Rdark ) is measured. The average for SK-IV is ∼4.5kHz.
all• NNalive: Dead PMT correction

There is a time variation in the number of dead PMTs, where Nall is the total number of
PMT, which is 11,129 for SK-IV. Nalive is the total number of alive PMTs. The correction
allis to multiply by the ratio of NNalive.

• S(θi1,φi ) : Photo-cathode coverage correction

The correction for direction dependence due to PMT photo-cathode coverage. The definitions of θi and φi are shown in Fig.4.7. S(θi , φi ) is the effective photocathode area for theangles (θi , φi ), which is shown in Fig.4.8.

48

Figure 4.8: The function of S(θi , φi ), photo-cathode coverage correction factor.

ri• exp( λ(run)): Water transparency correction

The correction for the effect of the water transparency, where ri is the distance between
the reconstructed vertex and the i-th PMT and λ(run) is the water transparency measured
with µ →e decays, which is described in Sec.5.3.

• QEi : PMT quantum efficiency correction

The correction for PMT quantum efficiency (QE). The QE for each PMT is described in
section 5.2.2.

The event energy is calculated using the function of Nef f . This ’Nef f to MeV’ convertedfunction is obtained by a Monte Carlo simulation. Fig.4.9 shows the obtained relationshipbetween Nef f and the reconstructed energy. The best fit result of the converted fitted function
is
2
−6
3
−9
4
0.82 + 0.13 × Nef f − 1.11 × 10−4 × Nef
× Nef
× Nef
f + 1.25 × 10
f − 3.42 × 10
f.

(4.7)

This function is applied below 25 MeV, which corresponds to Nef f < 189.8. Above 25 MeV, theNef f vs. energy relationship is fit with a one-demensional polynomial funciton, described as
25.00 + 0.138 × (Nef f − 189.8).

49

(4.8)

Figure 4.9: The relationship between Nef f and energy. The upper figure shows the fitted result
and the lower figure shows the ratio between the simulated point and the fitted function.

50

Chapter 5

Detector Calibration
The details of the detector calibration are described in this chapter. Super-Kamiokande
consists of about 11,000 20-inch PMTs and 50 kton of pure water. Many calibrations are done
for the PMTs and the pure water. As described in chapter 3, event reconstruction is basedon PMT response, especially the timing information. The PMT gain and quantum efficiency
measurements were done during the SK-III phase, and these quantities are essentially unchanged
with the replacement of the front-end electronics. However, several calibrations, such as PMT
timing calibration, water transparency measurements and absolute energy scale calibration wereredone after installation of the SK-IV electronics. Details of the calibration sources, input
parameters for Monte Carlo (MC), and trigger efficiency measurements are also described.

5.1

Calibration sources

To acquire accurate information from the PMTs and to measure and monitor water conditions and so on, the SK group has developed many calibration sources. In this section, detailsof these calibration sources are described.

5.1.1

Nickel source

The nickel source is a gamma ray source, which emits high energy gamma rays when a thermal
neutron is captured by the nickel through the reaction N in (n, γ)n+1 N i. As a neutron source,
the spontaneous fission source of 252 Cf, shown in left of Fig.5.1, is used. 252 Cf decays throughα−decay (∼ 96.9%) and spontaneous fission (∼ 3.1%) with a multiplicity of 3.8 neutrons. The
half life of 252 Cf is 2.65 years.
The 252 Cf source is set at the center of the so-called ’nickel ball’, shown in the right portion of
Fig.5.1. A pair of brass rods support the 252 Cf source within the ball, shown in the middle portionof Fig.5.1. The nickel ball consists of 3.5 kg of polyethylene and 6.5 kg of NiO. The accuracyof the uniformity of the nickel ball is within 1%. Table 5.1 shows the isotope abundance ratio,
neutron capture cross-section and the most frequently emitted gamma energies of the different
nickel isotopes.
The characteristics of the nickel source is very stable for weak Cherenkov light, emitting thislight uniformly and an ease of taking the data. It is useful for calibrations which need stable
light at the one photon level, such as determining the absolute gain and quantum efficiency of51

Reaction

isotope abundance[%]

capture cross-section[barns]Sum of all γ energy[MeV]

58 N i(n, γ)59 N i

67.88

4.4

9.000

60 N i(n, γ)61 N i

26.23

2.6

7.820

62 N i(n, γ)63 N i

3.66

15

6.838

64 N i(n, γ)65 N i

1.08

1.52

6.098

Table 5.1: Characteristics of neutron capture on a nickel nuleus.

Figure 5.1: The picture of the Cf source (left), the brass rod (middle), and the nickel ball (right)

each PMT. It is also used for checking the trigger efficiency, checking the water conditions andits position dependence and the quality of vertex reconstruction.

5.1.2

Xenon light source

The xenon light source was made by Hamamatsu Photonics, and has a stability of output
intensity better than 5%. The Xe lamp light goes through a UV filter, which passes only UV
light, and is then divided into two optical fibers. One goes into a scintillator ball set inside
the SK tank, while the other goes to a two inch PMT monitor, shown in Fig.5.2. Looking at
a spectrometer measurement, only light with wavelength less than 400 nm is transmitted to
the fibers. The characteristics of the xenon light source are its high stability and strong lightyield. Therefore, it is used for the PMT gain calibration and continuous monitoring of the waterconditions.

5.1.3

Laser light source for timing calibration

The characteristic of the laser light source is its very short pulse width. First, an N2 laser
generates 337 nm light, with a very short pulse (< 0.4nm), and using a dye it converts the
wavelength from 337 nm to 398 nm, which is near the peak of the Cherenkov spectrum seen in
SK. Then, the light is transported into the SK tank via optical fiber, and then semi-uniform
light is emitted with a diffuser ball.

52

Figure 5.2: Schematic view of the set up of the Xe light source.

USHO Laser
The USHO (Usho Optical Systems Co.) laser is used for making the TQmap. While thislaser requires N2 gas flow during operation time, it emits shorter pulses of light compared withother light sources. The characteristics of the USHO Laser are as shown in Table 5.2. The
USHO laser is used for the precise PMT timing calibration, which is described in section 5.2.3.
Although the pulse time width is very short, its longer time stability is not good. So other laser
light sources are needed for the long term stability check of PMT timing.
USHO N2 laser
Wevelength337.1nmMax. rate

20Hz

Pulse width(FWHM)

0.3ns

USHO Dye module
Wavelength

396nm

Spectrum Bandwidth(FWHM)

5ns

Pulse width(FWHM)

0.2ns

Max. rate

20Hz

Table 5.2: Characteristics of the USHO laser.

LSI Laser
The Laser Science Instruments (LSI) dye laser system is used for long term TQ calibration,
called ‘auto TQ calibration’. Because it runs every 1 second, it is also used for the PMT
nonlinearity calibration and the OD calibration. Since this laser system uses nitrogen as a gain
medium, continuous operation is easier compared to the USHO laser. The characteristics of the
LSI Laser are shown in Table 5.3.

53

LSI N2 laser
Wevelength337.1nmMax. rate

20Hz

Pulse width(FWHM)≤ 4ns

USHO Dye module
Wavelength

396nm

Spectrum Bandwidth(FWHM)

0.3ns

Pulse width(FWHM)≤ 4ns

Max. rate

20Hz

Table 5.3: Characteristics of the LSI laser.

5.1.4

Laser light source for water quality monitoring

The laser light source is also used for monitoring the water quality. There is 4 lasers, withdifferent wavelengths, to monitor the time variation and wavelength dependence of the water
quality. At the beginning of SK-IV, the 337 nm wavelength LSI laser converted the wavelengthby use of a dye. Since then a new laser, an LTB 337 nm laser, and a set of laser diodes were
newly installed to achieve stable intensity during data taking. The summary of used laser lightis shown in Table 5.4, with the laser characteristics is as follows.
laser

wavelength

Replace dateReplace laser

Replace wavelength[nm]LSI laser

337nm

Oct.2010

LTB laser

337nm

LSI+dye module

365nm

Jul.2009

Laser Diode

375nm

LSI+dye module

400nm

Jul.2009

Laser Diode

405nm

LSI+dye module

420nm

Mar.2009

Laser Diode

445nm

Table 5.4: Laser type for water quality monitoring, for each wavelength.

LTB laser
The Laser Technik Berlin (LTB) laser which is shown in Fig.5.3, is newly installed in Octoberof 2010. The characteristics of the LTB laser are shown in Table 5.5. Compared with the LSI
laser, the intensity of the light is much more stable.
Wevelength337.1nmMax. rate

100Hz

Pulse width(FWHM)

3ns

Table 5.5: Characteristics of the LTB laser.

54

Figure 5.3: Picture of LTB laser.

Figure 5.4: Picture of a single laser diode.

Laser diodes
To further improve water calibrations in SK-IV, the nitrogen/dye laser was further replacedwith laser diodes (LD)(Fig.5.4), which are NIM module types, because the intensity of thenitrogen laser is unstable and gradually its intensity decreases. The laser diodes are stable even
if environmental conditions, like temperature, are changed. At first, the 445nm LD was delivered
for testing in November of 2008, and then it was replaced to the 375 nm and 405 nm LD in Julyof 2009. Finally a 473 nm LD was introduced in Feb of 2010. The characteristics of the LD are
shown in Table 5.6.

5.1.5

LINAC

An electron beam produced by a linear accelerator (LINAC) is used for precise calibrationof the absolute energy. The LINAC is placed just beside the SK tank. The detailed information
55

Wevelength

445, 375, 405, 473nm

rate

Single∼1MHz

Select-able pluse duration

1,3,5ns

Intensity

tunableTable 5.6: Characteristics of the laser diodes.

Figure 5.5: Schematic view of the LINAC system.

about the LINAC is written in [33]. The momentum of the electron beam is accurately known(∼ 0.2% accuracy) using a well calibrated germanium detector. Fig.5.5 shows the schematic viewof the LINAC system. The electron beam which is accelerated by the LINAC is injected into
the SK tank through an evaculated pipe (less than 0.1Pa). The range of the beam momentum
is about 5-18 MeV/c, which covers the momentum range of recoil electrons coming from solar
neutrino interactions.

5.1.6

DT generator

The deuterium-tritium (DT) generator [31] is a calibration device which emits neutrons via
the following reaction:3

H +2 H →4 He + n.

(5.1)

16

(5.2)

The energy of the generated neutron is 14.2 MeV, which is enough energy to generate 16 N
via the following reaction:
O + n →16 N + p.

The 16 N decays with gamma-rays and/or electrons, with a half-live of 7.13 seconds. The
main decay mode is to produce a 6.1 MeV gamma and a 4.3 MeV electron (66%) or a 10.41
MeV electron (28%). Fig.5.6 shows the schematic view of the DT calibration.

56

Figure 5.6: Schematic view of the DT system.

5.2

PMT calibration

Super-Kamiokande consists of about 11,000 20-inch PMTs inside the inner detector. To
achieve precise measurements of solar neutrinos, PMT calibration has an important role, because
the PMT response affects the event reconstruction and so on. We have performed PMT gain
and quantum efficiency (QE) measurements during the SK-III period, and timing calibrations
during SK-IV. In this section, the PMT calibrations are described.

5.2.1

PMT gain calibration

We define two factors to describe the PMT charge response, QE and gain. QE is determinedas the probabilty of emitting a photo-electron when a photon hits the PMT photo-cathode. The
gain (A(i)) is defined as the amplification factor of each photo-electron by the PMT dynodes.
Furthermore, we split the gain factor into two factors, the ‘absolute gain‘ and the ‘relative gain’.
The absolute gain is a common gain factor for all PMTs, which is a conversion factor fromthe output charge [pC] to the number of photo-electron [pe]. The ’relative gain’ is the relative
difference of each individual PMT.
In the beginning of SK-III, the high voltage (HV) of each PMT was determined using thePMT charge response (Qobs ) from the xenon light source. The Qobs was determined as
Qobs (i) ∝ Nphoton (i) × QE(i) × A(i)(5.3)

where, i is the PMT sequential number and Nphoton is the number of photons which hit thephoto cathode. It was success to determine the HV to set the uniform charge output (Qobs )within 1.3% RMS.
After determination of the HV, the absolute gain was measured by the nickel source. Because
the nickel source emits 1 photo-electron level light, the charge distribution of one photo-electron
hits can be obtained. Fig.5.7 shows the output charge distribution of one photo-electron dataobtained by nickel calibration. The SK-IV distribution was multiplied by 0.844 in order thatthe peak of the SK-IV distribution matched the peak position of SK-III. Because the PMT gainincreased 2 ∼ 3% in every year, and with different performance of charge digitization due to newfront-end electronics, the peak value of SK-IV is higher than SK-III. The behavior difference57

Figure 5.7: Output charge distribution of one photo-electon hits for SK-III (black) and SK-IV(red). The SK-IV distribution was multiplied by 0.844 so that the peak of the SK-IV distribution
matched the SK-III peak.

of the lower charge between SK-III and SK-IV is caused by the difference of the front-end
electronics performance and the difference in charge discrimination. From this distribution, we
can get the absolute gain value for SK-IV as 2.645 [pC/p.e.]. Table 5.7 shows the absolute gain
value for each SK phase.
SK phase

Absolute gain

SK-I

2.055

SK-II

2.297

SK-III

2.243

SK-IV

2.645

Table 5.7: The absolute gain, conversion factor from output charge of PMT [pC] to number ofphoto-electrons. Because the gain increases as time passes, even with constant HV, the absolute
gain factor becomes larger with time, but new PMTs which were installed just before SK-III
have lower gain than those used during SK-II. The absolute gain was slightly lower in SK-III.
The relative gain was measured with two sets of laser calibration data. One with highlight output (∼50 photo-electrons/PMT) and another with a 1 photo-electron level in each
PMT. Using the former data, the observed charge (Qobs ) is obtained, while using the latter, thenumber of hit counts over many laser fires are counted, which is called Nhit (i).The ratio between
Qobs and Nhit gives the relative gain as
Qobs (i)∝ Relative Gain(i).
Nhit (i)58(5.4)

This method is free from the water transparency, geomagnetic field and so on.

5.2.2

PMT quantum efficiency measurement

As described in the previous section, quantum efficiency is defined as the probability ofphoto-electron production when a photon hits the photo cathode. Individual PMTs have theirown QE values, and so QE measurement for each PMT is needed. For the mearurement of theQE, the nickel source is used, mostly because this light source emits one photo-electron levellight and matches the spectrum of Cherenkov light.
The hit rate of each PMT is used. Details of the method are described here. Each PMT has
a sequential number, from 1 to 11146, with numbers 1-7650 placed in the barrel part of detector,
7651-9398 in the top, and 9399-11146 in the bottom. The hit rate of each PMT includes darknoise contrubution. To remove hits due to dark noise and only count hits from the nickel lightsource, ’on-time’ and ’off-time’ hits are defined. Fig.5.8 shows the hit timing distribution, withtime of flight subtracted, from the source to the reconstructed event vertex. Then counts of hits
in the windows of on-time and off-time are made. On-time and off-time are defined as 750-1050
nsec and 1200-1500 nsec, respectively. Substracting off-time hits from on-time hits, the true
hitrate is obtained. This quantity is defined as hit(i). In order to take into account position
dependence of PMT positions, a geometrical correction is applied and the corrected hit hitcorr
is defined as
hitcorr (i) = hit(i) × R2 (i)/a(θ(i)),(5.5)

where R(i) is the distance from the source position to PMT(i) and a(θ(i)) is the acceptance of
the PMT, which can be described as
F (θ) = 0.205 + 0.524 × cos θ + 0.390 × cos2 θ − 0.132 × cos3 θ.

(5.6)

This fuction was obtained from the simple geometry of the PMTs.
Finally the hitcorr is normalized by the total number of hitcorr and the number of PMTs
(NP M T =11,129 for SK-III and IV), and is described
hitrate(i) = hitcorr (i)/(Σhitcorr (i)/NP M T ).

(5.7)

Even though the upper corrections are applied, the position dependence of the PMT hit rate
remains. This is especially true in the case of photon reflections by neighboring PMTs or water
scattering. They can be reproduced by MC. Fig.5.9 shows the hitrate distribution of both data
and MC. In this MC, QE of all PMT is set to the same value. The hitrate ratio between thedata and MC is defined as the QE table. Data was taken when the detector water was well
mixed, and the position dependence of the water transparency must have been negligible. MC
was geneated with uniform water transparency.

5.2.3

PMT timing calibration

The PMT timing calibration is very important because event vertex reconstruction is basedon PMT timing information. Ideally, the timing response of each PMT is the same, but due tothe length of the PMT signal cable and response of the electronics, there is a PMT dependence
on the timing information. Due to time walk, which is shown in Fig.5.10, a correction due to the59

on time

off time

T-TOF [nsec]

Figure 5.8: Hit timing distribution of nickel light source events.

Figure 5.9: Position dependence of the PMT hit rate. The upper, bottom left and bottom rightfigures show the hitrate for the barrel, top and bottom PMTs, respectively. The horizontal axisof the upper figure is the z [cm] positon of PMTs. For the lower figures, the horizontal axis is
the square of the distance from the center [cm2 ] of the tank. The red and blue distributions
show the hitrate of data and MC, respectively.

60

Figure 5.10: Explanation about time-walk.

pulse height (integrated charge) is necessary. Time-walk occurs because if the observed charge islarger, the PMT signal exceeds the TDC discriminator threshold earlier, and if lower, it exceeds
later. To correct them, the timing-charge relation table, called the TQ map, should be madefor each PMT. In this section, details of the PMT timing calibration are described.
For the timing calibration, a laser light source is used because of its fast rise time and easeof light intensity control. At the beginning of SK-IV, data was taken using the USHO laser lightsource, with various light intensities. Fig.5.11 shows the schematic view of the USHO laser lightcalibration.
The method of making the TQmap is as follows; (1) Make the timing distribution plot foreach PMT, after subracting the time of flight from the light source position to each PMT.
Fig.5.12 shows an example of the timing vs charge distributions for one PMT. The charge value(Qobs ) is divided into 180 bins, with the following bin sizes.

∆Qbin ≡(

0.2 pC,
10

Qbin
50

− 10

Qbin −150,

for 1 ≤ Qbin ≤ 50 (0 pC ≤ Q ≤ 10 pC)

for 51 ≤ Qbin ≤ 180 (10 pC ≤ Q ≤ 3981 pC)(5.8)(2) The timing distribution has a binning effect, so it is necessary to smooth the timing histogram. This method is called the smearing method and Fig.5.13 gives an explanation. First
we apply an asymmetric gaussian fit from t-low to t-high, which is defined in Fig.5.13 as thevalue of horizontal axis of 0.05% from peak value. The asymmetric gaussian fit results are taken
into account into MC as a timing resolution of each observed charge. After getting the max
value for each Qobs bin, we apply fitting by 3rd-degree polynomial fuction for the low Q range,
5th-degree polynomial function for the middle Q range, and 7th-degree polynomial function forthe high Q range. Each charge range is defined in section 2. Timing information after the
TQmap correction is shown in Fig.5.14.

61

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

5-+&

8CTKCDNGHKNVGT&KHHWUGTDCNN
Figure 5.11: Schematic view of the USHO laser light calibration system.

Figure 5.12: Obtained TQ map by the USHO laser. The horizontal axis shows Qbin , defined inequation 5.8. Black, red and blue points are the QBEE charge ranges as defined in chapter 2.

62Figure 5.13: Schematic view of the TQ smearing method. Each bin has its binning effect,
expanded to a Gaussian distribution. The sigma of the Gaussian is defined for each Qbin value.
After smearing the timing distribution, we apply an asymmetric gaussian fit from the lower partto higher part, as shown in this figure.

Figure 5.14: Timing distribution after correcting with the TQ map.

63

5.3

Water transparency measurement

The water transparency measurement is also important because it affects the number ofphotons which are detected by the PMTs. This means water transparency affects energy determination. In this section, the two water transparency measurements, decay-e and N2 laser, aredescribed. Finally the input parameters to the MC are described.

5.3.1

Measurement by decay-e

As discussed in section 2, due to the improvements of the water circulation system, the watertransparency of the SK-IV detector is kept more stable at a higher quality. But still there is atime variation. In order to take into account the time variation, we always monitor the waterusing decay electrons/positrons from cosmic-ray muons, which have stopped in the detector (inthis thesis, these events are called decay-electron or decay-e). The selection criteria of decay-e
events are;
• The time difference between the parent muon event and the decay-e candidate event is
3.0µsec ≤ ∆T ≤ 8.0µsec.
• The reconstructed vertex of the decay-e candidate events is within the 22.5 kton fiducial
volume, i.e. 2 m from the wall.
• The distance between the stopping point of cosmic ray muon and decay-e candidate event
is within 250 cm.
• The number of hit PMTs within the 1.3µsec selects 120 ≤ Nhits ≤ 600.
The event rate of the decay-e events is about 1,500 events in one day. Using these decay-e
events, correlation between the log of the observed charge, log(Qobs (r)), of hit PMTs and the
distance between the decay-e vertex and hitPMTs can be obtained. Fig.5.15 shows the charge
versus distance distribution. For the hit PMTs, an opening angle of 32◦ < θdir < 52◦ was used.
Then we apply linear fitting to this distribution, and the inverse of the fitted slope becomes
the water transparency. From statistics, it needs to take a running average of ±7 days, using alinear function. Fig.5.16 shows time variation of the water transparency for the SK-IV period.
One point corresponding to one elapsed day.

5.3.2

Measurement by laser light injector

The coefficients of the water parameters are the absorption coefficent(αAbs ), the Raylieghscattering coefficient(αRaylie ) and the Mie scattering coefficients(αM ie ). Each coefficient is independently measured by either an N2 laser or a laser diode(LD). Fig.5.17 shows a schematic
view of the water coefficient measurement.
The lasers are fired every 1 minute. Using this data, the water coefficients are obtained by
the following method.
• Make [scattered Hit]/[Qtot ] distributions, Hit/Qtot hereafter, for each detector region (Top,
B1,..., B5, Bottom), as shown in Fig.5.17.

64log of charge(log(q))

1.5

1.4

1.3

1.2

1.1

1

0.9

0.8

500

1000

1500

2000

2500

3000

3500
distance(cm)Figure 5.15: Charge vs. distance between the decay-e vertex distribution and hit PMTs. The
vertical axis shows the log of the observed charge, log(Qobs ).

Figure 5.16: Time variation of the water transparency measured by decay-e events.

65

Figure 5.17: Schematic view of the measurements of the water coefficients. An N2 laser was usedfor 337nm measurements, while laser diodes were used for the other wavelength measurements.

• Generate Monte Carlo by varying water coefficients and make same Hit/Qtot distributions.
Fig.5.18 shows Hit/Qtot distributions of data (black) and MC (red).
• Compare each MC with data by χ2 defined as
χ2 = Σ

(DAT A − M C)2
.
2
2
σDAT
A + σM C(5.9)• Find minimum value for each water coefficient.
When the water coefficients were input to MC, the symmetric scatteing (Rayleigh+symmetricMie scattering) and the asymmetric scattering (forward Mie scattering), instead of Rayleighand Mie scattering are take into account. Fig.5.19 shows the definition of the symmetric andasymmetric scattering. So water transparency in the MC is described as
1.
(5.10)(αabs + αSym.scat. + αAsym.scat. )After getting the water coefficients for each wavelength, they are fit with an empirical function. Fitted functions for each water coefficient are as follows;
WT =



1
r2
αsym (λ) = r1 × 4 1.0 + 2
λλ4.623 × 106
2
×(λ−3992.4)αasy (λ) = m1 × 1.0 +λ410.94 
2.691 × 107
λαabs (λ) = a1 ×
+ a1 × 1.164
λ4
500.0
a2
= a1 × 4 + Popefry(λ),λ
66(λ ≤ 463.9)(λ ≥ 463.9)

Figure 5.18: Hit/Qtot distribution from the laser calibration. The horizontal axis shows time[ns], while the vertical axis shows Hits/total charge, for data (black) and MC (red). The firstpeak and second peak correspond to scattering of photons in the water and those reflected on
the bottom PMTs and black seats.

Figure 5.19: Definition of the symmetric and asymmetric scattering in the MC.

67where r1 , r2 , m1 and a1 are multiplicative factors which adjust the function to match the watertransparency and λ is the wavelength of Cherenkov light. The numbers, except for r1 , r2 , m1
and a1 , are used as fixed values because they were historical factors present in the equations.
The additional factor of the absorption function, above 463.9 nm, was defined by theoreticallycalculated data by Pope and Fry[34]. Fig.5.20 shows the observed water coefficents for eachwavelength, shown in circles with error bars, and the best fit value of function λ ≥ 463.9, using
April 2009 data. The symmetric and asymmetric parameters from this result are used during
the whole SK-IV period. Time and position dependences is described in the next section.

5.4

Input parameters to detector simulation

In this section, the input parameters which are used in the MC are introduced. Most of
the physics processes of the SK detector are based on GEANT 3.21, but the Cherenkov photon
propagation and light attenuation in water were originally developed by the SK woring group. As
already described along with PMT calibration, the one photon distribution, Quantum efficiency
table and TQ mapping are input parameters. Photon propagation, time and position dependenceof water transparency, and PMT response are described in this section.
Cherenkov photon propagation
The number of emitted Cherenkov photons dN by an electron is given by
d2 N =

12πα(1 − 2 2 )dxdλ,
2
n(λ)λ
n β

(5.11)

where n is the refractive index of the water (n=1.343 at λ=400 nm), α is the fine structure
constant (≃1/137) and β is the velocity of the electron, per light velocity, in vacuum(v/c). dx
and dλ are the track length and wave length, respectively. The opening angle of the emitted
Cherenkov photon is given bycosθ =

1.
nβ(5.12)

From the constraint on the value of cosθ, the energy threshold of the Cherenkov light emission(Ethr ) is
Ethr = √where m is a electron mass.

nm.
n2 − 1(5.13)

Time dependence of water transparency
Because the time variation of the scattering parameters is small, only the absorption parameter is variable in the MC and the symmetric and asymmetric scattering are set to fixed values.
The energy scale calibration, which is described in section 5.5, was done by LINAC data taken
in August of 2010, and needs the precise water coefficients from August 2010. The values of r1 ,r2 , m1 and a1 in August 2010 are calculated as68

2012/08/22 11.56

-1

10

-2

10

-3

10

-4

10
300

325

350

375

400

425

450

475

500

Figure 5.20: The fitted inverse water transparency as of April 2009. The black line shows thefitted result of the water transparency, the red is absorption, blue is the symmetric scattering,and pink is the asymmetric scattering.

69

1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
10000

11000

12000

13000

14000

15000

16000

Figure 5.21: The relationship between decay electron water transparency (horizontal[cm]) andthe absorption factor (vertical, scaled to 1 in August 2010).

r1 = 8.514 × 108
r2 = 1.138 × 106

m1 = 1.002 × 10−4

(5.14)

a1 = 0.7142.

The water quality in SK changes as a function of time, as shown in Fig.5.16. The relationship between the water transparency of decay electrons and the absorption parameter was
estimated to match the input electron energy and reconstructed electron energy. Fig.5.21 shows
the correlated function between water transparency and the absorption parameter. The verticalaxis is scaled so the absorption parameter is equal to 1 in August 2010.
Position dependence of water transparency
Because it was known that there is position dependence, especially z-position dependence, of
the water transparency. This position dependence mainly due to supplying purified water fromthe bottom of the detector and draining from the top. The position dependence of the watertransparency is continuously monitored by the auto-xenon calibration source (every 1 minute)and the nickel calibration source (once per month), since the beginning of the SK-III phase. This
position dependence is also checked by using the value of the top bottom asymmetry (TBA).
T BA = (< top > − < bottom >)/ < barrel >(5.15)For the nickel light source < top >, < bottom > and < barrel > show the mean hit PMTs
hitrate in each of the top, bottom and barrel parts of the detector. The situation is similarly
for the auto-xenon light source, but the mean PMT charge is used to define the value. From
the TBA value, there seems to be a 3% ∼ 5% level of z-dependence of the water transparency.
It accounts for ±0.5% of the systematic uncertainty of the directional dependence of the energy
reconstruction, in the SK-I solar neutrino analysis. Even though there is a z-dependence of
the water transparency, the original MC could not reproduce this position dependence, because
uniform attenuation length over the detector volume was used. When the quantum efficiency
70

temperature(degree)

13.3

13.27513.25

13.22513.2

13.17513.15

13.125

13.1
-15

-10

-5

0

5

10

15

position Z(m)Figure 5.22: Z-dependence of the detector water temperature. Below z=-1100 cm, the watertemperature is uniform because of convection.

was measured, the nickel data when the detector water transparency was uniform was used,and convection was forced to happen by increasing the supply water temperature. But, usuallyconvection does not happen and the non-uniforminity of water transparency exists. This should
be taken into account in the MC. So the simple model of the light absoption, as shown inequation 5.16, is introduced to MC.
(
αabs (λ)(1 + β × z),
for z ≥ −1100cm(5.16)
αabs (λ, z) =αabs (λ)(1 − β × 1100),for z < −1100cm
As discussed below, the variation of the water transparency is mainly due to the absorption
parameter, and so only the absorption parameter is varied along with the detector z position.
Below z=-1100 cm we assume a unifrom absorption parameter, because of convection of thewater. Fig.5.22 shows the z-dependence of the water temperature. It is clealy seen that there
is convection below z=-1100 cm. The parameter β shows the degree of the z-dependence of thewater transparency. This β parameter is tuned by the hitrate analysis.
In principle, the hitrate analysis is based on the method of measuring quantum efficiency.
To cancel the effects of the quantum efficency of the PMTs, QE(i), the equation 5.5 is changedtohitcorr (i) = hit(i) × R(i)2 /F (θ(i))/QE(i).

(5.17)

After generating the hitrate table for data and MC with the β value set between 0 to 0.016[1/m], we calcluate the χ2 for each β value of MC and determine the best β value from theminimum χ2 . Fig.5.24 shows an example of the hitrate for data and MC, β equal to 0 (noz-dependence) and to the best value (best z-dependence).
The TBA value and the β value are correlated according to71

Figure 5.23: Estimated χ2 for each β, with the minimum χ2 value setting the best β value.

β = {−0.163(T BA)2 × 3.676(T BA)} × 10−3 .

(5.18)

From the TBA value of the auto-xenon calibration, the time variation of the position dependence is reconstructed.
PMT response to photons
When generated photons arrive at the surface of the PMTs in the MC, one of the following
3 actions is selected; (1) absorbance by the photo-cathode (with some fraction of it causing
photo-electron production), (2) reflectance from the PMT surface, or (3) transmitance throughthe PMT, as explaind in Fig.5.25(b). Each process has an incident angle dependence. The final
probablity of the photo-electron genaration is defined as follows.
Probability = QE(λ) × Pobs (θi , λ) × COREPMT × qetable(i)Absorption(θi , λ)Pobs =
Absorption(0, λ)(5.19)(5.20)

QE(λ) is the incident wavelength (λ) dependence of the quantum efficiency of the 20 inch
PMTs, measured by Hamamatsu and shown in Fig.3.9, Absorption(θi , λ) is the summed fractionof the photo-electron generation (green) and absorption (black) in Fig.5.25(b), which is measured
by the laser light source calibration with each λ. COREPMT is a correction factor which is finetuned by LINAC calibration (described in detail in section 5.5), and qetable(i) is described in
5.2.2. After the photo-electrons are generated, the output charge, which is described in section
5.2.1, is simulated and if the output charge exceeds the threshold of the front-end electronics,
72

Figure 5.24: Hitrate of data (black) and MC (β=0 for green and β=best for red).

73

(a)

(b)

Figure 5.25: (a) Schematic view of PMT response for incident light. (b) PMT response forincident 420 nm unpolarized light, depending on incident angle. The black area shows thefraction of photons that are absorbed but don’t generate photo-electrons.
the photon is detected by the PMT. The timing resolution which is estimated in section 5.2.3 isalso taken into account in this simulation.

5.5

Absolute energy scale calibration

In this section the determination of the absolute energy scale and its position and directional
depnedences are described. The LINAC calibration system can produce an electron beam with
precisely tuned electron momentum, but can only take data with events in the down-going direction. The DT calibration system can produce uniform directional data, but emits continuous
momentum electrons along with gamma rays. Because of these differences, the absolute energy
scale, i.e. determination of COREPMT, is performed by the LINAC and the corresponding
energy scale cross check, which is position and directional dependence, is done by the DT calibration. The time variation of the energy scale, which is estimated by LINAC, DT and decay-e
calibrations, will be described in the final subsection.

5.5.1

LINAC calibration

LINAC data taking
The LINAC calibration was done in August of 2009 and August of 2010 during the SK-IV
phase. The water transparency was slightly unstable during the August 2009 data taking, so
the energy scale determination was done with the August 2010 data. The position and energy
mode is summarized in Table 5.8.
Beam momentum estimation
The precise electron beam energy is measured by a germanium detector. The characteristicsof the germanium detector is its good energy resolution and good output linearity for given input
particle energies. Fig.5.26 shows the output charge of the germanium detector as a function of
the gamma-ray energy of several calibrations, which are shown in the point. For the few MeV
region, the difference of the output of germanium detector count from the fitted line is within
0.1%.
74

Aug. 2009 data
position of beam pipeenergy mode(-1237,-70.7,1197)[cm]

18.9,13.6,8.8,4.8[MeV](-1237,-70.7,-6)[cm]

18.9,13.6,8.8,5.1[MeV](-1237,-70.7,-1209)[cm]

18.9,13.6,8.8,5.1[MeV](-388.9,-70.7,1197)[cm]

18.9,13.6,8.8,5.1,4.8[MeV](-388.9,-70.7,-6)[cm]

18.9,13.6,8.8,5.1,4.8[MeV](-388.9,-70.7,-1209)[cm]

18.9,13.6,8.8,5.1,4.8[MeV](-1237,-70.7,1197)[cm]

18.9,13.6,8.8,5.1,4.8[MeV]Aug. 2010 data(-1237,-70.7,1197)[cm]

13.6,7.0[MeV](-1237,-70.7,-6)[cm]

13.6,7.0[MeV](-1237,-70.7,-1209)[cm]

13.6,7.0[MeV](-388.9,-70.7,1197)[cm]

13.6,7.0[MeV](-388.9,-70.7,-6)[cm]

18.9,13.6,7.0[MeV](-388.9,-70.7,-1209)[cm]

13.6,7.0[MeV]10

Deviation (%)

Energy of γ-ray (MeV)

Table 5.8: Data sets of SK-IV LINAC calibration.

9
8
7

1
0.8
0.6
0.4

6

0.2

5

0

4

-0.2

3

-0.4

2

-0.6

1

-0.8

0

0

250

500

-1

750 1000 1250 1500 1750 2000

0

1

2

3

4

5

6

7

8

9

10

Energy of γ-ray (MeV)

ChannelFigure 5.26: The linearity of the germanium detector. The horizontal axis shows the output
from the germanium detector [count], while the vertical axis shows the energy of the calibration
gamma ray source [MeV]. Deviation of the output of germanium detector count from the fitted
line [%] for the solar neutrino energy region gives a difference within 0.1%.
75

Figure 5.27: Nef f for data (black) and MC (red) for 13.6 MeV (left) and 7.0 MeV (right).
LINAC calibration results
Fig.5.27 shows the Nef f distributions of both data and MC for the beam endcap positionof (-388.9,-70.7,-6) cm. The left figure shows the energy mode of 13.6 MeV and the right showsthat of 7.0 MeV. From the peak value of the Nef f distribution, we find the COREPMT. The
absolute energy scale which is described in section 5.4 is taken from the comparison of Data and
MC. The best value of COREPMT is obtained at 0.88 for the SK-IV phase. Fig.5.28 shows thepeak value difference between Data and MC.

5.5.2

DT calibration

DT data taking
The DT calibration is done every 3 months. The characteristics of the DT calibration are
as follows; the data taking is comparatively easier than that of the LINAC because many morepositions can be measured, and the event direction is isotopic.
DT calibration results
Fig.5.29 shows the position dependence difference between DT calibration data and thecorresponding MC, in August 2010. The position dependence can be reduced by introducing theposition dependence of the water transparency (originally the RMS of the difference was 0.88%),
but there still exists a position dependence making up 0.44% of the systematic uncertainty.
Fig.5.30 shows the directional dependence difference between DT data and the correponds
MC in August 2010. The directional dependence has been made much lower (0.1%) comparedwith SK-I (0.5% directional dependence).

76

Figure 5.28: The peak value difference of Nef f between data and MC for each beampipe position.
Vertical axis shows the ratio between MC and data[%].

Figure 5.29: The position dependence of DT calibration. Horizontal axis shows the positionz[m] and vertical axis shows the ratio between MC and data[%].
.

77

Figure 5.30: The directional dependence of DT calibration. Horizontal axis shows the cosine of
the zenith angle and vertical axis shows the ratio between MC and data[%].
.

78

5.5.3

Systematic uncertainty of absolute energy scale

As descibed in the DT calibration section, the position dependence and directional dependence of the absolute energy scale are 0.44% and 0.1%, respectively. The uncertainty of theelectron beam energy determination by the Ge detector is estimated as 0.21% by its calibration,and the water transparency variation during the LINAC calibration is estimated as 0.20%. The
total systematic uncertainty of the absolute energy scale is 0.54%, by adding all the contributionsin quadrature.

79

Chapter 6

Data Analysis
After installation of the new data acquisition system in September of 2008, debugging of
the system and calibrations were done and start of physics observations marked in October of2008. In this thesis, the data from October 2008 to March 2012, correponding to a livetime of1069.3 days, is used. The energy threshold of the SK-IV solar neutrino analysis is set to 4.5 MeV
total energy (4.0MeV in kinetic recoil electron energy). To achieve such a low energy threshold,several reduction steps are needed due to the large backgrounds which are listed below. They
are much higher event rates than that of the solar neutrino signals. The major background
events are as follows;• Electronics noise.
• PMT flashing.
• Cosmic ray muon.
• Decay-e events from incoming cosmic ray muons.
• Spallation products of cosmic ray muons .
• Radioactivity from the surronding rock, detector materials such as PMTs and their cases.
The solar analysis of SK is designed using total energy. However, other solar neutrino
experiments in the world use kinetic energy. So, when we present SK results outside the SK
collaboration, we shift the total energy by 0.511 MeV, and present the results using kinetic
energy. In this chapter, in order to describe details of the analysis method, total energy is used.

6.1

Run selection

The basic unit of the SK data set is a run. The maximum livetime of one run is set to 24
hours to prevent large data file size. For the SK-III period, the radioactive background levelchanged greatly, so different energy thresholds were set for different time periods (298.2 livetimedays of 5.0 MeV energy threshold and 547.9 livetime days of 5.5 MeV energy threshold). During
SK-IV, the background level was kept lower the other periods during the entire period, so allruns are used with a 4.0 MeV energy threshold. The quality of data is checked within each run
unit. The selection criteria of bad runs are as follows;80• The run time is less than 5 minutes.

If one run is very short, there is not enough pedestal data and so it is difficult to judge
whether it is a bad run or not. All runs less than 5 minutes are rejected.

• The runs within 15minutes after HV recovered.

For some calibration data taking, such as LINAC and DT calibrations, the HV needs to beturned off to all PMTs and the SK tank opened. But, soon after HV recovery, the PMTs
may still be noisy. So, runs within 15 minutes after the HV is tured on are rejected.

• Hardware and/or software problem happened.
• Calibration and detector maintenance run.

6.2

Pre reduction

If all data were stored, the total data size would be a few hundred GB per day. Almost all ofthese data is radioactive backgrounds, so it is nesessary to reduce the data size by about 1/100
using a pre reduction. The pre reduction is applied only to obvious background events. Further
reduction for solar neutrino analysis is described in the next section.
• Fiducial volume cut:Those background events which occurred near the wall of the detector and the reconstructed vertex is near the wall. In the pre reduction, the events which reconstruct within
2 m from the ID wall are cut.

• Gamma ray cut:Those background events from PMTs and ID wall inside the fiducial volume. The definitionof def f is the distance from the wall in the direction of the reconstructed event as shownin Fig.6.1. In the pre reduction, def f < 400cm cut is applied. More tight gamma ray cut
is applied below 8MeV.

• Lower energy cut:

If the reconstructed energy is below 3.5 MeV, these events are rejected in the pre reduction.

• Event quality cut:

There are background events due to mis-reconstructed events inside the fiducial volume.
This cut is to reject events with poor quality in the vertex and direction reconstruction.
The two dimensional vertex goodness (gv ) and direction goodness (ga ) distribution is usedin this cut. The vertex goodness (gv ) is defined as follows;
Phitall − 1 ( τi (~v)−t0 )2 − 1 ( τi (~v)−t0 )2
w
σ
e 2
e 2
,
gv =
Phitall − 1 ( τi (~v−t0 ) )2
w
e 2

(6.1)

where, τi (~v ) is a hit time after subtracting the time of flight, t0 is the fit peak time of theτ distrubution, w is the resolution of the τ diststribution and σ is the timing resolution of
the PMT. With this, the vertex goodness is the degree of quality of vertex reconstruction
81

Figure 6.1: Definition of def f82

number of events

efficiency

before cut

197,524100%

loose fiducial cut

28,161

14.2%

loose gamma ray cut

14,864

7.5%

lower energy cut

6,872

3.5%

loose event quality cut

2,057

1.0%

Table 6.1: The reduction step and efficiencies of pre reduction.

estimated by the hit PMTs timing information, the value of gv ranges from 0 to 1, and if
this value is close to 1, the quality of the vertex reconstruction is better. The direction
goodness (ga ) is defined as follows;max{∠unif orm (i) − ∠Data (i)} − min{∠unif orm (i) − ∠Data (i)}
,
(6.2)
2π
where, ∠unif orm (i) is the azimuthal angle of i-th hit PMT, assuming that the hit PMTs
are uniformly distrbuted along a Cherenkov cone, and ∠Data (i) is that of the real angle.
The value of ga ranges from 0 to 1, and if the event is a good event, this value is close
to 0. An example of a good event (low direction goodness value) and a bad event (highdirection goodness value) are shown in Fig.6.2 .
ga =Fig.6.3 shows the correlation between the square of the vertex goodness(gv2 ) and the direction goodness(ga2 ). The left figure shows the data just after the 2 m fiducial volume
cut and the 4 MeV cut, which contains a lot of badly reconstructed events, and the rightfigure shows the DT calibration data. The line shows the function of gv2 − ga2 = 0.25. Fromthese figures, good single electron like events can be separated from background events byusing the value of gv2 − ga2 . In the pre reduction, events with gv2 − ga2 < 0.1 are cut. Further
cuts will be described later.
The number of events after each pre reduction for one 24 hour run is summarized in Table
6.1. For these cuts, the data size can be reduced by 1/100. The reduction efficiency for the solar
neutrino MC simulation by these cuts is shown in Fig.6.17, and is about 5% above 6 MeV.

6.3

Reduction steps for the solar analysis

6.3.1

Total number of hit PMT cut

If the total number of hit PMTs of one event (NT OT ) exceeds 400, the energy will be muchlarger than that of solar neutrino signals. NT OT =400 corresponds to about 60 MeV for anelectron type event. By this cut, most of the cosmic ray muons are also rejected.

83

Figure 6.2: Example of a good event (top) and a bad event (bottom).

84

Figure 6.3: Correlation between gv2 (horizontal) and ga2 (vertical). The line shows the functionof gv2 − ga2 = 0.25. Left figure shows the background sample and right figure shows the solar MC.

6.3.2

Trigger cut

Events which are triggered by scheduled events are rejected. There are auto calibration
events 1 , which are used for timing and water calibration, and the so-called LED-bursts which
are used as a test for supernova burst events and happen once every ∼three weeks. Pedestal
events to adjust the pedestal of the electronics, happen once every 30 minutes. Those scheduled
events have external triggers and they are flagged by the electronics.

6.3.3

Flasher cut

When one PMT emits light due to arc discharge on the dynodes, other PMTs also get pickup the signal, and this is referred to as a ’flasher event’. The characteristics of flasher events
are that one PMT contains a large charge hit and there are many hit PMTs surrounding that
PMT. Fig.6.4 shows the two dimensional plot of the maximum charge of a PMT in an event andthe number of hits in the surrounding 24 PMTs. If the maximum charge of a PMT is greaterthan 50 [p.e.] and the number of hits in the surrounding 24PMTs is greater than 3, the event isregarded as flasher event and rejected.

6.3.4

Time difference cut

Events which occurred within 50 µsec from the previous LE triggered event are rejected. By
this cut, ringing events after high energy events, such as cosmic ray muons, and events which
are caused by after pulses and decay-e events are rejected.

6.3.5

Spallation cut

Some cosmic ray muons produce radioactive elements by breaking up an oxygen nucleus.
The reaction can be written as
1

calibration events which are automatically triggered during usual runs

85

Figure 6.4: Cut criteria for flasher events. The horizontal axis shows the maximum charge of a
PMT in an event and the vertical axis shows the number of hits in the surronding 24 PMTs.
The top figure shows one run which has flasher events and the bottom figure shows a typicalgood run.

µ +16 O → µ + X + · · · ,(6.3)

Lspa = f (∆T ) × f (∆L) × f (Qres).

(6.4)

where X is a radioactive nucleus. Also, hadrons (π ± , n, p, etc), which are produced by hadronic
cascades induced by muon interactions, are captured by other oxygen nucleus and radioactive
nucleus can be produced. They are summarized in Table 6.2. When those nuclei decay, βs
and/or γs are emitted with a lifetime in the range of 0.001 to 14 sec. These β and γ events arecalled spallation events. The energy of spallation events is similar to those from solar neutrinosignals, so the spallation event is one of the major backgrounds in the solar neutrino analysis.
To identify such spallation events, a likelihood method is used. The likelihood is defined as

The parameters for the likelihood function are as follows;• ∆T: Time difference between the spallation candidate event and the proceeding muon
event.
• ∆L: Distance from the reconstructed vertex of the spallation candidate event to the reconstructed track of the proceeding muon event.
• Qres : Residual charge of the proceeding muon event which is defined as
Qres = Qobs − Qunit × L,
86

(6.5)

Figure 6.5: Time variation of the observed charge per cm for muons.

where Qobs is the total number of photo-electrons from the proceeding muon, Qunit is
the average of the observed charge per cm and L is the reconstructed track length ofthe proceeding muon event. Because of the time variation of the water transparency and
the PMT gain depndence, Qunit changes time to time. Fig.6.5 shows time dependence ofQunit . If a muon make a spallation event, Qobs is often much greater than the expected
observed charge(Qunit ×L), i.e. Qres ≫0, because the spallation process often happens
with a hadronic casade shower.
Also in spallation events, the values of ∆T and ∆L would be smaller than random sample.
Those features are reflected in the likelihood method.
The likelihood function of f (∆T ) is generated by considering the lifetime of the produced
radioactive elements. The likelihood functions of f (∆L) and f (Qres ) are generated using two real
data samples, a spallation-like sample and a random sample. The spallation sample is composedof events with ∆T < 0.1sec and energy > 8 MeV, and the random sample is composed of eventswith energy < 5 MeV and assigned random vertex and direction, because most of the low energy
events are produced by radioactive sources from the ID wall and not by spallation. The bottom
figure of Fig.6.6 shows the calculated log likelihood distribution of the spallation-like sample(black) and the random sample (red). The point of the spallation cut is determined so that thedead time of the random sample is 20%. The cut point is determined to be log(likelihood)=4.52.
The position dependence of the dead time is shown in Fig.6.7. This position dependence of thedead time is taken into account when the expected number of solar neutrino events is calculated
in the Monte Carlo simulation.

6.3.6

gv2 -ga2 cut

The criteria of the event quality cut, called gv2 -ga2 cut, is determined by calculation of thesignificance which is defined as follows;87

Figure 6.6: The likelihood distribution of the spallation-like sample (black) and the random
sample (red). The horizontal axis shows the log(likelihood).

Figure 6.7: The position dependence of the dead time of the spallation cut. The horizontal
axis of the top figure is the detector position z[cm] and that of the bottom figure is the radiussquared(r 2 [cm2 ]).

88

Isotopeτ 1 (sec)

decay mode

Kinetic Energy (MeV)

8 He
2

0.119

β−
β− n

9.67 + 0.98(γ)
16%

8 Li
3
8B
3
9 Li
3

0.838

β−

0.77

β+

∼ 13

0.178

β−

9C
6
11 Li
3

2

13.9

β− n

13.6 (50.5 % )(∼ 50 % )

0.127

β+ n

3∼15

0.0085β−16∼20 (∼ 50 % )∼ 16 (∼ 50 % )

β− n
11 Be
4

13.8

β−

11.51 ( 54.7 % )9.41 + 2.1 (γ) ( 31.4 % )

11 Be
4
12 B
5
12 N
7
13 B
5
13 O
8
14 B
5
15 C
6

0.0236β−11.71

0.0202β−13.37

0.0110β+16.32

0.0174β−13.44

0.0086β+13.2 16.7

0.0138β−14.55+6.09 (γ)

2.449

β−

9.77 ( 36.8 % )4.47+5.30 (γ)

16 C
6
16 N
7

0.747

β− n

7.13

β−

∼4

10.42 ( 28.0% )
4.29+6.13 (γ) (66.2% )Table 6.2: Radioactive elements which are produced by Spallation in water89

Figure 6.8: gv2 − ga2 distriubtions of the data (black) and the solar neutrino MC (red) for 4.5-5.0
MeV (left figure) and 6.5-7.0 MeV (right figure). Black line shows the criteria of the gv2 − ga2 cut.

(Efficiency for solar neutrino)p.
(Backgrounds)

Fig.6.8 shows the gv2 -ga2 distriubtions of the data (black) and the solar neutrino MC (red) for
4.5-5 MeV (left figure) and 6.5-7.0 MeV (right figure). From the significance of these energy
dependence distributions, the criteria of the gv2 -ga2 cut is as follows:22 gv − ga > 0.29, 4.0 ≤ Energy < 5.0M eV(6.6)
gv2 − ga2 > 0.25, 5.0 ≤ Energy < 7.5M eV 2
gv − ga2 > 0.20, Energy ≥ 7.5M eV
.

6.3.7

Hit pattern cut

When an event which consists of multiple γ and/or β rays, i.e. spallation products of 16 N
decays give a Cherenkov ring image which is not a clear single ring pattern. To distinguish asingle electron event coming from the solar neutrino signal and not from such βs and/or γs, alikelihood of the hit pattern is used. The definition of the likelihood is as follows;
N

Lpattern (Energy, vertex, diretion) =50
1 X
log(P (i, Energy, cos θP M T , fwall )),
N50

(6.7)

i

where P is a probability density function of the hit pattern, made by a single electron MC. P is afunction of the angle between the reconstructed direction and the angle from the reconstructedvertex to each hit PMT (cosθP M T ), energy, and the effective distance between the wall and thereconstructed vertex point (fwall ). The schematic view of cosθP M T and fwall is shown in Fig.6.9.
Fig.6.10 shows Lpattern for data (black) and solar neutrino MC (red) for each energy range.
The criteria of the hit pattern cut is as follows;90

Figure 6.9: Definition of θP M T and fwall .

91

Figure 6.10: Lpattern of data(black) and solar neutrino MC(red) for 6.5-8.0 MeV (left), 8.0-12.0
MeV (middle) and >12 MeV (right). Bule line shows the criteria of hit pattern cut for eachenergy region.


 P > −1.88,
P > −1.86,

P > −1.95,

6.5 ≤ Energy < 8.0M eV
8.0 ≤ Energy < 12.0M eV
12.0M eV ≤ Energy(6.8)

Below 6.5 MeV, it is difficult to distinguish a single electron event and background events
because of the smaller number of hit PMTs, so this hit pattern cut is applied above 6.5 MeV.

6.3.8

Gamma ray cut

Fig.6.11 shows the def f distribution for 5.5-8.0 MeV and 8.0-8.5 MeV. The criteria of the
gamma cut is as follows;(def f > 650cm, 6.5 ≤ Energy < 8.0M eV(6.9)
def f > 400cm, 8.0M eV ≤ Energy
Because the radioactive background is not uniformly distributed, a tight fiducial volume cut,which is described in section 6.3.11, is needed. In the determination of the cut point of the
gamma ray cut below 5.5 MeV, events which are reconstructed in the central area (position z> -10[m] and r < 13[m]) are used. The criteria are individually determined depending upon
the nearest wall of the detector, which is either top, barrel or bottom, named poswall hereafterand defined in Fig.6.9. Fig.6.12 shows the def f distribution in the 4.5-5.0 MeV region, for eachposwall. The criteria of the gamma cut for the lower energy region (4.0-5.5MeV) are as follows;
poswall is top def f > 1000cm,(6.10)
def f > 1200cm,
poswall is barrel
def f > 1300cm,
poswall is bottom

92

Figure 6.11: def f distributons of data(black) and solar MC(red) for each energy region. These
distributions are scaled suitably. Blue line shows the criteria of gamma ray cut for each energy
region.

Figure 6.12: def f distributons of lower energy region for each poswal. The situation is the sameas Fig.6.11.

93

*KVVKOKPIYKVJQWVUWDVTWEVKPI61(%JCTIG=RG?

Figure 6.13: An example of a cluster hit event.

6.3.9

16

N cut

When lower energy µ− cosmic rays are captured by 16 O in the detector, 16 N can be produced.
16 N decays with gamma-rays and/or electrons with the half-life of 7.13 seconds. The main decay
mode is a 6.1 MeV gamma and a 4.3 MeV electron (66%) or a 10.41 MeV electron(28%). In orderto reject these events, the correlation between stopping muons in the detector and the remaining
candidate events are checked. In order to select only captured muons from the stopping muon
sample, stopping muons which did not accompany an event within 100 µsec are selected. Thenthe candidate events which satisfy the following criteria are rejected; (1) reconstructed vertex iswithin 335 cm from the stopping point of the muon, (2) the time difference is between 100 µsec
and 30 sec. The dead time of the 16 N cut is estimated to be 0.53% from accidental coincidence
between the selected stopping muons and the random sample events.

6.3.10Cluster hit event cut

This cut rejects events which are triggered by accidental coincidence of dark noise hits andsmall clusters of hits due to radioactive impurities on the detector wall. They are called cluster
hit events. Fig.6.13 shows one example of a cluster hit event. Solar neutrino events which occur
near the edge of detector also make small culsters, but compared with cluster hit events, theradius of the solar neutrino cluster is larger than these background events. So using the size ofthe cluster, one can distinguish solar neutrino events and cluster hit events.
There are two parameters to evaluate the size of cluster, ’R02’ and ’N 20rawT ’. R02 is thesize of the clustering hits in space and is defined to be the minimum radius such that more than
20% of hit PMTs are contained within a sliding 20 nsec window. N 20rawT is the clustering ofhits in time, which is defined as the maximum number of hits within a 20 nsec ’raw’ timing
window, without subtracting the time of flight from the reconstructed vertex. The reason for
94

Figure 6.14: Relationship between R02 and N 20rawT for background (left) and solar neutrino
MC (right).

this is that for backgrounds which hit PMTs outside of a cluster are randomly distributed. The
number of N 20rawT divided by total number of hits, N 20rawT /Nef f , is expected to be a smaller
than that of solar neutrino signal.
Fig.6.14 shows the relationship between R02 and N 20rawT /Nef f for background and solar
neutrino MC events, around the edge of fiducial volume. The definition of a parameter called
CLIK, which is used by this cluster hit event cut, is as follows;
CLIK = R02 × N 20rawT /Nef f .

(6.11)

And the criteria of the cluster hit event cut are as follows;(

6.3.11

CLIK < 75, r 2 > 155m2 , z < −7.5m,
CLIK < 75, r 2 > 120m2 , z < −3m or z > 13m

for 5.0 ≤E< 5.5 MeV
for 4.0 ≤E< 5.0 MeV(6.12)

Tight fiducal volume cut

Fig.6.15 shows the vertex distribution for the lower energy range. Even though several cuts
are applied, there still remain backgrounds in the bottom and edge of the barrel regions. The
uniformity of the vertex distribution is important because a non-uniform background shapes
makes large systematic uncertainty, especially for the day-night analysis of the solar neutrino
flux. Based on a significance calculation, the 1st fiducial volume cut is applied below 5.5 MeV
asR2 < 180m2 and z > −7.5m = 13.3kton,
95(6.13)

0.25

10

0.3

z[m]

z[m]

z[m]

rr-z distribution
0.3

15

15
10

0.3

15

0.25

10

0.25

5

0.2

5

0.2

5

0.2

0

0.15

0

0.15

0

0.15

-5

0.1

-5

0.1

-5

0.1

-10

0.05 -10

0.05

-10
-15
0

0.05

50

100

150

200
R^2[m^2]

0

-15
0

50

100

150

200
R^2[m^2]

0

-15
0

50

100

150
200
R^2[m^2]0

Figure 6.15: Vertex distribution of 4.0-4.5 MeV (left), 4.5-5.0 MeV (middle) and 5.0-5.5 MeV(right). The tight fiducial volume cut is shown with the red curve.

which is shown in the right portion of Fig.6.15. But a more oval like tight fiducial volume cut
is needed below 5.0 MeV. So the new tight fiducial volume shape is described as
150∗ |z − 4.25|4 ) ≤ 150,
(6.14)
11.754
is applied. The fiducail volume below 5.0 MeV is 8.8 kton. Above 5.5 MeV, no tight fiducial
volme cut is needed and the fiducial volume is 22.5 kton.
(x2 + y 2 ) + (6.4

Summary of reduction stepFig.6.16 shows the energy spectrum of each reduction step and Fig.6.17 shows the remaining
event efficiency of the 8 B solar neutrino MC. Both figures also show the SK-III final sample.
Above 6.5 MeV, the remaining efficiency of the final sample is almost the same level as SK-III,
while for 5.5-6.5 MeV the SK-IV remaining efficiency is better than SK-III, which is caused by alooser gamma ray cut and removal of the cluster hit cut. Below 5.5 MeV, the SK-III remaining
efficiency is better than that of SK-IV, caused by the additional tight gv2 − ga2 cut for 5.0-5.5 MeV
and a tighter fiducial volume cut for 4.0-5.0 MeV. Fig.6.18 shows the time variation of the final
data sample for SK-III and SK-IV. By improvements of the water circulation system, which is
described in Sec.3.3.2, a continuous lower background level is achieved in SK-IV period.

6.5

Signal extraction method

In the case of ν-e interactions of solar neutrino signals, the incident neutrino and recoil
electron directions are highly correlated. Fig.6.19 shows the cosθsun distribution for events
between 4.5-20 MeV, where cosθsun is the direction between the sun and the reconstructed
direction, which is defined in Fig.6.20. A clear peak due to solar neutrinos can be seen. The
flat component in the cosθsun distribution is due to remaining background, such as spallation
products and of surrounding rock and so on. In this section, the method of the solar neutrinosignal extraction is described.

96

Figure 6.16: Energy distribution for each reduction step. The 1st reduction of SK-IV corresponds
to the pre reduction and that of SK-III corresponds to a precut which is explained in this thesis.

Figure 6.17: Reduction efficiency for the solar neutrino MC simulation.

97

Event/bin

Figure 6.18: The event rate of the final data sample for SK-III (black) and SK-IV (red).

7000

6000

5000

4000

3000

2000

1000

0

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1
cosθsunFigure 6.19: cosθsun distribution for events between 4.5-20 MeV.

98

<

Ǒ\
G

5WP

ǰUWP

5-FGVGEVQT
Figure 6.20: The definition of cosθsun .

99

Probability

0.3

4.5MeV

0.25

8.0MeV

0.2

10.0MeV
0.15

0.1

0.05

0
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1
cosθ

Figure 6.21: The probability density function of the signal(sij ) for given energy.

In order to obtain the measured flux based on the SSM, an extended maximum likelihood
method is used. The likelihood function is defined as;P

L = e−(i Bi +S)

N
ni
bin Y
Y

(Bi · bij + S · Yi · sij ),(6.15)

i=1 j=1

where Nbin is the number of bins between 4.0 MeV and 20.0 MeV. For SK-IV, Nbin =23 isused, where the bin width is 0.5MeV from 4.0 to 14.0 MeV and 1.0MeV bin width is usedfrom 14.0 to 16.0 MeV. The last bin goes from 16.0 to 20 MeV. ni is the number of observed
events in the i-th energy bin. S and Bi are the free parameters of this likelihood function andare the total number of observed solar neutrino events and the number of background in thei-th energy bin, respectively. Yi is the expected fraction of signal event in the i-th energy bin,
estimated by solar neutrino MC. The probability density functions bij and sij are the expectedshapes of the background and solar neturino signal, respectively. The background shape, bij ,is based on the zenith and the azimuthal angular distribution of real data. Namely, thesedetector oriented angular distributions are fit with polynomial functions, and then converted
to the cosθsun coordinate. bij is not a completely flat shape because of the effect of cylindricalshape of the detector and the effect of the zenith and azimuthal dependence of the background
distribution. The expected solar neutrino signal shape, sij is obtained by the solar neutrino MC
by fitting the cosθsun distributions with a triple exponential plus a Gaussian. Fig.6.21 shows theobtained sij probability density function for 4.5 MeV, 8.0 MeV and 10.0 MeV, while Fig.6.22
shows the bij probability density function. Finally, the values of S and Bi are obtained by the
maximum likelihood method.

100

probaility

0.52
0.515
0.51
0.505
0.5
0.495
0.49

4.5MeV

0.485

8.0MeV

0.48

10.0MeV

0.475
0.47

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

cosθsunFigure 6.22: The probability density function of the background(bij ) for a given energy.

6.6

Multiple scattering goodness

There are still remaining backgrounds in the lower energy region, even after applying reductions which were described in this chapter. The main background comes from radioactive
impurities in the detector, with the most likely source coming from 214 Bi decays into beta rays
which have a 3.3 MeV endpoint energy. Because of the finite (about 20% for 4.0MeV electron)
energy resolution, those beta rays can be reconstructed above 4.0 MeV. However, they shouldhave incurred more multiple scattering in water than true 4.0 MeV electrons. The degree ofmultiple scattering is evaluated using a ’multiple scattering goodness (MSG)’ which is calculatedby the following method.
• Select the pairs of hit PMTs within a 20 nsec window from the initial event time, after
time of flight subtraction. The two vectors from the vertex to each hit PMTs, ri and rj
are considered ‘hit direction’s.
• For each pair of hit PMTs, 2 cones which have 42◦ opening angles from the vertex are
projected and the counted number of intersecting points of the two cones on the detector
surface gives zero, one or two. Fig.6.23 shows the case of the two intersection case.
• For pairs of PMTs with two intersection points, the vector from the vertex to each of theintersection points is taken as a ’unit vector’, which has one unit length. we find all the
candidates of ’unit vectors’ by selecting all the possible pairs of hit PMTs within a 20 nsec
window.
• For all candidates of ’unit vectors’, the ’sums vector’ is calculated by forming vector sumsof the ’unit vectors’ which are within 50◦ of one candidate ’unit vector’. The longest ’sums
vector’ is defined as the ’best direction vector’ and we then sum the ’unit vectors’ which
101

Figure 6.23: The schematic view of cones which have 42◦ opening angles from the vertex andthe number of intersection points.

Figure 6.24: The schematic view of the ’unit vector’ and the ’best direction vector’ for more andless scattering events.

forming the ’best direction vector’ along the ’best direction’. The schematic view of the’unit vector’ and the ’best direction vector’ for the more or less scattering cases are shownin Fig.6.24• The MSG value is defined as follows;
Length of best direction vector(6.16)
number of unit vector
The MSG value ranges from 0 to 1, with higher real energy electrons having a higher MSG
value.
M SG =The events below 7.5 MeV are divided into three MSG groups, low MSG (MSG<0.35),
middle MSG (0.35≥MSG≥0.45) and high MSG (MSG>0.45), and then the signal extraction is
executed. By dividing into 3 groups, the effective signal extraction can be improved because themiddle and high MSG groups have an improved signal to noise ratio, and it makes the statistical102

error smaller. The systematic uncertainty and signal extraction results is described in the next
chapter.

103

Chapter 7

Systematic Uncertainties
In this chapter, the systematic uncertainties for the total flux between 4.5-20.0 MeV and forthe energy spectrum shape are described. The systematic uncertainty for the day/night analysis
is described in section 9.5.

7.1

Energy scale

The systematic uncertainty on the flux due to the energy scale is estimated taking into
account the inaccuracy of the abolute energy scale (0.54%). In order to do this, the signal
extraction was repeated with the energy scale shifted. Yiscale+ and Y scalei , which are the obtained
total flux after shifting the energy scale by ±0.54%, are calculated by changing the reconstructedenergy of each solar neutrino MC event. The Yi is the expected fraction of signal event in thei-th energy bin which is defined in Sec.6.5. With this method the 0.54% energy scale uncertainty
makes ±1.2% change in the total flux for 4.5-20.0 MeV and is taken as the uncertainty resulting
from the energy scale.

7.2

Energy resolution

The systematic uncertainty due to energy resolution is obtained based on the difference ofthe energy resolution between data and MC, obtained from the LINAC calibration as shown in
Fig.7.1. The difference of the energy scale (∆σ(Eres )) is 2.53%(E < 4.77M eV )∆σ(Eres ) =1.0%(E > 6.75M eV )interpolation(4.77 ≤ E ≤ 6.75M eV )(7.1)

To estimate the systematic uncertainty of the energy resolution, the predicted energy spectrum was distorted with a shifted energy resolution. The predicted spectrm(F) is calculatedbyZ EmaxF (Ee )R(Ee , Eobs )dEe ,F(Eobs ) =
(7.2)
0

104

Figure 7.1: The difference of the energy resolution between data and MC for the LINAC calibration.

where Ee is the total energy of the recoli electron, F(Ee ) is the recoil energy spectrum, calculated
in equation 3.4, R(Ee , Eobs ) is the detector response function, which takes into account energy
resolution(σ(Ee )) and is described as;(Ee − Eobs )2
1
exp −
.
(7.3)
R(Ee , Eobs ) = pσ(Ee )22πσ(Ee )2

Fig.7.2 shows the energy distribution (top) and the energy resolution function (bottom) obtainedby single electron MC. The black points show the standard deviation of the Gaussian fit, while
the red line shows the fit result;σ(Ee ) =p1(−0.123 + 0.376 Ee + 0.0349Ee ).
Ee

(7.4)

To replace σ(Ee ) of equation 7.3 with σ + (Ee )=σ(Ee )(1 + ∆σ(Eres )) or σ − (Ee )=σ(Ee )(1 −∆σ(Eres )), the F + (Eobs ) or F − (Eobs ) are obtained. Finally, the ratio of F + (Eobs ) or F − (Eobs )to the original F is used as a weighting factor of Yi , and hence the YiEres+ or YiEres− is obtained.
Using this method, the obtained flux uncertainty due to the energy resolution is estimated tobe ± 0.15%.

7.3

8

B spectrum

The uncertainty due to the 8 B spectrum shape is estimated in paper [25], which includes
both experimental and theoretical uncertainties. To estimate this effect, the original spectrum
8
φ(Eν ) is deformed over the range of the spectrum shape uncertainty. Then F B+(or-) is made888with equation 3.4. By using F B+(or-) in equation 7.3, Yi B+ and Yi B− are obtained and the105

SK-IC MC enelf v4

9000
8000
7000
6000
5000
4000
3000
2000

Energy resolution

1000
0
0.3 0
0.275
0.25
0.225
0.2
0.175
0.15
0.125
0.1
0.075
0.05
0

5

10

15

20

25

30

35

40

SK-IV:( -0.0839379 + 0.349061 sqrt(E) + 0.0396896 E) / E

5

10

15

20

25

30
35
40
Total energy (MeV)

Figure 7.2: Top; Reconstruction energy distribution (black) of electron MC simulation with
Gaussian fit (red). Bottom; Black point shows the obtained one standard deviation of theGaussian fit with the fittng result by the polynominal function.

106

deviation from the original Yi is taken. Thus the obtained flux uncertainty due to the 8 B
spectrum shape is ±0.35%.

7.4

Trigger efficiency

The trigger efficiency depends on the vertex position, water transparency, hit PMTs and soon. The systematic uncertainty of the trigger efficiency is estimated by comparing the nickel
calibration data and MC simulation. For 4.0-4.5 and 4.5-5.0 MeV, the difference between data
and MC is 4.5% and 0.37%, respectively. Above 5.0 MeV, trigger efficiency is 100%, so its
uncertainty is negligible.

7.5

Reduction

7.5.1

Fiducial volume cut

The systematic uncertainty of the fiducial volume cut is estimated using the difference ofthe reconstructed vertex from the nickel calibration, between data and MC, which we call the’vertex shift’. Fig.7.3 shows the vertex shift at each position for SK-IV (left) and SK-III (right).
The origin of the arrow shows the nickel source position, while the direction indicates how thereconstructed vertex is shifted. The length of the arrow shows the magnitude of the vertex shift.
The vertex shift for SK-IV is improved compared with that of SK-III. The reason is not fully
understood, but the main contributions could be better linearity of the new front-end electronics
and a more precise timing calibration.
To estimate the systematic uncertainty, the reconstructed vertex positions of solar neutrino
MC events are artificially shifted, following the arrows in Fig.7.3, and the fraction of events
which are rejected by the fiducial volume cut is estimated. Fig.7.4 shows the energy dependenceof the fraction of rejected events. The energy dependence below 5.5 MeV comes from the tightfiducial volume cut, not by the energy dependence of vertex shift. The systematic uncertainty
on the total flux is estimated to be ±0.17%.

7.5.2

Gamma ray cut

The vertex shift and angular resolution difference between data and MC can cause a bias
to the gamma ray cut efficiency. We shift the reconstructed vertex and direction artificially as
shown in Fig.7.5, and estimate the fraction of events which are rejected by gamma ray cut. And
the systematic uncertainty of the gamma ray cut is obtained. The systematic uncertainty on
the total flux is estimated to be ±0.2%.

7.5.3

gv2 − ga2 cut

The systematic uncertainty of the gv2 − ga2 cut is estimated using the difference of the gv2 − ga2
cut efficiency between LINAC data and the correponding MC, as shown in Fig.7.6. The gv2 − ga2
cut systematic uncerteinty on the total flux is estimated to be ±0.3%.

107Fiducial volume sys.

Figure 7.3: The vertex shift which is measured by the nickel calibration for SK-IV (left) and SKIII (right). The origin of the arrow shows the nickel source position and the direction indicatesthe average of the reconstructed position. The length of arrow shows the magnitude of thevertex shift.

0.01

0.008

0.006

0.004

0.002

0
4

6

8

10

12

14

Total electron energy(MeV)
Figure 7.4: The energy dependence of the fraction of events which are rejected by (tight) fiducial
volume cut, due to the vertex shift.

108

5JKHVGF
FKTGEVKQP
1TKIKPCN
FKTGEVKQP

F GHH
UJKHV
5JKHVGFRQUKVKQP
1TKIKPCNRQUKVKQP

Figure 7.5: The method to artificially shift the reconstructed vertex and direction. The shifted
direction is determined based on the original direction and reconstructed direction.

Figure 7.6: The difference of the gv2 − ga2 cut efficiency between data and MC obtained by the
LINAC calibration.

109

Figure 7.7: The difference of the hit pattern cut efficiency between data and MC for the LINAC
calibration.

7.5.4

Hit pattern cut

The systematic uncertainty on the total flux by hit pattern cut is also estimated by thedifference of the cut efficiency between LINAC data and the correspponding MC, as shown in
Fig.7.7. The hit pattern cut systematic uncertainty on the total flux is estimated to be ±0.3%.

7.5.5

Cluster hit event cut

The systematic uncertainty on the total flux by cluster hit event cut is estimated by thedifference of its efficiency between DT data and the correponding MC. DT events taken at theregion where the cluster hit event cut is enforced (6.12) are used. Fig.7.8 shows the obtained
CLIK distribution of the DT data (black) and MC (red). The difference of the cluster hit
event cut efficiency for 5.0-5.5 MeV, 4.5-5.0 MeV and 4.0-4.5 MeV is 0.44%, 0.64% and 1.1%,
respectively. The systematic uncertainty on the total flux is estimated to be 0.3%.

110Figure 7.8: The difference of the CLIK cut efficiency between data and MC obtained by the DT
for 5.0-5.5MeV(left), 4.5-5.0MeV(middle) and 4.0-4.5MeV(right).

Figure 7.9: The time variation of the dead time of the spallation cut.

7.5.6

Spallation cut

The systematic uncertainty coming from the spallation cut is estimated by the difference inthe dead time due to the spallation cut between data a random sample and MC, which takes
into account the position dependence of the dead time function, and is shown in Fig.6.7. Fig.7.9
shows the time variation of the dead time. The spallation cut systematic uncertainty on the
total flux is estimated to be 0.2%.

7.6

Angular resolution

The angular resolution is obtained using the LINAC. The angular resolution is defined as
the angle which includes 68% of the reconstructed directions within the distribution from the
LINAC beam direction. The systematic uncertainty of the angular resolution is obtained fromthe difference of the angular resolution between the LINAC data and the correponding MC,
which is shown in Fig.7.10. The result is comparable to that for SK-III in the lower energy
111

Figure 7.10: The difference of the angular resolution between data and MC from the LINAC
calibration.

region, and slightly worse in the higher energy region, but the same level as that for SK-I.
In the SK-III solar neutrino analysis, the Yiang+ or Yiang− was made by artificially shifting
the reconstructed direction of the solar neutrino MC events. But this estimation method seems
to over-estimate the uncertainty because the difference between Yiang+ or Yiang− and Yi includes
the systematic uncertainty of the reduction steps, which is described in the last section. For SKIV, the estimation of the expected shape of the solar neutrino signal, which takes into accountthe angular resolution systematic uncertainty(sang+or sang−), is used. This method is the sameij
ij
as that for SK-I. The systematic uncertainty on the total flux is estimated to be ±0.36%.

7.7

Background shape

The systematic uncertainty of the background shape prediction is estimated as follows. As
described in section 6.5, the solar angular distribution is fit with a non-flat background shape.
To get the non-flat background shape, not only the distribution of the z-direction, but also the
distribution of the phi direction is fit. To estimate the systematic uncertainty of the backgroundshape, the background shape assuming a flat phi distribution, and taking into account only
the zenith angle distribution, is made. The difference of the fit results from these two types ofbackground shapes are taken as the systematic uncertainty on the total flux, which is 0.1%.

7.8

Signal extraction method

The systematic uncertainty coming from the signal extraction method is estimated by applying the signal extraction to dummy data, which have a known number of signal and background
events. The systematic uncertainty of signal extraction method is estimated to be ±0.7%.

112Source
Energy scale
Energy resolution
8 B spec.

trigger efficiency
Vertex shift
gv2 − ga2

Hit pattern cut
Spallation
Gamma-ray cut
Cluster hit cut
Ang. resolution
Background shape
Signal ext. method
Cross section
Total

Flux(4.5-20.0MeV)

Reference±1.2%

Sec.7.1±0.33%

Sec.7.3[25]±0.17%

Sec.7.5.1±0.3%

Sec.7.5.4±0.2%

Sec.7.5.2±0.36%

Sec.7.6±0.7%

Sec.7.8±0.15%±0.1%

Sec.7.2
Sec.7.4±0.3%

Sec.7.5.3±0.2%

Sec.7.5.6±0.3%

Sec.7.5.5±0.1%

Sec.7.7±0.5%

Sec.7.9[37]±1.7%Table 7.1: Summary of the systematic uncertainty on the total flux. The combined total systamatic uncertainty is ±1.7%.

7.9

Cross section

The systematic uncertainty of the solar neutrino interaction cross section is estimated fromthe paper [37], in which the radiative corrections parameters for neutrino-electron scattering, ρand κ, have 1σ theoretical uncertainties. The errors of ρ and κ are mainly due to the lack ofprecise knowledge of top quark and Higgs mass and QCD origin. The parameters are shifted by±1σ and the flux was obtained. The obtained systematic uncertainty on the total flux is ±0.5%.

7.10

Summary of systematic uncertainty

The systematic uncertainty of the total flux is summarized in Table 7.1. The combined
systematic uncertainty is calculated as the quadratic sum of all components and found to be
1.7%. This is the best value compared with SK-I and III. The spectrum systematic uncertainty
is described in the next section and the day/night systematic uncertainty is described in 9.5 .

7.11

Spectrum systematic uncertainty

The systematic uncertainty of the spectrum shape consists of two components, which areenergy-correlated and energy-uncorrelated errors.

113

0.15

0.1

0.05

0

-0.05

-0.1

-0.15
4

6

8

10

12

14

16

18

20

Wed Nov 7 16:18:53 2012

Figure 7.11: The energy-correlated systematic errors for each energy bin. The red, pink and blue
distributions correspond to systematic errors of the 8 B spectrum shape, the energy resolution
and the absolute energy scale, respectively.

7.11.1

Energy-correlated systematic uncertainty

The energy-correlated systematic uncertainty comes from the absolute energy scale, the energy resolution and the 8 B spectrum shape. This systematic uncertainty is estimated using
equation 7.2, with individually artifically shifted energy scale, energy resolution and 8 B spectrum. The result is shown in Fig.7.11, with these values taken into account in the spectrum
shape analysis and the oscillation analysis.

7.11.2

Energy-uncorrelated systematic uncertainty

Multiple scattering goodness
The systematic uncertainty of the signal extraction using MSG, below 7.5 MeV, is estimatedusing the MSG distribution from LINAC data and MC. Fig.7.12 shows an example of a MSG
distribution of data (black points) and MC (blue), for the 4.8 MeV LINAC (top) and the ratioof the MC to the data for 4.8 MeV (red), 8.8 MeV (black) and 18.6 MeV (blue). This ratio isused as a scaling factor to bring the MC into better agreement with the data. After applying
this ratio to the MC, the signal extraction is applied again. The difference in the number ofsolar neutrino events is taken as the systematic uncertainty. The systematic uncertainty of the
MSG signal extraction is summarized in Table 7.2.
Summary of the energy-uncorrelated systematic error
The energy-uncorrelated energy spectrum uncertainties are summarized in Table 7.2. In thistable, the common offset of the uncertainties is omitted in order to estimate the true effect on

114

0.04
0.03
0.02
0.01
0

0

0.2

0.4
0.6
0.8
1
Multiple Scattering Goodness

1.3
1.2
1.1
1
0.9
0.3

0.4
0.5
0.6
Multiple Scattering Goodness

Figure 7.12: Top; MSG distributions of 4.8 MeV LINAC data (black points) and MC (red).
Bottom; Ratio of the MC to the data for 4.8 MeV (red), 8.8 MeV (black) and 18.6 MeV (blue).

115

Energy (MeV)

4-4.5

4.5-5

5-5.5

5.5-6

6-6.5

6.5-7.5

7.5-8.5

8.5-20

Trig eff
gv2 − ga2
Hit patternGamma Cut
Vertex shift
BG shape
Sig.Ext.
Cross section
MSG

±4.5%
±0.8%
±0.1%
±0.5%
±2.9%
±2.1%
±0.2%
±0.4%

±0.37%
±0.8%
±0.1%
±0.5%
±1.0%
±2.1%
±0.2%
±0.3%

±0.8%
±0.1%
±0.3%
±0.8%
±2.1%
±0.2%
±0.3%

±0.8%
±0.1%
±0.17%
±0.2%
±0.7%
±0.2%
±0.3%

±0.64%±0.1%±0.17%
±0.1%
±0.7%
±0.2%
±1.7%

±0.64%±0.64%±0.1%±0.17%
±0.1%
±0.7%
±0.2%
±1.7%

±0.1%
±0.28%±0.1%±0.17%
±0.1%
±0.7%
±0.2%
±1.7%

±0.1%
±0.28%±0.1%±0.17%
±0.1%
±0.7%
±0.2%
-

Total

±6.0%

±2.6%

±2.5%

±1.2%

±2.0%

±2.1%

±1.9%

Table 7.2: Energy uncorrelated systematic uncertainty on the spectrum shape.
the energy spectrum shape.

116

±0.8%

Chapter 8

Results of the SK-IV solar analysis
In this chapter, the results of the solar neutrino analysis of SK-IV are described. The unitof energy in this chapter is kinetic energy. In order to explicity describe so, it is written as’MeV(kin)’.

8.1

8

B neutrino flux

Fig.8.1 shows the solar angle distribution of the SK-IV final sample, with a recoil electron
kinetic energy range from 4.0 to 19.5 MeV(kin). The horizontal axis shows cosθsun , which isdefined in Fig.6.20 and is divided into 80 bins. The vertical axis shows the event rate in unitsof [Event/day/kton/bin]. The best fit signal and background shapes are also shown in the solidand dash red lines, respectively. The total number of extracted solar neutrino signals is 19,811+220−218 (stat.) ±337(syst.).
By comparing with the total number of expected solar neutrino signals obtained from thesolar neutrino MC, the observed 8 B solar neutrino flux is calculated.
First, the expected number of solar neutrino interactions in SK is calculated as
Z Eνmaxφ(Eν )σtotal (Eν )Ne Tlive dEν ,

(8.1)

0

where Eν is the incident neutrino energy, φ(Eν ) is the solar neutrino flux of 8 B neutrinos forEν , σtotal (Eν ) is the total scattering cross section, Ne is the total number of target electrons
in the ID detector (1.086×1034 electrons) and Tlive is the livetime in units of seconds. For the
8 B neutrino spectrum, the calculated spectrum based on the alpha spectrum measurement by
Winter et al. [25] is used. Using a total flux of 5.25 [×106 /cm2 /s], which is the central value ofthe flux measured by the SNO NC analisys [7], the calculated event rate from equation (8.1) is
3.4×10−3 events/second, for 8 B neutrinos.
Second, the solar neutrino MC is generated. The position is set uniformly and the energy
and direction of the recoil electrons are determined by equations (3.4) and (3.6), respectively.
To reduce the statistical error of the calculation, the generated event rate of the Monte Carlosimulation is 100 per minute, for the full operation time. The generated MC events are processed
through the same analysis procedure described in section 6.3, and the expected number of events
for the assumed flux (5.25×106 /cm2 /s) is found.
Finally, the observed 8 B solar neutrino flux is found to be117

Event/day/kton/bin

0.3
SK-IV 1069.3days 3.898-19.489MeV(kin)0.2

Signal = 19811 +220 -218 (stat.) events
8
B Flux(ES mode) =62
2.34 +/-0.03(stat.) +/-0.04(syst.) x 10 /cm /s
8(with Winter06 B supectrum)

0.1

0
-1

-0.5

0

0.5

1
cosθsunFigure 8.1: The solar angle distribution of 4.0-19.5 MeV(kin) events. The black point show thedata. The solid and dashed red lines show the fit signal and background shapes, respectively.

2.34 ± 0.03(stat.) ± 0.04(±syst.) × 106 cm−2 sec−1 .

8.2

(8.2)

Day/Night flux asymmetry

The day and night times are defined ascosθz ≤ 0cosθz > 0

Day livetime=506.5 days

(8.3)

Night livetime=562.8 days

(8.4)

θz is the solar zenith angle, which is defined in Fig.6.20. Fig.8.2 shows the angular distributionof the day and night times. The observed fluxes areΦday = 2.27 ± 0.04(stat.) ± 0.05(sys.) × 106 cm−2 sec−1
6Φnignt = 2.39 ± 0.04(stat.) ± 0.05(sys.) × 10 cm

−2

−1

sec

(8.5)
.

The straight day/night flux asymmetry ADN (straight) is defined as

ADN (straight) =Φday − Φnight1
2 (Φday + Φnight )= −0.052 ± 0.023(stat.) ± 0.014(sys).
118

(8.6)

Event/day/kton/bin

0.3

day Signal = 9131 +150 -149(stat.) events
8

6

2

day B Flux = 2.27 +/-0.04 (stat.) (x 10 /cm /s)
0.2
night Signal = 10680 +161 -159(stat.) events
8

6

2night B Flux = 2.39 +/-0.04(stat.) (x 10 /cm /s)

0.1

0
-1

-0.5

0

0.5

1
cosθsunFigure 8.2: The solar zenith angle distribution for the day time and the night time.

Fig.8.3 shows the summary of fluxes for each cosz value, which is divided into 10 bins. For
each given oscillation parameter, the time variation during the night time can be predicted. So,
using such a constraint, the day/night asymmetry can be more precisely measured. Details ofsuch a day/night asymmetry is described in section 9.5.

8.3

Energy spectrum

The spectrum of recoil electrons is measured in the same method as the measurement of
the total flux. In the case of the spectrum measurement, the final sample of solar neutrino
candidates is divided according to the reconstructed electron kinetic energy, and then the flux is
calculated using the extended maximum likelihood method described in section 6.5. As describedin section 6.6, the final sample below 7.5 MeV(kin) is divided into three MSG groups and aretaken into account in the flux calculation. Fig.8.4 shows the solar zenith angle distribution for
the three MSG groups. For each MSG group and energy bin, the signal extraction was donewith a given bij (Ekin ,MSG) and sij (Ekin ,MSG), which are probability density functions of the
background and solar neutrino signal, respectively. Table 8.1 shows the observed and expected
rates in each energy bin with units of [event/kton/year]. The error value of the observed rate
is a statistical error only. Fig.8.5 shows the statistical error differences between the original
signal extraction and the signal extraction by taking into account MSG groups. The statisticalerror can be reduced by about 10% at maximum. Fig.8.6 shows the event rate of both solar
neutrino MC (red) and observed solar neutrino events, for each energy bin, and Fig.8.7 shows theenergy spectrum of SK-IV. The vertical axis is [(observed rate)/(expected rate) with the expect
rate=5.25×106 cm−2 s−1 ]. The obtained spectrum, normalized by the expectation, is consistent
with a flat shape. More detailed analysis of the energy spectrum combining other SK phases isdiscussed in section 9.4.2

119

SK-IV Winter6

2

Flux (x10 /cm /s)

2.8

2.6

2.4

2.2

Day
Night

2

All

Day

-1

Night

0

1
cosθz

Figure 8.3: The solar zenith angle dependence of the solar neutrino flux. The error bars show
only statistical errors.

Figure 8.4: The solar zenith angle distribution for each MSG groups for 3.5-5.5 MeV(kin).

120

Events/day/22.5kt/bin

Figure 8.5: Ratio of the statistical errors of the method taking into account the MSG groups
and the original (i.e. w/o MSG) methods. The red and black points are for the positive andnegative errors, respectively.

10

1

-1

10

-2

10

-3

10

-4

10

4

6

8

10

12

14

16

18

kinetic energy of recoil electronΜεΧ(κιν)

Figure 8.6: The event rate of the solar neutrino MC (red) and observed solar neutrino events.
The reduction efficiencies are corrected.

121

Energy(MeV(kin))
3.5 − 4.0
4.0 − 4.5
4.5 − 5.0
5.0 − 5.5
5.5 − 6.0
6.0 − 6.5
6.5 − 7.0
7.0 − 7.5
7.5 − 8.0
8.0 − 8.5
8.5 − 9.0
9.0 − 9.5
9.5 − 10.0
10.0 − 10.5
10.5 − 11.0
11.0 − 11.5
11.5 − 12.0
12.0 − 12.5
12.5 − 13.0
13.0 − 13.5
13.5 − 14.5
14.5 − 15.5
15.5 − 19.5

ALL
−1 ≤ cos θz ≤ 1
102.1+13.6−13.3
75.7+6.5−6.3
85.2+4.3−4.2
69.7+2.6−2.6
58.9+2.0−1.9
55.0+1.9−1.8
48.1+1.7−1.6
40.8+1.5−1.5
37.0+1.4−1.3
29.5+1.2−1.1
24.9+1.0−1.0
18.0+0.9−0.8
15.0+0.8−0.7
10.9+0.6−0.6
8.70+0.55−0.52
6.82+0.47−0.44
4.27+0.37−0.34
2.84+0.30−0.27
2.31+0.27−0.24
1.49+0.22−0.19
1.68+0.23−0.20
0.447+0.133−0.1010.224+0.108−0.073Observed rateDAY
−1 ≤ cos θz ≤ 0
103.0+21.1−20.4
66.3+9.9−9.4
87.5+6.5−6.3
64.0+3.8−3.7
56.1+2.9−2.8
50.5+2.6−2.5
46.4+2.4−2.3
40.5+2.2−2.1
37.9+2.0−1.9
29.6+1.7−1.6
25.4+1.5−1.5
17.6+1.2−1.2
15.7+1.1−1.1
10.6+0.9−0.8
7.52+0.77−0.70
6.40+0.68−0.61
3.39+0.49−0.43
2.90+0.44−0.37
2.59+0.41−0.35
1.42+0.32−0.26
1.79+0.35−0.29
0.439+0.191−0.1280.296+0.176−0.109NIGHT
0 < cos θz ≤ 1
101.4+18.0−17.4
83.1+8.8−8.4
83.3+5.8−5.6
74.7+3.6−3.5
61.5+2.8−2.7
59.1+2.6−2.5
49.6+2.3−2.2
41.2+2.1−2.0
36.2+1.9−1.8
29.4+1.6−1.6
24.5+1.4−1.3
18.3+1.2−1.1
14.3+1.0−1.0
11.1+0.9−0.8
9.79+0.81−0.75
7.21+0.68−0.62
5.13+0.56−0.50
2.78+0.44−0.38
2.03+0.37−0.31
1.55+0.32−0.26
1.55+0.31−0.25
0.457+0.206−0.1410.148+0.146−0.074

Expected rate
8B
hep
196.7
183.0
167.7
153.2
138.0
122.0
106.7
92.2
78.0
65.0
53.4
42.8
33.7
26.0
19.54
14.30
10.19
7.08
4.75
3.10
3.19
1.117
0.460

0.345
0.336
0.325
0.314
0.303
0.287
0.272
0.256
0.238
0.220
0.204
0.185
0.167
0.150
0.133
0.117
0.102
0.087
0.074
0.062
0.092
0.059
0.068

Table 8.1: The observed and expected event rates in each energy bin at 1 AU. The errors are
statistical errors only. The reduction efficiencies are corrected and the expected event rates arefor a flux of 5.25×106 /cm2 /s.

122

Figure 8.7: The energy spectrum of SK-IV. Below 7.5 MeV(kin), signal extraction with MSG is
used.

123

Chapter 9

SK-I,II,III,IV Combined Analysis
9.1

8

B neutrino total flux

The total flux values of 8 B neutrinos obtained from SK-I [6], II [35], III [36] and IV are as
follows;
2.38 ± 0.02(stat.) ± 0.08(sys.) × 106 cm−2 s−1
6−2 −1
2.41 ± 0.05(stat.)+0.16−0.15 (sys.) × 10 cm s
6
−2 −1

2.40 ± 0.04(stat.) ± 0.05(sys.) × 10 cm
62.34 ± 0.03(stat.) ± 0.04(sys.) × 10 cm

SK-I(4.5-19.5MeV(kin))
SK-II(6.5-19.5MeV(kin))s

SK-III(4.5-19.5MeV(kin))−2 −1

SK-IV(4.0-19.5MeV(kin)).

s

For the SK-III flux calculation, the energy dependence in the ν-e differential cross section(equation 3.2) was accidentally eliminated in our previous report [36]. This problem was fixedin this thesis.
The 8 B flux in each SK phase is summarized in Fig.9.1. The vertical axis is in units of[(SK-rate)/MC(unoscillated)], where MC(unoscillated) corresponds to 5.25×106 cm−2 s−1 . The
statistical average flux from the SK-I, II, III, IV results is 0.451±0.007, and the χ2 to this averagevalue is 0.58/3D.O.F. So, the fluxes of all SK phases are consistent.
The combined solar zenith angle distribution is shown in Fig.9.2. The total number ofobserved solar neutrino signals is counted to be about 57,700 events during the SK-I, II, III,IV total livetime of 3904 days. The day/night asymmetry analysis and the oscillation analysis,
which are described in the next sections, can be accomplished with such a large number of solar
neutrino events.

9.2

Energy spectrum

The energy spectra for all phases, with systematic uncertainties, are shown in Fig.9.3. As
described in 6.6, the flux measurement with MSG is used below 7.5 MeV(kin) for the SK-III and
SK-IV spectrum. The trigger efficiency of the 3.5-4.0 MeV(kin) energy bin in the SK-IV phaseis about 86%. By the SK experiment rule, the energy bin in which obtained trigger efficiency
was below 90% should not be used, so we did not use this energy bin in the oscillation analysis.

124

SK-rate/MC(unoscillated)

0.6

0.55
0.5

0.45
0.4

0.35

1

2

3

4
SK phase

Events/bin

Figure 9.1: The 8 B flux in each SK phase.
The vertical axis is in units of ([SKrate)/MC(unoscillated)], where MC(unoscillated) corresponds to 5.25×106 cm−2 s−1 . The solidline shows the statistical averaged flux of SK-I, II, III, IV and the dashed line shows the 1σ error
from the average.

20000

SK-I,II,III,IV 3904 days
SIGNAL = 57721.3 events

15000

10000

5000

0
-1

-0.5

0

0.5

1
cosθsunFigure 9.2: The combined solar zenigh angle distribution from SK-I, II, III, IV with livetime of3904 days.

125Data/MC(unoscillated)

Data/MC(unoscillated)

Winter 06 Spectrum
Blue:Total uncertainty
Red:Energy-uncor. Sys.
: Energy-cor. Sys.

0.7

0.6

0.5

0.4

0.6

0.5

0.4

0.3

0.2

Winter 06 Spectrum
Blue:Total uncertainty
Red:Energy-uncor. Sys.
: Energy-cor. Sys.

0.7

0.3

SK-I Energy Spectrum
Green : Average Flux(4.5-19.5MeV(kin))
862
MC B Flux = 5.25 x 10 /cm /s
4

6

8

10

12

14

16

0.2

18

SK-II Energy Spectrum
Green : Average Flux(6.5-19.5MeV(kin))
862
MC B Flux = 5.25 x 10 /cm /s
4

6

8

10

Winter 06 Spectrum
Blue:Total uncertainty
Red:Energy-uncor. Sys.
: Energy-cor. Sys.

0.7

0.6

0.5

14

16

18

Winter 06 Spectrum
Blue:Total uncertainty
Red:Energy-uncor. Sys.
: Energy-cor. Sys.

0.7
0.6
0.5
0.4

0.4

0.3

0.2

12

kinetic energy(MeV)

Data/MC(unoscillated)

Data/MC(unoscillated)kinetic energy(MeV)

0.3

SK-III Energy Spectrum
Green : Average Flux(4.5-19.5MeV(kin))
862
MC B Flux = 5.25 x 10 /cm /s
4

6

8

10

12

14

16

0.2

18

kinetic energy(MeV)

SK-IV Energy Spectrum
Green : Average Flux(4.0-19.5MeV(kin))
862
MC B Flux = 5.25 x 10 /cm /s

4

6

8

10

12 14 16 18
kinetic energy(MeV)

Figure 9.3: The energy spectrum for each phase, with systematic uncertainty.

126

9.3

Oscillation analysis method

9.3.1

Neutrino oscillation predictions

The predicted energy spectrum which takes into account neutrino oscillations is calculated
as follows. As described in section 2.4, the oscillation parameter sin2 (θ13 ) is fixed to 0.0242,
with a three flavor oscillation analysis used in this analysis.
In the following, two flavor oscillation method is described for simplicity. The survival
probability of the electron neutrino(pe ) is predicted by three steps.
1. From the production point of the solar neutrino to the surface of the Sun.
2. From the surface of the Sun to the surface of the Earth.
3. From the surface of the Earth to the detector.
In practice, since the distance between the Sun and Earth is much larger than the vacuum
oscillation length of the favored parameter ∆m2 /E ∼ 10−9 eV 2 /MeV, the propagation of ν1 andν2 at step 2 can be assumed to be incoherent. In this case, the net electron survival probability
pe can be described by the following equation, using pi (νe which is produced in the core toappear as νi , where i is the mass eigenstate) and pie (νi at the surface of the earth to appear as
νe at the detector).
pe = p1 × p1e + p2 × p2e = 2p1 p1e + 1 − p1 − p1e(9.1)

For the solar portion (step1), the production point distribution is calculated using the standard solar model. The value of p1 is obtained by averaging over the production points. For the
Earth portion (step3), p1e is obtained as a function of neutrino energy and zenith angle of thesun (cosΘz ). For various oscillation parrameter sets, ∆m221 , θ12 , and pe (∆m221 , θ21 , Eν , cosΘz )are obtained.
Second, from the survival probability obtained by equation (9.1), the predicted neutrino
interaction rate, with neutrino oscillations, for the i-th energy bin(Elow,i <E< Ehigh,i ), can be
written asri =Z Ehigh,iElow,idσνµ ,ντdσνe+ (1 − pe )dEe R(Ee , E) pe
dEν φ(Eν )
dE
dEe
dEe
Ee
Eν
Z



Z



,

(9.2)

νe
where R(Ee , E) is the energy resolution function, which is defined in equation(7.3), dσ
dEe anddσνµ ,ντdEeis the difference in cross section of the elastic scattering for νe and νµ,τ of energy Eν ,with a recoil electron energy Ee (shown in Fig.3.2), and φ(Eν ) is the solar neutrino flux as afunction of neutrino energy, which is shown in Fig.3.3 for 8 B neutrinos.
In this oscillation analysis, the zenith angle (cosθz ) of the nighttime (0< cosθz ≤1) is divided
into 1000 bins and 1 bin during the daytime (-1< cosθz ≤0). This binning is defined as the cz bin(cz=1∼1001 bin). With this binning, the predicted observed event rate of 8 B and hep neutrinos
for the i-th energy bin are described as127

livetime fraction

1.4

1.2

1

0.8

0.6

0.4

0.2

0

0

200

400

600

800

1000
cz binFigure 9.4: The livetime fraction in bin cz from 1 to 1000. cz bin = 1 and 1000 corresponds tocosΘz = 0 and 1, respectively.

1001
X

Biosc =

τ (cz) 8 B· ri (cz)
τtot
cz=1

Hiosc

1001
X

=

cz=1

τ (cz) hep· ri (cz),
τtot

(9.3)

where τ (cz) is the livetime of the cz bin, which is obtained from the run profile of the real dataand shown in Fig.9.4 (only for night part, corresponding cz bin from 1st to 1000th bin), and τtot
is the total livetime.

9.3.2

Spectrum shape fitting

The comparison between the observed and predicted spectra is done by fitting their energy
spectra in a χ2 analysis. The χ2 value is calculated for each SK phase individually, because each
SK phase has unique energy correlated and uncorrelated errors, as shown in Fig.9.3.
A simple χ2 caluclation is given byχ2simple =

N
bin
X

(di − βbi − ηhi )2,2
σi,stati=1(9.4)

where Nbin is defined in section 6.5 and is the number of energy bins (21st bin for SK-I, 17th
bin for SK-II and 22nd bin for SK-III,IV). di is the data spectrum term, with statistical error of128

σi,stat and β and η are free parameters to allow for the arbitrary fit of the total neutrino fluxes.
The di , bi and hi in equation 9.4 are as follows;Di
Bi + H i
Bioscbi =Bi + H i
Hiosc
,
hi =
Bi + H i

(9.5)

di =

(9.6)
(9.7)

where Bi and Hi are the predicted rates in case of no neutrino oscillations, which are shownin Table 8.1. So, di is the observed energy sepctrum, which is shown in Fig.9.3, and Biosc andHiosc are the predicted rates, which take into account neutrino oscillations and are calculated inequation (9.3).
Systematic uncertainty treatment
The χ2 value described in equation (9.4) takes into account only statistical errors. In thissubsection, the treatment of the systematic uncertainty, both energy correlated and uncorrelated
are described.
The definition of the χ2 which takes into account the systematic uncertainties is written as
2

χ =

N
bin
X

(di − (βbi + ηhi ) × f (Ei , δB , δS , δR ))22
2+ δB+ δS2 + δR
.
2
σ
i
i=1

(9.8)

The energy correlated systematic uncertainties consist of uncertainties on the absolute energy
scale, the energy resolution and the 8 B spectrum, which are shown in Fig.7.11. Since these
uncertainties directly impact the shape of the SK energy sepctrum, the χ2 must reflect these
possible changes. The energy correlated systematic uncertainties are applied to the spectrum as
multiplicative energy shape factors (f), in order to reflect an actual one sigma shift. However,since the data may better agree with a lesser or greater shift from the one sigma assigned value,
the factors are varied until the χ2 gives a minimum value.
The combined energy shape factor is defined as below;
f (Ei , δB , δS , δR ) = fB (Ei , δB ) × fS (Ei , δS ) × fR (Ei , δR ),(9.9)

where B, S and R denote the 8 B spectrum shape, the absolute energy scale and the energy
resolution, respectively. The energy shape factors (fx (Ei .δx )) are written asfx (Ei , δx ) =1,
1 + δx × ǫ±
x (Ei )(x = B, S, R),(9.10)±
where ǫ±x is the 1σ energy correlated uncertainty and δx is the unit-less parameter which scales2±ǫx . δx is varied until the χ gives the minimum value for a given βm and ηm .
The σi in equation (9.8) is an uncertainty which combines the statistical and energy uncorrelated systematic uncertainty by quadrature with these errors. To take into account the energy
uncorrelated systematic uncertainty on the total flux , a Taylor expansion to the second order129

around the minimum βm and ηm is used. The χ2 using the Taylor expansion terms can be
written as2χ (β, η) = χ2m (βm , ηm ) +β − βmη − ηm!TC0

β − βmη − ηm

!

(9.11)

where C0 is the curvature matrix which is defined as

 2
bi
bi hi
N
bin
X
2
2
σi 
 σi
.
C0 =
2
bi hi
σi2

i=1

(9.12)hi
σi2

The C0 is inversely proportional to the uncertainties of the fitting parameters β and η.
Therefore it is possible to scale the curvature matrix by αsys to allow the addition of the totalenergy uncorrelated systematic uncertainty to the total flux scale factors. The αsys is defined asσ02,
2
σ02 + σsyswhere

χ2α (β, η) = χ2m (βm , ηm ) +β − βmη − ηmαsys =σ02 =and so, χ2α can be wriiten as
!T

N
bin
X

1

,
σ2
i=2 stat,iαsys · C0β − βmη − ηm

(9.13)!

.

(9.14)

Finally, the χ2 for the energy spectrum shape is expressed as22
χ2spec(β, η) = Min(χ2α (β, η, δB , δS , δR ) + δB
+ δS2 + δR),

(9.15)

with the number of fitting parameters at five; β, η, δB , δS and δR .

9.3.3

Time variation fitting

The time variation analysis is a different approach from the spectrum fit analysis, as it cantake into account the earth matter effects.
The likelihood function that was used for extracting the solar signal (equation (6.15)) was
modified using a scaling function which reflects the deviation from the average number of events,
arising from time variations. Using a suitable substitution for the zenith angle, to take into
account its time dependence, the oscillation rate prediction of earth (step3 in section 9.3.1) canbe used to form the ratio ri (cz)/riave , where ri is a predicted solar zenith angle time variation fora given cz bin and energy bin i, which is obtained from equation (9.2), where riave is the averaged
variation of ri . The calculated time variation (ri ) for ∆m221 =4.84× 10−5 eV2 and sin2 θ12 =0.314
is shown in Fig.9.5.
The modified likelihood function can be written as follows.
P

Lt.v. = e−(i Bi +S)

N
ni
bin Y
Y

(Bi · bij + S · Yi · sij · ri (cz)/riave ).

i=1 j=1

130(9.16)

0.425

4.5-5 MeV
6-5.5 MeV
5.5-6 MeV
6-6.5 MeV
6.5-7 MeV

Oscillated MC/MC

Oscillated MC/MC

0.43
7-7.5 MeV
7.5-8 MeV
8-8.5 MeV
8.5-9 MeV
9-9.5 MeV

0.42

0.45
0.445
0.44
0.435

9.5-10 MeV
10-10.5 MeV
10.5-11 MeV
11-11.5 MeV
11.5-12 MeV
12-12.5 MeV

12.5-13 MeV
13-13.5 MeV
13.5-14.5 MeV
14.5-15.5 MeV
15.5-19.5 MeV

0.43
0.425

0.415

0.42
0.41

0.415
0.41

0.405

0.405
0.4
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.4
-1

0.8
1
Cosθz

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8
1
Cosθz

Figure 9.5: The predicted solar zenith angle time variation assuming a typical LMA solution,∆m221 =4.84× 10−5 eV2 and sin2 θ12 =0.314.

If we set ri (cz)/riave =1, this likelihood funciton becomes the original one. The effect of the earth
matter oscillations can be tested by comparing the modified likelihood (Lt.v ) with the original
likelihood. ∆χ2t.v. is obtained from the calculation of∆χ2t.v. = −2(logL − logLt.v. ).

(9.17)

χ2SK = χ2spec + χ2t.v. .

(9.18)

Then, the full SK χ2 becomes

9.3.4

Flux constrained fitting

In the spectrum shape method, the 8 B and hep rate factors β and η are free to take any
value, in order to minimize the difference between data and MC oscillated predictions. For therate constrained fit, the rate factors β and η are constrained to the total rate of all active flavorsof neutrinos on Earth. For the results presented later, we adopted the NC flux measurement by
SNO.
In the flux constrained fitting the terms (β − 1)2 /σφ8 B and (η − 1)2 /σφhep are added to theχ2 of equation (9.18). σφ8 B =0.04 and σφhep =2.0 correspond to the SNO NC 8 B flux uncertainty
and the hep flux limit(φhep <2.9SSM) [7].

9.4

Oscillation analysis results

9.4.1

Oscillation parameter constraints from SK

The results of the two χ2 fitting methods, the spectrum shape fitting (equation 9.15) and time
variation fitting (equation 9.17), are described. After minimizing the free parameters describedabove, the χ2 value is calculated for each oscillation parameter set, i.e. sin2 θ12 and ∆m221 . A
χ2 map is obtained on the plane of these oscillation parameters, and the minimum χ2 point isfound. From the minimum χ2 point, allowed/excluded regions were obtained using a standard∆χ2 method. The minimum χ2 point is obtained at

131

∆χ

2

8 3σ
6
4 2σ
2
1σ

2

10

∆m in eV

2

-3

10

-4

-5

10

-6

10

-7

10

-8

10

1σ

-9

10

-4

10

-3

10

-2

10

-1

10

1

10

2σ

3σ

2

10 2 4 6 822tan (Θ)∆χ

Figure 9.6: The excluded region from the SK combined analysis with KamLAND 3σ allowed
region(purple). The χ2 contours are shown for 1σ steps. The brown region corresponds to the3σ excluded region.

sin2 (θ12 ), SK = 0.339+0.028−0.024∆m221 , SK=

−52
4.69+1.80−0.83 × 10 eV .

(9.19)(9.20)

Fig.9.6 shows the obtaned SK-I,II,III,IV combined excluded/allowed regions by the spectrum
shape fitting and the time variation fitting methods. SK data now excludes SMA and solution
above the 5σ level and LOW solution above the 4.5σ level. Fig.9.7 shows the SK flux constrained
allowed regions (green), showing the LMA region from KamLAND (blue) and the SK and
KamLAND combined result (purple). The best fit parameters obtained by the SK combined fit
with the flux constrained aresin2 (θ12 ),SK = 0.322+0.024−0.020∆m221 ,SK=−52
7.49+0.2−0.20 × 10 eV .

132

(9.21)(9.22)

∆χ

2

8 3σ
6
4 2σ
2
1σ

2

10

∆m in eV

2

-3

10

-4

-5

10

-6

10

-7

10

-8

10

1σ

-9

10

-4

10

-3

10

-2

10

-1

10

1

10

2σ

3σ

2

10 2 4 6 822tan (Θ)∆χ

Figure 9.7: The allowed region from SK combined analysis with the flux constrained. The χ2
contours are plotted from the 1σ to 5σ levels. The green painted region corresponds to the 3σ
allowed region, the blue painted region corresponds to the KamLAND 3σ allowed region andthe purple painted region corresponds to the all SK+KamLAND 3σ allowed regions.

133

oscillation par’sχ2 (d.o.f=81)

Φ8 B [/cm2 /s]

Φhep [/cm2 /s]

LMA MSW osc. prediction1

79.02

5.34·10617.4·103

LMA MSW osc. prediction2

76.54

5.47·10616.0·103

flat probability

75.29

5.25·10614.7·103

flat probability, dσ ratio

75.23

5.53·10614.4·103

Table 9.1: The comparison of χ2 value, best fit 8 B neutrino flux and best fit hep neutrino flux
for each prediction of given oscillation parameters.

9.4.2

Results of the energy spectrum fit

Fig.9.8 shows the observed energy spectrum and the predicted energy spectrum distortions
under four different assumptions. In the spectrum shape fitting, we calculated the χ2 value foreach SK phase, and then added the χ2 ’s of all of them to give a final combined χ2 . This plotshows the SK combined energy spectrum for descriptive purposes, because it is not statisticallycorrect to add spectra which have different systematic errors, and so this plot itself is not usedin the oscillation analysis.
The assumptions of the predicted energy spectrum are as follows;
1. LMA MSW oscillation prediction1 (blue):
Oscillation parameters are sin2 θ12 =0.304 and ∆m221 =7.44×10−5 eV 2 , and MSW effect istaken into account.
2. LMA MSW oscillation prediction2 (green):
Oscillation parameters are sin2 θ12 =0.310 and ∆m221 =4.86×10−5 eV 2 , and MSW effect istaken into account.
3. Flat probability (black):
Assuming that the unoscillated 8 B spectrum is decreased by a factor independent of energy. In the spectrum analysis, the ’flat’ corresponds to the case that neutrino oscillation
is assumed as energy independent. To obtain the flat probability shape, the pe in theequation(9.1) is set to constant value, and calculate the χ2 value. The best fit factor fromthe SK energy spectrum is obtained by pe =0.317. But because of the different shape ofthe differential cross section between νe and νµ,τ (dσνe /dEe vs. dσνµ,τ /dEe ), the energy
spectrum is slightly distorted.
4. Flat probability, dσ ratio (purple);
Assuming that dσνµ,τ /dEe has the same energy dependence as dσνe /dEe .
The χ2 values and the best fit 8 B and hep flux (Φ8 B , Φhep ) for each assumption are summurized in Table 9.1. The fourth assumption, ’the flat probability, dσ ratio’, gives the minimum χ2among them, and is favored over the MSW oscillation assumptions by 1.1-1.9σ.

134Figure 9.8: The SK combined energy spectrum (red point are statistical errors only) with thepredicted distorted shape of sin2 θ=0.304 and ∆m2 =7.44×10−5 eV 2 (blue), sin2 θ=0.310 and∆m2 =4.86×10−5 eV 2 (green) and flat shapes (best quadratic fit (black) and flat prob of dσ
ratio (purple)).

9.5

Day/Night asymmetry analysis

Due to the matter effect from the Earth, the observed solar neutrio flux is affected while thesun is below the horizon. This effect is called ’the day/night effect’ and it increases the νe flavor
content as shown in Fig.9.5.
To evaluate the day/night effect, two methods are used, one is called the straight day/nightflux asymmetry, defined in equation (8.7)and the other uses a day/night amplitude fitting method
described later. In this subsection, these two methods are explained and the results are described.

9.5.1

Straight day/night asymmetry

A straight day/night asymmetry method separately measures the flux during the day(φday )and night(φnight ) times and forms the day/night asymmetry ADN . The definitions of φday ,φnight
and ADN were described in section 8.2
The values of ADN for SK-I,II,III and IV were found to be−2.1 ± 2.0(stat.) ± 1.3(sys.)[%]−6.3 ± 4.2(stat.) ± 3.7(sys.)[%]−5.6 ± 3.1(stat.) ± 1.3(sys.)[%]−5.2 ± 2.3(stat.) ± 1.4(sys.)[%]

135

SK-I(4.5-19.5MeV(kin))
SK-II(6.5-19.5MeV(kin))
SK-III(4.5-19.5MeV(kin))
SK-IV(4.0-19.5MeV(kin)).

The negative sign on each value corresponds to the fact that the night flux is larger thanday flux. Taking just the statistically weighted average of those values,
ADN , SK combined = −4.0 ± 1.3(stat.)[%](9.23)

is obtained. Only the combined statistical uncertainty is quoted here, because a more sophisticated method is necessary to combine the systematic uncertainties. The method is described in
the next subsection.

9.5.2

Day/night amplitude fit

Day/night amplitude fit method
The day/night amplitude fit method [38] is a more sophisticated method which employsa maximum likelihood fit to the amplitude of the time variation, constrained by the predicted
manner of the variation. A modified likelihood function from the original signal extraction (6.15)
is defined as follows:
N
ni
bin Y
P
Y
−( i Bi +S)

L=e(Bi · bij + S · Yi · sij · zi (α, t)),(9.24)

i=1 j=1

where zi (α, t) is a new signal factor which includes any time variable parameters, such as solar
zenith angle (day/night effect).
To take into account the time variation of the solar zenith angle, the predicted solar zenith
angle variation rate of the i-th energy bin (ri (cz)), which is shown in Fig.9.5, is used. From
the ri (cz) distribution and livetime distribution (l(cz)), which is shown in Fig.9.4, the livetime
averaged total rate(riav ), day rate(riday ), night rate(rinight ) and predicted day/night asymmetry(Ai ) of the i-th energy bin are calculated as follows;
riday =Zdayri (cz) · l(cz)d(cz) ,
riave

=

Ai

=

Z
ri (cz) · l(cz)d(cz)rinight =nightZri (cz) · l(cz)d(cz)allriday − rinight

1/2(riday + rinight).

(9.25)

From the day livetime (LD ) and the night livetime (LN ), the livetime asymmetry (LDN ) and theeffective asymmetry parameter (ai ) are also calculated. The LDN and ai are defined as follows;
LD − LN
1/2(LD + LN )11
ai = Ai × LDN .
2
2

LDN =

(9.26)(9.27)

Then, the zi (α, t) which was introduced in equation (9.24) is defined aszi (α, t) =1 + α((1 + ai )ri (t)/riav − 1),
1 + αai
136(9.28)

where α is a free parameter which is called the day/night scaling factor. The signal (S), thebackgrounds (Bi ) and the α which give the maximum likelihood is computed and for the resulting
α and σα , which is a statistical uncertainty of the α value, the day/night asymmetry is calculatedby
ADN = Ai × α ± Ai × σα (stat.).

(9.29)

This day/night amplitude method fit depends on the assumed shape of ri (t), and the predicted day/night asymmetry value Ai also depends on the oscillation parameters (sin2 θ, ∆m2 ).
For this day/night amplitude method, events which have energy above 4.5 MeV(kin) for SK-I,III
and IV and above 6.5 MeV(kin) for SK-II were used.
Systematic uncertainty
Almost all the systematic uncertainties which were described in chapter 7 can be canceled
for the day/night asymmetry analysis because the difference between such short time scales isdiscussed. However, there still remain some systematic uncertainties for the day/night analysis,
especially coming from direction dependence of the signal and background.
• Direction dependence of the energy scale:

The direction dependence of the energy scale may cause a systematic shift in the day/night
asymmetry. To quantify the direction dependence of the energy scale, DT data and thecorresponding MC were compared. For this purpose, a polynomial fit to the detector zenith
angle dependence of the energy scale was made. To translate this function to a direction
dependence of the energy scale systematic uncertainties, solar neutrino MC event energies
were scaled based on this function. Fig.9.9 shows a fit result of the direction dependence ofthe energy scale for solar neutrino events for SK-IV, which varies by 0.3% from downward
to upward going events. The fitted function was then applied to all events while runningthe unbinned amplitude fit, and the 0.08% change in the day/night asymmetry taken
as the systematic uncertainty for SK-IV. The systematic uncertainty method was donefor each SK phase, and obtained to be 0.8% for SK-I and II and 0.2% for SK-III. The
systematic uncertainty for SK-III and IV is lower than that for SK-I and II because of theintroduction of the position dependence of the water transparency to the MC, which was
described in detail in section 5.4.

• Background shape:

The systematic uncertainty from the background shape should be considered. The methodof the estimation is similar to that for the flux uncertainty which is described in section
7.7, comparing the day/night amplitude factor between the corresponding two backgroundshapes and the difference taken as the systematic uncertainty. The total SK-IV uncertainty
is estimated as 0.6%, and is the largest contribution to the day/night systematic uncertainty.

• Gamma ray cut:Because of the tight fiducial volume cut used for 4.5-5.0 MeV(kin) and the poswall dependence of the lower energy gamma ray cut (equation (6.10)) for SK-III and SK-IV, there is acontribution to the systematic uncertainty from the gamma ray cut. Following the method137

Figure 9.9: The zenith angle dependence of the energy scale for SK-IV.

energy scale
background shape
gamma ray cut
Total

SK-I

SK-II

SK-III

SK-IV

±0.8%

±0.8%

±0.2%

±0.08%

-

-

±0.2%

±0.1%

±0.6%
±1.0%

±0.6%
±1.0%

±0.6%
±0.7%

±0.6%
±0.7%

Table 9.2: Systematic uncertainties for the day/night asymmetry.

of estimation of systematic uncertainty described in chapter 7, the estimated systematic
uncertainty is ±0.1% for SK-IV and ±0.2% for SK-III.
The total systematic uncertainties for each SK phase are summarized in Table 9.2
Day/night amplitude fit results
As described above, when we obtain the day/night scaling factor(α) and the day/night
asymmetry(ADN , which is defined in equation 9.29), the predicted solar zenith angle time
variation (ri ) is used. Because ri has dependence on oscillation parameters, the day/night
asymmetry also has oscillation parameters dependence. By assuming sin2 (θ12 ) = 0.339 and∆m221 = 4.69 × 10−5 eV 2 , the day/night asymmetries resulting from the day/night amplitude fit
for each SK phase are obtained as follows;−2.0 ± 1.7(stat.) ± 1.0(sys.)[%]−4.3 ± 3.8(stat.) ± 1.0(sys.)[%]−4.3 ± 2.7(stat.) ± 0.7(sys.)[%]−2.8 ± 1.9(stat.) ± 0.7(sys.)[%]

138

SK-I(4.5-19.5MeV(kin))
SK-II(6.5-19.5MeV(kin))
SK-III(4.5-19.5MeV(kin))
SK-IV(4.5-19.5MeV(kin)),Figure 9.10: Energy dependence of the fit day/night asymmetry for ∆m2 =4.84·10−5 eV 2 andsin2 (θ12 )=0.314. The red line shows the expected energy dependence of the day/night asymmetry.

and the SK combined day/night amplitude is obtained as
ADN , SK combined = −2.8 ± 1.1(stat.) ± 0.5(syst.)[%].

(9.30)

Fig.9.10 shows the energy dependence of the day/night asymmetry along with the predicted
day/night asymmetries (Ai , red line). The error bar shows only the statistical uncertainty.
Fig.9.11 shows the ∆m221 dependence of the predicted and observed day/night asymmetry forsin2 (θ12 ) fixed to 0.314. The allowed ∆m221 range of the solar experiments (green band) andKamLAND (purple band) are overlaid. Fig.9.12 shows the sin2 (θ12 ) dependence of the day/night
asymmetry when ∆m221 is fixed to 4.69×10−5 eV 2 . The day/night asymmetry is sensitive to∆m212 , but gives no information about sin2 (θ12 ).

139

Figure 9.11: Dependence of the day/night asymmetry on ∆m2 for sin2 (θ12 )=0.314. The lightgray band shows the statistical error and the blue band shows the statistical + systematic error.
The red line shows the expected day night asymmetry and the green and purple bands show the1σ allowed region for the solar experiments and the KamLAND ∆m2 ,respectively.

Figure 9.12: Dependence of the day/night asymmetry on sin2 (θ12 ) for ∆m2 =4.88·10−5 eV 2 . The
light gray band shows the statistical error and the blue band shows the statistical + systematic
error. The red line shows the expected day night asymmetry.

140

Chapter 10

Discussion
In this chapter, the results of the neutrino oscillation parameters(θ12 , ∆m221 ) obtained by
SK are compared with that of other experiments. Also, combined results with them are alsopresented.

10.1

Comparison/Combine with SNO result

The SNO collaboration fit the neutrino signal based on an average Φ8 B for the day and nightday
time, a νe survival probability as a function of neutrino energy(Eν ), during the day(Pee(Eν ))
night
and the night(Pee (Eν )), and a day/night asymmetry survival probability(Aee (Eν )), which isdefined byAee (Eν ) = 2

night
day
Pee
(Eν ) − Pee
(Eν )

night
day
Pee
(Eν ) + Pee
(Eν )

.

(10.1)day
The Pee(Eν ) and Aee (Eν ) are parameterized byday
Pee(Eν ) = c0 + c1 (Eν [MeV] − 10) + c2 (Eν [MeV] − 10)2

Aee (Eν ) = a0 + a1 (Eν [MeV] − 10),(10.2)(10.3)

where c0 , c1 and c2 are parameters defining the νe survival probability and a0 and a1 areparameters defining the relative difference between the night and day νe survival probability [7].
For the combined fit to all SNO phases (I, II, III) using the maximum likelihood technique,each obtained parameters are summarized in Table 10.1.
day
Fig.10.1 shows the Pee(Eν ) and Aee (Eν ) with RMS spread, taking into account the parameter uncertainties and correlations between each parameter. The red band represents the result
from the maximum likelihood fit, and the blue band shows the cross check from the Bayesianfit.
Then a χ2 was defined to compare between the oscillation parameters from the predicted
neutrino oscillation hypothesis and the SNO observations, and contours of oscillation parameters
were obtained and seen in Fig.10.2. Comparing Fig.10.2 and Fig.9.7, the results of SK and SNO
are consistent with each other. The SK and SNO results are then combined by adding their

141

Φ8 B

Best fit

+Stat

-Stat

+Syst

-Syst

5.25

0.16

0.16

0.11

0.13

c0

0.317

0.016

0.016

0.009

0.009

c1

0.00390.00650.00670.00450.0045c2-0.00100.00290.00290.00140.0016

a0

0.046

0.031

0.031

0.014

0.013

a1

-0.016

0.025

0.025

0.010

0.011

[×106 /cm2 /s]

Table 10.1: results from the maximum likelihood fit

χ2 ’s. At this time, the total flux of 8 B flux is a common parameter. Fig.10.3 shows the obtained
combined contours of SK-I,II,III,IV and SNO-I,II,III. The best fit oscillation parameters are
sin2 (θ12 ),SK,SNO = 0.317+0.014−0.016∆m221 ,SK,SNO=−52
4.85+1.44−0.54 × 10 eV .

(10.4)(10.5)

The contour of KamLAND is also shown in the figure. There is a slight tension in the ∆m221
between KamLAND and the SK/SNO combined result.

10.2

Combined with other solar neutrino experiments

The results from the Chlorine experiment (Homestake), Gallium experiments (SAGE, Gallex/GNO)
and Borexino (only 7 Be neutrino flux) are added to the χ2 of the SK and SNO combined oscillation analysis. The χ2 values of those experiments are based on two equivalent ways of defining
the χ2 function [39] which is described by
N (=3)

χ2rate =

X

2 −1
expt
theor
(Rnexpt − Rntheor )[σnm
] (Rm
− Rm
),

(10.6)

n,m=1

where N is the number of experiments, in this case N=3 (Cl, Ga, Borexino experiment). The2
is a matrix of squared uncertainties which are sums of the correlated and uncorrelated
σnm
uncertainties among experiments, Rexpt is the observed rate and Rtheor is the SSM predicted
event rate. Fig.10.4 shows the contour plot of all solar combined allowed regions (green at 3σ
allowed region) and with KamLAND (blue at 3σ allowed region). The best fit values and 1σvalues of SK, SNO, radiochemical and Borexino combined analysis are
sin2 (θ12 ),all solar = 0.310+0.014−0.015∆m221 ,all solar=−52
4.86+1.44−0.52 × 10 eV(10.7)(10.8)

Finally, combining the results of all solar neutrino experiments and the KamLAND reactor
neutrinos,

142

day
Figure 10.1: The observed Pee(Eν ) (top) and Aee (Eν ) (bottom) with the RMS spread in the
SNO combined result. The red band represents the results from the maximum likelihood fit andthe blue band represents the cross check by the Bayesian fit.

143

x10
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1

2

∆m21 in eV

2

∆χ

2

8 3σ
6
4 2σ
2

-5 1σ
sin2(Θ12)=0.309+0.040-0.029sin2(Θ12)=0.321+0.017-0.0272sin (Θ12)=0.309±0.016-52∆m221=(7.49+0.21-0.19) 10 eV-52∆m221=(4.93+2.72-1.03) 10 eV-52∆m221=(7.48+0.21
)
10
eV
-0.19

1σ

0.1

0.2

0.3

0.4

0.5
2
sin (Θ12)2σ

3σ

2 4 6 82∆χFigure 10.2: The allowed region from the SNO analysis and KamLAND. The χ2 contours areplotted from 1σ to 5σ levels. The green painted region corresponds to the SK+SNO 3σ allowed
region, the blue painted region corresponds to the KamLAND 3σ allowed region and the purple
painted region corresponds to the SK+SNO+KamLAND 3σ allowed region.

144

sin2 (θ12 ),all solar,KamLAND = 0.304+0.013−0.013−5
2∆m221 ,all solar, KamLAND = 7.44+0.20−0.19 × 10 eV

10.3(10.9)(10.10)The future prospects of solar neutrino analysis

As described in Sec.9.4.2 and Sec.9.5, the energy spectrum analysis with the combined SKI,II,III,IV data showed that the flat probability spectrum was favored at 1.1-1.9σ over MSWLMA oscillation prediction shapes and the day/night asymmetry analysis showed a non-zerovalue with 2.3σ significance. In this section, the future prospects of the energy spectrum analysis
and day/night asymmetry analysis are described.

10.3.1

Future Energy spectrum analysis

To observe the energy spectrum distortion due to the MSW effect, it is necessary to achievea lower energy threshold(<4.0MeV(kin)), and to reduce the statistical uncertainty in the lower
energy region. For this purpose, (1) instllation of another data acquisition system and (2)reducing the background and suppressing statistical uncertainty due to background fluctuation
are considered.
• Additional data acquisition system
The current trigger threshold for data acquisition is limited by the performance of disk
access. We are now preparing another data acquisition system, which is called WIT (Wideband Intelligent Trigger). Fig.10.5 shows a schematic view of the data acquisition system
including WIT. Details of the WIT system are described in [40]. In the WIT module, fast
online event reconstruction is performed to apply fiducial volume cut. This can reduces
backgrounds, in paticular the radioactive background originating in the PMT glass andthe fiberglass backings of the PMT enclosures. That is, the WIT system can take lower
energy data without introducing more backgrounds. After installation of WIT, the energy
threshold will only be limited by the event reconstruction performance, and a 3.0MeV(kin)threshold will be possible.
• Reducing the backgroundAs shown in Fig.6.15, there are still backgrounds remaining near the wall. The main
backgrounds would be a radioactive source from the wall, but we haven’t completely
understood the origin of those backgrounds. If we understand those backgrounds andremove them, the sensitivity of energy spectrum analysis would be improved.

10.3.2Future day/night asymmetry analysis

Due to improvements in the solar neutrino MC, the day/night asymmetry result is statistically limited, so by taking more data, the significance will improve. For 10 more years of data,
the significance of a non-zero day/night asymmetry will be more than 3σ.
Another prospect for day/night asymmetry analysis is using solar neutrinos at HyperKamiokande (HK)[41]. The HK detector is proposed as a next generation underground water
145

x10
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1

2

∆m21 in eV

2

∆χ

2

8 3σ
6
4 2σ
2

-5 1σ
sin2(Θ12)=0.309+0.040-0.029sin2(Θ12)=0.317+0.014-0.0162sin (Θ12)=0.309±0.014-52∆m221=(7.49+0.21-0.19) 10 eV-52∆m221=(4.85+1.44-0.54) 10 eV-52∆m221=(7.44+0.21-0.19) 10 eV

1σ

0.1

0.2

0.3

0.4

0.5
2
sin (Θ12)2σ

3σ

2 4 6 82∆χFigure 10.3: The allowed region from the SK and SNO combined analysis and KamLAND. The
χ2 contours are plotted from 1σ to 5σ levels. The green painted region corresponds to the
SK+SNO 3σ allowed region, the blue painted region corresponds to the KamLAND 3σ allowed
region and the purple painted region corresponds to the SK+SNO+KamLAND 3σ allowed
region.

146

x10
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1

2

∆m21 in eV

2

∆χ

2

8 3σ
6
4 2σ
2

-5 1σ
sin2(Θ12)=0.309+0.040-0.029sin2(Θ12)=0.310+0.014-0.0152sin (Θ12)=0.304±0.013-52∆m221=(7.49+0.21-0.19) 10 eV-52∆m221=(4.86+1.44-0.52) 10 eV-52∆m221=(7.44+0.20
)
10
eV
-0.19

1σ

0.1

0.2

0.3

0.4

0.5
2
sin (Θ12)2σ

3σ

2 4 6 82∆χFigure 10.4: The allowed region from all solar combined analysis and KamLAND. The χ2
contours are plotted from 1σ to 5σ levels. The green painted region corresponds to the all solar
3σ allowed region, the blue painted region corresponds to the KamLAND 3σ allowed region andthe purple painted region corresponds to the all solar+KamLAND 3σ allowed region.

147

Figure 10.5: Schematic view of the data acquisition system including the WIT system.

Figure 10.6: Schematic view of the proposed HK detector.

Cherenkov detector and R&D studies are on going. Fig.10.6 shows the schematic view of HK.
The proposed total mass of the detector will be 0.99 million metric tons, which is about 20
times larger than SK, and the photo-cathode coverage will be about 20%; similar coverage as
SK-II. The HK observation will help the precise measurement of day/night asymmetry withhuge statistics.

148

Chapter 11

Conclusion
The solar neutrino analysis of the fourth phase of the Super-Kamiokande(SK-IV) detector
was carried out. With the improvement of the water circulation system, lowering the backgroundlevel in the central region was achieved and enabled a lowering of the energy threshold of theanalysis down to 4.0 MeV(kinetic energy) from 4.5MeV(kin). The position dependence of thewater transparency was introduced to the detector simulations for SK-III and IV, and a newparameter to evaluate multiple scattering goodness was introduced in the data analysis. Becauseof those improvements, statistic and systematic uncertainties were greatly reduced.
The systematic error of the total flux is estimated as ±1.7% in the SK-IV analysis, which is8
reduced compared with SK-I (+3.5%−3.2% ) and SK-III (2.2%). The B flux obtained from the data inthe energy range of 4.0 to 19.5 MeV(kinetic energy) in SK-IV is
2.34 ± 0.03(Stat.) ± 0.04(Syst.) × 106 /cm2 /s,(11.1)

which is consistent with the SK-I,II,III results.
The neutrino oscillation analysis combining all SK phases and other solar neutrino experiments (Chlorine, Gallium, SNO and Borexino), give the best fit oscillation parameters as
sin2 (θ12 ),all solar = 0.310+0.014−0.015∆m221 ,all solar

=

+1.44
4.86−0.52
× 10−5 eV 2 .

(11.2)(11.3)The sin2 (θ12 ) values of the solar global analysis and the KamLAND reactor experiment areconsistent, but there is a slight tension in the ∆m221 they measure. Combining the results of all
solar neutrino experiments and the KamLAND reactor experiment, the oscillation parameters
are obtained assin2 (θ12 ),all solar,KamLAND = 0.304+0.013−0.013−5
2∆m221 ,all solar, KamLAND = 7.44+0.20−0.19 × 10 eV .

The χ2 of the spectrum fit to all SK data with MSW predictions for the two best fit oscillation
parameters described above and for an assumption of a flat shape (constant reduction factor
independent of energy) are 76.54, 79.2 and 75.29 for 81 d.o.f, respectively. The flat shape isfavored at the 1.1-1.9σ level compared with the best fit oscillation parameter predictions.
149

By introducing the day/night amplitude fit method and with improved systematic uncertainties, the SK combined day/night amplitude was obtained as ADN (=(day-night)/ 21 (day+night))= -2.8 ± 1.1(stat.) ±0.5(syst.) [%], and a hint of a non-zero day/night flux asymmetry, at the2.3σ level, is seen.

150

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