Precise Measurement of Solar Neutrinos
withSuper-Kamiokande III

Motoyasu Ikeda
December 16 2009

Abstract
New solar neutrino measurements with the Super-Kamiokande detector are reported. The main
motivation of this thesis is to observe the spectrum distortion of solar neutrinos caused by thematter eﬀect of neutrino oscillation in the Sun (MSW eﬀect).
The data for this thesis were taken between August 2006 and August 2008, during the third
phase of Super-Kamiokande (SK-III). Two neutrino samples are used in this thesis. The ﬁrst
one with total electron energy between 6.5 and 20MeV has a total livetime of 547.9 days. The
second, with total electron energy between 4.5 and 6.5MeV has a total livetime of 298.2 days
after rejecting high background periods caused by radioactive impurities accidentally injected
into the detector.
With improved detector calibrations, a full detector simulation, and analysis methods, thesystematic uncertainty on the total neutrino ﬂux is estimated to be ±2.3%, which is about
two thirds of the systematic uncertainty in the ﬁrst phase of Super-Kamiokande (SK-I). The
observed 8 B solar ﬂux in the 5.0 to 20MeV electron energy region is 2.28± 0.04 (stat.) ± 0.05
(sys.) ×106 cm−2 sec−1 , in agreement with previous measurements. The day-night asymmetry
is measured to be ADN = −0.057 ± 0.031(stat.)±0.013(sys.). In the 4.5-5.0 MeV region, the6−2−1 and is consistent with the ﬂux in the 5.0-20MeV
observed ﬂux is 2.14+0.56−0.54 (stat) ×10 cm sec
region.
A global oscillation analysis is carried out using SK-I, II, and III, and is combined with theresults of other solar neutrino experiments. The best-ﬁt oscillation parameters are obtained with
+1.21
2
−5
2
the world’s best accuracy as sin2 θ12 = 0.29+0.024−0.011 and ∆m12 = 6.03−1.67 × 10 eV . Combinedwith KamLAND result, the best-ﬁt oscillation parameters are found to be sin2 θ12 = 0.304+0.017−0.016−5 eV 2 . This parameter region corresponds to a 8 B ﬂux of 5.08+0.10 ×and ∆m212 = 7.59+0.12
×
10
−0.39
−0.07
106 cm−2 sec−1 .
The χ2 value of spectrum ﬁt with the solar plus KamLAND best-ﬁt prediction is 26.7/20d.o.f.
which is slightly better than 27.7/20d.o.f. with a ﬂat shape. Although, this result is notstatistically signiﬁcant, it is estimated that the improved calibration and analysis methods will
give a sensitivity of 3σ level discovery of the spectrum distortion within a few years, together
with re-analysis of the SK-I data.

Acknowledgements
First of all, I would like to express my great gratitude to Prof. Y. Takeuchi for giving me theexcellent opportunity of studying the solar neutrino at Super-Kamiokande. This thesis would
never exist without his support and encouragement.
I would like to appreciate Prof. Y. Suzuki for giving me the opportunity of studying in the
Super-Kamiokande experiment.
I would like to extend my gratitude to LOWE members, Prof. M. Nakahata, Prof. Y.
Koshio, Prof. A. Takeda, Prof. H. Sekiya, Prof. S. Yamada, Prof. H. Watanabe, Prof. M.
Sakuda Prof. H. Ishino Prof. Y. Fukuda Prof. M. Vagins, Prof. M. Smy, Prof. K. Martens,
Prof. C. Shaomin, T. Iida, K. Ueno, T. Yokozawa, K. Bays, Andrew, Jordan, Dr. J. P. Cravens,
Dr. L. Marti, Dr. J. Schuemann, H. Zhang, and B. Yang.
I would like to thank Kamkoka members who encouraged me for all the time, Prof. K. Abe,
Prof. Y. Hayato, Prof. J. Kameda, Prof. K. Kobayashi, Prof. M. Miura, Prof. S. Mine, Prof. S.
Moriyama, Prof. S. Nakayama, Prof. Y. Obayashi, Prof. H. Ogawa, Prof. M. Shiozawa, Prof.
S. Tasaka, Prof. M. Yamashita, Dr. A. Minamino, Dr. Y. Takenaga, Dr. G. Mitsuka, Dr. H.
Nishino, Dr. R. Wendell, Dr. M. Litos, Dr. J. L. Raaf, Dr. O. Simard, Dr. K. Hiraide, Dr. A.
Minamino, K. Ueshima, C. Ishihara, N. Okazaki, D. Ikeda, T. Tanaka, Y. Furuse, Y. Idehara,
D. Motoki T.F. McLachlan, S. Hazama, Y. Nakajima, Y. Yokosawa, Y. Kozuma, T. Hokama,
H. Nishiie, A. Shinozaki, K. Iyogi, Maggie, Patrick, Y. Heng, P. Mijakowski, M. Dziomba, K.
Connolly, and Prof. C. K. Jung.
Finally, I would like to express my greatest appreciation and gratitude to my family, relatives,
all my friends, and K. Nagaoka.

1

Contents
1 Introduction

4

2 Physics Background
2.1 Solar neutrino and the Standard Solar Model . . . . . . . . . . . . . . . . . . . .
2.2 Energy spectrum of 8 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Neutrino Oscillation of Solar Neutrinos . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Solar Neutrino Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Motivation and strategies of this thesis . . . . . . . . . . . . . . . . . . . . . . . .

5
5
10
12
14
16

3 Super-Kamiokande Detector
3.1 Detector outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 20-inch PMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Data acquisition (DAQ) system . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Water puriﬁcation system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20
20
20
21
27

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

30
30
32
32

5 Simulation
5.1 Outline of detector simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 New modeling of water condition . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Tunable input parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37
37
39
41

6 Calibration
6.1 Outline of detector calibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Timing calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Water Transparency Measurement . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4 TBA tuning and Q.E. measurement by Ni calibration . . . . . . . . . . . . . . .
6.5 DT calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6 LINAC Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.7 Energy scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.8 Energy resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.9 Angular resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43
43
44
51
52
53
54
57
63
63

2

7 Data Analysis
7.1 Monitoring of Radon level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Run deﬁnition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Noise reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4 Reduction for solar ν analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5 Summary of Reduction step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67
67
67
69
72
86

8 Signal extraction
8.1 Solar angle ﬁtting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 How to get Yi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89
89
91

9 Systematic Uncertainties
9.1 Energy scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2 Energy resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3 8 B spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4 Angular resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.5 Vertex shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.6 Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.7 Spallation cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.8 Gamma ray cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.9 Background shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.10 Signal extraction method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.11 Cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.12 Further quality cut for the lowest energy region . . . . . . . . . . . . . . . . . . .
9.13 Summary of systematic uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . .

94
94
94
95
95
95
96
96
97
97
97
99
99
99

10 Result102

11 Oscillation Analysis
111
11.1 Spectrum ﬁt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
11.2 Time-Variation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
11.3 Oscillation constraint from SK . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
12 Discussion
116
12.1 Global oscillation analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
12.2 Comparison of Winter and Ortiz 8 B spectrum . . . . . . . . . . . . . . . . . . . . 119
12.3 Comparison with other results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
12.4 Treat reduction uncertainties as energy correlated uncertainties . . . . . . . . . . 122
12.5 Sensitivity to upturn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
13 Conclusion

128

A Glossary129

3Chapter 1

Introduction
The Sun is the source of life, the source of neutrinos. The number of solar neutrinos passing
through our body is 6.6 × 1010 per cm2 per sec. Since the beginning of 1960s, many experiments
have succsessfully observed the neutrino signal from the Sun.
Until 1990s, all the results reported from the solar neutrino experiments showed that theobserved solar neutrino ﬂux was signiﬁcantly smaller than the ﬂux calculated by the Standard
Solar Model (SSM). This conﬂict between experiments and the SSM was called ”solar neutrino
problem”. In 2001, the high statistical measurement done by Super-Kamiokande in Japan, which
is a water Cherenkov detector with the ﬁducial volume of 22.5kton, together with SNO experiment in Canada, which can measure the all types of active neutrinos from the Sun, answeredthat the solar neutrino problem can be understood by considering the eﬀect of the neutrino
oscillation.
The neutrino oscillation was at ﬁrst studied by Z.Maki, M.Nakagawa, S.Sakata, V.Gribov
and B.Pontecorvo [1]. Their main subject at that time was the neutrino oscillation in vacuum.
As Wolfenstein, Mikheyev and Smirnov discussed the neutrino oscillation in matter [2, 3], it
was found that the adiabatic transition of neutrinos can occur under some special conditions inmatter, such as the Sun and the Earth. This is call ’MSW eﬀect’. By taking into account theMSW eﬀect, the results of all the solar neutrino observations are consistent with the predictionof SSM, and the experimental uncertainties of the solar neutrino ﬂux (8 B neturino) has beenreaching the size of theoretical uncertainty. However, the direct conﬁrmation of the MSW eﬀect
has not been done.
From this standpoint, this thesis is focused on the direct direct veriﬁcation of the MSW
eﬀect in the Sun. This is done by a precise measurement of energy spectrum of 8 B neturinos.
Since the oscillation probability of electron neutrinos has energy dependence due to the MSW
eﬀect, the spectrum is expected to be distorted from the original shape of 8 B neturinos.
In this thesis, the results of solar neutrino analysis using Super-Kamiokande detector arereported after the 547.9 days of observation from August 2006 to August 2008 (SK-III). In
Chapter 2, introductions of SSM, the neutrino oscillation, and the solar neutrino experiments
are presented. The strategies to observe the energy spectrum is also explained in Chapter 2. In
Chapter 3, an overview of Super-Kamiokande experiment is given. From Chapter 4 to 9, themethod of solar neutrino data analysis in SK-III is explained. The results of data analysis andoscillation analysis with Super-Kamiokande are presented in Chapter 10 and 11. Finally, thediscussion and the conclusion is given in Chapter 12.

4

Chapter 2

Physics BackgroundIn this chapter, an overview of solar models, the neutrino oscillation and solar neutrino
experiments are shown. In Section 2.1, the introduction of Standard Solar Model will be given,and current problems of the theory will be shown. After a brief explanation of 8 B neutrino
spectrum in Section 2.2, a short discussion of neutrino oscillation will be given in Section 2.3.
Results of all solar neutrino experiments are summarized in Section 2.4. The main motivationand strategies of this thesis is explained in the last section.

2.1

Solar neutrino and the Standard Solar Model

How does the Sun shine? (J.N.Bahcall) This is the ﬁrst question in his book ”Neutrino
Astrophysics” [4]. To answer this quite astronomical question, we had to wait until the theoryof relativity and the quantum mechanics come out. It was in 1939 that Hans Bethe ﬁrst discussed
that the nuclear fusion is the source of the energy produced in a star including the Sun[6].

2.1.1

Nuclear reactions in the Sun

The nuclear fusion reaction can be written as:
4p −→ α + 2e+ + 2νe(2.1)

This reaction releases an energy of 26.7 MeV. Equation 2.1 is a form of a net reaction which proceeds via two diﬀerent reaction systems; proton-proton chain (pp chain) and Carbon-NitrogenOxygen cycle (CNO cycle) which are shown in Figure 2.1 and 2.2. Here, neutrinos are generated
by the following reactions;
In pp chain:p + p −→ 2 H + e+ + νe

(pp)

(2.2)

−

p + e + p −→ H + νe

(pep)

(2.3)

Be + e− −→ 7 Li + νe

(7 Be)

(2.4)

8

( B)

(2.5)

(hep)

(2.6)

7

2

8
3

∗

B −→ Be + e + νe
8+

He + p −→ α + e+ + νe

5

Figure 2.1: The pp-chain reactions

In CNO cycle13

N −→ 13 C + e+ + νe

(2.7)

15

O −→

N + e + νe

(2.8)

F −→ 17 O + e+ + νe

(2.9)

17

15

+

The CNO sycle is predicted to contribute about 2% to the total solar luminosity, and other
98% contribution is due to pp chain [5].

2.1.2

Overview of the Standard Solar Model

The Standard Solar Model (SSM) predicts the solar neutrino ﬂux which has been developed
by J.N.Bahcall, who took over Bethe’s calculation. Some of the input parameters of SSM arenuclear reaction cross section, the solar luminosity, the solar age, elemental abundances, radiative
opacities. There are recent solar models which are not based on SSMs. One of them is provided
by Turck-Chieze et al. [7]. Their calculation is based on the standard theory of stellar evolution
which is tuned especially for the Sun using seismic measurement (called the seismic model).
The seismic model predicts the neutrino ﬂuxes which agree with the SSM prediction within
their uncertainty ∼ 10%. In this thesis, the BP04 SSM [8] is used to calculate the solar neutrino
ﬂux. Figure 2.3 shows the expected energy spectrum of solar neutrinos. Our results will be also
compared to BP05(OP) and BP05(AGS,OP) [9] to see our sensitivity to the diﬀerent SSMs. To
understand diﬀerence of the SSMs, some key aspects for the SSM are explained in remaining
part of this section. Then, in the next section, a current problem of SSM will be explained.
S-factor (Cross section factor)
The energy dependence on the nuclear fusion cross section is deﬁned by a conventional
form [4]:S(E)σ(E) =exp(−2πη)(2.10)
E
6

Figure 2.2: The CNO-cycle reactions

Figure 2.3: The solar neutrino energy spectrum predicted by BP04SSM.

7

wheree2(2.11)~v
E is the total energy of the interaction, Z1 , Z2 are the atomic numbers of interacting particles, and v is the relative velocity of the incoming particle. 1/E is called the geometricalfactor which is proportional to the the De Broglie wavelength squared ( πλ2 ∝ 1/E ). The
exp(−2πη) is the probability factor of tunnelling through a Coulomb potential barrier iscalled Gamow penetration factor. The function S(E) carries a pure intrinsic property of
the nuclear interaction which varies smoothly in the absence of resonances. The value ofS(E) at zero energy is known as the cross section factor, S0 , which is measured experimentally. The 8 B neutrino ﬂux ϕ(8 B) has a dependence on the S-facotrs such asη = Z1 Z2−2.6 −0.4 0.81 1.0
ϕ(8 B) ∝ S11
S33 S34 S17(2.12)

where the notation of the S-factor is listed in Table 2.1
Reaction

S-factor

1 H (p, e+ ν ) 2 H
e
3 He (3 He, 2p) 4 He

S11

3 He (4 He, γ) 7 Be

S34

7 Be (p, γ) 8 B

S17

S33

Table 2.1: Notation of S-factor
Radiative opacity
The radiative opacity plays a key role in the SSM since the photon radiation is the main
contribution of the energy transport in the central part of the Sun. The calculation of the
opacity depends on the chemical composition and on the modeling of the atomic reactions.
Heavy metal abundance
The initial mass ratio of elements heavier than helium (Z) relative to hydrogen (X), Z/X
is a very important input parameter in the calculation of SSM. The fractional abundancesof each element determines the stellar opacity, which is closely related to the neutrino
ﬂuxes.
Helioseismology
Helioseismology studies how the wave oscillation, particularly acoustic pressure wave (pmode), propagates in the Sun. The calculation of SSMs can be checked by the comparison
with the seismic measurements of the depth of the convection zone, sound speed, anddensity in the Sun.

2.1.3

Current problems of the solar models

In 2005, the input parameters such as S-factor, Opacity, and surface abundances of heavy
element(Z)are updated [10]. The largest variation was found in the surface abundances which
is about 35% lower than the previous compilation [11]. The variation comes from diﬀerent8Figure 2.4: Relative sound speed diﬀerences, δc/c = (cheli −cSSM )/cSSM , and relative densities,δρ/ρ, between solar models and helioseismological results from Michelson Doppler Imager data
[9]. The solid line shows the calculation by BP04SSM which is used in this thesis. BS05(OP)uses the previous version of the abundances with the reﬁned opacity calculation done by the
Opacity Project [13]. BS05(AGS,OPAL) uses the new abundances with the previously used
opacity calculation done by OPAL group [14]. BS05(AGS,OP) uses the new abundances withthe opacity of OP group.

improvements and one of them is installing 3D solar atmosphere modeling. The new abundances
are in better agreement with the gaseous medium between stars in our neighborhood of thegalaxy, and the agreement between photospheric and meteoritic abundances becomes very good.
However, the sound speed and the density calculated by the new SSM conﬂicts with the seismic
measurement. On the other hand, if the previous abundances [12] are used, the agreement
between SSM calculation and the seismic measurement is very good as shown in the Figure 2.4
Considering this situation, the two set of SSMs were published; one includes all the latest
information (hereafter BS05(AGS,OP)), and the other one includes all updates except for theabundances (hereafter BS05(OP)). Speciﬁcally, BS05(AGS,OP) uses Z/X = 0.0165 (so called”low Z model”) while BS05(OP) uses Z/X = 0.0229 (so called ”high Z model”). OP means thatthe latest version of the opacity calculation which is done by Opacity Project is used [13]. This
conﬂict between the SSM calculation and the helioseismology measurement has not been solved
yet, and is under discussion [15].

What can we say from solar neutrino experiments?
It is suggested that a precise measurement of 8 B neutrino ﬂux may tell what the real abundance
9

Neutrino type

BP04

BS05(GS98)

BS05(AGS05)

pp

5.94

5.99

6.06

pep

1.40

1.42

1.45

7 Be

4.86

4.84

4.38

8B

5.79

5.69

4.59

hep

7.88

7.93

8.25

N

5,71

3.07

2.01

O

5.03

3.33

1.47

F

5.91

5.84

3.31

Table 2.2: Total ﬂux for each neutrino type predicted by the SSMs. This table presents thepredicted ﬂuxes, in units of 1010 (pp), 109 (7 Be), 108 (pep, 13 N, 15 O), 106 (8 B, 17 F) , and 103 (hep)cm−2 s−1
SSM

+σ(%)

−σ(%)

BP04

23

23

BS05(OP)

17.3

14.7

BS05(AGS,OP)

12.7

11.3

Table 2.3: The uncertainty of the SSMs prediction on the 8 B ﬂux.
is. Since the dependence of the abundances on the 8 B neutrino ﬂux ϕ(8 B) is given by( )1.3Z
8
ϕ( B) ∝
X

(2.13)

which is relatively large dependence, the 8 B neutrino ﬂux value predicted by BS05(AGS,OP)
is smaller than BS05(OP) by ∼ 30%. Therefore, if experiments can measure the ﬂux within
5%, then the result will improve the understanding of the solar interior [10]. This is one of the
motivations of this thesis.

2.1.4

Solar Neutrinos predicted by SSM

In this section, some basic results from the SSM prediction are shown. Table 2.2 shows
the total ﬂux for each neutrino source predicted by the diﬀerent versions of the SSM. andthe uncertainties of the 8 B neutrino ﬂux predictions are shown in Table 2.3. The electron
density proﬁle and production point of each neutrino source are shown in the Figure 2.5 and
2.6, respectively.

2.2

Energy spectrum of 8 B

In this thesis, I used the neutrino energy spectrum from the β + decay of 8 B calculatedby Ortiz et.al.[17]. The neutrino spectrum can be obtained from the measurement of energy
10

&KUVCPEG'CTVJ4CFKWU

EQUΘ\Figure 2.5: Upper: Electron density calculated by SSM. Lower: Electron density in the Earth(PREM) [16] which is used in the calculation of neutrino oscillation in the Earth.

Figure 2.6: Production point for each solar neutrino. Horizontal axis shows the distance fromthe center of the Sun in unit of the solar radius. Vertial axis shows the solar neutrino ﬂux as afunction of the distance.

11+βFGEC[α decay

Figure 2.7: Decay scheme of 8 B.

spectrum of the α particle from 8 Be which is the daughter nucleus of 8 B (see Figure 2.7). In theirestimation of uncertainty, however, a theoretical uncertainty to calculate the neutrino energy
from the alpha spectrum which are considered in [18] is not fully taken into account. Thus
the uncertainty of 8 B spectrum from reference [18] is used. In 2006, Winter obtained the 8 B
spectrum after a very precise measurement of the alpha spectrum. Because of the improvement
on both the measurement and the calculation method, the uncertainty of the spectrum becomes
smaller. The Winter spectrum and the diﬀerence between Ortiz and Winter are shown in Figure
2.8. Thus, I also studied the eﬀect of the uncertainty of Winter to the uncertainty of the total
ﬂux of our measurement for future analysis improvements. The detail of estimation of systematic
uncertainty will be discussed in Chapter 9.

2.3

Neutrino Oscillation of Solar Neutrinos

The neutrino oscillation is the only experimental observation which cannot be explained by
the standard model of elementary particle physics. A brief explanation of the essence of theneutrino oscillation is given in this section.

2.3.1

Neutrino Oscillation in vacuum

For simplicity, let us consider the neutrino oscillation between only two ﬂavors, for example
νe and νx . The mixing of the two-neutrino oscillation in vacuum is
)
) (
)(
(
cos θ sin θνeν1(2.14)=− sin θ cos θ
νx
ν2

12

Figure 2.8: 8 B neutrino energy spectrum [19]. Upper plot shows the neutrino energy spectrum,and lower plot shows the comparison with the Ortiz spectrum [17]. Dashed lines show the ratio
between the Ortiz spectrum and ±1σ experimental uncertainty, to the Winter spectrum. Blackband represents the ±1σ experimental uncertainty of the Winter spectrum.

where θ is called the mixing angle. This ﬂavor mixing gives an oscillating survival probability
for a neutrino with energy E(M eV ) traveling a distance L(m) as
)
(
L
2
2
(2.15)
pe→x = px→e = sin 2θ sin π
Loscwhere Losc is deﬁned by
Losc ≡πE
1.27 ∆m2(2.16)and ∆m2 = m22 − m21 (eV2 ) is the squared mass diﬀerence of mass eigenstate 1 and 2.

2.3.2

Neutrino Oscillation in matter

In matter, neutrinos feel the potential of weak interaction. While νe has a chance of bothcharged current interaction and neutral current interaction, the other type of neutrinos have
only neutral current interaction. The Hamiltonian H in the two neutrino case can be expressedin the following form, omitting the common multiple factor.
H = Hv + HM(
) (− cos 2θ sin 2θ
V
∆m2
=
+
4E
sin 2θ cos 2θ
0

0
0

)

(2.17)(2.18)

where Hv is the Hamiltonian in vacuum, and the HM is that of in matter. V is the potential of
the charged current interaction,√
V = 2GF Ne(2.19)
13

where GF is Fermi coupling constant and Ne is the electron density in the matter. By putting
the Hamiltonian to the Schroedinger equation, the eigenvector is given as
)(
)
(
) (
νe
ν1M
cos θMsin θM=(2.20)− sin θM cos θMνxν2M
where ν1M and ν2M are mass eigenstates of neutrinos in matter, and θM is the mixing angle inmatter which is∆m2
E sin 2θ,(2.21)
tan 2θM =2
2V + ∆m
E cos 2θThe survival probability can be obtained by replacing θ in equation 2.15 with θM , and Loscwith the oscillation length in matter LM which is
√

LM =
1.27

2.3.3

π
2

(2.22)2∆m
2
2(2V + ∆m
E cos 2θ) + ( E sin 2θ)The eﬀect of neutrino oscillation

As described in [20], in the case of solar neutrino propagation, matter eﬀects in the Sun andthe Earth are taken into account. The calculation of the survival probability of a solar neutrino
at our detector has in principle 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, and 3)
from the surface of the Earth to the detector. In practice, since we already know from theprevious experimental results that the distance between the Sun and the Earth is much lager
than the vacuum oscillation length of the favored parameter ∆m2 /E > ∼ 10−9 eV 2 /M eV , thepropagation of ν1 and ν2 at 2) can be assumed to be incoherent. This makes the calculation
more simple, so that we only need the probability of νe to be produced in the core of the Sun
to appear as ν1 (or ν2 ), which is written as p1 (or p2 ), and the probability of ν1 (or ν2 ) at thesurface of the Earth to be observed as νe in the detector, which is written as p1e (or p2e ). The
total survival probability of the νe , pe is then obtained bype = p1 × p1e + p2 × p2e = 2p1 p1e + 1 − p1 − p1e(2.23)

where p1 and p1e are numerically calculated, andp2 = 1 − p1p2e = 1 − p1e
The details can be found in [21]2.4

Solar Neutrino Experiment

In this section, introductions of the solar neutrino experiments (except for the Super-Kamiokande)
are given. A summary of all solar neutrino experimental results is shown at the end of this section.

14

2.4.1

Radiochemical experiment

Cl target experiment
The ﬁrst experiment which tried to observe solar neutrinos is the chlorine experiment in
Homestake, USA. The neutrino target was 2.2 × 1030 atoms (133 tons) of 37 Cl in the form of
3.8 × 105 liters (615 tons) of liquid perchloroethylene, C2 Cl4 . The tank was located at theHomestake mine at 1478m depth(4100m of water equivalent (m.w.e)). The experiment period
is from 1967 to 1994 [22]. The neutrino rate is measured byνe +37 Cl → e− +37 Ar (threshold 814keV)(2.24)The 37 Ar atoms produced by solar neutrino are extracted from the tank by purging it with He
gas with 95% eﬃciency. With half life(τ1/2 ) of 34.8 days, the decay of 37 Ar through orbital
electron capture produces Auger electrons. The number of produced 37 Ar is obtained fromcount of Auger electrons in the proportional counter. Given the threshold value(=814keV), 8 B
neutrinos contribute about 77% of predicted event rate.
Ga experiment
GALLEX(1991-1997)/GNO(1998-2003) [23, 24] in Gran Sasso, Italy and SAGE(1900-2001)[25] in Baksan, Russia use 71 Ga as the neutrino target. The detectors of GALLEX/GNO andSAGE locate at 3800m.w.e. and 4700 m.w.e. underground, respectively. The target volumes of
GALLEX/GNO and SAGE are 30tons(GNO used 60 tons since 1991) and 60 tons. The neutrino
rate is measured byνe +71 Ga → e− +71 Ge (threshold 233keV)(2.25)
Like the Cl experiment the 71 Ge atoms are chemically extracted and introduced to a proportionalcounter to count the Auger electrons (half live of 71 Ge is 11.43 days). The low threshold enables
to observe all the sources of solar neutrinos including the pp neutrinos.

2.4.2

Kamiokande

Kamiokande (1987-1995)[26] is a water Cherenkov detector with 3kton of pure water in
Kamioka mine, Japan. The solar neutrinos are detected using the reaction:
ν + e− → ν + e−

(2.26)

where the angular correlation between the incoming neutrino and the recoil electron is verystrong. The Cherenkov photons emitted by the recoil electron are detected by 20-inch photo multiplier tubes. The ﬁrst real-time observation of solar neutrino was carried out by Kamiokande
showing that the direction of recoil electron is pointing in the direction of the Sun. This observation principle is basically the same as in Super-Kamiokande.

2.4.3

Borexino

Borexino is currently running low background liquid scintillator detector since 2007 [27].
The detector is located at 3800m.w.e. in Gran Sasso, Italy. The ﬁducial volume is 87.9 ton.
The solar neutrinos are detected by means of their elastic scattering of electrons. The light yield15

of the scintillator which is measured as 500 p.e./MeV enables the real-time detection of 7 Be
neutrinos, mainly. The neutrino rate can be obtained after statistical subtraction of radioactive
background.

2.4.4

SNO

The Sudbury Neutrino Observatory (SNO, 1999-2006) [28, 29, 30] uses 1000 tons of D2 O
contained in a spherical vessel. The detector is located at 6010m.w.e. in Sudbury, Canada. SNO
mearsures 8 B neutrinos via the reactions
Charged Current(CC) :νe + d → p + p + e − 1.442M eV(2.27)

Neutral Current(NC) :νx + d → p + n + νx − 2.224M eV(2.28)

Elastic Scattering(ES) :−−νx + e → νx + e(2.29)

where x = e, µ, τ . SNO has 3 phases of the experiment, and each phase has diﬀerent methodof the neutron tagging. In Phase I, only pure D2 O is contained, and the neutrons are captured
by D2 O with the emission of 6.25MeV γ which is used to tag the NC event [28]. In Phase II,
2tons of NaCl were added to enhance both the neutron capture eﬃciency and the total γ energy(8.6MeV) [29]. In Phase III, an array of 36 strings of proportional counters ﬁlled with 3 He are
installed to detect the neutrons [30]. Since CC is sensitive only to νe and NC is sensitive to alltypes of neutrinos, SNO can actually measure how many fraction of νe oscillate to νx by taking
ratio of CC rate to NC rate, which is the most signiﬁcant result from SNO.

2.4.5

Summary of solar neutrino experimental results

Figure 2.9 summarize the results of solar neutrino experiment. The neutrino oscillation is nottaken into account for the SSM predictions in that ﬁgure. If the neutrino oscillation is considered,then the SSM and the experimental results agree very well as shown in Figure 2.10 In the Figures,the unit of SNU for Ga, Cl experiment stands for ”Solar Neutrino Unit” which is deﬁned as 10−36
captures per target atom per second. To obtain the expectation ﬂux with neutrino oscillation in
Figure 2.10, the favored Large Mixing Angle (LMA) area (∆m2 = 6 × 10−5 eV 2 , tan2 θ12 = 0.4)
is assumed.

2.5

Motivation and strategies of this thesis

The biggest remaining problem is to ﬁnd the MSW eﬀect in solar neutrinos. Figure 2.11
shows the survival probability of νe as a function of neutrino energy. Super-Kamioande canobserve the energy spectrum in terms of the total electron energy which is deposited by a recoil
electron. Figure 2.12 shows the spectrum observed in the ﬁrst phase of Super-Kamiokande
experiment (SK-I) and predicted spectrum shapes with diﬀerent oscillation parameters. As
shown in Figure 2.12, it is still diﬃcult to decide whether the distorted spectrum is correct or
not because the sizes of the statistical uncertainty in lower energy region and the systematic
uncertainty in higher energy region are large. Likewise, no other experiment has observed the
distorted spectrum successfully. Therefore, the ﬁnal goal of this thesis is to observe the distorted
spectrum. The following improvements are necessary to observe the spectrum distortion as soon
as possible.
16

Figure 2.9: Comparison between various experimental results and SSM. The neutrino oscillation
is not taken into account.

Figure 2.10: Comparison between various experimental results and SSM. The neutrino oscillation
is taken into account.

17

5WTXKXCN2TQDCDKNKV[2 νGGψν

VCPθ

∆OG8

0GWVTKPQGPGTI[=/G8?
Figure 2.11: Survival probability of electron neutrino with the solar best ﬁt 2 ﬂavor oscillation
parameters.

㧗‫ޓ‬5-+FC[
'PGTI[EQTTGNCVGFU[U

6QVCN'NGEVTQP'PGTI[=/G8?

Figure 2.12: Observed energy spectrum/SSM. The color histograms show expected value fordiﬀerent oscillation parameters, and the gray band shows the energy correlated systematic uncertainty.

18

'ZRGEVGFTCVGYKVJQUEKNNCVKQPYKVJQWVQUEKNNCVKQP










6QVCNGNGEVTQPGPGTI[=/G8?

Figure 2.13: Expected energy spectrum after 5 years observation. The black histogram showsthe expect value with expected statistical uncertainty with 70% background reduction below 5.5
MeV region. The blue band shows half size of energy correlated uncertainty as SK-I.

First, the number of background must be reduced to make the statistical uncertainty smaller
especially in the lower energy region (<5.5MeV). This is because the size of statistical uncertainty
is approximately the size of the square root of (statistical uncertainty of background)2 plus
(statistical uncertainty of signal)2 and the statistical uncertainty of background is dominant for
the low-energy region. The numerical goal is to reduce 70% of background from SK-I level inthe lower-energy region. It was known from radioactive measurement of detector water that
40% of remaining low energy background around 5MeV was due to radioactive impurities in thewater during SK-I, thus 40% reduction was expected because the water puriﬁcation system andcirculation system had been upgraded since SK-I. The other 60% of background was considered
to be miss-reconstructed events which originally occurred in the edge of the detector but werereconstructed inside the ﬁducial volume. Such miss-reconstructed events could be reduced byhalf if new reconstruction tools and reduction tools are developed.
Second, the energy correlated systematic uncertainty must be reduced by 50% compared
to SK-I. To reduce the systematic uncertainty, the most important thing is to understand thedetector response. Thus, careful calibrations and tuning of the detector simulation are key tomake the reduction possible. Another strategy is to update the theoretical calculation of 8 B
spectrum as already discussed in Section 2.2.
Figure 2.13 shows predicted energy spectrum after 5 years observation with reduced background and systematic uncertainty. By comparing with Figure 2.12, we can see the upturn ofthe distorted spectrum with the reduced size of uncertainties. In this thesis, because the total
livetime is about one third of SK-I, it is diﬃcult to conﬁrm the upturn statistically, however,these improvements which will be discussed in the following chapters must set the foundation
for the future observation of the MSW eﬀect in the Sun.
19

Chapter 3

Super-Kamiokande Detector
3.1

Detector outline

Super-Kamiokande is a cylindrical water Cherenkov detector with 50kton of ultra-pure water.
The diameter of the detector is 39.3m, and the height is 41.4m. The detector location is 1000m
underground (2700 m.w.e.) in the Kamioka mine, Gifu Pref., Japan, The latitude and longitude
are 36◦ 25’ N and 137◦ 18’ E, respectively. The downward-going cosmic-ray muon rate is about
2Hz which is 10−5 times smaller than that at the ground level. The outside 2m of the detector iscalled the Outer Detector (OD) corresponding 18kton of volume, and the inside 32kton is called
the Inner Detector (ID). ID and OD are separated by plastic black sheet and stainless steel
structure on which ID and OD photo-multiplier tubes (PMTs) are assembled. The purpose of
OD is to veto events from outside rock and reject cosmic-ray muon events. The ﬁducial volume
is 22.5kton, 2m inside from the ID wall.
Figure 3.1 is a brief description of the detector in each phase of the experiment. As shownin the ﬁgure, SK has three phases of the experiment which are called SK-I, II, and III1 Afterthe full reconstruction in 2006, 11129 20-inch PMTs in ID, and 1885 8-inch PMTs in OD are
installed. The photodetector coverage is 40% in ID. The detail information about the detector
can be found in [31].
The update items from SK-I(or II) are• ID PMTs are covered by FRP and acrylic case.
• Water puriﬁcation system is upgraded (see the section 3.4)
• Detector calibration is improved.
• OD is segmented by Tyvek which improved the Cosmic-ray muon identiﬁcation (see Figure
3.9).

3.2

20-inch PMT

R3600-06 HAMAMATSU PMT are used for ID PMT. Figure 3.2 shows a schematic view of
a 20-inch PMT and the bleeder circuit. In Figure 3.3, the basic speciﬁcations of those PMTs
are listed. Figure 3.4 shows basic distributions measured by HAMAMATSU [32].
1

SK-IV has started since Sep.6 2008 with new electronics and DAQ system.

20

Figure 3.1: Time-line of each phase of Super-Kamioande experiment.

2

Figure 3.5 shows the transparency of the acrylic cover for photons with normal incidence
in water. The transparency is more than 96% for longer wavelength than 350nm, which isreasonably good considering the quantum eﬃciency of 20-inch PMT (see Figure 3.4).

3.3

Data acquisition (DAQ) system

Figure 3.6 shows a schematic view of the DAQ system. The signal from each ID PMT is sent
to each channel of front-end electronics called Analog Timing Module (ATM) [33, 34] througha 70m coaxial cable. The ATM is developed based on the TKO standard [35]. 12 PMTs areconnected to one ATM, which records and digitizes the integrated charge and the arrival timing
information with a 12-bit ADC. Each ATM channel has 2 buﬀer channels (a and b) to process
successive events, such as a muon and the decay electron after the muon. Thus, one ATM has
24 channels (0a to 11b). The ATM characteristics is summarized in Table 3.1
The analog input of the ATM is made following the block giagram which is shown in Figure
3.7. If a hit exceed a threshold (0.25p.e.), the signal charge is integrated at Charge to Analog
Converter (QAC) with a 400 nsec gate, and Time to Analog Converter (TAC) starts to integrate
charge with a constant current to measure the time interval between the start time generated by
2
Radiant sensitivity is the photoelectric current from the photocathode, divided by the incident radiant power
at a given wavelength, expressed in A/W (amperes per watt). Quantum eﬃciency (QE) is the number of photoelectrons emitted from the photocathode divided by the number of incident photons. Quantum eﬃciency isusually expressed as a percent. Quantum eﬃciency and radiant sensitivity have the following relationship at agiven wavelength.

21

Figure 3.2: Schematic veiw of the 20-inch PMT and the bleeder circuit [32].

Number of channels

12 ch/board

One hit processing time∼ 5.5µsec

Charge dynamic range∼ 400 to 600pC (12bit)

Timing dynamic range∼ 1300nsec (12bit)

Charge resolution (LSB)

0.2 pC/LSB

Charge resolution (RMS)

0.2 pC (RMS)

Timing resolution (LSB)

0.3 ∼ 0.4nsec/LSB

Timing resolution (RMS)

0.4 nsec (RMS)

Temperature dependence (QAC)

3 Count/deg. ↔ 0.6 pC/deg.

Temperature dependence (TAC)

2 Count/deg. ↔ 0.8 nsec/deg.

Envent number

8 bit

Data size of one hit

6 ByteFIFO

2 kByte (∼ 340 hits)

Table 3.1: Speciﬁcations of the ATM module22

Figure 3.3: Speciﬁcations of the 20-inch PMT [32]

C

E

D

Figure 3.4: Basic distributions of 20-inch PMT (a) Cathode radiant sensitivity and Quantum
eﬃciency. (b) 1p.e. charge distribution (c) Transit time distribution [32]23Figure 3.5: Transparency of acrylic cover as a function of the photon wavelength.

the hit signal and the stop signal which will be generated by a global trigger. After the global
trigger arrived, the integrated charge in QAC and TAC are digitized in ADC. (Note that theoutput of a hit timing is the time interval between the hit and the global trigger, and thus, thebigger value of the output timing corresponds to earlier hit.) On the other hand, if there is no
global trigger within 1.3µsec, the charges in QAC and TAC are cleared.
Every 16 events, the digitized data in the ATM are sent to VME memory module called
Super Memory Partner (SMP). One SMP handle 20 ATMs and there are 8 slave computers eachreading out 6 SMP memories. One sequence of data ﬂow ends with the online host computer towhich the slave computer sends its data.
More detailed information about the DAQ system is described in [31].
Trigger
The global trigger is made in the following process. At each ATM, in parallel with theprocesses in QAC and TAC, a hit signal is converted to a square pulse, then the ATM generate
one HITSUM signal of 200nsec width by taking analog-sum of all the square pulses from 12
PMTs. The height of the summed HITSUM is proportional to the number of hit PMTs in theATM. All HITSUMs are collected and AC-coupled into a discriminator to subtract oﬀset due todark noise hits. At this point, one hit corresponds to -11mV of the pulse height. If the summed
HITSUM signal exceed a discriminator threshold, a global trigger is generated and then it fed
to a VME TRiGger module (TRG) which issues the global trigger and the event number to be
distributed to ATMs.
There are three types of the threshold values depending on the analysis region: Super Low
Energy trigger (SLE), Low Energy trigger (LE), and High Energy trigger (HE). Table 3.2 shows
the summary of each trigger setting. There also exists OD trigger, which has similar to that of
ID and independent of the ID system. The OD trigger threshold is set to 19 OD hits in 200
nsec.
Since most of the SLE triggered events are caused by gamma-rays from the surrounding
rock, and radioactive decay in the PMT and FRP, the vertex positions of such low energy

24Analog sum of 12PMT signals
20-inch PMT

:

:
Sparc

KPVGTHCEGSCH(NCUJ#&%ATM

8ch(12x8=96PMT)/board

online CPU(slave)

x 15
x 20
x 240

(NCUJ#&%

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

interface

Super Memory Partner

VME
VME

interrupt reg.
TRG

20-inch PMT
SCH
TRIGGERATM

TRIGGER

x 20
x 240

PROCESSORHIT INFORMATION(HITSUM)

ATM
GONG

Analog Timing ModuleTKO

PMT x 11129

ATM x ~1000

SMP x 48

online CPU(slave) x 17

FADC x 120

Figure 3.6: Schematic view of DAQ system

25

to PMTSUM
HITOUTDISCRI.

to HITSUM

ONE-SHOT HIT
W:900ns

CURRENT SPLITTER&AMPLIFIER

PMT INPUT

W:200ns

SELF GATESTART/STOP
TAC-A

TAC-A
CLEAR
QAC-A

GATE

QAC-A
to ADC
TAC-B

START/STOP

TAC-B
CLEAR
GATE

QAC-B

QAC-B

THRESHOLD

DELAY

GATE
TRIGGERPED_START

Figure 3.7: Block diagram of ATM.

Trig. TypeThre. (mV)

Hits

Trg. Rate (Hz)

SLE1

-212

19

∼ 1.3k

SLE2

-186

17

∼ 3.5k

LE

-302

27

∼ 30

HE

-320

29

∼ 10

Table 3.2: Summary of trigger settings in SK-III. SLE1 and SLE2 are set in diﬀerent periods.
SLE1 is from Jan 2007 to Apr 2008, and SLE2 is from Apr 2008 to Aug 2008. LE and HE are
used in whole SK-III period. See also Table 7.1. Hits shows corresponding number of hits atthe threshold value namely Hits = (Thre. mV)/(-11 mV). About 6 hits correspond to 1MeV ofthe electron energy.

2641C 41D
Figure 3.8: Upgraded water puriﬁcation system.

background events are concentrated in the edge of ID. To reduce the number of events to be
stored, a fast vertex ﬁtter, which is a realtime vertex ﬁtter and is applied to only SLE triggered
events, was installed to cut events outside the 22.5kton ﬁducial volume. This software ﬁlteringand associated online computer hardware are called the Intelligent Trigger (IT) [20].

3.4

Water puriﬁcation system

The puriﬁed water in the detector is constantly circulated through the water puriﬁcation
system with a ﬂow rate of 60 ton/hour. Figure 3.8 shows the upgraded water puriﬁcation
system. Reverse Osmosis ﬁters (RO) remove particles, and RO3-a was installed during SKI after supplying water, but it was used from the begining of SK-III, and a new RO3-b was
installed during SK-III. In addition, a new heat exchanger was installed, so that the watertemperature can be controlled with 0.1◦ C accuracy. A brief description of the puriﬁcation
system components is summarized in Table 3.3.
Figure 3.9 shows the circulation mode before and after the improvements of the system in
SK-III. This ﬁnal setting has been running since August, 2007. Before the improvements, wetested many circulation modes to achieve a stable and low background rate. It accidentally
happened that sometimes radon-rich water contaminates even deep inside the ﬁducial volume.
Such high background periods are eliminated from the ﬁnal data sample for the very low energy
region from 4.5MeV to 6.5MeV. (see Section 7.1).

27

(KPCNUGVVKPI5GIOGPVCVKQP6[XGEM
UJGGVKPUVCNNGFKP5-+++Figure 3.9: Water circulation mode before and after upgrade. OD is also separated by the
Tyveck sheet.

Component

Function

1µm ﬁlter

Remove dust

Heat exchanger

Cool water heated by pump

Ion exchanger

Remove ions

UV sterilizer

Kill bacteria

Vacuum degasiﬁer

Remove dissolved oxygen and radon gas

Cartridge Polisher

Remove U and Th

Ultra ﬁlter

Remove small particle (>∼100nm)

Membrane degasiﬁer

Remove radon gas

Reverse osmosis

Remove small particles (226 Ra etc.) (>∼10nm)

Table 3.3: Functions of the components in the water puriﬁcation system

28

After the improvements, the background level in the central region of the ﬁducial volume
become signiﬁcantly lower . This will be discussed in Chapter 10.

29

Chapter 4

Event reconstruction
The event reconstruction for solar neutrino events will be explained in this chapter. SuperKamiokande observes the solar neutrino by detecting Cherenckov photons emitted from therecoil electron of the ν − e elastic scattering:ν + e− −→ ν + e−

(4.1)

Since the recoil electron from a solar neutrino can travel only ≤∼10cm in water, it can beassumed that the Cherenkov photons are emitted from point-like source. First item is vertex
reconstruction which ﬁnds the interaction point. After the vertex reconstruction, the directionand energy of recoil electron are reconstructed. Figure 4.1 explains the basic variables in the
SK detector.

4.1

Vertex reconstruction

The event vertex reconstruction for solar neutrino analysis performs a maximum likelihood
ﬁt to the timing residuals of the Cherenkov signal as well as the dark noise background for eachtesting vertex. This vertex ﬁtter is called BONSAI [36]. The deﬁnition of the likelihood isL(⃗x, t0 ) =

N
hit
∑

log(P (t − tof (⃗x) − t0 ))

(4.2)

i=1

Here, ⃗x and t0 are the testing vertex position and the peak of hit timing t subtracted by Time Of
Flight(tof ) from the testing vertex ⃗x to each PMT position. N hit is the number of hit PMTs,
and P is a probability density function (pdf) which describes timing distribution of a signal
event as a function of the hit timing. The pdf is obtained from the shape of the timing residual(t − tof − t0 ) distribution from LINAC calibration data which is shown in Figure 4.2, and PMT
dark noise is taken into account by constructing the likelihood.
The testing vertex with the largest likelihood is chosen as the reconstructed vertex. However,
the accidental coincidence of dark noise hits after time-of-ﬂight subtraction could produce local
maxima of the likelihood at several positions which are far away from the true vertex. The
search for the true global maximum is complicated and time consuming due to the large volumeof the detector. To improve both the speed performance and the number of mis-reconstructions,the likelihood is then maximized from a vertex search from a list of vertex candidates calculated
30

'ZRCPUKQPRNCPGQH5-FGVGEVQT

5-FGVGEVQT

Z
\

VQR

0

[

TZ
[XGTVGZFKTGEVKQP
Z

\
[

\
[

DCTTGN0Z

ϕ0

DQVVQO[Z.KMGNKJQQF

Figure 4.1: Basic variable deﬁnitions used in this thesis.

661(6=PUGE?
Figure 4.2: Likelihood function of BONSAI vertex ﬁtter from LINAC calibration data. The
peaks around the 40nsec and 110nsec are caused by the after pulses.

31

from PMT hit combinations of 4 hits each. The four-hit combinations each deﬁne a uniquevertex candidate given their timing constraints. Thus, any events with four hits or more isreconstructed. The likelihood values for each vertex candidate are then calculated. To avoid a
local maximum point, iterations of grid search (from 7.8m to 1cm)around each candidate aredone until it ﬁnds a vertex which gives lager and more stable likelihood values at the surrounding
grid points. We also use the result from the second vertex ﬁtter which is described in [20] as across check and a noise reduction.

4.2

Direction reconstruction

To reconstruct the direction of a low energy event, a maximum likelihood method is used toscan directions which correspond with Cherenkov ring patterns. The deﬁnition of the likelihood
isN20∑
cos θi
⃗ ≡
(4.3)
L(d)
log(fdir (cos θdir,i , E))i ×a(θi )iwhere N20 is the number of hit PMTs within a 20 ns time window around t − tof − t0 = 0.
fdir describes distributions of opening angles between a candidate particle direction d⃗ and thereconstructed vertex ⃗v to hit PMT position h⃗i (⃗v − h⃗i ). This function is made by Monte Carlo(MC) simulation. An example for 10 MeV electrons is shown in Figure 4.4. Since the electron
multiple scattering angle depends on the electron momentum 1 , the function fdir is constructed
as a function of not only the opening angle but also the electron energy.
The factor multiplied to fdir in Equation 4.3 is to correct the eﬀect the photon incident angle
to PMT (θi ) weighted by the PMT acceptance function a(θi ) (see Figure 4.3 for the deﬁnitionof θdir,i and θi ). Equation 4.3 is maximized by carrying out a grid search. The size of a grid is20◦ at ﬁrst and 1.6◦ at the end. An absolute angular resolution is then deﬁned as the maximum
angular diﬀerence between 68% of the reconstructed and true directions. The angular resolution
is checked by LINAC event (see Section 6.9). The diﬀerence of angular resolution between MC
and data results in systematic uncertainty on the total solar neutrino ﬂux which will be discussed
in Chapter 9.

4.3

Energy reconstruction

The reconstruction of the energy of recoil electron is explained here. The energy reconstruction depends on the intensity of that light. That is, the number of Cherenkov photons isapproximately proportional to the electron energy and in turn proportional to the number ofgenerated p.e.. However, because of the poor charge resolution (∼ 50% at 1p.e.) of the PMTs
and the fact that electrons below 20 MeV produce roughly one photon per a hit PMT, thenumber of hit PMTs is used to reconstruct the energy of an event. Another reason why the1

The approximate multiple scattering angle θmsc (radian) can be calculated as√
13.6M eVxθmsc =
βcp
X0

(4.4)

where p and βc are momentum and velocity, and x/X0 is the thickness of the scattering medium in the unit of
the radiation length. In the case of 10 MeV electron in water (the radiation length is 36.1 cm), θmsc ∼ 27◦ . Thus,
the angular resolution for solar neutrino is limited by the multiple scattering.

32

㪪㫌㫅
ν

SK detector

θiθdir,iG

KVJ2/6

4GTCVKXG2TQDCDKNKV[EQUθFKT

Figure 4.3: Deﬁnition of θdir,i and θi .

GNGEVTQPGPGTI[=/G8?
Figure 4.4: Angular likelihood of 10 MeV
electron

Figure 4.5: Likelihood of angular reconstruction depending on energy.
33number of hits is used for energy calculation is that the number of hits does not strongly dependon the gain of PMTs compared to the total p.e.
N50 is deﬁned as the maximum number of hits found by 50nsec sliding time window searchof T − T of − T 0 distribution. Since N50 is aﬀected by several factors, such as accidental hits
from dark noise of PMTs, the eﬀective number of hits (Nef f ) is used to estimate the energy ofan event. The deﬁnition of Nef f is following;
}
N50 {
∑
Nall
1
ri
(Xi + ϵtail − ϵdark ) ×
Nef f =
×× exp() × Gi (t)Nalive S(θi , ϕi )λ(run)(4.5)

iwhere the explanation of each correction are following;
Xi : Multi-photo-electron hit correction
This correction is needed to estimate the eﬀect of multiple photo-electrons in the i-th hit
PMT due to the fact that if an event occurs close to a detector wall and is directed towards
the same wall, the Cherenkov cone does not have much distance to expand then the observed
number of hits becomes smaller. The correction Xi is deﬁned as
1
 log 1−xi,xi < 1
xi
(4.6)
Xi =
 3.0,
xi = 1
where xi is the ratio of hit PMTs in a 3×3 PMTs surrounding the i-th PMT to the total numberof live PMTs in the same area. The − log(1 − xi ) term is then the estimated number of photons
per one PMT in that area and is determined from Poisson statistics. When xi = 1, 3.0 is
assigned to the value.
edark : Correction for dark noise hits
This factor is for hits due to dark noise in the PMTs.
ϵdark =

50nsec × Nalive × Rdark
N50

(4.7)

where Nalive is the number of all live ID PMTs and Rdark is the measured dark rate for a given
run.
etail : Correction for late hit
Some Cherenkov photons being scattered for reﬂected arrive late to the PMT, and make late
hit outside the 50nsec time window. To correct the eﬀect of the late hits, the term
ϵtail =

N100 − N50 − Nalive × Rdark × 50nsec
N50

(4.8)

is added where N100 is the maximum number of hits found by 100nsec sliding time window
search.

34

Figure 4.6: Correction factor of photocathode coverage.

Nall
Nalive : Dead PMT correction

This factor is for the time variation of the number of dead PMTs. Nall is total number of
PMTs in SK-III and is 11129.
1
S(θi ,ϕi ) : Correction for photo-cathode coverage

This term is to account for the direction-dependent photcathode coverage. S(θi , ϕi ) is theeﬀective photocathode area of the i-th hit PMT as viewed from the angles (θi , ϕi ) which is shownin Figure 4.6. The deﬁnition of (θi , ϕi ) is shown in Figure 4.7.
riexp( λ(run)):Water transparency correction

The water transparency is accounted for by this factor where ri is the distance from thereconstructed vertex to the i-th hit PMT. λ(run) is the measured water transparency for agiven run.
Gi (t): PMT gain correction
This factor is to adjust the relative gain of the PMTs at the single photo-electron level. The
diﬀerences in gain depend on the fabrication date of the PMTs.

After determining Nef f , an event’s energy in terms of MeV can be calculated as a functionof Nef f . The relation between Nef f and MeV is obtained by uniform electron MC events , andthe systematic uncertainty of the reconstructed energy is checked by LINAC and DT calibration(see Section 6.7.2)

35

Figure 4.7: Deﬁnition of (θi , ϕi ).

36

Chapter 5

Simulation
5.1

Outline of detector simulation

The detector response can be understood by Monte Carlo(MC) simulation of the detector. In
solar neutrino event simulation, the tracking of the recoil electron and the emission of Cherenkov
photons are done in the ﬁrst step. In the second step, the Cherenkov photons are propagated.
Finally, the response of the detector is simulated. Most of the physics processes such as multiple
scattering or bremsstrahlung are simulated by GEANT3.21 [37], except for the production andthe propagation of Cherenkov photons and the light attenuation in water, which are developed
by SK group [38].
Cherenkov photon production
The number of Cherenkov photons dN emitted by an electron per wavelength interval dλper track length dx is given by
)
(
1
1
dxdλ
(5.1)
dN = 2πα 1 − 2 2n βλ2where n is the refractive index of water (n = 1.334), α is the ﬁne structure constant, and β
is the velocity of the electron in unit of the light velocity in vacuum. The opening angle θ ofCherenkov photons is given by1
cos θ =
(5.2)
nβBased on these formulae, Cherenkov photons are produced in the simulation.
Propagation of photons in water
The velocity of Cherenkov photons depends on its wavelength. The velocity is given as agroup velocity (vg ) of the wave packet:
c
vg =
(5.3)
n(λ) − λ dn(λ)
dλwhere n(λ) is the refractive index of water as a function of the wavelength such as√a1n(λ) =+ a2 + a3 λ2 + a4 λ3 + a5 λ6 .
2
λ − λ2a37(5.4)

The unit of λ is µm, and parameters are
λ2a = 0.018085,

a1 = 5.7473534 × 10−3

a2 = 1.769238,

a3 = −2.797222 × 10−2

a4 = 8.715348 × 10−3 ,a5 = −1.413942 × 10−3

These are obtained by measurements.
There are three processes which are considered during the photon propagation: Rayleighscattering, Mie scattering, and absorption. The attenuation length (Lattn. ) of light in water isthen given by1
Lattn. =(5.5)
αabs (λ) + αRayleigh (λ) + αM ie (λ)
where αabs (λ), αRayleigh (λ), αM ie (λ) are coeﬃcients for the absorption, Rayleigh scattering, andMie scattering as a function of the wavelength of photons (λ). The fraction of these processes
depends on the wavelength and the purity of water. The wavelength dependence of each process
is given by A1
λ ≤ 350nm
 λ4 ,AA
1
4
αabs (λ) =(5.6)+ A2 (λ/A3 ) ,350 < λ ≤ 415nm
λ4

 A1
+ f (λ),415nm < λλ4
R2
R1
× (1 + 2 )(5.7)
αRayleigh (λ) =
4
λ
λ
M1
(5.8)
αM ie (λ) =λ4where Ax , Rx , Mx are tuning parameters, and f (λ) is measured by [39]. The measurement isdone by a periodical laser injection calibration. When a photon arrives at the detector wall (theacrylic cover, the PMT surface or the black sheet), the reﬂection, absorption and transmission
are considered. An example of the fraction of these processes at the PMT surface is shown in
Figure 5.1.
Detector response
When a photon reaches at the PMT surface, the detector response is simulated. The probability to produce one p.e. from the photon arriving at each PMT surface has four components:
P robability = QE(λ) ×

AorP (λ, θi )× COREP M T × qetable(i)AorP (λ, 0)(5.9)

where QE(λ) is the quantum eﬃciency of 20-inch PMT depending on the wavelength of theincident photon λ, which is measured by HAMAMATSU, AorP (λ, θ) is summed fraction of theabsorption and p.e production (black + green in ﬁgure 5.1), and is measured by a calibration(λ,θi )
with injected laser light. Thus, AorP
AorP (λ,0) is a correction depending on the incident angle θifor the photon with λ. COREP M T corrects the average quantum eﬃciency (Q.E.) which iscommon to all PMTs and tuned by LINAC (see Section 6.6.2). Finally, qetable(i) is the relative
Q.E. for the i-th PMT measured by Ni calibration (see Section 6.1). After one p.e. is producedwith the probability, the output charge from PMT is simulated according to the measured 1
38

(TCEVKQP

transmission

Reflection

absorption
p.e. production(QE)+PEKFGPVCPINGQHRJQVQP=FGITGG?

Figure 5.1: Fraction of reﬂection, absorption, or transmission at PMT surface (the wavelengthof the incident photon is λ = 420nm)

p.e. charge distribution. If the output charge exceeds the ATM threshold within the ADC gate,
the hit information is stored. The hit timing is simulated according to the timing distribution
obtained by LINAC. Finally, one event is generated by adding random dark hits.

5.2

New modeling of water condition

It was found that the PMT hit-rate measured by Ni calibration at top region are systematically lower than that of bottom region by 3 ∼ 5% (called top-bottom asymmetry (TBA)).
Figure 5.2 shows the time variation of top-bottom asymmetry measured by Ni calibraion and
Xe calibration (see Section 6.1). However, MC simulation cannot reproduce the top-bottom
asymmetry with the uniform attenuation length in space. To solve this problem, a simple modelof the light absorption is introduced to take into account the position dependence of the attenuation length in Equation 5.5. In this modeling, only z-dependence of the absorption parameteris considered. The reason is that by the laser injection calibration, the dominant contribution to
the time variation of the water transparency is measured to be the absorption. Figure 5.3 shows
the absorption parameter measured by laser light injectors at diﬀerent z positions in February
2008. As suggested by the measured z-dependence, the z-dependence can be modeled as{
αabs (λ)(1 + β · z),
for z ≥ −1200cm
αabs (λ, z) =(5.10)
αabs (λ)(1 − β · 1200),
for z < −1200cm
where the absorption factor is assumed to be uniform below z=-1200cm due to the convectionof water. This β is called Top-Bottom Asymmetry parameter (TBA parameter), and is tuned
so that the top-bottom asymmetry becomes smallest. The tuning result is shown in section 6.4

39

6$# VQRDQVVQOCU[OOGVT[

0.15

6$#D[0KECNKD
6$#D[:GECNKD

0.1

0.05

0

-0.05

-0.1

-0.15

,CP


,WN


,CP


,WN


Figure 5.2:
Time variation of TBA measured by Ni and Xe calibration The
vertical axis show the top-bottom asymmetry (TBA) value which is deﬁned as(Hirate(top)-Hitrate(bottom))/Hitrate(barrel) for Ni calibration, and (Charge(top)Charge(bottom))/Charge(barrel) for Xe calibration.

#DUQTRVKQPHCEVQT

0.8

0.6041E-03/ 2
0.42040.9256E-04

P1
P2

0.7

0.6

0.5

0.4

0.3

0.2
-1500

-1000

-500

0

500

1000

1500

<RQUKVKQPQHNCUGTNKIJVKPLGEVQT=EO?

Figure 5.3: Z dependence of the absorption factor αabs measured by laser light injector. The
data was taken in February 2008. The Ni calibration for the tuning of TBA parameter was
carried out in February 2008. The line shows the result of linear ﬁtting of the four points. This
slope is consistent with the tuned TBA parameter value (see Section 6.4)

40

1/aenuaon length (1/m)

Abs.+Sca.
Absopron
Ray. sca.
Mie sca.

Wavelength(nm)Figure 5.4: Tuned water parameters measured by laser injection calibration
Makers show the measured parameters and lines show the ﬁtted functions.

5.3

Tunable input parameters

The tunable parameters for MC simulation are listed here. The tuning results are shown in
Chapter 6.
Water parameters
The absorption and scattering parameters are measured by laser light injected from top, barrel, and bottom. Figure 5.4 shows the measured water attenuation length and the contributionof the absorption, Rayleigh scattering, and Mie scattering as a function of photon wavelengthfor SK-III.
TBA parameter
TBA parameter is tuned so that the hit-rate of MC simulation reproduces that of calibration
data taken by Ni calibration. The time variation of TBA parameter is also implemented byperiodical measurements of the top-bottom asymmetry factor.
Timing resolution of PMT
Timing resolution of PMT is calibrated by LINAC. The hit timing is smeared by the measured
timing resolution in MC simulation.
After pulse
PMT hit timing has some characteristic peaks after the main peak (called after pulse). A
possible explanation of the cause of the after pulse is that if a photo-electron is back scattered at
the ﬁrst dynode, it takes some time until it loses its velocity against the electric ﬁeld then comesback to the ﬁrst dynode, producing a delayed hit. The fraction of the after pulse is measured
by LINAC calibration. (see Section 6.6.2)
41

Average quantum eﬃciency (Q.E.) of PMT
While the relative Q.E. of each PMT is measured by the Ni calibration, the average Q.E. istuned so that the Nef f of MC simulation reproduces LINAC data.

42

Chapter 6

Calibration
6.1

Outline of detector calibrations

An outline of the detector calibration will be given in this section. Some important calibrations for the solar neutrino measurement such as timing of PMT, water transparency, quantum
eﬃciency(Q.E.) of PMT, and LINAC are explained in the subsequent sections. Especially, theimprovement to the timing calibration is newly achieved in this thesis. The detail explanationof the detector calibration is in [31].
First, High Voltage (HV) value which is applied to each PMT is adjusted in order to achieve
the same Q.E.× gain value for all PMTs becomes same within ±1% accuracy. This HV determination was done by light emission of Xe lamp guided through optical ﬁber (hereafter Xecalibration). A scintillator ball at the end of the optical ﬁber absorbs the UV light and emits
light isotropically with a peak at 440nm wavelength.
After the HV determination, the conversion factor of ADC charge to p.e. is measured foreach PMT. The conversion factor is given by (average PMT gain)×(relative PMT gain). The
average PMT gain is obtained from an average 1 p.e. distribution of PMTs measured by Cf-Ni
γ-ray source (peak energy ∼9MeV, average number of hits is 50PMTs, hereafter Ni calibration).
The obtained value of average gain is 2.243 pC/p.e.. The relative PMT gain for each PMT is
obtained by calibration data using high and low intensity laser light (398nm). The laser light
was put into the detector through optical ﬁber and diﬀuser ball. The relative PMT gain (Gi : idenotes PMT ID) is the ratio of the mean charge (Qi ) in the high intensity data to the numberof hits (Hiti ) at the low intensity data for each PMT; Gi = Qi /Hiti . This is because the Qi
is proportional to Q.E. × gain, and the Hiti is proportional to only Q.E., thus, only the gain
factor can be extracted by taking the ratio of Qi to Hiti .
The charge of each PMT hit is corrected by taking into account the ADC non-linearity. This
non-linearity is measured by obtaining deviation of measured p.e value from the expected p.e.
value. The data for this calibration is taken by laser light source.
In parallel to these calibration related to the charge of each hit, timing calibration, watertransparency measurement, and Q.E. measurement were done. Then LINAC and DT (deuterium
and tritium neutron generator) calibration were done especially for the low-energy event analysis.
In the following sections, the explanation about these calibrations which are closely related tothe solar neutrino analysis will be given.

43

6.2

Timing calibration

The timing of all PMTs must be calibrated so that the event vertices and directions arereconstructed accurately. To obtain correct timing, the followings must be considered; the lightintensity of each PMT, the length of cable between each PMT and electronics, and diﬀerent rangeof hit timings. Diﬀerent light intensity makes diﬀerent triggered timing which is so called ”timewalk”. The time walk exists because larger charge hits exceed TDC discriminator thresholdearlier than those with less charge even though the hit PMT gets photons in the same timing.
The timing calibration for SK-III consists of two measurements; one is to make a correction
table for the time walk and cable length, and the other one is to make a correction function for
hit timings in diﬀerent time ranges. It should be mentioned that the former one is a main part oftiming calibration which is basically same as the one in the previous phase of experiments, but
the latter one is newly introduced in SK-III to decrease the systematic shift of the reconstructedvertex position.
Conventional timing calibration
This calibration is to correct the time walk of each PMT and the overall process time foreach PMT (cable length and electronics. etc.). The time walk and the overall process time
can be measured by putting a fast pulsing light source in the center of the detector. Figure 6.1
shows a schematic view of the timing calibration system. First, a N2 laser pulse is divided in the0.CUGT

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5-+&

T [ns]64)/QP2/6

990
980

8CTKCDNGHKNVGT
970
960
950
940
930
920
910
900
890



&KHHWUGTDCNN
Figure 6.1: Schematic view of timing calibration system














3DKP

Figure 6.2: Example of TQ distribution.
Horizontal axis shows charge of each hit, andVertical axis shows hit timing without correction.

laser module. One goes to a trigger PMT which makes an external trigger for timing calibration
44

event. The other goes to dye laser module to produce a dye laser pulse. The dye laser pulse(wavelength; 398±5nm(FWHM), pulse width 0.2nsec) then goes through variable ﬁlter which
is used to adjust the laser intensity. After traveling 70m optical ﬁber, the dye laser pulse isemitted in the detector from a diﬀuser ball.
The calibration data are taken by changing the intensity of the light from the diﬀuser ball.
Using the variable ﬁlter, the light intensity can be adjusted from 0.0 p.e. level to high intensity
level (250p.e.) which causes saturation of electronics. Figure 6.2 shows a typical relation between
timing and charge. The vertical axis shows the hit timing without correction, and the larger value
corresponds to earlier hit timing. This two dimensional histogram is called ”TQ distribution”.
The purpose of this calibration is to ﬁt the TQ distribution by a proper function. This function
is called ”TQmap”.
The TQmap is made for each channel (11129 PMTs × 2 ATM buﬀer channels). The method
to generate TQmap is following. In the ﬁrst step, after making the timing distribution plot foreach channel, selection of laser hit is done by setting 100 nsec time-window around the peak of
the timing distribution with TOF(time of ﬂight from the diﬀuser ball to each PMT) ”added”.
The ”+” sign for TOF means that before the timing correction the vertical axis corresponds totime interval between the hit and the global trigger (see the DAQ section 3.3). Since the smaller
value of vertical axis shows the later hit, TOF must be added to get the time when a pulse lightis emitted from the diﬀuser ball.
A selection is needed to remove pre-pulses which occur at around 50nsec before the main
peak by a photon hitting the dynode of PMT, and after-pulses which occur at about 100 nsec
after the main peak by an electron back scattered at the ﬁrst dynode. After the selection oflaser hits, for each of the 140 bins in charge (Qbin ), the peak timing and the standard deviationof timing distribution is obtained by asymmetric Gaussian ﬁtting which takes into account theeﬀect of the scattered or reﬂected hits. The bin width of each charge bin is deﬁned as{
0.2 pC,for 1 ≤ Qbin ≤ 50 (0 pC ≤ Q ≤ 10 pC)(6.1)∆Qbin ≡
Qbin −1
Qbin
for 51 ≤ Qbin ≤ 140 (10 pC ≤ Q ≤ 630.95 pC)
10 50 − 10 50 ,Finally, the peak timings with respect to the charge is ﬁtted by a seventh order polynomial
function.
An improvement of SK-III method is following. In SK-I and II, instead of the peak value,
the timing of each charge bin was determined from an average timing. The timing obtainedby the previous method, therefore, was aﬀected by late scattered or reﬂected hits especially at
small charge regions. 1 Since the amount of reﬂected hits are large in the edge of ID, such eﬀect
gives systematic timing shift depending on PMT positions. Figure 6.3 shows one example of thediﬀerence between the two methods. As shown in Figure 6.3, the hit timing of the new method
is faster than that of the old method at small charge region, which means that the new method
is less aﬀected by the late hits.

1

In the previous method, to reduce the eﬀect of timing of late hits, First, an average is obtained from theselected hits within 100nsec for each Qbin. Then, for each Qbin, a narrow timing window is set +2.0RMS-1.5RMS
around the average timing within the 100 nsec time-window. Finally, the timing of each Qbin is determined fromthe average timing in the narrow time window.

45

JKVVKOKPIYKVJQWVEQTTGEVKQP=PUGE?

970
965
960
955
950
945
940
935

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925

1NF࡯0GY=PUGE?

920
0

20

40

60

80

100

120

0

20

40

60

80

100

120











%JCTIGQHJKV=3DKP?
Figure 6.3: Example of comparison between the old and new methods of TQmap.

46

&KHHWUGTDCNN
5-+&Figure 6.4: Setup for additional timing calibration

Additional timing calibration
The other calibration which has been carried out is to improve the performance of reconstruction. It was observed that the reconstructed position of Ni calibration events was shifted
inward direction from the real source position to the tank center (see Figure 6.7-(a)).
This direction of the vertex shift indicates that the linearity of relative timing within a wide
range (∼100nsec) is not perfect due to characteristics of electronics. For source positions awayfrom center and near the ID wall, the spread of hit times is larger (∼100nsec) if the TOF isnot subtracted. In this case, the earlier hits seem to be shifted later by ∼0.5nsec. This timing
diﬀerence between far side hits and near side hits can create the bias to reconstructed position.
I have measured this timing shift by the following setup. Figure 6.4 shows a schematic
diagram of trigger system of this calibration. In order to measure the timing shift between
late hit and early hit, a calibration trigger signal (common stop signal) from the monitor PMT
is artiﬁcially delayed by putting additional cables (from 15m to 80m), and the time delay ismeasured by a independent system which is pre-calibrated within 0.1% accuracy. For example,
a 20m cable can make the arrival time of the global trigger at each ATM by 50nsec later, whichcorresponds to make hit timing of each PMT 50nsec faster. By changing the cable length 15m
up to 80m, we can check the relation of the timing measured by ATM and the timing measured
by the independent system within the range of about 300 nsec.
Figure 6.5 shows an example of the result for a PMT. The horizontal axis shows hit timemeasured by ATM, and vertical axis is delayed time measured by independent system. Whilethe horizontal axis and vertical axis show good linearity within ±1nsec as shown in the top plotof Figure 6.5, the diﬀerence has a clear dependence on the value of hit time. This tendency
is common to all channels which is shown in Figure 6.6. The average of ﬁtted slope values is-0.67nsec/100nsec, thus correction function is
T = T − slope × (T − 1000.0) + of f set(ch)(6.2)

where T is hit time, 1000.0nsec is selected as normalization point, and of f set(ch) is an oﬀset
value depending on ATM ch (0a ∼ 11b). After applying this correction, the vertex shift is47

6KOGFGNC[OGCUWTGFD[5-=PUGE?

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
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


6KOGFGNC[OGCUWTGFD[6&%=PUGE?
Figure 6.5: Example of timing shift as a function of hit timing range for one channel

48

*KVVKOKPI=PUGE?
Figure 6.6: timing shift as a function of absolute hit timing. Diﬀerent colors show timing shiftof the diﬀerent ATM channels (0a to 11b). The marker show mean timing shift of the sameATM channel PMTs with respect to the absolute hit timing. The lines show the results of linear
ﬁtting for all ATM channels.

49

D<RQUKVKQP=EO?

C

∆r
10cm

:RQUKVKQP=EO?
Figure 6.7: Vertex shift of Ni calibration events before(a) and after(b) correction. The origin ofarrows shows the true Ni source position and the direction indicates the averaged reconstructed
position. The length of arrow shows the length of vertex shift. The red line is the scale of 10cm
vertex shift.

reduced as shown in Figure 6.7-(b). The remaining vertex shift gives a systematic uncertainty
of ﬁducial volume which is estimated as 0.54%. The corresponding value in SK-I was 1.3%. So,
more than factor 2 improvement is achieved in this analysis.

50

NQI S T

1.5
P1
P2

1.424
-0.8768E-04

1.4

1.3

1.2

1.1

1

0.9

0.8

1000
1500
2000
2500
3000
3500
&KUVCPEGHTQOFGEC[GXGTVGZVQJKV2/6=EO?

Figure 6.8: Charge vs. distance from the decay-e vertex. Vertical is log of hit charge (p.e.) andline shows the result of linear ﬁtting.

6.3

Water Transparency Measurement

The water transparency is measured by using decay electrons and positrons (hereafter decaye) from cosmic-ray muons stopped in the detector (hereafter stopping µ). The selection criteriaof decay-e are following:• The time diﬀerence between the stopping µ and the decay-e candidate (∆T ) must be
2.0µsec < ∆T < 8.0µsec.
• The reconstructed vertex of the decay-e candidate must be within the 22.5-kton ﬁducial
volume (=2m from ID wall).
• The number of hit PMTs within 1.3µsec, Nhit ≥ 50
About 1500 decay-e events are selected with this criteria every day, which is statistically enoughfor this weekly transparency measurement.
Figure 6.8 shows the correlation between log of the charge (q(r)) of hit PMTs (vertical)
and the distance from the decay-e vertex to each hit PMT (horizontal). The charge of PMT iscorrected considering the acceptance depending on the incident angle of photon. To eliminate
scattered hits or reﬂected hits, hit PMTs are selected by the following criteria:• Hit PMT must be one of N50 .
• The opening angle of a hit PMT θdir must be 32◦ < θdir < 52◦
After ﬁtting the histogram with linear function, the water transparency is obtained by the inverseof the ﬁtted slope.
Figure 6.9 shows the time variation of the water transparency. Each point represent a watertransparency for 6 days.

51

YCVGTVTCPURCTGPE[=EO?

16000

14000

12000

10000

8000

6000

07/01/06 10/01/06 01/01/07 04/01/07 07/01/07 10/01/07 01/01/08 04/01/08 07/01/08 10/01/08&#6'
Figure 6.9: Time variation of water transparency. Each point is determined from 6day data
period.

6.4

TBA tuning and Q.E. measurement by Ni calibration

As already described in the previous chapter, the TAB parameter is newly introduced to SKIII detector simulation to model the z-dependence of the absorption parameter. This parameteris tuned so that the PMT hit-rate of data and MC simulation shows the best agreement. Withthe tuned TBA parameter, the relative Q.E. of individual PMT is obtained by taking the ratioof the hit-rate of data to that of MC simulation. The usage of Ni-Cf γ-ray source is favoredbecause of following reasons• Easy to operate and good isotropy.
N hit• Low occupancy ( Total #of PMTs ∼ 1%) is needed to avoid multi-photon hit.

• Q.E. should be measured at the Cherenkov wavelength range.
Figure 6.10 shows a comparison of the hit-rate between data and MC simulation with TBA=0
and TBA=tuned value respectively. Hit-rate is deﬁned as number of hits in each PMT in a unitof the averaged number of hits for all PMT during the calibration run2 . As shown by the ﬁgure,data and MC shows better agreement with TBA=tuned value (8.85×10−5 ). Then, the relative
Q.E. to be put into MC is obtained as the ratio of data to MC.

2

The acceptance correction as a function of incident angle of photon to each PMT and the distance correction
between the source position and each PMT position are taken into account to obtain the number of hits for each
PMT.

52

*KVTCVG$CTTGN*KVTCVG*KVTCVG<RQUKVKQP=EO?
6QR


T=EO?

$QVVQOT=EO?

Figure 6.10: Hit rate of data(Black) and MC simulation (Green: TBA=0,Red: TBA=tuned)

6.5

DT calibration

6.5.1

DT neutron generator

Deuterium-Tritium Generator [40] produce neutrons by colliding deuterium and tritium ionswith a ﬁxed metal hydride target with a reaction;3

H +2 H −→ 4 He + n(6.3)

The deuterium and tritium are ionized by Penning ion source. The energy of neutrons is 14.2MeV
which is high enough to make 16 N by the (n,p) reaction on 16 O [41] in water. The 16 N thendecays with a half-life of 7.13 second with gamma-ray and/or electrons. The dominant decay
mode (66%)produces a 6.1MeV γ with an electron of 4.3MeV endpoint energy, while the second
dominant decay mode (28%) produce a 10.41MeV electron. The advantage of this calibration isfollowing; First, as shown in the Figure 6.11, this calibration is easier to operate compared to
LINAC. Second, the data can be taken in many places. Third, the event direction is isotropic.
Taking advantage of DT neutron generator, the data are uesd to study the uncertainty of energy
scale depending on position and direction (see Section 6.7.2).

6.5.2

Trigger eﬃciency

In addition to data taking with the normal LE and SLE trigger threshold, special calibration
data was taken with the threshold value at -150mV. With the special data, the trigger eﬃciency
is deﬁned as follow ;NLE,jEf fLE,j =(6.4)
Nspecial,j53

Figure 6.11: Schematic view of DT neutron generator calibration setup. (a) The generator isinserted in the tank, (b) Neutrons are created, (c) The generator is lifted 2 meters, so that we
can make the shadow eﬀect of the the generator as small as possible.

Ef fSLE,j =NSLE,jNspecial,j(6.5)

where NLE(SLE),j is the number of events in j-th energy bin triggered by LE (SLE) threshold,and Nspecial,j is that of the special threshold. Figure 6.12 shows the trigger eﬃciency measured
by DT calibration compared with trigger simulation of MC. The diﬀerence between data andMC is 2.0% in 4.5-5MeV, 1.0% in 5-5.5MeV, 0.5% in 5.5-6MeV, and 0.1% in 6-6.5MeV (SLE2
period). The trigger eﬃciency in higher energy region is 100%, thus the diﬀerence is negligible.

6.6

LINAC Calibration

6.6.1

the LINAC system

An introduction of the LINAC system will be given. The detail information can be found in[42]. Figure 6.13 and 6.14 show the schematic view of the LINAC system and structure of theend cap. The LINAC is a model ML-15MIII from Mitsubishi. During the operation, the beam
pipe is evacuated to less than 0.1Pa, and the beam size and momentum is tuned by collimatorand magnets. The energy range of the electron is selected from 4.4-18MeV, so that we can makea similar event as the recoil electron event. Other speciﬁcations are summarized in Table 6.1
Due to the small systematic uncertainty of the system [42], we can use the LINAC data not only
for the absolute energy scale calibration but also for various systematic uncertainty studies.

54

'HHKEKGPE[

1.2

1

&CVC
/%

0.8

0.6

0.4

0.2

0

2

3

4

5

6

7



'PGTI[=/G8?
Figure 6.12: An example of SLE trigger eﬃciency obtained by DT data and MC

Figure 6.13: The LINAC system

Figure 6.14: The structure of end cap

55

Energy range
Intensity
Number of bunches
Beam energy spread
Beam size
Vacuum

4.4 - 18 MeV∼ 0.1 electron/bunch
adjusted by the intensity of electron gun
up to ∼ 90 bunches/sec< ±0.3%1 ∼ 2cm at end cap∼ 10−4 Pa in accelerator tube
<0.1 Pa in beam pipe

Table 6.1: LINAC speciﬁcation
Beam energy calibration
The beam energy is measured by a Ge detector. The crystal size of the Ge detector is 57.5mm
in diameter and 66.4mm in length. The energy resolution is 1.92keV for 1.33 MeV γ from 60 Co.
At the end of data taking for each position and each beam momentum, the Ge detector is placed
just behind of D3 magnet whose current set to 0A. There is the same titanium window at theposition. The energy evaluation by Ge is done by comparison with Ge detector MC simulationconsidering the energy losses in the Ti window, the Be window of the Ge detector, and inactive
region of the Ge detector. The total energy in SK detector is obtained from the measured beam
momentum. The energy loss of the trigger counter and the Ti window is taken into accountin that calculation. The uncertainty of the beam energy is evaluated as ±20keV at 9.52MeV(0.21%).
Data set
LINAC data was taken in March 2008. Beam energies taken in that time are 5.1, 8.8, 13.6,
18.9 MeV. The beam positions are shown in Figure 6.13.

6.6.2

Tuning parameter results

Average Q.E.
The average Q.E. (COREPMT) is obtained by the following steps. For this study, 8.8MeV
mode is used.
1. Make MC of all positions with diﬀerent COREPMTs
2. Get peak value of Nef f
3. Compare the peak values with that of DATA.
4. Find COREPMT which gives minimum position dependence.
As the result, obtained value of COREPMT is 0.888. Figure 6.15 shows the tuning result. Figure
6.16 shows the energy scale diﬀerence between data and MC simulation. The position averaged
diﬀerence between LINAC data and MC is within 0.5% for all energies.
56

Figure 6.15: Tuning result of COREPMT Diﬀ in the right plot shows the diﬀerence between the
maximum and minimum of R(=MC/DATA).

After pulse
If a photo-electron is back scattered at the ﬁrst diode, it takes some time until it loses its
velocity against the electric ﬁeld then comes back to the ﬁrst dynode. In such case, the hit timing
delays a certain amount and makes an after pulse. The fraction and the delay time are tuned
by LINAC hit timing distribution. Tuning result shows that the characteristic components of
the after pulses are at 111.5 nsec (1.4%) and 39.0 nsec (1.3%) after the main pulse (see Figure
4.2).
Timing resolution
Timing resolution determines the vertex resolution, and thus it aﬀects the vertex resolution
directly. Therefore, the tuning of timing resolution is important in terms of reducing systematic uncertainty of MC. In this analysis, I have performed a very precise tuning of the timing
resolution.

After these tuning done, the timing distribution of LINAC data and MC agree very well asshown in Figure 6.17.

6.7

Energy scale

6.7.1

Nef f to Energy

After the all detector simulation tuning, the conversion function from Nef f to energy is
obtained by uniformly generated electron MC events. This energy is electron total energy. The

57

100

Neff of MC

140
120
100

80

80

60

60

40

40
4

6

8

10

12

14

16

18

20

20

0.05

(MC-DATA)/DATA

Neff of DATA

120

20
(MC-DATA)/DATA

X = -1237cm, Z = 1197cm
X = -1237cm, Z = -6cm
X = -1237cm, Z = -1209cm
X = -388.9cm, Z = 1197cm
X = -388.9cm, Z = -6cm
X = -388.9cm, Z = -1209cm

140

0.04
0.03
0.02
0.01
0
-0.01

14

16

18

-0.05

20

16

18

20

4

6

8

10

12

14

16

18

20

0

-0.04
12

14

-0.01
-0.03

10

12

0.01

-0.04
8

10

0.02

-0.03

6

8

0.03

-0.02

4

6

0.04

-0.02

-0.05

4

0.05

expected energy in tank
Figure 6.16: Energy scale diﬀerence of LINAC data and MC simulation . The left bottom plotshows the diﬀerence of each position (makers correspond to the position of Figure 6.13 ) The
right bottom plot shows the position averaged diﬀerence.

58

0QTOCNK\GFD[PWODGTQHGXGPVU

&CVC
/%

6KUM61(6=PUGE?
Figure 6.17: Timing distribution after tuning MC. LINAC 5.1MeV (x,z)=(-12m,12m)

relation between the energy and Nef f is given
E = p1 + (p2 + (p3 + (p4 + p5 Nef f )Nef f )Nef f )Nef f(6.6)

where pi (i = 1..5) are ﬁtting parameter. Figure 6.18 shows the ﬁtting result. The obtained
parameters are
p1 = 0.82221
p2 = 0.12897
p3 = −9.49202 × 10−5
p4 = 1.09289 × 10−6
p5 = −2.91088 × 10−9

6.7.2

Energy scale uncertainty

Position dependence of Energy scale
To estimate the position dependence of energy scale, the z-dependence and phi-dependence
should be taken into account. The z-dependence is measured by LINAC, and the ϕ-dependence
is measured by DT because the large y position is not included in the calibration positions for
LINAC.
Figure 6.19 shows the diﬀerence of energy scale between LINAC data and LINAC MC depending on the LINAC position. By averaging over all positions, the z-dependence is estimatedas 0.06% (diﬀerence between r=4m and r=12m is also included here.)
59

6000

5000

4000

3000

2000

1000
0

0

10

20

30

40

50

60

70

80

90

100

Generated total energy (MeV)

4EQPUVTWEVGFVQVCNGPGTI[ /G8
80
70
60
50
40
30
20
10
0

0

100

200

300

400

0GHH

500

600

700

Figure 6.18: Energy distribution of electron MC simulation (upper), and Nef f to energy function(lower) In lower plot, the maker shows the relation of the mean of Nef f and generated electron
total energy, and the line shows the ﬁtted polynominal function.

-

M C L IN A C /L IN A C

䃂 5 M eV
䂓 8 M eV
䂥 1 3 M eV
䂯 1 8 M eV

X: -12 -12 -12 -4
Z: 12 0 -12 12

-4
0

-4
-12

Figure 6.19: Diﬀerence of Nef f mean between LINAC Data and MC simulation. Black markers
show each energy, and red markers show energy averaged values.

60

&CVC/% &CVC CXGTCIGQHUCOG<RQUKVKQP< O<O<O



[


\


  


Z

&GVGEVQT㨪㨪 㨪 

2QUKVKQPQH&6PGWVTQPIGPGTCVQT

Figure 6.20: Diﬀerence of Nef f mean between DT Data and MC simulation. The deﬁnition ofhorizontal axis is shown in the ﬁgure.

Figure 6.20 shows the diﬀerence of energy scale between DT data and DT MC the horizontal
axis is normalized by the average of same z positions. The mean value of deviation from theaverage is taken as a phi-dependence of energy scale which is 0.35%.
Angular dependence of energy scale
The directional uniformity of the energy scale can be obtained by DT calibration. Since
the LINAC beam is only downward-going, the directional uncertainty of energy scale should bechecked and added to the total energy scale uncertainty. This is also important when studying
the energy spectrum distortion and the day/night eﬀect caused by the MSW eﬀect.
Figure 6.21 shows energy scale diﬀerence between DT data and DT MC depending on theevent direction. The diﬀerence between LINAC direction (the left most bin of the Figure 6.21)
and the average of other bins is 0.02%. Conservatively, the size of error is taken as the angulardependence of energy scale which is 0.25%.
Eﬀect of Water transparency accuracy to energy scale
The uncertainty of the energy scale is also depending on the uncertainties of water transparency measurement when the LINAC calibration data was taken. The uncertainty of thewater transparency measurement is evaluated by ﬁtting error distribution. Figure 6.22 shows a
distribution of statistical uncertainty of water transparency of every 6day period which is obtained from the ﬁtting error of the transparency slope. The average value of uncertainty for 6day
measurement is taken as the uncertainty of water transparency during LINAC run period. The
average value 159cm is correspond to 0.2% of energy scale which is obtained from MC generated
with artiﬁcially increased water transparency.
61

0GHH/%&#6#&#6#CXGTCIG





0QVWUGFFWG
VQUJCFQYGHHGEVFKT<Figure 6.21: Diﬀerence of Nef f mean between DT Data and MC simulation depending on
direction.

25

20

15

10

5

0
0

50 100 150 200 250 300 350 400 450 500(KVWPEGTVCKPV[=EO?
Figure 6.22: Fit uncertainty of water transparency measurement.

62

Position dep.
Direction dep.
WT. uncertainty
LINAC energy sys.

0.35 %
0.25 %
0.20 %
0.21 %

Total

0.53 %

Table 6.2: Systematic uncertainty of energy scale
Electron beam energy determination
Uncertainty of electron beam energy determination of LINAC system is estimated as 0.21 %[42].
Total energy scale systematic uncertainty
Adding all the contributions in quadrature, the total uncertainty of SK-III energy scale isestimated as 0.53% (see Table 6.2). The corresponding value for SK-I is 0.64%. About 20% of
improvement can be seen in SK-III.

6.8

Energy resolution

Energy resolution function is obtained from MC which is same as the one the energy function
is made from. Figure 6.23 shows the energy resolution as a function of the reconstructed energy.
The ﬁtted energy resolution function is:
σ(E) =√
1(−0.123 + 0.376 E + 0.0349E)
E

(6.7)

where E is the energy of an event (MeV). The diﬀerence of the energy resolution between data
and MC is checked by LINAC, and it is shown in Figure 6.24. The observed shift seemed on
direction, but I took the diﬀerence between data and MC is ±2.5%.

6.9

Angular resolution

An absolute angular resolution is deﬁned as an angle which include 68% of the reconstructed
directions around the beam direction. The angular resolution as a function of electron energy
is shown in Figure 6.25. The diﬀerence between data and MC is 0.7% at maximum.
This angular resolution shows about 10% improvement in 5MeV region if it is compared tothat in SK-I [20].

63Energy resolution

0.3

SK-III:( -0.12253 + 0.376394 sqrt(E) + 0.0349229 E) / E

0.25
0.2

0.15
0.1
0.05

0

5

10

15

20

25

30
35
40
Total energy (MeV)

Figure 6.23: Energy resolution function obtained by electron MC simulation . Black points show
the one standard deviation of Gaussian ﬁt of MC simulation (see top plot of Figure6.18), while
the line shows the ﬁtting by the polynominal function.

64

22
20
18
16

Resolution of MC(%)

24

26
24
22
20
18
16

14

14

12

12

10
(MC-DATA)/DATA

X = -1237cm, Z = 1197cm
X = -1237cm, Z = -6cm
X = -1237cm, Z = -1209cm
X = -388.9cm, Z = 1197cm
X = -388.9cm, Z = -6cm
X = -388.9cm, Z = -1209cm

4

6

8

10

12

14

16

18

10

20

0.2

(MC-DATA)/DATA

Resolution of DATA(%)

26

0.15
0.1
0.05
0
-0.05

10

12

14

16

18

14

16

18

20

4

6

8

10

12

14

16

18

20

-0.05

-0.2

20

12

0

-0.15
8

10

0.05

-0.15
6

8

0.1

-0.1

4

6

0.15

-0.1

-0.2

4

0.2

expected energy in tank

Figure 6.24: Energy resolution of LINAC data and MC simulation The left bottom plot shows
the diﬀerence of each position (makers correspond to the position of Figure 6.13 ) The rightbottom plot shows the position averaged diﬀerence.

65

#PIWNCTTGUQNWVKQP FCVC
=FGI?

=FGI?

#PIWNCTTGUQNWVKQP /%5[U EQU 4GU/% EQU 4GU&CV  EQU 4GU/%
0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05

0

0

−0.05

−0.05

−0.1

−0.1

−0.15

−0.15

−0.2

4

6

8

10

12

14

16

18

−0.2

20

2QUKVKQPCXGTCIGQH5[U5-+5-+++

4

6

8

10

12

14

16

18

20

'ZRGEVGFGPGTI[KPVCPM=/G8?
Figure 6.25: Angular resolution of LINAC data and MC simulation

66

Chapter 7

Data Analysis
The data for SK-III solar neutrino analysis were taken from August 5, 2006 to August 17,
2008. SK-III has three periods with diﬀerent hardware threshold settings. The explanation ofeach period is summarized in Table 7.1

7.1

Monitoring of Radon level

A major background for the lower energy region (total energy is less than 6.5 MeV) is
radioactive impurities such as radon which are mixed in the pure water. To achieve the low
energy threshold at 4.5 MeV, it is needed to reject such high radon concentration periods that
happened after water system maintenance or detector calibration, etc.. In order to do this,the radioactivity at diﬀerent positions of the detector has been monitored by checking low
energy event rate. Figure 7.1 shows how the low-energy event rate changes at the diﬀerent
detector positions. The high event rate periods coincide with the water system maintenance anddetector calibration. These high rate are caused by remaining radon in water. After removing
the high radioactivity periods, the sample for low-energy threshold analysis is obtained. The
corresponding livetime is 298.2 days. Since the radon event only aﬀect the low energy region
the periods are not removed for sample with 6.5 MeV threshold whose livetime is 547.9 days.

7.2

Run deﬁnition

A basic unit of dataset is a run. Each run is at most 24 hours long, and divided into subruns
which have usually 1 to 5 minutes long depending on the trigger rate.
Date (From Y/M/D to Y/M/D)

Hardware thr. (mV)

100% eﬀ. above: (MeV)

livetime (days)

2006/08/05 - 2007/01/24

-302

6.5

121.7

2007/01/24 - 2008/04/17

-212

5.0

331.5

2008/04/17 - 2008/08/18

-186

4.5

94.8

Table 7.1: Summary of run periods with diﬀerent hardware threshold values .

67

Figure 7.1: Low energy event rates in diﬀerent Z positions. The horizontal axis is date
(mm/dd/yy). The noise cut and quality cuts (not the ﬁnal tune) are applied.

cts/day/kt/(5.5-4.5MeV)

cts/day/kt/(5.5-4.5MeV)

cts/day/kt/(5.5-4.5MeV)

cts/day/kt/(5.5-4.5MeV)

2m fid cut
cut -402.5<-402.5<Z<0
5m fid cut
cut -402.5<-402.5<Z<0

01/
01/08
11/01/0701/01/0803/01/08
09/01/08

09/01/0711/01/0701/01/0803/01/0805/01/0807/01/08
09/01/08
07/
07/01/07
09/01/07
09/
01/07
01/
01/08
01/0807/
01/08
07/
07/01/

2m fid cu
cutt -1207-1207.5<Z<-80-1207.5<Z<-805.5<Z<-805.5<Z<-80
5m fid cu
cutt -1207-1207.5<Z<-80-1207.5<Z<-805.5<Z<-805.5<Z<-80

2m fid cu
cutt -805<-805<Z<-402.5-805<Z<-402
Z<-402
Z<-402.5
Z<-402.
5m fid cu
cutt -805<-805<Z<-402.5-805<Z<-402
Z<-402
Z<-402.5
Z<-402.

&GCFVKOG

1000
900
2m fid cu
cutt -1610-1610<Z<-1207.5-1610<Z<-12<Z<-12<Z<-1207.5<Z<-1207.
800
700
600
500
400
300
200
100
0
01/01/0703/01/0705/01/0707/01/07 09/01/0711/01/0701/01/0803/01/0805/01/0807/01/08 09/01/08

900
800
700
600
500
400
300
200
100
0

0

100

200

300

400

500

600

0
01/01/0703/01/0705/01/0707/01/07
09/01/08
01/01/
01/01/07
01/07
05/01/
01/07
07 07/
07/01/07 09/01/0711/01/0701/01/0803/01/0805/01/0807/01/08
09/01/07
09/
01/07
01/
01/08
01/0807/
01/08
07/
07/01/

50

100

150

200

250

cts/day/kt/(5.5-4.5MeV)
cts/day/kt/(5.5-4.5MeV)
cts/day/kt/(5.5-4.5MeV)
cts/day/kt/(5.5-4.5MeV)

68

2m fid cu
cutt 1207.5<Z<1610
1207.5<Z<1610

09/01/08

2m fid cu
cutt 0<Z<402
0<Z<402.5
5m fid cu
cutt 0<Z<402
0<Z<402.5%WVETKVGTKC

001/01/0703/01/0705/01/0707/01/07 09/01/0711/01/0701/01/0803/01/0805/01/0807/01/08 09/01/08

100

200

300

400

500

600

700

11/01/0701/01/08

07/01/07
09/01/0711/01/0701/01/0803/01/0805/01/0807/01/08
09/01/08
07/
09/01/07
09/
01/07
01/
01/08
01/0807/
01/08
07/
07/01/

2m fid cu
cutt 402.5<Z<8
402.5<Z<805055m fid cu
cutt 402.5<Z<8
402.5<Z<80505

03/01/07

2m fid cut
cut 805<Z<1
805<Z<1207.5
805<Z<1207.55m fid cut
cut 805<Z<1
805<Z<1207.5
805<Z<1207.5

001/01/0703/01/07
01/01/
01/01/07
01/07

50

100

150

200

250

300

350

0

50

100

150

200

250

300

350

0
01/01/07
05/01/
01/07 07/
07/01/07 09/01/0711/01/0701/01/0803/01/0805/01/0807/01/08
09/
09/01/07
01/08
01/0807/
07/01/ 09/01/08
01/01/0703/01/0705/01/0707/01/07

50

100

150

200

250

300

The data quality is checked with subrun or run unit. The criteria for bad run or subrun are
following;• The run length is less than 5 min.
• The subrun length is less than 30 sec.
• The one of event reduction rates of a subrun is higher than the allowed rate.
• Hardware and/or software problems happen• Calibration run

7.3

Noise reduction

7.3.1

Total charge cut

Events with the total charge more than 2000 p.e. are rejected. This cut is to remove cosmic
ray muons. 2000 p.e. corresponds to about 300 MeV for an electron, so no solar neutrino eventsare cut by this criterion.

7.3.2

Fiducial volume cut

Events which have vertex position within 2m from the ID wall are rejected. This cut is to
reduce background events coming from the surrounding rock or the ID structure. 2m is set
considering the resolution of vertex position (∼100cm at 6.5MeV). The volume 2m of the ID
wall is called ﬁducial volume.
A tighter ﬁducial volume cut is applied for the event with energy less than 5.5 MeV as
explained later in this chapter.

7.3.3

Pedestal/Uncompleted event cut

The pedestal data was taken at the beginning and 30 min after the beginning of every run.
After that, the 8 online computers, each handles 6 TKO crates, check temperatures of 6 cratein every 30min. If more than or equal to one of 6 TKOs have a diﬀerent temperature by ±0.15 degree from the previous pedestal time, the online computer takes pedestal data for the 6TKOs. If pedestal data is not taken for more than 6 hours by a online computer, the online
computer takes pedestal data for the 6 TKOs.
Other than the pedestal data taking, some parts of the detector happen to become oﬀ due
to hardware problem. The data taken in such period (2 ∼ 3%)is not included in this analysis.

7.3.4

Time diﬀerence cut

Events are rejected if they occurred within 50µsec after the previous event. This is to remove
electrons from the decay of cosmic ray muons. Figure 7.2 shows the distribution of time interval
in a typical data. The fractional dead time due to this cut is ∼ 2 × 10−4 , which is taken into
account in the solar neutrino simulation.

69

10

5

10

4

10

3

10

2

EWV

10

1
2

3

4

5

6

7

.QI F6

8

9

10

PUGE

Figure 7.2: Time diﬀerence to previous low energy triggered event. The ﬁrst peak around 1µsec
is due to ringing events after a high energy cosmic muon, and the second peak around 15µsec
is due to after pulse events, and the third peak around 30µsec is so called ”after trigger” event,which is issued after HE triggered event.

7.3.5

OD event cut

Events with an OD trigger are rejected. An OD trigger is issued if the OD hits are morethan 19 his within a 200 nsec time window (see Figure 7.3). This is to remove cosmic ray muon
event with total p.e. less than 2000 p.e.

7.3.6

Noise cut

Event with high fraction of noise hits are rejected. This is to remove events due to electronicnoise. The fraction of noise hits is deﬁned as follows;
N Sratio = Nnoise /Ntotal(7.1)

where Nnoise is the number of hit PMTs with charge less than 0.5 p.e., and Ntotal is the total
number of hit PMTs. If N Sratio is greater than 0.4, the event is rejected. Figure 7.4 shows
N Sratio distribution of a typical good run.

7.3.7

Goodness of vertex reconstruction cut

Events with poor vertex reconstruction are rejected. Figure 7.5 shows the goodness of thesecond vertex ﬁtting [20]. If a event has goodness < 0.3, the event is rejected. Some broken
PMT made ﬂash light. These ﬂasher events are usually clustered in goodness < 0.3 region.

70

0WODGTQH1&JKVU
Figure 7.3: Number of OD hits distribution of a typical good run. Solid histogram includes
events with OD trigger on. Dashed histogram corresponds to the events with OD trigger oﬀ.
Events in the right side of the red line are rejected.

Figure 7.4: N/S ratio in a typical good run. Events in the right side of the red line are rejected.

71)QQFPGUU
Figure 7.5: Goodness distribution in linear (top) and log (bottom) scale of a typical good run.
Events in the left side of the red line are rejected.

7.4

Reduction for solar ν analysis

In this section, I will explain a series of further reductions which select a good single electron
events. The total number of events after each reduction step and the eﬃciencies of each reduction
will be summarized at the end of this section, and the systematic uncertainties due to each itemof reductions are discussed in Chapter 9

7.4.1

Spallation Cut

This cut is to reject events caused by cosmic ray muons. When a muon interacts with an
oxygen, radioactive elements are produced;µ +16 O −→ µ + X(7.2)

where X represents radioactive nuclei. Possible radioactive nuclei are summarized in Table
7.2. The γ and β decay of such a radioactive nuclei trigger the detector and make background
events so called ”Spallation events”, which are main background source in energy region 6.5 to
20 MeV. Here, I explain an overview of spallation event cut. The details of muon selection,
muon track reconstruction, and how to built a likelihood function to identify a spallation event
was described in [20].
For each low energy event, a likelihood value is calculated for every preceding muon within
100 sec. The likelihood is a product of three functions;
Lspa = f (∆t) × f (∆l) × f (Qres )(7.3)

where ∆l is a distance from the low energy event vertex to the reconstructed muon track.
∆t is a time diﬀerence between the low energy and a preceding muon event. Qres is a residual72

Isotopeτ 1 (sec)

decay mode

Kinetic Energy (MeV)

2

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−

∼ 13

0.77

β+

13.9

0.178

β−
β− n

13.6 (50.5 % )(∼ 50 % )

0.127

β+ n

3∼15

0.0085β−16∼20 (∼ 50 % )∼ 16 (∼ 50 % )

9C
6
11 Li
3

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

∼4

7.13

β−

10.42 ( 28.0% )
4.29+6.13 (γ) (66.2% )Table 7.2: Possible Spallation products in water [20]73

&GCFVKOG%WV

5RCNNCVKQPNKMGNKJQQF

Figure 7.6: Spallation likelihood distribution. Blue histogram shows spallation like events, andred histogram shows random sample (The areas are normalized).

charge of the preceding muon which is deﬁned as
Qres = Qtotal − Qunit (t) × Lµ

(7.4)

the Qtotal is the total charge (p.e.) of a muon event, Qunit (t) is the average number of photoelectrons per unit length of a muon track observed in a time period t .
The spallation likelihood for a low energy event then obtained as the maximum value of thelikelihood values, and the muon which gives the maximum value is called a parent muon. Figure
7.6 shows the distribution of the spallation likelihood in spallation like event and randomly
sampled events. For spallation like event, events with ∆t < 0.1sec and energy> 8MeV are
selected. For a random sample, events with energy <5MeV are selected because most of such
low energy events are produced near the detector wall due to radioactive impurities and are notto spallation event. A random vertex is assigned to the vertex position of the random sample
event. The cut point for the spallation likelihood is 4.52, so that the ineﬃciency of the random
sample (dead time) becomes 0.20. The position dependence of the dead time is shown Figure7.7. This position dependence is taken into account when the expected number of solar neutrino
events is calculated using MC.

7.4.2

Ambient background reduction

Vertex and Angular Reconstruction Quality check
There still remain many background events due to mis-reconstruction inside the ﬁducial
volume after noise reduction. To check the quality of vertex and angular reconstruction andremove the mis-reconstructed events, the 2 dimensional vertex and angular goodness cut isapplied. Since the mis-reconstruction usually occurs for low energy background (<∼ 6MeV),
such events has small number of hits. Thus, there is no hit selection to calculate goodness ofvertex reconstruction, so that we get information as much as possible from all hit PMTs. The

74

URCNNCVKQPEWVFGCFVKOG

\=EO?


T=EO?

Figure 7.7: Spallation cut dead time depending on the detector position.

goodness vertex reconstruction is calculated using all hit PMT timing as follows,∑ − 21 ( τi (⃗vω)−t0 )+( τi (⃗vσ)−t0 ) 2eGV =∑ − 1 ( τi (⃗v)−t0 )2
ω
e 2

(7.5)

where τi (⃗v ) = ti − |⃗v − ⃗hi |/c is a TOF subtracted hit time, and t0 is the ﬁtted peak time ofτ distribution. The sums are over all hits.
The goodness of angular reconstruction can be tested by checking how the hit PMTs distribute in space. In the case of a real electron event, the hit PMTs should reside uniformly
along with Cherenkov ring toward the reconstructed direction. To check the uniformity, theKolomogorov-Smirnov test is applied. Figure 7.8 shows examples of event displays for a single
electron event and a background event. Figure 7.8 also shows the distribution of azimuth angleof hit PMTs with respect to the reconstructed direction. The deﬁnition of goodness is the full
width of azimuth angle deviation which is given byGA =max {∠unif orm (i) − ∠Data (i)} − min {∠unif orm (i) − ∠Data (i)}
2π

(7.6)

where ∠unif orm (i) is the azimuth angle of i-th hit PMT among the N50 assuming that the hit
PMTs are uniformly distributed along with a Cherenkov cone (the dashed line in Figure7.8),
and ∠Data (i) is that of real event (the closed circles in Figure7.8). As shown in the ﬁgures, agood event has a small value and a bad event has a large value.
The correlation between the vertex goodness GV and the angular goodness GA is shown inﬁgure 7.9. Figure 7.9-(a) is a plot of real data which is just after the 1st reduction and stillcontains a lot of badly reconstructed events, and ﬁgure 7.9-(b) is that of solar neutrino MC. As
shown in the ﬁgure, we can separate a good single electron like event from background eventsby checking a quality factor which is a combination of the vertex goodness and the angulargoodness ovaQ (One dimensional variable of Vertex and Angular reconstruction Quality);
75

Figure 7.8: Examples of direction KS test for good event (top) and bad event (bottom).

76

C

D

Figure 7.9: Correlation of G2V (horizontal) and G2A (vertical). Line indicate G2A = G2V − 0.25

ovaQ ≡ G2V − G2A(7.7)

Distributions of the quality factor for several energy are shown in ﬁgure 7.10. The cut criteria
is following; ovaQ > 0.29, 4.5 ≤ Energy < 5.0M eV(7.8)
ovaQ > 0.25, 5.0 ≤ Energy < 7.5M eV
ovaQ > 0.20, Energy ≥ 7.5M eV
Flasher cut
If there is a PMT which emits light because of arch discharge on dynodes, such a PMT
causes background events so called ”ﬂasher events”. The ﬂasher event is characterized with alarge charge hit and many hits in the surrounding PMTs.
Figure 7.11 shows the maximum charge of a PMT in an event for the horizontal axis, andthe number of hits in the surrounding 24 PMTs. The cut criteria are also shown in the Figure
7.11.
Hit Pattern cut
This cut is to select a clean single electron event. If additional γ-rays in an event, the hit
pattern does not show a clear Cherenkov ring, then the reconstructed direction might not becorrect. To select a single electron event, a likelihood function is made from cos θdir,i distributionof the uniform electron MC event; Figure 7.14 shows examples of the Cherenkov ring probability

77

4.5-5MeV

0.1

5-5.5MeV

0.1

Data
B8MC
0

0

0.2 0.4 0.6 0.8
ovaq

Data
B8MC
0

1

5.5-6MeV

0.1

0

0.2 0.4 0.6 0.8
ovaq
6-6.5MeV

0.1

Data
B8MC
0

0

0.2 0.4 0.6 0.8
ovaq

Data
B8MC
0

1

6.5-7MeV

0.1

0

0.2 0.4 0.6 0.8
ovaq

0

0.2 0.4 0.6 0.8
ovaq

Data
B8MC
0

1

7.5-8.5MeV

0.1

0

0.2 0.4 0.6 0.8
ovaq

0

0.2 0.4 0.6 0.8
ovaq

1

8.5-9.5MeV

0.1

Data
B8MC
0

1

7-7.5MeV

0.1

Data
B8MC
0

1

Data
B8MC
0

1

0

0.2 0.4 0.6 0.8
ovaq

Figure 7.10: ovaQ distribution of diﬀerent energy region.
78

1

Figure 7.11: 2 dimensional cut of ﬂasher events. The top ﬁgure contains ﬂasher sample, while
the bottom ﬁgure shows a typical good run.

RQUYCNN
FGHH
XGTVGZFKTGEVKQP

HYCNN

SK detector
Figure 7.12: Deﬁnition of variables which appear in hit pattern cut and gamma-ray cut.

79

functions for several energy regions. The deﬁnition of Cherenkov ring probability is50
1 ∑P (Energy, vertex, diretion) =log(f (Energy, cos θdir,i , f wall))
N50

N

(7.9)

i

where f wall is the distance from the event vertex to the detector wall (See Figure 7.12) along withthe reconstructed direction, and probability functions f (Energy, cos θdir.i , f wall) are obtained
from single electron MC with diﬀerent energies. To determine the cut point of P , the signiﬁcanceis calculated as∫P
M C(P ′ )dP ′(7.10)
Signif icance(P ) = √P∫0P′ )dP ′
BG(PP0
where M C(P ) and BG(P ) are distributions of the Cherenkov ring probability obtained fromsingle electron MC and control sample of real data and P0 is the likelihood value at the left edgeof Figure 7.13. The obtained cut criteria which give maximum signiﬁcance for diﬀerent energy
regions are P > −1.88, 6.5M eV ≤ Energy < 8.0M eV(7.11)
P > −1.86, 8.0M eV ≤ Energy < 12.0M eV
P > −1.95, 12.0M eV ≤ Energy
Since, for an event whose energy is less than 6.5 MeV, the distributions of likelihood for dataand solar neutrino MC become the same (see Figure 7.14), I decided not to use this cut for theenergy region.
Fiducial volume cut using the second vertex ﬁtter
The result of second vertex reconstruction is checked here. If the reconstruction returns avertex position outside of the ﬁducial volume (2m inside of the ID wall), the event is rejected.

7.4.3

Gamma-ray cut

This cut is to reject events induced by gamma-ray which are from the PMTs or the detector
wall. To eliminate the background, events with direction inward from the wall are rejected. The
criteria for event energy ≥ 6.5M eV are{
def f > 650cm, 6.5M eV ≤ Energy < 8.0M eV(7.12)
def f > 400cm, 8.0M eV ≤ Energy
where def f is the distance from the wall in the direction of the reconstructed event (see Figure7.12), and this cut points are obtained by calculation of signiﬁcance as in Equation 7.10 but thesigniﬁcance here is a function of def f .
Figure 7.15 shows the distribution of def f for diﬀerent energy regions. Because the radioactive background (energy <6.5 MeV ) is not uniformly distributed in the detector (see ??), atighter ﬁducial volume cut is needed. Considering the tighter ﬁducial volume which will betuned in the later step, events in the central area (z > −5m,r< 13m) are selected to tune the
gamma-ray cut point of def f for energy < 6.5M eV region. Then, for this energy region, diﬀerent

80

0.1

0.05

12-20MeV

0
-2.5
0.1

0.05

0.05

-2

-1.5

-1

.KMGNKJQQF

0
-2.5

-2

-1.5

-1

significance

8-12MeV1

best=-1.86
Data(62%)
B8MC(87%)

0
-2.5
0.1

1

best=-1.95
Data(91%)
B8MC(98%)

-2

-1.5

-1

.KMGNKJQQF

0
-2.5

-2

-1.5

-1

significance

6.5-8MeV1

best=-1.88
Data(71%)
B8MC(89%)

0
-2.5

-2

-1.5

-1

0
-2.5

-2

.KMGNKJQQF

-1.5

-1

significanceFigure 7.13: Hit pattern likelihood distribution of data and solar neutrino MC in E>6.5MeV.

0.05
0.045
0.04
0.035
0.03
0.025


&#6#
$/%

0.02
0.015
0.01
0.005
0

−2.4

−2.2

−2

−1.8

−1.6

−1.4

−1.2

−1

.KMGNKJQQF
Figure 7.14: Hit pattern likelihood distribution 4.5≤E<6.5MeV.

81

CTGCPQTOCNK\GF

/G8&#6#$/%
RQUYCNNVQR

/G8&#6#$/%/G8&#6#$/%
RQUYCNNDCTTGN

/G8&#6#$/%/G8&#6#$/%
RQUYCNNDQVVQO 
FGHH=O?
Figure 7.15: def f distribution for diﬀerent energy regions. The cut criteria are indicated in theline.

cut criteria are applied depending on where the original position at the detector wall (poswall)is (see Figure 7.15);

poswall is top def f > 1000cm,(7.13)
def f > 1100cm,
poswall is barrel
def f > 1300cm,
poswall is bottom

7.4.4

Cosmogenic 16 N cut

This cut is to reject events caused by decay of 16 N The 16 N is produced when cosmic rayµ− is captured by 16 O in water;µ− +16 O −→16 N + ν(7.14)The most of 16 N produces a 6.1 MeV γ with a β whose end point energy is 4.3 MeV; its halflife is 7.13 second.
In order to reject this background, the correlation between all captured muons and all remaining low energy event are checked. The ﬁrst step is to select only captured muon event which
are characterized as following; Muon with no exit point(stopping muon) and is not followed by adecay electron event in 100µsec. After the selection, a low energy event within 250cm from the82

*KVVKOKPIYKVJQWVUWDVTWEVKPI61(%JCTIG=RG?

Figure 7.16: A typycal event display of low-energy background.

stopping point of the captured muon and a time-window of 100msec to 30sec following the muonis rejected. The total number of the rejected events 4750 events. The dead time due to this cut
is estimated accidental coincidence between the selected muons and random event sample which
is 0.53%.

7.4.5

Further quality cut for the lowest energy region

Clustering hits background
This cut is newly developed in this analysis to reduce the low energy background in edge
region. Figure 7.16 shows one example of remaining background. This kind of event is supposed
to be triggered by a coincidence of dark hits and the small clustering hits due to radioactive
sources in the FRP or the structure of the detector wall.
We can separate this background from the solar neutrino signals by searching a small cluster
in both space and time. A real neutrino signal at the edge region also has similar characteristics,
but it causes a bigger cluster compared to the background events. Thus, the key is to evaluate
the size of clustering hits.
In order to do this, two variables are used. One is to check the size of clustering hits in space;
the minimum distance from any of N20 PMTs to another N20 PMT within which more than 0.2
number of N20 are included (called ”R02”). The other variable is to check the clustering hits intime; the maximum number of hits in 20nsec sliding time window search ”without” subtracting
TOF from the reconstructed vertex (called ”N 20rawT ”). The reason why TOF is not subtracted
is that for the background case, except for the small clustering, hit PMT positions should be
randomly distributed while for the real signal in the edge of ﬁducial volume, hit PMT positions
should be concentrated close to the wall. Thus, timing distribution without subtracting TOF
makes the diﬀerence between the background and the signal clearer.
Figure 7.17 shows correlation between R02 and N 20rawT for background sample data andsolar neutrino MC events. The background sample is selected from the data after the gamma
cut applied around the edge of ﬁducial volume (r>13m). To compare the background with MC
events, the same cuts are applied. As expected, the background sample shows smaller R02
and smaller N 20rawT (which is indicated by the orange area in Figure 7.17 ). The cut point83

Figure 7.17: Correlation between R02 and N 20rawT for background sample (left) and solar
neutrino MC events (right).

is obtained from one dimensional distribution of R02 × N 20rawT /Nef f . The signiﬁcance as afunction of R02 × N 20rawT /Nef f becomes maximum at R02 × N 20rawT /Nef f = 75 (see Figure7.18). Following is a summary of cut criteria of this new cut;22
 R02 × N 20rawT /Nef f > 75 and r > 180m for 5.5 ≤E< 6.5 MeV(7.15)
R02 × N 20rawT /Nef f > 75 and r2 > 155m2 for 5.0 ≤E< 5.5 MeV
R02 × N 20rawT /Nef f > 75 and r2 > 120m2 for 4.5 ≤E< 5.0 MeV
After these cut applied, the vertex distributions in low-energy events are shown in Figure
7.19.
Tighter ﬁducial volume cut
This cut is to reject the remaining background in the edge region. As shown in Figure 7.19,
background events in the bottom region still remain after the cluster cut. This nonuniformityof the vertex distribution of background makes the angular distribution of the backgrounddistorted, which causes a large systematic uncertainty for day-night asymmetry of solar neutrino
ﬂux.
To set a tight ﬁducial volume, the signiﬁcance as a function of detector radius is calculated.
Based on the signiﬁcance calculation, the ﬁnal value of ﬁducial volume is obtained for eachenergy region which is shown in Figure 7.19:
4.5 − 5M eV : (r2 < 180m2 andz > −3m)or(r2 < 120m2 and − 7 < z < 3m) = 12.3kton
5 − 5.5M eV :(r2 < 180m2 andz > −7.5m) = 13.3kton

5.5 − 20M ev :no tight ﬁducial volume cut = 22.5kton
84

Figure 7.18: Distribution of R02 × N 20rawT /Nef f and the signiﬁcance.

Figure 7.19: Vertex distribution of low-energy events. The tight ﬁducial volume is shown by the
red box: 12.3kton for 4.5-5MeV, 13.3kton for 5-5.5MeV and 22.5kton 5.5-20MeV.

85

5-+ 5-+++Figure 7.20: Number of event after each reduction step as a function of energy. Precut in SK-I
(open circle) means noise reduction. For and SK-III(blue) there still remained lots of low-energy
events due to FRP. Thus, ovaQ > 0.2 is applied for all events, and this is called precut in SK-III.

7.5

Summary of Reduction step

Figure 7.20 shows the energy spectrum after each step of the reductions and Figure 7.21 shows
the remaining eﬃciency of 8 B solar neutrino MC with respect to the reconstructed energy. Whilethe event rate in the real data as a function of energy is the same as SK-I, the cut eﬃciencies
are improved by 10% in the ﬁnal data sample. For 4.5-5MeV region, while the eﬃciency gets
worse by about 30% compared to SK-I, the background is reduced by about 80%, which enables
to observe the solar neutrino signals in that energy region with better signal to noise ratio. The
number of events after each reduction step is summarized in Table 7.3 (real data) and 7.4.

86

5-+5-+++‫ ع‬#HVGT6TKIIGTUKO
ً #HVGT0QKUGEWV
ٕ #HVGT5RCNNCVKQPEWV
٤ #HVGT#ODKGPVEWV#HVGT)COOCTC[EWV‫( غ‬KPCNUCORNG

Figure 7.21: Reduction eﬃciency for the solar neutrino MC simulation. Color lines show SK-III
and Makers show SK-I (From top; after SLE trigger, noise reduction, spallation cut, ambient
BG reduction, and ﬁnal data sample)

Reduction step

4.5-20.0 MeV

5.0-20.0 MeV

6.5-20.0 MeV

PrecutSpallation cut
ovaQ
Second ﬁdv. cut
Hit pattern cut
Gamma ray cut
16 N cut
Cluster cut
Tight ﬁdv. cut

10162637
8324066
179587816094652381464
1861770866835734843

89096
89096
67774
13346

54820
54351
42916
24311

669641114350100470
89808
58187
43146
39879

13346

24311

Final

Table 7.3: Reduction results of data

87

39879

Reduction step

4.5-5.0 MeV

5.0-6.5 MeV

6.5-20.0 MeV
1932146
16721141670429
1670429
1310258
1295878129587810293651020542998017889857815156Total
Bad run cut
Trigger condition
Goodness cut
Fiducial volume cut
PrecutFlasher events cut
Spallation cut
ovaQ
Secont ﬁdv cut
Hit pattern cut
Gamma ray cut
Cluster cut
Tight ﬁdv cuts

502745428206225121225119211920196572
196572155869136319135223

1277710109888092042792042474835272425972425957467953752252754592326.887390.745109.5355474341162308757

Final

45109.5308757

Table 7.4: Reduction results of MC

88815156

Chapter 8

Signal extraction
8.1

Solar angle ﬁtting

The strongest information which is used to extract solar neutrino signal from the ﬁnal sample
is the direction of the recoil electrons because the direction of the recoil electron is strongly
correlated with the direction of the incident neutrino. Thus, by checking the cosine of thedirection from the Sun and the event direction (cos θsun ), the solar neutrino signals can beextracted. The signal extraction is done by an extended maximum likelihood ﬁt [43] with data
and the solar neutrino MC events distributions. The likelihood function is∑

L = e−(i Bi +S)

N
ni
bin ∏
∏

(Bi · bij + S · Yi · sij )(8.1)

i=1 j=1

where Nbin = 22 is the number of bins between 4.5 MeV and 20.0 MeV: 19 energy bins of
0.5 MeV between 4.5 and 14.0 MeV, two energy bins of 1 MeV between 14.0 and 16.0 MeV,
and one bin between 16.0 and 20.0 MeV 1 . and ni is the number of observed events in thei-th energy bin. S and Bi are free parameters to be ﬁtted and represent the total number ofsolar neutrino events and the number of background in the i-th energy bin. Yi is the expected
fraction of signal events in the i-th energy bin which will be explained in the next section. The
probability density functions bij and sij describe the expected shapes of the background andrecoil electron signal respectively, and are deﬁned for each energy bin as a function of the solar
angle.
bij = u(cos θij )(8.2)
sij = p(cos θij , Eij )(8.3)

where θij and Eij are a solar angle and energy of j-th event in i-th energy bin. The deﬁnition of
the solar angle (θj ≡ θSun ) is shown in Figure 8.1. The background shape is not completely ﬂatdue to the cylindrical shape of the detector. To estimate the shape of background, at ﬁrst, thezenith angle and azimuthal distributions of real data (cos θsun < 0.7 is imposed to avoid solar
signal) are ﬁtted with an eighth-order polynomial. Then, the expected solar angle distribution
can be obtained from the ﬁtted zenith and azimuthal distributions by generating a toy MC. The
1

If the threshold is 5.0 MeV, the number of bins are 21 bins: 18 energy bins of 0.5 MeV between 5 and 14.0
MeV, two energy bins of 1 MeV between 14.0 and 16.0 MeV, and one bin between 16.0 and 20.0 MeV. That is,i = 2 ∼ 22 is for 5MeV threshold, and i = 5 ∼ 22 is for 6.5MeV threshold.

89

<

Ǒ\
G

5WP

ǰUWP

5-FGVGEVQT

Figure 8.1: Deﬁnition of Θz and θsun .

C

D

Figure 8.2: Distributions of cosθsun for the expected background (a) and the solar neutrino MC
events (b).

90signal shape is obtained from the known neutrino-electron elastic scattering which is smearedby electron multiple scattering in water and the detector’s angular resolution. Figure 8.2 showssome examples of the expected background shapes and solar neutrino signal shapes. Finally, thevalues Bi and S are chosen so the likelihood has a maximum value.

8.2

How to get Yi

As denoted in the previous section, Yi is the expected fraction of signal events in the i-thenergy bin. Yi can be calculated as follows:
1. Based on the 8 B spectrum in section 2.2, calculate the recoil electron energy distributionof the elastic scattering:νe + e− −→ νe + e−
(8.4)
2. Obtain the total electron energy distribution to be observed considering the detector response.
3. Calculate the expected total number of signal events and the expected number of signal
events in each energy bin.
To obtain the recoil electron energy distribution, the cross section of the reaction in equation
8.4 is given [44][])(G2F medσ
Te 2
me T e=A0 + B0 1 −+ C0 2
(8.5)
dTe
2π
Eν
Eν
where GF , me , Eν , Te , are the Fermi coupling constant, the electron mass, the incident neutrinoenergy and the kinetic energy of the recoil electron, respectively (see Figure 8.3-(b) ). The
A0 , B0 , C0 are deﬁned by
2
A0 = (gV + gA )2 , B0 = (gV − gA )2 , C0 = (gA− gV2 ){
gV = 2 sin2 θW + 12 , gA = + 21
for νe112gV = 2 sin θW − 2 , gA = − 2for νµ , ντ

(8.6)
(8.7)

where θW is the Weinberg angle (sin2 θW = 0.2317 ) 2 . We have applied the radiative correction[45] to the equation 8.5. If the corrections are considered, they contribute to reduce the cross2

The total cross section is given:∫ Tmaxdσ
σtotal =
dTe
dT
e
0
[
{


G2F meEν
Tmax=
A0 Tmax + B0
1− 1−
2π
3
Eν

where
Tmax =

3}

2
me Tmax
+ C0
2Eν2

Eν
me
1 + 2Eν(see ﬁgure 8.3-(a)) For electron neutrino, the total cross section is1) νe e− → νe e−
Eν (M eV )σνe e− → νe e− = 9.478 × 10−44
10(M eV )

91

]
(8.8)

(8.9)

(cm2 )(8.10)C

D

E

Figure 8.3: (a)ν − e total cross section,(b)diﬀerential cross section, (c) total recoil energy for 8 B
solar neutrino.

section by ∼ 2% for the 8 B energy region [45]. The recoil electron energy distribution is thencalculated as:}{∫ Eν,maxdσνe (Eν , Te )ϕ(Eν )dEν dTe(8.14)F (Te )dTe =dTe0
where ϕ(Eν ) is the solar neutrino ﬂux as a function of neutrino energy. Figure 8.3-(c) shows theenergy distribution of the recoil electron for 8 B solar neutrinos.
In the second step, in order to take into account the detector response in the observed energy
distribution prediction, we actually carry out the full detector simulation. In that simulation,
electrons which obey the recoil energy distribution are generated, and the real run time is also
simulated so that the solar neutrino MC events can be directly compared with the data for thetime variation or the day-night asymmetry study.
Finally, the same reconstruction tools and reduction steps are applied to the solar neutrino
MC events, and the MC ﬁnal sample is made. The expected fraction in each energy bin, Yi is
obtained from the energy distribution of the solar neutrino MC ﬁnal sample. Figure 8.4 shows
the predicted energy distribution of the MC ﬁnal sample.

For muon and tau neutrinos, the total cross section is2) νµ,τ e− → νµ,τ e−
σνµ,τ e− → νµ,τ e− = 1.559 × 10−44

Eν (M eV )10(M eV )(cm2 )(8.11)

which is about 1/6 smaller than that of electron neutrino. The scattering angle is determined by kinematics, such
as:
e
1+ m
Eν
cos θ = √(8.12)e
1 + 2mTein case of me ≪ Teme
(8.13)Te
which shows a strong correlation between the incident neutrino direction scattering angle of the recoil electron.
cos θ > 1 −92

'XGPVTCVG GXGPVFC[

9KPVGT
1TVK\

6QVCNGNGEVTQPGPGTI[=/G8?
Figure 8.4: Energy spectrum of the solar neutrino MC ﬁnal sample. The 8 B ﬂux from BP04 SSM
and Winter spectrum is used for red histogram and Ortiz spectrum is used for black histogram.
The reduction eﬃciency in Figure 7.21 is considered. They agree within 5%.

93Chapter 9

Systematic Uncertainties
9.1

Energy scale

In order to study how the energy scale systematic uncertainty (0.53%) aﬀect to the ﬂux
value, the diﬀerence of the Yi is obtained ﬁrst. By changing the energy of each solar ν MC eventby + or - 0.53% after the energy reconstruction, Yiscale+ and Yiscale− are calculated. Using the
Yiscale+ and Yiscale− , the solar angle ﬁttings are carried out. Then it turns out that the 0.53%
scale shift makes ±1.4% diﬀerence on the total ﬂux in the 5.0-20.0 MeV electron total energy
region.
For the day-night asymmetry study, I only take into account the angular dependence ofenergy scale uncertainty (0.25%) because the other systematic uncertainties of energy scale
would be canceled out in the calculation of the asymmetry factor ((ΦD − ΦN )/0.5(ΦD + ΦN )).
In the same way, YiDN + and YiDN − are calculated with + or - 0.25% scale shift. As the resultof the solar angle ﬁttings, the systematic uncertainty of the day-night asymmetry is estimatedas ±1.3% in the 5.0-20MeV region.
Since the water transparency varies in time, the energy scale uncertainty depending on thewater transparency (0.21%) should be contributed to the uncertainty to the seasonal study. It
is estimated as ±1.3%.

9.2

Energy resolution

In order to estimate the eﬀect of the systematic uncertainty of the energy resolution (±2.5%)
on the total ﬂux, the prediction of observed energy spectrum is obtained not from the full detector
simulation but from the calculation:{∫ Emax}F(Eobs ) =F (Ee )R(Ee , Eobs )dEe dEobs(9.1)
0

where Eobs , F(Eobs ) are observed energy and the calculated observed energy spectrum, Ee istotal energy of recoil electron, and F (Ee ) is the recoil energy spectrum calculated in equation
8.14. R(Ee , Eobs ) is the detector response function:()(Ee − Eobs )2
1
exp −
(9.2)
R(Ee , Eobs ) = √σ(Ee )22πσ(Ee )294

where σ(Ee ) is the energy resolution function given by Equation 6.7. Then, the eﬀect of theenergy resolution systematic uncertainty on the observed energy is calculated by replacing σ(E)with σ + (E) = σ(E)(1 + 0.025) or σ − (E) = σ(E)(1 − 0.025) to get F + (Eobs ) or F − (Eobs ),respectively. Finally, the ratio of F +(or-) to the original F:+(or -)
weres(Eobs ) =F +(or -) (Eobs )F(Eobs )(9.3)

is used as a weight for a each solar neutrino MC event, then the Yieres+ and Yieres− are obtained.
Using the Yieres+ and Yieres− , the solar angle ﬁttings are carried out. As the result, ±0.12%
uncertainty on the total ﬂux in 5-20MeV energy region is estimated from the 2.5% energy
resolution uncertainty.

9.3

8

B spectrum

Winter et.al. calculate 8 B spectrum from a very precise measurement of spectrum of 8 Be
α decay [19]. The uncertainty of the 8 B calculation is taken from the paper which include
both experimental and theoretical uncertainties. To estimate the eﬀect of this uncertainty, the8original spectrum ϕ(Eν ) is changed by ± uncertainty, then F B+(or-) is made by equation 8.14.
8 B−
8 B+
8 B+(or-)are made in the same way as Yieres+and YiBy using the F
to Equation 9.1, Yi
eres−
and Yi. After the solar angle ﬁtting, the systematic uncertainty due to the 8 B shape isestimated as ±0.2% for the Winter 8 B spectrum. For the Ortiz 8 B spectrum, the same method
in [20] is used, then the estimated as +1.1
−1.0 %.

9.4

Angular resolution

In the same way as energy scale, Yiang+ and Yiang− are made by artiﬁcially shifting thereconstructed direction of the solar neutrino MC events. The shifted direction is calculated as
the reconstructed direction ± the systematic uncertainty of angular resolution (section 6.9) withrespect to the generated direction of the recoil electron. After the solar angle ﬁtting with Yiang+
and Yiang− , the systematic uncertainty due to the uncertainty of angular resolution is estimatedas ±0.7% on the total ﬂux in 5.0-20MeV region.

9.5

Vertex shift

The vertex shift makes the systematic uncertainty of the ﬁducial volume. To estimate thetotal ﬂux uncertainty due to the vertex shift, The reconstructed vertex positions of the solar
neutrino MC events are artiﬁcially shifted toward outward of the detector. Then, it is estimatedhow many fraction of events are rejected by the ﬁducial volume cut due to the shift. As theresult, 0.54% of the MC events are rejected and this contributes to the total ﬂux systematic
uncertainty.

95

.KPCE/%&#6#/%'PGTI[=/G8?
Figure 9.1: The diﬀerence of the ovaQ cut eﬃciency (MC-DATA)/MC of LINAC.

9.6

Reduction

9.6.1

ovaQ cut

Figure 9.1 shows the diﬀerence of the eﬃciency of ovaQ cut using the LINAC data and
MC. To estimate the systematic uncertainty of the total ﬂux due to the ovaQ cut, this energy
dependence of the eﬃciency is taken into account. As the result, ±0.4% uncertainty to the total
ﬂux in 5.0-20MeV energy region is found.

9.6.2

Hit pattern cut

Figure 9.2 shows the diﬀerence of the ineﬃciency of the hit pattern cut using the LINAC data
and MC. As seen in the ﬁgure, the eﬃciency diﬀerence between data and MC agree within 0.25%.
This maximum discrepancy 0.25% is taken as the uncertainty to the total ﬂux, conservatively.

9.6.3

Second vertex cut

The reconstructed vertex shift using the second ﬁtter is also taken into account to estimate
the systematic uncertainty on the ﬂux. The estimation is done in the same way as the BONSAI
vertex ﬁtter, resulting 0.45% of uncertainty should contribute to the total ﬂux.

9.7

Spallation cut

The same uncertainties as SK-I [20] is taken in this thesis. This is because the origin of thesystematic uncertainty of spallation cut is mainly due to the characteristics of the muon suchas rate, energy, arrival direction, and thus, the uncertainly should not depend on each phases ofthe experiment. The estimated valuses are 0.2% on the total ﬂux in 5.0-20MeV region and 0.1% for the time dependence part.

96

.KPCE/%&#6#/%

'PGTI[=/G8?
Figure 9.2: The diﬀerence of hit pattern cut eﬃciency (MC-DATA)/MC of LINAC.

9.8

Gamma ray cut

The vertex shift and the diﬀerence in the angular resolution measured by LINAC make somebias to the gamma cut eﬃciency. To estimate the eﬀect, ﬁrst, the reconstructed vertex anddirection of solar neutrino MC events are shifted with the diﬀerence of the vertex and angularresolution systematic uncertainties between data and MC as shown in Figure 9.3. Second,
the gamma cut eﬃciencies of the shifted MC sample and the original MC sample are compared.
Then, the systematic uncertainty due to the gamma cut is estimated as ±0.25% for the total ﬂux,
and ±0.2% for the day-night asymmetry. For the uncertainly of spectrum, ±0.1% uncertainty
is estimated.

9.9

Background shape

The systematic uncertainty due to the background shape prediction is estimated. As in thesection 8.1, the solar angle distribution is ﬁtted with the non-ﬂat background shape. To get the
non-ﬂat background shape, not only the distribution of z-direction but also the distribution ofphi direction is ﬁtted. To estimate the systematic uncertainty, the background shape assuming
ﬂat ϕ distribution is made and then the solar angle ﬁtting is carried out with the backgroundshape. The diﬀerence of the ﬁtting results between these two types of background shapes aretaken as the systematic uncertainty, which is 0.07% for the total ﬂux and 0.4% for the day-night
asymmetry.

9.10

Signal extraction method

To check how the signal extraction method biases the ﬂux value, the solar angle ﬁt is applied
to dummy data which have known number of signal and background events. As the result,±0.7% diﬀerence is found between the input and the output number of signal events for the
total ﬂux, and ±2% diﬀerence is found especially in the 5.0 < E < 5.5MeV region.

97

F GHH

5JKHVGF
FKTGEVKQPUJKHV5JKHVGFRQUKVKQP

1TKIKPCN
FKTGEVKQP

1TKIKPCNRQUKVKQP

Figure 9.3: Gamma cut systematic uncertainty check. the amount of vertex shift and direction
shift are measured by calibration. The shifted direction is determined based on the original
direction and reconstructed direction.

FC[PKIJV 

4GNCVKXG

 /CZKOWOFGXKCVKQP

6QVCNGPGTI[=/G8?

5WPFKTGEVKQP<=/G8?

Figure 9.4: Gamma cut systematic uncertainty check. diﬀerence of ineﬃciency before and after
the vertex and direction shift as a function of energy (left) and z component of sun direction(right)98

9.11

Cross section

The systematic uncertainty due to cross section is estimated from [45], in which cross section
parameters ρ, κ has 1σ theoretical uncertainty. Shifting the parameters by ±1σ uncertainty
gives the diﬀerence in cross section which corresponds to ±0.5% uncertainty on the total ﬂux.

9.12

Further quality cut for the lowest energy region

Clustering hits cut
Using DT data and MC simulation at the position (x=-12m,y=0m,z=0m), the diﬀerence of
cut eﬃciency between DT data and MC events is estimated as 2%. Assuming the all ﬁducial
volume and energy region of solar neutrino signal, this diﬀerence corresponds to 0.5% uncertainty
on the total ﬂux in the 5.0-20MeV region. 2% is assigned to the spectral shape uncertainty inthe 5.0 to 6.5 MeV bins.
Tighter ﬁducial volume cut
Assuming the vertex shift, the tighter ﬁducial volume cut makes 0.9% uncertainty on theﬂux in 5.0 to 5.5 MeV region. This is negligible for the total fulx uncertainty, but is assigned tothe spectral shape uncertainty in the 5-5.5 energy bin.

9.13

Summary of systematic uncertainty

9.13.1

Flux,time variation,Day-Night asymmetry

The systematic uncertainty of total ﬂux, time variation and day-night asymmetry is summarized in the Table 9.1 In this thesis, the systematic uncertainty on the total ﬂux is estimated as
2.3% when we use Ortiz 8 B spectrum. This is about two thirds of the corresponding SK-I value.
For comparison, SK-I systematic uncertainty is shown in Table 9.2. The main contributions forthe improvement are vertex shift, angular resolution, and reduction step which are reestimatedin this thesis.

9.13.2

SpectrumThe systematic uncertainty of the spectral shape consists of two components.
Energy correlated
The energy correlated systematic uncertainties are given from Equation 9.1 with artiﬁciallyshifted energy scale, energy resolution and 8 B spectrum. The results of the calculations areshown in Figure 9.5 These correlations are taken into account in the oscillation analysis.
Energy uncorrelated
The energy uncorrelated energy spectral uncertainties are listed in Table 9.3 In this table,
the common oﬀset of the uncertainties are omitted in order to estimate the eﬀect on the energy
spectral shape.
99

Source

Flux

Time variation

Day/Night

Energy scale
Energy resolution
8 B spec.
Trig. eﬀ.
Ang. resolution
Vertex shift
Reduction
ovaQ
Hit pattern cut
Second vertex
Spallation
Gamma-ray cut
Cluster hit cut
Background shape
Signal ext. method
Cross section

±1.4%
±0.2%
±0.2/1.0%±0.5%±0.67%±0.54%

±1.3%

±1.3%

Total

±2.1/2.3%±0.4%±0.25%±0.45%±0.2%±0.25%
±0.5%
±0.1%
±0.7%
±0.5%

±0.1%

±0.1%

±0.1%
±0.2%
±0.4%

±1.3%

±1.4%

Table 9.1: Summary of the total systematic uncertainty. The total value 2.1% is for Winter 8 B
spectrum and 2.3% is for Ortiz

SourceFlux

Time variation

Day/Night

Energy scale, resolution
8 B spec.
Ang. resolution
Vertex shift
Reduction
Spallation
Gamma-ray cut
Background shape
Cross section
Livetime calculation

±1.6%
±1.0%
±1.2%
±1.3%
+2.1
−1.6 %
±0.2%
±0.5%
±0.1%
±0.5%
±0.1%

+1.2
−1.1 %+1.2−1.1 %±0.1%±0.25%

±0.1%

Total

+3.5
−3.2 %

±0.4%
±0.1%

±0.1%

±1.3%

+1.3
−1.2 %

Table 9.2: Summary of the total systematic uncertainty in SK-I.

100

0.15

0.1

0.1

UJKHVQTKIKPCNUJKHVQTKIKPCN

0.15

0.05

0.05

0

Scale 0.53%
Resolution 2.5%
Ortiz B8 spec,Bahcall err.

-0.05

-0.1

0

Scale 0.53%
Resolution 2.5%
W inter 06 B8 spec & err.

-0.05

-0.1

-0.15

-0.15
4

6

8

10

12

14

16

18

20

4

6QVCNGNGEVTQPGPGTI[=/G8?

6

8

10

12

14

16

18

20

6QVCNGNGEVTQPGPGTI[=/G8?

Figure 9.5: Energy correlated systematic uncertainties for Ortiz (left) and Winter (right) 8 B
spectrum.

Energy(MeV)

5-5.5

5.5-6

6-6.5

6.5-7

7-7.5

7.5-20

Trig eﬀ
ovaQ
Hit pat.
Cluster cut
Gamma Cut
Vertex shift
BG shape
Sig.Ext.
Cross section

±2.4%
±2.%
±2.%
±0.1%
±0.5%
±0.2%
±2.1%
±0.2%

±0.9%
±1.75%
±2.%
±0.1%
±0.1%
±0.8%
±0.7%
±0.2%

±0.1%
±1.5%
±2.%
±0.1%
±0.1%
±0.2%
±0.7%
±0.2%

±1.25%±0.25%
±0.1%
±0.1%
±0.2%
±0.7%
±0.2%

±1.0%
±0.25%
±0.1%
±0.1%
±0.2%
±0.7%
±0.2%

±0.25%
±0.1%
±0.1%
±0.2%
±0.7%
±0.2%

Total

±4.3%

±3.0%

±2.6%

±1.5%

±1.3%

±0.8%

Table 9.3: Energy uncorrelated systematic uncertainty on the spectrum shape.

101

Chapter 10

Result
The results of solar neutrino signal ﬁtting are shown in this chapter.
Total Flux and time variation
Figure 10.1 shows the solar angle distribution of the ﬁnal sample. The energy range of thissample is 5.0 to 20.0MeV. The ﬁtted signal and background shape are also plotted in the ﬁgure.
The observed total number of solar neutrino events is 8098+132−131 (stat.) ± 186(sys.), the resulting
ﬂux is2.28 ± 0.04(stat.) ± 0.05(sys.) × 106 cm−2 sec−1(10.1)
where Ortiz 8 B neutrino spectrum is used, and thus the total systematic uncertainty is ±2.3%.
This value is consistent with the the ﬂux values obtained in SK-I and II:
2.35 ± 0.02(stat.) ± 0.08(sys.) × 106 cm−2 sec−1

SK-I

6
−2
−1
2.38 ± 0.05(stat.)+0.16−0.16 (sys.) × 10 cm sec

SK-II

In this thesis, since the angular resolution becomes better than SK-I due to the new ﬁtter, thepeak at cos θsum = 1 is gotten higher resulting in a better S/N ratio.
If the Winter spectrum is used instead of Ortiz’s one, the ﬂux value in SK-III is found to be
2.31 ± 0.04(stat.) ± 0.05(sys.) × 106 cm−2 sec−1(10.2)

This diﬀerence between Equation 10.1 and 10.2 is expected from the diﬀerence between the two
8 B spectrum shapes in Figure 2.8
The day time and night time ﬂuxes are deﬁned bycos Θz ≤ 0

Day(10.3)cos Θz > 0

N ight(10.4)

where Θz is the solar zenith angle (see Figure 8.1). Figure 10.2 shows the angular distribution
in day-time and night-time. The observed ﬂuxes areΦday = 2.22 ± 0.05(stat.) ± 0.05(sys.) × 106 cm−2 sec−1
Φnignt = 2.35 ± 0.05(stat.) ± 0.05(sys.) × 10 cm
6

102

−2

sec

−1

(10.5)

D ataB est-fit
B ackgroundFigure 10.1: Angular distribution of ﬁnal sample 5-20MeV.

ADN =(Φ Day − Φ Night )(Φ Day + Φ Night ) / 2= −0.057 ± 0.031(stat.)

Figure 10.2: Angular distribution on day (red) and night (blue) time.

103

SK-III SLE 298d + LE 548d

8

6

2

B Flux (10 /cm /s)

5
4
3
2
1
Errors: stat. only
Line: SK-III average

0

5

10
15
20
Total energy of recoil electrons (MeV)

Figure 10.3: Energy spectrum of day(red) and night(blue). Black line shows SK-III average ﬂux.

&C[PKIJVCU[OOGVT[ #&0 XCNWG

0.1

0.05

0

-0.05

-0.1

-0.15

-0.2
4

5

6

7

8

9

10

6JTGUJQNFGPGTI[=/G8?
Figure 10.4: Day night asymmetry values as a function of energy threshold. The upper limit ofenergy is 20MeV.

104

Predicted

Figure 10.5: Time variation of observed ﬂux. Green line shows the predicted value

The day-night asymmetry ADN is deﬁned by
ADN=

Φday − Φnight1
2 (Φday + Φnight )(10.6)= −0.057 ± 0.031(stat.) ± 0.013(sys)
The ADN in SK-I and II are:ADN

= −0.021 ± 0.020(stat.) +0.013−0.012 (sys.)SK − IADN= −0.63 ± 0.020(stat.) ± 0.037(sys.)SK − II(10.7)The ADN in SK-III is about 1.7σ away from 0, and the expected value from the oscillation
parameters in favored Large Mixing Angle (LMA) solution is -1.5% [20]. To study this diﬀerence
between ADN and 0, ADN values in the diﬀerent energy regions are calculated, and it is conﬁrmedthat those ADN values in the diﬀerent energy regions are consistent with the 5-20MeV value (seeFigure ?? and 10.4). Further more, using the event sample before spallation cut above 6.5MeV(most events in a such sample are due to a single β ray which is emitted by spallation products ),it is also conﬁrmed that energy scale of upward going events and that of downward going eventsare consistent within the assigned systematic uncertainty (±0.25%). Thus, any systematic bias
has not been found to explain the large ADN value so far.
Figure 10.5 shows the time variation of observed solar neutrino ﬂux. Each horizontal bincorresponds to 45 days. The correction of the elliptical orbit of the Earth is not taken into
account in this analysis. The ﬂuctuation in SK-III seems large, but it would be statisticallyconsistent with the average value obtained by SK-I. Figure 10.6 shows the seasonal variation of
the solar neutrino ﬂux. The χ2 value is 13.6(d.o.f. = 7) which corresponds to 5.95% probability.
(Cf. The χ2 value of SK-I is 4.7(d.o.f = 7) which corresponds to 69.9% probability.) Combining
105

Figure 10.6: Seasonal variation of observed ﬂux. Solid line shows the predicted value from theelliptical orbit of the Earth.

106

Figure 10.7: Solar angle distributions of diﬀerent energy regions from 4.5-7.5 MeV.

SK-I,II,and III statistically, the χ2 value is 3.6(d.o.f.=7) which corresponds to 89% probability.
Thus, seasonal variation in SK-III is statistically consistent with SK-I.
Figure 10.7 shows the solar angle distribution in the diﬀerent energy regions. The solar angle
distribution in the 4.5-5MeV region is shown in the upper left plot. The background level isalmost the same as that of SK-I 5-5.5 MeV region (green histogram in middle top plot). The
solar angle ﬁtting result looks good, and the ﬁtted number of signal events and the ﬂux in the
4.5-5MeV region are
4.5 − 5M eV :232. ± 59.event(10.8)
and4.5 − 5M eV :6−2−1
2.14+0.56−0.54 (stat.) × 10 cm sec(10.9)

This ﬂux value is consistent with the 5-20MeV ﬂux. Since the trigger eﬃciency in the SLE1
period is not clear, the systematic uncertainty for this energy region was not estimated.
Energy spectrum
Figure 10.8 shows the expected and measured electron total energy spectrum. The expected
rate is calculated by the detector simulation which uses BP04 SSM. The solid line is expected
spectrum by SSM. The summary of observed and expected event rate is summarized in Table
10.1. Figure 10.9 shows the ratio of observed energy spectrum to the SSM prediction. As shown107

Events/day/22.5kt/0.5MeV

10Solar neutrino MC
Observed solar neutrino events

1

10

10

10

-1

-2

-3

(reduction efficiencies are corrected)

5

10
15
20
Total energy of recoil electrons (MeV)

Figure 10.8: Energy spectrum of the solar neutrino signal. Horizontal axis is the total energy ofrecoil electrons. Vertical is the event rate.

in the ﬁgure, no signiﬁcant up-turn was not observed. And the average value is consistent with
SK-I.

108

Ennergy (MeV)

ALL

DAY

NIGHT

8B

hep

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 − 14.0
14.0 − 15.0
15.0 − 16.0
16.0 − 20.0

64.4+8.0−7.7
52.6+5.0−4.8
49.1+3.8−3.7
55.0+2.7−2.6
53.7+2.5−2.4
40.3+2.2−2.1
36.5+1.9−1.8
30.6+1.7−1.6
22.4+1.4−1.3
19.3+1.2−1.2
14.4+1.0−1.0
13.9+1.0−0.9
9.54+0.80−0.74
5.69+0.64−0.58
4.96+0.57−0.51
3.08+0.44−0.38
1.95+0.36−0.30
1.35+0.30−0.24
2.19+0.36−0.31
0.854+0.240−0.1800.115+0.134−0.06673.2+12.2−11.6
58.1+7.6−7.2
43.2+5.4−5.1
51.0+3.9−3.7
55.5+3.6−3.5
39.6+3.1−3.0
37.3+2.7−2.6
28.6+2.4−2.2
19.8+1.9−1.8
17.9+1.8−1.6
15.1+1.5−1.4
15.1+1.5−1.3
9.59+1.20−1.06
5.28+0.91−0.77
4.16+0.80−0.67
2.71+0.62−0.50
1.61+0.48−0.36
1.16+0.40−0.28
2.06+0.53−0.41
0.389+0.295−0.1620.248+0.242−0.11956.9+10.6−10.1
47.6+6.6−6.2
54.9+5.5−5.2
58.7+3.8−3.7
52.0+3.5−3.3
41.0+3.1−2.9
35.7+2.6−2.5
32.7+2.4−2.3
25.0+2.1−1.9
20.7+1.8−1.7
13.7+1.5−1.3
12.9+1.3−1.2
9.47+1.11−0.99
6.11+0.96−0.82
5.71+0.85−0.73
3.44+0.66−0.54
2.28+0.56−0.44
1.51+0.48−0.36
2.32+0.54−0.42
1.249+0.398−0.2840.000+0.128−0.415

150.9
138.2
125.3
140.4
122.6
106.5
90.5
75.6
62.3
50.0
39.6
30.6
23.07
17.07
12.30
8.60
5.68
3.78
3.84
1.326
0.592

0.260
0.250
0.241
0.289
0.271
0.257
0.240
0.223
0.205
0.186
0.169
0.151
0.134
0.118
0.102
0.088
0.074
0.062
0.092
0.059
0.068

Table 10.1: Observed and expected rates in each energy bin at 1AU. The unit of the rate is
event/kton/year. The errors in the observed rate are statistical only. The reduction eﬃciencies
are corrected, and the expected event rates are for the BP2004.

109

SK-III 548days (>6.5MeV)
289days (<6.5MeV)
4
3

8

6

2

B Flux (10 /cm /s)

5

2
1
0

SK-I 1496days PRD73,112001(2006)
Errors: stat. only
Line: SK-I average

5

10
15
20
Total energy of recoil electrons (MeV)

Figure 10.9: Energy spectrum of SK-I and III. Horizontal axis is the total energy of recoil
electrons. Vertical is data/SSM×5.79 × 106 cm−2 s−1 (8 B ﬂux of BP2004).

110

Chapter 11

Oscillation Analysis
11.1

Spectrum ﬁt

11.1.1

Spectrum shape prediction

The method to calculate the expected energy spectrum of the recoil electron by solar neutrino
is following. Using the survival probability pe in Equation 2.23, the neutrino interaction rate in
the observed energy range Elow to Ehigh with the neutrino oscillation is written as∫∫∫ EhighdE
dEν ϕ(Eν )dEe R(Ee , E) (pe Se (Eν , Ee ) + (1 − pe )Sµ,τ (Eν , Ee )) (11.1)
rosc =
Elow

Eν

Ee

where Se,µ,τ (Eν , Ee ) describe the probability of the elastic scattering of a νe,x of energy Eν with
an electron of recoil energy Ee . R(Ee , E) is the detector response function given by the equation
9.2. ϕ(Eν ) is the solar neutrino ﬂux (8 B or hep) as a function of neutrino energy. For the null
oscillation case, the rate rnull is given by setting pe = 1.
As described in Section 2.3, the pe consists of the Sun propagation part, p1 , and the Earthpropagation part, p1e . Since we do not know the path of the observed neutrino in the Sun, p1
is obtained by averaging over the production points. On the other hand, for the earth part, if
the event time and the position of the Sun and the detector at the time are given, a uniquepath of the observed neutrino can be calculated for an oscillation parameter set, (∆m2 , θ), andeventually, pe is obtained as a function of neutrino energy (Eν ) and zenith angle of the Sun,cz ≡ cos Θz , pe (Eν , cz). The interaction rate with the neutrino oscillation in Equation 11.1
correspondingly has a form of rosc (E, cz).
In this analysis, cz of the night time (0 < cz ≤ 1) is divided into 1000 bins and 1 bin is for
day time (−1 ≤ cz ≤ 0). With this binning, the interaction rate of 8 B and hep neutrinos are
B

osc

(E) =

1001
∑
j=1

H osc (E) =

1001
∑
j=1

τ (czj ) 8 B· rosc (E, czj )τtot(11.2)

τ (czj ) hep· rosc (E, czj )τtot(11.3)

where τ (czj ) is the livetime in the j-th zenith angle bin which is obtained from the real data
(see Figure 11.1), and τtot is the total livetime. The predicted rate of the i-th bin ( Elow,i <111

NKXGVKOGHTCEVKQPE\DKP
Figure 11.1: livetime fraction in each cz bin 1 to 1000. cz bin = 1 and 1000 correspond to cosΘz= -1 and 1, respectively.

E < Ehigh,i ) is given as
Biosc∫ Ehigh,i=dE · B osc (E)(11.4)

dE · H osc (E)(11.5)

Elow,i

Hiosc =

∫ Ehigh,iElow,i11.1.2

Constructing χ2

In this calculation, the 5.0MeV energy threshold is used. The deﬁnition of χ2 is
χ2 =

N
bin
∑

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

(11.6)

where i is the energy bin as explained in Section 8.1. di , bi , and hi are ratios of the data andoscillated solar neutrino rate to the unoscillated SSM rate:
di =
bi =
hi =

Di
B i + Hi
Biosc
B i + Hi
Hiosc
B i + Hi

(11.7)(11.8)(11.9)

Di is observed solar neutrino observed rate, and Bi and Hi are the expected rate without theneutrino oscillation, and β and η are free parameters to allow the arbitral ﬁt of the total neutrino
ﬂuxes. f in Equation 11.6 is the energy shape factor:f (Ei , δB , δS , δR ) = fB (Ei , δB ) × fS (Ei , δS ) × fR (Ei , δR )112

(11.10)wherefx (Ei , δx ) =1,
1 + δx × ϵ±x (Ei )(x = B, S, R)(11.11)and B, S, R denote the 8 B spectrum, energy scale, energy resolution, respectively. δx is the unitless parameter to scale the energy correlated uncertainty, ϵ±x (Ei ) in Figure 9.5. The δx is varied2
until the χ gives the minimum value with βm and ηm . βm and ηm are calculated analyticallyby setting the partial ﬁrst derivative of χ2 with respect to β and η to zero.
σi in equation 11.6 is combined statistical-systematic uncertainty, in which the energy uncorrelated systematic uncertainties are combined in quadrature with the energy spectrum’s statistical uncertainties. To account for the total ﬂux energy uncorrelated systematic uncertainty,
Taylor expanding of χ2 is carried out around the χ2 minimum (χ2m ) with:(
χ2 (β, η) = χ2m (βm , ηm ) +where C0 is
C0 =

N
bin
∑




i=2

β − βmη − ηm
b2i
σi2
bi hi
σi2

)T

(
C0

bi hi
σi2
h2i
σi2

β − βmη − ηm)(11.12)(11.13)

Then C0 is scaled by αsys to add the total energy uncorrelated systematic uncertainty σsys . αsys
is deﬁned as
N
bin
∑
σ2
1
αsys = 2 0 2 ,
where
σ02 =
(11.14)
σ0 + σsys
σ2i=2 i
σsys is obtained √as the total ﬂux × (energy uncorrelated sys. on the total ﬂux) / SSM ﬂux,
which is 2.28 × 0.0232 (total) − 0.0142 (escale) − 0.0022 (eres.) − 0.012 (8 B)/5.79 = 0.6% for
SK-III. Then, the χ2 becomes(
χ2α,SK3 (β, η) = χ2m (βm , ηm ) +β − βmη − ηm)T(αsys · C0β − βmη − ηm)(11.15)The χ2 for the energy spectrum shape is eventually expressed as: excluded22
χ2specSK3 (β, η) = Min(χ2α,SK3 (β, η, δB , δS , δR ) + δB
+ δS2 + δR)

(11.16)

where the number of ﬁtting parameters is ﬁve; β, η, δB , δS , δR . Inside of line is excluded. Figure11.2-(a) shows the exclude region by the SK-III spectrum obtained from the Equation 11.16.

11.2

Time-Variation Analysis

In the time-variation analysis, 6.5MeV energy threshold is used. In order to take into accountthe time variation of the rate due to the matter eﬀects of the Earth, the sij of Equation 8.1 is
modiﬁed with a scaling function ri (tj )/riave . Here, tj is the event time of the j-th event andri (tj ) is the predicted time dependence (that is, cz dependence) of the expected oscillated solar

113

C

D

%.

%.

Figure 11.2: Excluded region only from the SK-III result (inside area of the line is excluded).
(a) Spectral shape only, (b) Spectral shape and time variation

neutrino rate in energy bin i, and riave is the average of the oscillated rates over all the zenith
angle bins. The likelihood is for the signal extraction (Equation 8.1) is modiﬁed as:
Lt.v.SK3 = e−(∑i Bi +S)

N
ni
bin ∏
∏

(Bi · bij + S · Yi · sij

i=5 j=1

ri (tj ))riave(11.17)The eﬀect of the matter oscillation can be tested by comparing the likelihood Lt.v with theoriginal L (ri (t)/riave = 1). In this calculation, the Yi is now determined from the predictionof the oscillated spectrum. This spectrum is given by using the 8 B and hep neutrino ﬂuxes
and energy spectral shape factors from the χ2specSK3 ﬁt. The combination of χ2specSK3 and thelikelihood of the time-variation is done by converting the likelihood value in terms of χ2 . Thatis:
χ2SK3 = χ2specSK3 + ∆χ2t.v.SK3(11.18)where∆χ2t.v.SK3 = −2(log Lt.v.SK3 − log L).

(11.19)Figure 11.2-(b) shows the excluded region by the spectral shape and time variation only from
SK-III data.

114

C

D

%.

%.

Figure 11.3: Excluded region from the SK combined analysis (inside area of the line is excluded).
Black shows only SK-I and II, Red shows SK-I,II and III. (a) Spectral shape only . (b) Spectral
shape and time variation.

11.3

Oscillation constraint from SK

The combined χ2 to be minimized is following:χ2specSK= M in(χ2α,SK1 (β, η, δB , δS1 , δR1 )+χ2α,SK2 (β, η, δB , δS2 , δR2 )+χ2α,SK3 (β, η, δB , δS3 , δR3 )
2
2
2
2
2
2
2
+δB
+ δS1
+ δR1
+ δS2
+ δR2
+ δS3
+ δR3
)

(11.20)where the energy threshold is 5.0MeV (i=2 to 22) for SK-I and 7.0MeV (i=6 to 22)for SK-II[46]. In Equation 11.20, seven parameters are ﬁtted. The full SK χ2 is obtained by adding thetime variation terms in each phase.
χ2SKf ull = χ2specSK + ∆χ2t.v.SK1 + ∆χ2t.v.SK2 + ∆χ2t.v.SK3(11.21)Figure 11.3 show the exclusion region obtained from the spectral ﬁt (a), and the spectral ﬁt plus
the time variation (b).

115

Chapter 12

Discussion
12.1

Global oscillation analysis

The combined analysis of other solar neutrino experiments and KamLAND experiment is
discussed in this section. The numbers which are used in this global analysis are based on the
results summarized in Section ??.

12.1.1SNO

The results of the CC total ﬂux measured by D2 O phase(CC) and salt phase(CC) are
included in this thesis. For the NC ﬂux, the latest low-energy threshold analysis (LETA) [47] isalso included. The latest value of NC ﬂux is+0.1326−2 −1
ΦN C = 5.14+0.160−0.158 (stat.)−0.117 (sys.) × 10 cm s

(12.1)

This represents 3.1% statistical uncertainty and 2.4% systematic uncertainty.
The deﬁnition of χ2 is following:χ2SN O =SN O1 − (βB SN O1 + ηH SN O1 ))2
SN O2 − (βB SN O2 + ηH SN O2 ))2
(DCC
(DCC
CC
CC
CC
CC
+
SN O1 )2
SN O2 )2
(σCC
(σCC

+

(DN C − (βBSSM + ηHSSM ))2 (ADNCC − ADNpred (β, η))2+
(σN C )2(σADNCC )2(12.2)

where ADNpred (β, η) is the predicted day-night asymmetry parameter of pure D2 O phase:
ADNpred (β, η) = 2 ×

SN O1,day
SN O1,day
SN O1,night
SN O1,night(βBCC+ ηHCC) − (βBCC+ ηHCC)SN O1,day
SN O1,day
SN O1,night
SN O1,night(βBCC+ ηHCC) + (βBCC+ ηHCC)(12.3)

The combined χ2 to be minimized is then given:χ2SK+SN O (β, η) = χ2specSK (β, η) + χ2SN O (β, η)(12.4)

The common ﬂux scale factors are used in this calculation. So, the number of ﬁtted parameters
are 7, same as in Equation 11.20.
116

12.1.2

Radiochemical and Borexino

The Homestake, GALLEX/GNO, SAGE and Borexino (only 7 Be neutrino ﬂux [48] is usedin this thesis) rates are added to the χ2 . The calculation of the χ2 from these experiments are
based on [49].
N (=3)
∑
2
2 −1
expt
theor
χGaClBore =(Rnexpt − Rntheor )[σnm
] (Rm
− Rm
)
(12.5)
n,m=1

where GALLEX/GNO, and SAGE are merged into one Ga experiment and N is the number ofexperiments. According to [25], the merged result from Ga experiment is66.1 ± 3.1

SNU (All Ga experiment)(12.6)

2 is a matrix of squared uncertainty [50] which are the sum of the correlated uncertainty
The σnm(such as uncertainties of SSM) and uncorrelated uncertainty among these experiments, , Rnexpt
is the observed solar neutrino rate for the n-th experiment, and Rntheor is the predicted solar
neutrino rate by the SSM. In this thesis, correlation of β, η among Radiochemical experiments
are omitted because the dependence could be negligible in Large Mixing Angle (LMA) solutionarea.

12.1.3

KamLAND

KamLAND experiment (2002∼)is a liquid scintillator experiment in Japan. measuring theinverse beta decay reaction:ν¯e + p −→ e+ + n(12.7)KamLAND detects antineutrinos from surrounding nuclear reactors. According to the latest result from 2002 to 2007, the total exposure is 2.44 × 1032 proton-yr(2881 ton-yr). The
+0.15
−5
2
best ﬁt parameters are obtained as ∆m212 = 7.58+0.14−0.13 (stat)−0.15 (sys) × 10 eV and tan θ12 =+0.10
0.56+0.10−0.06 (stat)−0.06 (sys) [51].
Adding the χ2 which is given in [51], the combined result is shown in the ﬁlled contour inFigure 12.2.

117

5- 501 )C%N

5- 501

%.
%.
%.

5QNCT

5- 501 )C%N $QTGZKPQ
sin 2 θ12 = 0.29 +−00..024
011
2
21
∆m21
= 6.03+−11..67
×10 −5 eV 2

-CO.#0&Figure 12.1: Allowed region of LMA (inside area of the line is allowed)

This result2

sin θ12 = 0.304+−00..017
016
2
12
∆m21
= 7.59+−00..39
×10−5 eV 2

Solar

KamLAND

Figure 12.2: Allowed region of LMA, Solar + KamLAND (inside area of the line is allowed)

118

4


9KPVGT$URGEVTWO
1TVK\$URGEVTWO

3

8

6

2

B Flux (10 /cm /s)

5

2
1
Errors: stat. only
Line: SK-I average

0

5

10
15
20
Total energy of recoil electrons (MeV)

Figure 12.3: SK-III data /SSM (BP2004) with Winter and Ortiz 8 B spectrum. Green shows
SSM with Winter 8 B spectrum, and red shows SSM with Ortiz 8 B spectrum.

12.2

Comparison of Winter and Ortiz 8 B spectrum

In this section, the eﬀect of the diﬀerence between Winter 8 B spectrum and Ortiz 8 B spectrum to the oscillation contour is discussed.
Figure 12.3 shows the SK-III energy spectra divided by SSM with Winter 8 B spectrum andOrtiz 8 B spectrum. Since the Ortiz spectrum has more tail in the higher energy region comparedto the Winter spectrum (see Figure 2.8), the energy spectrum over the SSM prediction becomeslarger in high energy region if the Winter spectrum is used for the SSM prediction. Therefore,
if the Wineter spectrum is used, this makes the χ2 value of the spectrum ﬁtting worse in the
LMA solution area because the LMA predicts upturn of the energy spectrum in the lower energy
region, and thus, the LMA area is excluded slightly lager. In Figure 12.4, we can see this eﬀect.
For the spectrum ﬁtting, I use diﬀerent energy correlated systematic uncertainties for Winterand Ortiz spectrum which are shown in Figure 9.5.

12.3

Comparison with other results

Figure 12.5 shows the comparison of the results on the mixing angle measurements. The
reason why only the mixing angle results are compared is that the uncertainty of ∆m2 is mainly
controlled by KamLAND, and thus, for the solar neutrino experiments, the contribution to the∆m2 measurement is smaller than the contribution to the mixing angle measurement. As shownin Figure 12.5, even though only the CC and NC rate information from SNO are used in thisthesis, the size of uncertainty is about the same. This indicates that if more information of
the SNO results, for example CC spectral shape, is added to the SK result, the global analysis
results will be improved.
Using these oscillation parameters obtained from the solar and reactor experiments, the ﬂux
value of 8 B neutrino can be extracted. As in Equation 11.20, the β is a free parameter tominimize the χ2SK+SN O and there is no constraint from the SSM prediction in the χ2SK+SN O .
119

-3

2

∆m in eV

2

10

-4

10

LMA
-5

10

-6

10

-7

10

-8

10

-9

10
-4
10

Winter8 B spectrum
Ortiz 8B spectrum
95% C.L
-3

10

-2

10

-1

10

1

10

2

10
2
tan (Θ)Figure 12.4: Excluded region by SK-III data /SSM (BP2004) with Winter and Ortiz 8 B spectrum shapes. Green shows SSM with Winter 8 B spectrum, and red shows SSM with Ortiz 8 B
spectrum.

120

6JKUYQTM

θ 12 [deg.]

40
38
36

3flavor (T.Schwetz)

SK-I,II,III,SL+ KL

26

SNO-III,LETA, SL+KL

28

SNO-I,II, SL+KL

30

SNO-I,II,SL

32

SNO-I,II,III, SL+KL

34

Figure 12.5: Comparison of the mixing angle measurement between SNO results and the results
obtained by this thesis. the error size corresponds to 1 σ uncertainty of (stat.+sys.). SNOI,II,SL use SNO phase I and II and other solar neutrino data (SL) [29] and SNO-I,II,SL+KL
added KamLAND(2004) [29]. SNO-I,II,III,SL+KL add the phase III [30] and updated KamLAND(2008) [51]. SNO-III,LETA+KL is latest SNO result [47]. Result of this work is shown
as SK-I,II,III SL+KL. The 3 ﬂavor analysis is from the analysis of latest solar + reactor+
atmospheric results [52]Thus, the ﬂux value scaled by the βm does not depend on the SSM so much. Figure 12.6 shows
the scaled 8 B ﬂux values by using the βm at the best ﬁt point obtained by the global solar
analysis and the global solar + KamLAND analysis and they are compared with the 8 B ﬂux
obtained by SNO NC rate. The size of error corresponds to the maximum and minimum ﬂux
values among the 1σ oscillation parameter region. The obtained 8 B ﬂux values are;Global solar analysis:
Global solar+KamLAND analysis:

6−2 −1
5.27+0.16−0.13 × 10 cm s

6−2 −1
5.08+0.10−0.07 × 10 cm s

As shown in Figure 12.6, the ﬂux obtained by the solar neutrino experiments agree with the all
the SSM predictions (BP04, BS05(OP) and BS05(AGS,OP)), and the seismic model prediction[7] within their uncertainties. However, the size of the experimental uncertainty (∼ 2%)is muchsmaller than that of SSMs. Thus, our results of 8 B ﬂux measurement can be a strong tool totest the future improvement of SSM.

121

Solar Global (This work)

SNO-III

SNO-I

Solar+KamLAND(This work)

5.5

SNO-LETA

6

SNO-II

6.5

5GKUOKE/QFGN

7$5 12

7.5

$2

8

$5 #)512

8

B Flux [X106cm-2 sec-1]

8.5

5
4.5
4

Figure 12.6: Comparison of the 8 B ﬂux between SNO results and the results obtained by this thesis. the error size corresponds to 1 σ uncertainty of (stat.+sys.). Square mark shows theoreticalpredictions and cross marks shows experimental results.

12.4

Treat reduction uncertainties as energy correlated uncertainties

In Chapter 9, the uncertainties of ambient cuts (ovaQ cut and hit pattern cut) are treated asenergy uncorrelated uncertainty. However, if MC simulation gives systematically better goodness
than the real data, for example, the uncertainties for each energy bins move to the same direction.
That is, the uncertainties of ambient cuts have some energy correlation. To estimate how thisenergy correlation aﬀects the ﬁtting of the spectrum, those uncertainties (ovaQ cut and hit
pattern cut) are treated as energy correlated uncertainties here.
Figure 12.7 shows the uncertainties of ovaQ cut and hit pattern cut which are obtained fromthe comparisons between LINAC data and MC simulation (see Figure 9.1 and 9.2 of Section
9.6). These uncertainties are assigned as ±1σ uncertainties of the ambient cut eﬃciency (seethe purple histogram of Figure 7.21), and treated as ϵ±
x of Equation 11.11. Hence, the deﬁnition2in χspecSK3 of Equation 11.16 becomes222
2
χ2specSK3 (β, η) = Min(χ2α,SK3 (β, η, δB , δS , δR , δovaQ , δHitP at ) + δB
+ δS2 + δR
+ δovaQ+ δHitP
at )(12.8)
where the free parameters to be ﬁtted are β, η, δB , δS , δR , δovaQ , δHitP at . To avoid double countingof uncertainties, the energy uncorrelated systematic uncertainties are modiﬁed as in Table 12.1.

By taking them as energy correlated uncertainties, all data points shift to the same direction
when they are ﬁtted. This treatment gives a slightly tighter allowed region than when the data
points are uncorrelated (see Figure 12.8). The best-ﬁt point is unchanged after this modiﬁcation.
122

7PEGTVCKPV[=?

5

QXC3EWVWPEGTVCKPV[*KVRCVVGTPEWVWPEGTVCKPV[

4
3
2
1
0
-1
-2
-3
-4
-5
6

8

10

12

14

16

18

20

'PGTI[=/G8?
Figure 12.7: Energy correlated systematic uncertainties of ovaQ and hit pattern cut.

Energy(MeV)

5-5.5

5.5-6

6-6.5

6.5-7

7-7.5

7.5-20

Total

±3.8%

±2.4%

±2.1%

±0.8%

±0.8%

±0.8%

Table 12.1: Energy uncorrelated systematic uncertainty on the spectrum shape. The contributions from ovaQ and Hit pattern cut are removed from Table 9.3

123

2

2 -4
x10
QXC3CPF*KVRCVVGTPEWVWPEGTVCKPVKGUCTGVTGCVGFCUGPGTI[WPEQTTGNCVGFU[U

2∆m in eV∆m2 in eV2

-3

10

-4

10

%.
%.
%.

-5

10

QXC3CPF*KVRCVVGTPEWVWPEGTVCKPVKGUCTGVTGCVGFCUGPGTI[EQTTGNCVGFU[U

-6

10

1

%.
%.
%.

-7

10

-8

10

%.
-9

10
-4
10

-3

10

-2

10

-1

10

1

2

10
10
2
tan (Θ)0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
2sin (Θ)Figure 12.8: Excluded region (inside area of blue line) from SK-III spectrum ﬁt (left) and allowed
region from all solar neutrino experiments (right). The systematic uncertainties of ambient cut
are treated as energy correlated uncertainties. Black line in left plot shows the result in which
the uncertainties have no correlation. The best ﬁt point of global analysis is not changed whether
the uncertainties are energy correlated or not.

124

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0.8
9KVJQWVFKUVQTVKQP

0.7

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0.6

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0.5
0.4
0.3
0.2
0.1
0

6

8

10

12

14

16

18

20

6QVCNGNGEVTQPGPGTI[=/G8?

Figure 12.9: SK-III spectrum with expected distorted shapes and ﬂat shape. The error shows
only statistical uncertainty.

Even if this treatment gives better results, it is not clear how much the uncertainty band shownin Figure 12.7 is energy correlated. Thus, in the ﬁnal results on parameter region of solar
neutrino oscillation, the uncertainties of the ambient cuts are treated as energy uncorrelatedconservatively.

12.5

Sensitivity to upturn

Figure 12.9 shows SK-III energy spectrum with solar best ﬁt, solar plus KamLAND best
ﬁt, and ﬂat expectations. The χ2 value with each case is following: 26.8/20d.o.f. (solar bestﬁt: ∆m2 = 6.03 × 10−5 eV 2 , sin2 θ = 0.29), 26.7/20d.o.f. (solar plus KamLAND best ﬁt:∆m2 = 7.59 × 10−5 eV 2 , sin2 θ = 0.34), and 27.7/20d.o.f. (ﬂat shape). When the ambient cut
uncertainties are treated as energy correlated systematic uncertainty, the χ2 value with eachcase is following: 26.9/20d.o.f. (solar best ﬁt), 27.0/20d.o.f. (solar plus KamLAND best ﬁt),and 27.5/20d.o.f. (ﬂat shape).
Figure 12.10 shows spectral shape of each phase with solar best ﬁt expectation. Although it
is not statistically signiﬁcant, the spectral shape of SK-III might show a little hint of the upturn.
Finally, a sensitivity of the observation of upturn is estimated. Followings are assumed forthis estimation;
1. Background level is same as SK-III.
2. Energy correlated systematic uncertainty is half of SK-I
3. 4 MeV threshold is accomplished.

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0.8

5-+

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5-+++

0.7
0.6
0.5
0.4
0.3
0.2
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0

5

10

15

20
5

10

15

20
5

10

15

20

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Figure 12.10: SK-I,II, and III spectrum with expected distorted shapes and ﬂat shape. The
error shows only statistical uncertainty.

126

If we use Winter06 8 B spectrum, energy correlated uncertainty of SK-III already achieves 30%
improvement since SK-I. Further 20% improvement would be possible if the position and direction dependence of energy scale becomes half of current dependence. This is done by more
precise calibrations and a better modeling of water condition. Item 3 would be possible because the new electronics and DAQ system, which are already installed in 2008, can store all
hit information without hardware trigger. Under this assumption, about 2σ level observation
within 3 years, and about 3σ level within 10 years could be expected. In addition to new data,
re-analysis of SK-I data is desirable because the calibration method and analysis tools which
are developed in this thesis can be applied to SK-I data and expected to reduce uncertainties.
Since we already have 6 years of total livetime during SK-I and III, A 3σ level discovery of the
upturn might be possible within few years if the SK-I data are re-analyzed.

127

Chapter 13

Conclusion
In conclusion, the measurement of solar neutrino with SK-III was carried out aimed at theobservation of the spectrum distortion due to MSW eﬀect. The strategies of this thesis to achieve
this goal are to reduce the background level by 70% from SK-I in the lower-energy region, andto reduce the energy correlated systematic uncertainty by 50% compared to SK-I.
With improved detector calibrations, a full detector simulation, and analysis methods, thenumber of background events is reduced by about 50% from SK-I. Meanwhile, the energy correlated systematic uncertainties are reduced by 30%, and the energy uncorrelated systematic
uncertainties are reduced by 60% from SK-I. Thus, the goal is almost achieved and acceptable
for the future observation of the spectrum distortion.
The systematic uncertainty of the total ﬂux is estimated as ±2.3%, and is about two thirdof SK-I. The 8 B ﬂux in SK-III is
2.28 ± 0.04(stat.) ± 0.05(sys.) × 106 cm−2 sec−1
which is the most precise result of previous results from SK-I and II. A day-night asymmetry is
obtained to be
ADN = −0.057 ± 0.031(stat.) ± 0.013(sys.)
After combining the previous phase of SK and other solar neutrino experiments, the best-ﬁt
parameters of the solar neutrino oscillation are determined with the world’s best accuracy;sin2 θ12 = 0.29+0.024−0.011−5
2∆m212 = 6.03+1.21−1.67 × 10 eV

By adding the result from KamLAND, the obtained best-ﬁt parameters are
sin2 θ12 = 0.304+0.017−0.016(θ12 = 33.46+1.05−1.00 )

−52
∆m221 = 7.59+0.12−0.39 × 10 eV

which are consistent with the latest analysis by SNO group.
The χ2 value of spectrum ﬁt with the solar plus KamLAND best-ﬁt prediction is 26.7/20d.o.f.
which is slightly better than 27.7/20d.o.f. with a ﬂat shape. Although, this result is notstatistically signiﬁcant, it is estimated that the improved calibration and analysis methods will
give a sensitivity of 3σ level discovery of the spectrum distortion within a few years, together
with re-analysis of the SK-I data.
128

Appendix A

Glossary
ADC Analog to Digital Converter
ATM Analog Timing Module; front end electronics based on TKO standard.
BP04 One of the SSM series published in 2004, which is used in this thesis.
BONSAI Branch Optimization Navigating Succesive Annealing Iterations, the standard vertex
ﬁtter for this analysis.
BS05(AGS,OP) One of SSM series published in 2005, which uses the latest physics information, called the ”low Z” model including updated opacity calculation by Opacity Project.
BS05(OP) One of SSM series published in 2005, which used the previous heavy metal abundance, called the ”high Z” model. Other updates including the opacity calculation by OP
are implemented.
COREPMT Average Q.E. determined by LINAC 8.8MeV. This in one of input parametersfor MCDT Duteron Tritium generator (neutron source)
HE High Energy Trigger (-320mV)HITSUM summation of hit square pulses
ID Inner Detector
LMA Large Mixing Angle solution of solar neutrino oscillation
LE trigger Low Energy trigger (-302mV)
LINAC LINear ACcelerator
LMA Large Mixing Angle solution of the solar neutrino oscillation. 10−5 < ∆m2 < 10−4 eV2and θ12 ∼ 30 − 40◦
Nef f Eﬀective number of hits in a event.
Nhit The number of hits in one event.
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Ni calibration Calibration using Cf-Ni gamma-ray source
Nx The number of hits within x nsec time window. The number is found after applying thesliding x nsec time-window search to the t − tof − t0 distribution.
OD Outer Detector
ovaQ One dimensional variable for Vertex and Angular Quality: G2V − G2A
PMT Photo Multiplier TubeQAC Charge to Analog Converter
Q.E. Quantum eﬃciency of PMT
Seismic model One of solar models. This model is based on a standard stellar evolution andthe input information is adjusted for the Sun by the seismic measurements.
S-factor Cross section factor of nuclear fusion reactions in the Sun.
SLE1 trigger Super Low Energy trigger period 1(-212mV)
SLE2 trigger Super Low Energy trigger period 2(-186mV)SMP Super Meory Partner; VME memory modules after ATM
Spallation cut Cut to reject sapallation products
SSM the Standard Solar Model. A series of solar model ﬁrst developed by J.N.Bahcall.
TAC Timing to Analog Converter
TBA Top Bottom Asymmetry of the detector
Tisk hit timing of each PMT (without subtruction of TOF)
TKO Tristan KEK Online standard
TOF Time of ﬂight
TQmap Hit timing correction function as a fuction of charge
TRG TRiGger module which distribute the global trigger and the event number

130

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