U  C,I

Solar Neutrino Measurement from the Second Phase of the Super-Kamiokande
ExperimentD

submitted in partial satisfaction of the requirements
for the degree ofD  P
in Physics

byJohn Parker Cravens

Dissertation Committee:
Professor Henry W. Sobel, Chair
Professor Steven Barwick
Professor Arvind Rajaraman2008

c 2008 John Parker Cravens

The dissertation of John Parker Cravens
is approved and is acceptable in quality and form for
publication on microfilm and in digital formats:Committee Chair

University of California, Irvine
2008

ii

D

For Mina.

iii

T  C
L  F



L  T



A



C V

A   D



1N
1.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Formalism of Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Neutrinos in the Standard Model . . . . . . . . . . . . . . . .
1.2.2 Neutrino Mass . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.3 Majorana Mass . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.4 See-saw Mechanism . . . . . . . . . . . . . . . . . . . . . . .
1.3 Neutrino Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Weak Interaction . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.2 Neutrino Flavor Change . . . . . . . . . . . . . . . . . . . . .
1.3.3 Neutrino Flavor Change in Matter . . . . . . . . . . . . . . .

1
1
3
3
4
5
6
9
9
11
13

2

S N
2.1 The Standard Solar Model . . . . . . . . . . . . . . . . . . . . . . . .
2.2 The Solar Neutrino Problem . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 The Oscillation Hypothesis . . . . . . . . . . . . . . . . . . .
2.3 Solar Neutrino Experiments . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Homestake . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Kamiokande . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 GALLEX/GNO & SAGE . . . . . . . . . . . . . . . . . . . . .
2.3.4 Super-Kamiokande . . . . . . . . . . . . . . . . . . . . . . . .
2.3.5 Sudbury Neutrino Observatory . . . . . . . . . . . . . . . . .
2.3.6 Summary of Current Solar Neutrino Data . . . . . . . . . . .

17
17
22
22
23
27
27
27
28
29
29
30

3

T S-K-II D
3.1 Experiment Overview . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Detection Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Neutrino Interaction . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Cherenkov Radiation . . . . . . . . . . . . . . . . . . . . . . .
3.3 Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34
34
37
37
37
38
38

iv

3.3.2 Inner Detector . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Outer Detector . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.4 Photomultiplier Tubes . . . . . . . . . . . . . . . . . . . . . .
Purification Systems . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Data Acquisition System . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 ID DAQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.2 OD DAQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.3 Trigger System . . . . . . . . . . . . . . . . . . . . . . . . . .
Water Transparency Measurement . . . . . . . . . . . . . . . . . . .
3.6.1 Laser Light Injection . . . . . . . . . . . . . . . . . . . . . . .
3.6.2 Time Variation . . . . . . . . . . . . . . . . . . . . . . . . . .
Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7.1 PMT Calibration . . . . . . . . . . . . . . . . . . . . . . . . .
3.7.2 Energy Calibration . . . . . . . . . . . . . . . . . . . . . . . .

39
39
41
41
41
45
47
47
49
49
51
51
52
55
55
61

4

SK-II S E
4.1 Event Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.3 Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Data Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Noise Reduction . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Spallation Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Muon Track Reconstruction . . . . . . . . . . . . . . . . . . .
4.3.2 Selection of Spallation Events . . . . . . . . . . . . . . . . . .
4.4 Timing & Hit Pattern Cut . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Gamma-ray Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Final Data Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7 Monte Carlo Event Simulation . . . . . . . . . . . . . . . . . . . . .
4.7.1 Neutrino Rate and Interaction . . . . . . . . . . . . . . . . .
4.7.2 Detector Response . . . . . . . . . . . . . . . . . . . . . . . .
4.7.3 Energy Resolution Function . . . . . . . . . . . . . . . . . . .
4.8 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8.1 Signal Extraction . . . . . . . . . . . . . . . . . . . . . . . . .
4.8.2 Day-Night and Seasonal Variation . . . . . . . . . . . . . . .
4.8.3 Energy Spectrum . . . . . . . . . . . . . . . . . . . . . . . . .
4.9 Summary of Systematic Errors . . . . . . . . . . . . . . . . . . . . .

68
68
68
70
71
75
77
82
83
84
86
88
88
89
91
93
95
98
98
101
104
106

5

O A
5.1 Analysis of the SK-II χ2 . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Oscillated Rate Predictions . . . . . . . . . . . . . . . . . . .
5.1.2 SK-II Systematic Error Treatment . . . . . . . . . . . . . . . .
5.1.3 Time-Variation Analysis . . . . . . . . . . . . . . . . . . . . .

109
109
111
114
118

3.4

3.5

3.6

3.7

v

5.2

5.3

5.4

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 SK-II Spectrum & Time-Variation Fits . . . . . . . . . . . . .
5.2.2 SK-II Rate Constrained Fit . . . . . . . . . . . . . . . . . . . .
Combining with Other Solar Experiments . . . . . . . . . . . . . . .
5.3.1 Sudbury Neutrino Observatory . . . . . . . . . . . . . . . . .
5.3.2 Radiochemical Experiments & KamLAND . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

B

119
120
122
123
123
126
128
130

vi

L  F
2.1
2.2
2.3
2.4
2.5
2.6
2.7

The CNO cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The pp chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The solar electron density profile . . . . . . . . . . . . . . . . . . . .
The MSW effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Oscillation parameter solutions allowed by solar neutrino data . . .
BP2004 SSM solar fluxes . . . . . . . . . . . . . . . . . . . . . . . . .
BP2004 SSM compared with experimental data . . . . . . . . . . . .

19
21
24
25
26
31
33

3.1 SK in Japan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 The SK detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Photograph of the ID . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Diagram of an ID PMT. . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 The PMT quantum efficiency and Cherenkov spectrum . . . . . . .
3.6 Water purification system . . . . . . . . . . . . . . . . . . . . . . . .
3.7 Radon levels at SK . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8 Air purification system . . . . . . . . . . . . . . . . . . . . . . . . . .
3.9 ATM schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.10 ID DAQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.11 SK-II laser system . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.12 Laser light timing distributions . . . . . . . . . . . . . . . . . . . . .
3.13 SK-II light scattering coefficients . . . . . . . . . . . . . . . . . . . .
3.14 PMT acceptance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.15 Water transparency . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.16 Xenon lamp calibration system . . . . . . . . . . . . . . . . . . . . .
3.17 Relative gain by xenon lamp calibration . . . . . . . . . . . . . . . .
3.18 The Ni-Cf source apparatus . . . . . . . . . . . . . . . . . . . . . . .
3.19 The PMT charge distribution . . . . . . . . . . . . . . . . . . . . . .
3.20 The N2 laser calibration system . . . . . . . . . . . . . . . . . . . . .
3.21 Timing-charge map . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.22 The LINAC calibration system . . . . . . . . . . . . . . . . . . . . .
3.23 LINAC energy scale and resolution differences in LINAC data andMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.24 DTG setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.25 DTG energy scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35
36
40
42
43
45
46
46
48
50
53
54
55
56
57
58
59
60
60
61
62
63

4.1
4.2
4.3
4.4
4.5
4.6
4.7

70
72
72
76
78
78
79

Vertex resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Directional likelihood function . . . . . . . . . . . . . . . . . . . . .
Directional resolution . . . . . . . . . . . . . . . . . . . . . . . . . . .
SK-II energy scale as a function of time . . . . . . . . . . . . . . . . .
Time difference cut . . . . . . . . . . . . . . . . . . . . . . . . . . . .
OD trigger cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ambient electronic noise cut . . . . . . . . . . . . . . . . . . . . . . .
vii

65
66
67

4.8 Flasher cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.9 Fiducial volume cut . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.10 Muon track reconstruction . . . . . . . . . . . . . . . . . . . . . . . .
4.11 PMT timing and hit pattern cut . . . . . . . . . . . . . . . . . . . . .
4.12 Gamma-ray cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.13 Data reduction summary . . . . . . . . . . . . . . . . . . . . . . . . .
4.14 Data reduction efficiency summary . . . . . . . . . . . . . . . . . . .
4.15 Neutrino distributions, ν − e− elastic scattering cross sections, andexpected recoil electron spectra . . . . . . . . . . . . . . . . . . . . .
4.16 The SK-II light absorption coefficient . . . . . . . . . . . . . . . . . .
4.17 Energy resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.18 Predicted recoil electron spectrum . . . . . . . . . . . . . . . . . . .
4.19 Solar angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.20 The expected signal shape . . . . . . . . . . . . . . . . . . . . . . . .
4.21 The extracted signal in the solar direction . . . . . . . . . . . . . . .
4.22 The extracted signal by energy bin . . . . . . . . . . . . . . . . . . .
4.23 Flux time variation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.24 Flux time variation compared with solar activity . . . . . . . . . . .
4.25 SK-II energy spectrum . . . . . . . . . . . . . . . . . . . . . . . . . .

92
94
96
97
99
100
101
102
104
106
107

5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8

113
113
115
121
122
124
127
129

Live time and solar zenith rates . . . . . . . . . . . . . . . . . . . . .
Electron densities of the Sun and Earth . . . . . . . . . . . . . . . . .
Energy correlated systematic errors on predicted spectrum . . . . .
Spectrum χ2 map . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Spectrum and time variation χ2 map . . . . . . . . . . . . . . . . . .
8B flux constrained χ2 map . . . . . . . . . . . . . . . . . . . . . . . .
SK-SNO combined χ2 map . . . . . . . . . . . . . . . . . . . . . . . .
Global χ2 map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

viii

80
82
85
87
89
90
90

L  T
2.1
2.2

BP2004 SSM fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Summary of experimental flux values . . . . . . . . . . . . . . . . .

20
32

4.1
4.2
4.3

SK-II observed and expected rates . . . . . . . . . . . . . . . . . . .
SK-II systematic error . . . . . . . . . . . . . . . . . . . . . . . . . . .
Energy correlated systematic errors for each energy bin . . . . . . .

105
108
108

ix

A
I wish to thank Dr. Michael Smy for his tremendous help on the analysis and thedrafting of this thesis. He was a great model to follow and I continually admired
his abilities as an experimentalist.
I wish to thank Prof. Hank Sobel for his support. I had many chances to attend and speak at conferences as well as traveling to Irvine to work with Dr. Smyon the analysis. He never restricted my path to this degree.
I wish to thank Prof. Masayuki Nakahata for letting me work with the solar
neutrino group in Japan and also letting me participate as a presenter in local conferences. My time in Japan was unique and I hope to spend more time there as ascientist.
These three men are only the principle people during my graduate career. There
are so many more that I want to thank but are too numerous to list here. To all
those I had the honor of knowing and working with, I will always remember you
fondly.

x

C V
J P C
1998

B.S. in Physics, Southern Methodist University, Dallas

2003

M.S. in Physics, University of California, Irvine

2008

Ph.D. in Physics, University of California, Irvinexi

A   D
Solar Neutrino Measurement from the Second Phase of the Super-Kamiokande
Experiment
By
John Parker Cravens
Doctor of Philosophy in Physics
University of California, Irvine, 2008
Professor Henry W. Sobel, Chair

The second phase of the Super-Kamiokande experiment aimed at the continuation of the solar neutrino measurement after the 1496-day first phase. However,the second phase operated with a photocathode coverage 47% of the first phase’s.
This reduction in sensitivity prompted the development of new analysis tools andcreated larger estimations of systematic errors. Despite these changes, the second
phase solar neutrino data showed consistency with the first phase and no indication of systematic tendencies between the two phases was present. An oscillation
analysis of the second phase resulted in reduced exclusion power of the neutrino
mixing angle and mass difference parameter set. However, a rate constrained combined oscillation analysis of both phases continues to favor the Large Mixing Angle
solution at 95% confidence level.

xii

C 1
N
The study of neutrinos is currently an active topic. Both theoretical and experimental research is being conducted to unlock the secrets of neutrinos and neutrino
mass. The phenomenon of neutrino oscillation is currently the only way experimentalists can conclude that neutrinos have mass but are still unable to reveal theirdefinite quantities. Theoreticians, however, have laid the groundwork that could
possible describe why neutrinos have the masses they do and the consequence thatis oscillation.

1.1

H

The neutrino was the first particle to be proposed that was not constituent of atomic
matter. This proposal came in the last month of 1930 from Wolfgang Pauli in attempt to reconcile the canon of conservation of energy with the results of James
Chadwick’s 1914 214 Pb and 214 Bi decay experiments. In these experiments, Chadwick demonstrated the decaying β particles were continuous in energy, not discreet
as expected in the burgeoning quantum theory at the time. The introduction of a
weakly interacting neutral fermion seemingly solved the discrepancies although
presented a greater challange: how to detect it.
The newly proposed neutral fermion was dubbed ”neutrino” by Enrico Fermi
and almost all serious scientific efforts aimed at the neutrino were through theoretical research, mainly by Fermi and his theory of β-decay. It wasn’t until 1956 that
Frederick Reines and Clyde Cowan announced their detection of reactor antineutrinos from the Savannah River reactor in South Carolina. Only one neutrino type1

(or flavor) was believed to exist.
Earlier in 1947, with the observation of the decay π+ → µ+ , energy conservation
was again seemingly violated but this could now be easily remedied by the assumption that an invisible neutrino carried off the remaining energy. It would not
be shown until 1962 (after the detection by Reines and Cowan) in an experiment
by Leon Lederman, Melvin Schwartz and Jack Steinberger, working at the thenpowerful AGS, that the neutrinos in pion decay were in fact distinguishable fromthose in β-decay. The new neutrino was named the muon neutrino and the original one the electron neutrino due to the interactions with their respective charged
leptons.
This brought rise to the idea of lepton family in which each charged lepton has
a chargeless neutrino as its partner. Also with this grouping came conservation of
lepton family number. Lepton number conservation (where leptons would have
L = 1 and antileptons L = −1 and their total value was conserved during a reaction)
was already proposed in 1953 to account for missing decay modes. But othermodes, such as µ → e− + γ, while allowed by lepton number conservation, had
experimental cross section limits that were exceedingly small. With two families, it
was possible to assign Le and Lµ for each the electron and muon flavors respectively.
The lepton family numbers were postulated to be conserved quantities in addition
to global lepton number. This eliminated the possibilities of numerous processes.
A third family would soon appear with the discovery of the tau lepton in 1975
by Martin Perl. It was again assumed that energy conservation in tau decays wasmediated by a neutrino. On July 21st, 2000, the neutrino beam experiment DONUT
at Fermilab announced that the tau neutrino had successfully been detected.
Thirty-two years earlier, neutrinos from the Sun were detected by Ray Davis andhis Homestake experiment making it the first detection of non-terrestial neutrinos.
In the abandoned Homestake gold mine in South Dakota, USA, electron neutrinos

2were detected through the reaction 37 Cl + νe →37

Ar + e− . This was ironic since

Davis’ first attempt was to detect reactor antineutrinos through the reaction 37 Cl +
ν̄e →

37

Ar + e− [1], a process which violates global lepton number. Homestake

operated continuously from 1967 until 1994.
What Homestake’s results concluded were that the observed solar neutrino
flux was three times lower than what is predicted by theoretical solar models. A
possible explanation was the solar models were flawed. However, after constantrevision of solar theory and with the measurements of proceeding solar neutrino
experiments, this discrepancy only got worse [2]. Another possibility was that theelectron neutrinos from the Sun may transition to another flavor where the detectors
are not sensitive or have limited sensitivity. This is possible if neutrinos wereascribed some mass. The theory of massive neutrinos ”oscillating” was alreadydeveloped by Z. Maki, M. Nakagawa, and S. Sakata in 1962 and independently by
B. Pontecorvo in 1967. But previous neutrino measurements gave no indication ofmass, nor is it required by the Standard Model. What Homestake and eventually
other solar, atmospheric, and accelerator neutrino experiments did was present thepossibility of neutrino mass.

1.2

F  N

1.2.1

N   S MIn the fermion group of the Standard Model (SM) of particles and interactions,fundamental particles are divided into three generations labeled simply by order.
Each generation contains 2 quarks and 2 leptons. The particles in the lepton sectorof the 3 generations can be grouped into electron, muon, and tau flavors. They
consist of the electron and electron neutrino in the first generation (electron flavor),muon and muon neutrino in the second generation (muon flavor), and tau and tau
3neutrino in the third generation (tau flavor).
With the absence of charge, neutrinos interact with themselves and other matter
only through the weak and gravitational interactions. There are two classes of
weak interactions characterized by their exchange bosons. The charged current(CC) interaction is mediated by the massive W ± boson and the neutral current(NC) interaction is mediated by the massive Z boson.
Neutrinos are assumed to be massless in the SM. Along with their ~/2 spin
angular momentum, this constrains neutrinos to be in a state of definite helicity.
Furthermore, in agreement with experimental results and the accepted form of the
weak interaction theory, neutrinos have negative helicity while their antineutrino
partners have positive helicity. No positive (negative) neutrino (antineutrino) states
exist in the SM.

1.2.2

N M

Although the SM does not account for massive neutrinos, mass can be introduced
into its current framework. This requires the inclusion of a chirally right-handed
neutrino field which could give rise to a positive helicity neutrino state dependingon an observer’s inertial reference frame. The procedure is analogous to the way
the charged leptons acquire their masses: through coupling with the Higgs field.
The relevent term in a massive neutrino-Higgs Lagrangian density is
Lmass = −yφ0 ν0 R ν0L + h.c.,(1.1)

where y is the Yukawa coupling constant, φ0 is the vacuum expectation value, andh.c. denotes the hermitian conjugate. The neutrino mass is recognized as mν = yφ0 .
The vacuum expectation value φ0 is 174 GeV and, as the same for the charged
leptons, y is determined experimentally.

4

Unfortunately, there is no direct experimental measurement of definite neutrino
mass. There only exists mass squared differences (∆m2 ) between neutrinos ofdifferent generations. But these limited results do suggest a definite mass aroundthe value of 0.05 eV. This makes the order of magnitude of y 10−13 .

1.2.3

M MThe Yukawa coupling constant y is inexplicably small, especially given the sizes of
the charged lepton couplings. In an effort to give better reason for small neutrino
masses, a second mass term can be introduced into the mass Lagrangian. The
nature of this term assumes that νL and νR are not independent fields but are
somehow related. This comes in the form of the Majorana Condition and is given
explicitly as
Tν0R(L) = Cν0 L(R) ,(1.2)

where C is the charge-conjugate matrix and T represents transpose. Equation 1.2
can be made slightly clearer by recognizing the right side is the definition of thecharge conjugate field, νc . This is the first major departure from the SM. For aneutrino field with both chirally left- and right-handed fields ν = νL + νR , charge
conjugation returns the same field (νc = ν). This is identical to saying a particle
is its own antiparticle. If neutrinos have this property, they are called Majorana
neutrinos. When a neutrino and its antineutrino have distinction, they are known
as Dirac neutrinos.
Using the Majorana Condition, the mass term of 1.1 can be rewritten as
1
Lmass = − mR (ν0 R )c ν0R + h.c.,
2

(1.3)

where the factor 1/2 is introduced to avoid double contributions of the same field.
The mass mR is known as a right-handed Majorana mass. (The same can be5accomplished with νL and mL but is omitted for brevity in the following discussion.)
The mass of Equation 1.1 is known as a Dirac mass (mD = yφ0 ).

1.2.4

S- M

For simplicity, only a single flavor will be considered and will regard the two otherflavors as nonexistent. The new Majorana mass term can be added to the Diracmass Lagrangian to obtain
Lmass = −mD ν0 R ν0L +mR 0 c 0(ν R ) νR + h.c.
2

(1.4)

Using the Majorana Condition for the fields associated with the Dirac mass mD , theLagrangian can be written in matrix form:

Lmass = −12

(ν0 L )c  0ν0 R 
 m

D



0




mD   νL  
 + h.c.
m   (ν0 )c 
R

(1.5)

R

From Equation 1.2, it can be seen that the left and right field vectors in the above
expression each have two identical components making the leading factor of 1/2
necessary. For the mass matrix M (the matrix in the above equation with elements
0, mD , and mR ), it is no longer diagonal due to the presence of the Dirac mass mD .
To find the mass of the neutrino, M must be diagonalized and can be done so by
the common method of eigen decomposition. The resulting diagonal elements are
m1,2 =q1
mR ± (mR )2 + 4m2D .
2

6

(1.6)

To simplify the eigenvalues, the assumption mR  mD is made and results in mass
eigenvalues
m1 =

m2D
mR

,

(1.7)

m2 = mR .

(1.8)

By Equation 1.6, m2 is negative, but since the eigenvalues represent real, physicalmasses, m2 is made positive.
To avoid forcing m2 to be positive, eigenvalues greater than zero can be obtainedby a suitable choice of rotation matrix V to be operated on the mass matrix M [3]:

 



 1 −iρ   0 mD   1 ρ   m2 /mR 0    D

 

 
 =  
 ,VT MV = 







 ρ 1   m m   −iρ 1  
0
mR 
D
R

(1.9)

where ρ = mD /mRM and again assuming mR  mD .
With the matrix notation of Equation 1.9, it is possible to write the mass Lagrangian in term of mass eigenvalues and their corresponding eigenstates:

Lmass = −

= −

1
2
1
2



(ν0 L )c ν0 R




 0

VV−1 
 m

D






0


 ν 
mD 
 VV−1  L  + h.c.|V 0 c 
m 
(ν )
R

R

 2


  mD 0   ν  mR
  1 
 
 ,
ν2 
 0 m   ν R2

ν1

where h.c.|V signifies that the V rotation is applied to the hermitian conjugate. The
components ν1 and ν2 are recognized as neutrino mass eigenstates with masses
m2D /mR and mR , respectively. They are defined in terms of ν0L and ν0R as






 ν1  ν0  (ν0 )c 







 = V−1  L  + V−1  L  .
 0 c  0 
ν
(ν )
ν
2

R

R

7

(1.10)

What has happened is by assuming one neutrino field ν0 = ν0L + ν0R and two
mass terms in the Lagrangian mD and mR , two mass eigenstates appear. With thecondition that mR be considerably larger than mD , the eigenstate masses are likewisevery small and very large. Furthermore, it is exactly due to mR ’s relatively large
value that m1 is small. This is known as the see-saw mechanism and is an attractive
explanation of the observed smallness of neutrino mass. A small Yukawa coupling
is not necessary and mD can be assumed to be of the same scale as the charged
leptons and quarks. There exists no constraints on the size of mR and it can bechosen accordingly. As an example [4], suppose mD is 175 GeV, the same mass as
the top quark. If mR is given an order of magnitude of around 1015 , m1 becomes∼ 0.03 eV, a very likely mass for a light neutrino.
Several questions arise from this formulation. One is the prediction of a heavy
neutrino which is as-of-yet unobserved. By dividing ν1 and ν2 into left and righthanded components, Equation 1.10 can be rearranged toν0L = V11 ν1,L + V12 ν2,L ,(1.11)

ν0R = V21 ν1,R + V22 ν2,R .

(1.12)The left-handed component which participates in weak interactions is mostly ν1,L
with mass m1 since V12 = mD /mR is a small parameter. On the other hand, ν0R is
mostly ν2,R with mass mR . But ν0R does not take place in weak interactions and isregarded as sterile. Therefore, if a heavy neutrino does exist, it might be impossible
to detect.
A second concern is that of a Majorana-type neutrino. By charge-conjugating
Equation 1.10, it can be seen that ν1 and ν2 are indeed equal to their antiparticles.
But it has repeatedly been observed that weak interactions favor different outcomes
depending on whether a neutrino or antineutrino contributed to the process. This8can be rectified by realizing that a massive neutrino can be in two different helicity
states and that the differences in processes involving neutrinos or antineutrinos
depend only upon the Majorana neutrino’s helicity.
Thirdly, since a Majorana neutrino is its own antiparticle, lepton number conservation is clearly violated. There may not be a solution for this but given the
new physics possible with massive neutrinos, lepton number violation could justas well be tolerated, explained, or discarded in a broader theory. The exceptions to
the rules of the SM that massive neutrinos bring can be far-reaching and worthyof new experiments to test such theories as Majorana neutrinos and the possible
existence of a heavy sterile neutrino. But in this section, only one neutrino field wasconsidered. When extending the argument for three neutrinos flavors of the threefermion generations, striking new behavior not present in the SM is uncovered.

1.3N O

1.3.1

W I

The one flavor model in the previous section was seen to be a combination of two
massive neutrinos, an active one and a sterile one. Given the small contributionof the sterile neutrino in the weakly interacting field component ν0L , ν2 will bediscarded. Equation 1.11 now reads ν0L = V11 ν1,L = ν1,L and is purely light in mass.
It also states the mass eigenstate ν1 is also an eigenstate of the charged current weakinteractiong
LW = − √ Wρ− `¯L0 γρ ν0L + h.c.,2(1.13)

where Wρ− is the massive exchange boson and `L0 is the left-handed charged lepton
partner of ν0L . This can be extended to the three known flavors by rewriting the9flelds `L0 and ν0L as vectors composed of their mass eigenstates: ν 
 L,1 ν0L = Z  νL,2  .




 νL,3  ` 
 L,e `L0 = AL  `L,µ  ,




 `L,τ (1.14)The left side represents the left-handed field in terms of the charged leptonic mass
eigenstates `e , `µ , and `τ . The 3 × 3 hermitian matrix AL [5] diagonalizes the mass
matrix for the charged leptons in much the same way as V did for the one flavor
neutrino case in Equation 1.9. The resulting masses are me , mµ , and mτ .
The right side of Equation 1.14 is the left-handed neutrino field in terms of threelight neutrino mass eigenstates with masses m1 , m2 , and m3 . The 3 × 3 matrix Z is
analogous to the matrix element V11 in Equation 1.11. By writing ν0L as the combination of three light neutrino mass eigenstates, it is implied that the diagonalizationof the mass matrix resulting from the see-saw mechanism for three active neutrinos
produces three additional heavy neutrinos ν4 , ν5 , and ν6 . They are discarded forthe same argument given at the beginning of this section.
The charged current interaction Lagrangian can now be written for three flavors
as


T


 ν  `¯  L,e  L,1 




g




LW = − √ Wρ−  `¯L,µ  γρ A†L Z  νL,2  + h.c.,




2




 `¯L,τ  νL,3 or more compactly as
3
X X


g
−
¯ γρ A† V νL,i + h.c.
`L,α
LW = − √ WρLαi
2α=e,µ,τ i=1(1.15)

Here, the charged lepton `L of flavor α interacts via the charged boson W − withnot one neutrino of mass-type i but a mixture of all three neutrinos. This mixture10

is defined by the leptonic mixing matrix U = A† Z. The mixed neutrinos can thenbe labeled as new states to correspond with the charged lepton they interact with:Pνα = 3i=1 Uαi νL,i where α = e, µ, τ for electron, muon, and tau respectively.

1.3.2N F C

With three mass-type neutrinos now written in the flavor basis να , it becomes clearthat the creation and annihilation of a neutrino by weak interactions does nottake place when the neutrino is in a definite mass state. As a consequence, sincea neutrino of flavor α is a superposition of mass eigenstates, the time evolutionof a flavor state can induce a transition to another flavor state. This can best be
demonstrated by starting with the expressions of flavor states and mass states inbraket notation:|να i =

3X

|νi i =

Uαi |νi i,

3
X

U∗βi |νβ i,(1.16)

β=1

i=1

where the handedness subscript L has been dropped for brevity.
A neutrino is produced weakly and is initially (t = 0) in flavor state α. By
applying the time evolution operator, it is possible to render the time-dependent
expression|να (t)i =3XUαi e−iEi t |νi i,(1.17)

i=1

from which the probability that the neutrino at some later time t will be in anotherflavor state β:Pνα →νβ = |hνβ |να (t)i| =
2

3
X

2

U∗βi e−iEi t Uαi.

(1.18)

i=1

For relativistic neutrinos, the energy component Ei can be approximated byEi =p|p|2 + m2 ' |p| +

11

m2i
2|p|

'E+

m2i
2E

.

(1.19)

Along with t = v/L ' c/L where L is the distance from the neutrino source to thepoint of observation and c is the speed of light (c = 1 in current units), this makes
the transition probability

Pνα →νβ = |hνβ |να (t)i| =
2

3
X

m2
i

2

U∗βi e−iEt e−i 2E L Uαi .

(1.20)

i=1

Before expanding the absolute square, it will benefit the discussion by simplifying to the case of two mass-type neutrinos (i = 1, 2). The leptonic mixing matrix
can then be written as cos θ sin θ 


 .
U = 
 − sin θ cos θ 

(1.21)


2 L
Pνα →νβ = sin 2θ sin ∆m
4E(1.22)

Equation 1.20 becomes2

2

for two neutrinos. The survival probability (the chance a neutrino of flavor α willremain flavor α after some time t) is likewise given byPνα →να = 1 − Pνα →νβ = 1 − sin 2θ sin ∆m
2

2

2 L

4E



,

(1.23)

where ∆m2 = m21 − m22 . The sin2 2θ term describes the amount of ”mixing” between
the two mass eigenstates and θ is hence known as the mixing angle. The masssquared difference ∆m2 in the second sine term, along with E, determines thefrequency of flavor change. It can therefore be seen why flavor change is also
referred to as neutrino oscillations.
It is interesting to note that neutrino oscillation is purely a consequence ofmassive neutrinos. By setting m1,2 = 0, the chances a neutrino will not oscillate are
100%, making any in-flight transition impossible.

12The mixing angle and mass-squared difference are quantities that are currently
experimentally determined. By measuring only neutrino rates in disappearance
experiments (where a lower-than-expected rate is obtained for a given neutrino
flavor) or appearance experiments (where a non-negligible rate is obtained for aflavor not expected when considering no oscillations), individual masses cannot
be determined. In addition, an experiment’s L/E (L-over-E) value specifies to what
sensitivity ∆m2 can be measured [6]. This sensitivity is the value ∆m2 for which
∆m2 L
∼ 1.
2E

(1.24)

Also dependent on L/E and ∆m2 is the ability to detect neutrino oscillations atall. Given a length L0 = 4πE/∆m2 , the survival probability becomesLPνα →να = 1 − sin 2θ sin π
.
L0
2

2



(1.25)

If L  L0 , the period of oscillation becomes large, driving sin2 (πL/L0 ) to zero andthereby making detection extremely difficult. On the other hand, if L  L0 , thefrequency becomes high and only the mixing angle values can be probed. Since∆m2 is determined by nature, care must be taken when choosing detector location
for a given source energy in baseline oscillation experiments. For natural sources
such as cosmic ray-induced atmospheric neutrinos and solar neutrinos, a little bitof luck is also helpful.

1.3.3N F C  M

The formulism above only concerns neutrinos traveling in vacuum. When matter istraversed, neutrinos interact with the ambient particles via charged W and neutral
Z boson exchange. This can lead to forward elastic scattering where an interaction13

potential Vint gives an added contribution. Ordinary matter (matter composed ofelectrons, neutrons, and protons) will be considered making the W boson-mediatedprocesses only affect the electron neutrino. The Z boson-mediated processes affect
all flavors equally. To write an explicit form of Vint , the leptonic mixing matrix will
be assumed to describe mixing between mass eigenstates |ν1 i and |ν2 i and flavor
states |νe i and |νµ,τ i. The actual form of U in Equation 1.21 does not change.
Before constructing Vint , the Hamiltonian in the flavor basis will be obtained. Theenergies Ei of the previous section are energy eigenvalues of the time-dependentneutrino mass eigenstates |νi (t)i. Therefore, it is possible to write the Schrödinger
equationid|νi (t)i = H|νi (t)i = Ei |νi (t)i,dt(1.26)

where H is diagonal with elements Ei . To express this in the flavor basis, one needs
only to apply the leptonic mixing matrix to the above expression to yieldidU|νi (t)i = UHU† U|νi (t)i = H(α) |να (t)i,dt(1.27)

where H is now rotated to

m
∆m 

H(α) = EI +I+
4E
4E  sin 2θP2  − cos 2θ


sin 2θ  .
cos 2θ (1.28)

Now it is possible to add the matter dependent potential Vint to H(α) . As statedearlier, the electron neutrino is exclusive in its interactions with ambient electrons
via W exchange. And, obviously, the traversed matter’s electron density is a factor
in the potential. Therefore, Vint can be written as 1 0 
√
√
√
√
 ,
Vint = 2GF Nn I + 2GF Np I + 2GF Ne I + 2GF Ne 
 0 0 14

(1.29)

where GF is the Fermi constant and Nn , Np , and Ne are the neutron, proton, andelectron number densities respectively. Dropping identity matrix I terms in H(α)
and Vint (they do not contribute effects to neutrino flavor change), the Hamiltonian
in matter becomes





 1 0 
sin 2θ  √ .
 + 2GF Ne 
cos 2θ
0 0 


∆m 

H0 = H(α) + Vint =4E  sin 2θ

2 
 − cos 2θ(1.30)

This Hamiltonian can then be diagonalized back into the mass basis but witheigenvalues∆m2 A ∆m2
EM,1,2 =
±
2E
4E

q

(cos 2θ − A)2 + sin2 2θ.

(1.31)

An effective mass squared difference in matter can be identified as∆m2M = ∆m2

q(cos 2θ − A)2 + sin2 2θ.

(1.32)√The term A = 2 2EGF Ne /∆m2 is a unit-less parameter that is regarded as a gaugeof the influence matter has on oscillation. By taking the limit A → 0, vacuum
oscillation is restored. An effective mixing angle in matter is likewise written as
sin2 2θM =sin2 2θ
.
sin2 2θ + (cos 2θ − A)2(1.33)The effective mixing angle and mass squared difference can replace their vacuum
counterparts sin2 2θ and ∆m2 in Equation 1.23 to describe the survival probabilityof νe propagating through matter of constant electron density Ne . Matter effects
play an important role in the way neutrinos behave in the dense environments ofthe solar and terrestrial interiors.
The possibility exists when A = cos 2θ which can arise when the electron number
density reaches a critical value Ne,crit . In this case, the mixing angle in vacuum

15could be small but propagation through matter promotes the mass eigenstates tomaximal mixing (θM = π/4). This is known as the MSW effect and was first realized
by Mikheev, Smirnov, and Wolfenstein [7]. It will soon be explored in the context
of propagation in the Sun.

16

C 2
S N
The detection of solar neutrinos first occurred in 1967 with the Homestake experiment of the late Raymond Davis. Since then, solar neutrinos have been detected
almost continuously by various other experiments. Neutrinos are the only particles
which allow researchers to peer inside the Sun to decipher its mechanisms. Thishas been the historical basis of early solar neutrino detection. Nowadays, solar
neutrinos are being studied themselves to determine their properties and, on alarger scale, the possibility of new fundamental physics. This is precisely the aimof the latest research in solar neutrinos including Super-Kamiokande. Includedin this chapter is a general description of the production of neutrinos inside theSun as theorized by the Standard Solar Model, a brief history of the solar neutrino
problem, how neutrino oscillation from the previous chapter can be applied tosolar neutrinos, and a synopsis of the solar neutrino experiments included in thisanalysis.

2.1

T S S MThe late John Bahcall, the eminent solar neutrino physicist and author of several
well-regarded solar models, gives his definition of the Standard Solar Model (SSM):
By the Standard solar model, we mean the solar model which is constructed with the best-available physics and input data. All of the solarmodels we consider ... are required to fit the observed luminosity andradius of the sun at the present epoch, as well as the observed heavy
element to hydrogen ratio at the surface of the sun [8].
17One of the principle goals of the SSM is to accurately model the energy productionthat occurs in the core of the Sun. Since this energy is released in the form of photons
and neutrinos, understanding the production mechanisms can yield predictions of
the solar neutrino flux. Comparisons can then be made with experimental data totest the accuracy of the model or, conversely, the completeness of neutrino theory.
The SSM considered in this work is BP2004 [9]. To begin constructing a solarmodel, three assumptions are made: 1.) the Sun evolves in hydrostatic equilibrium,
2.) solar energy is transferred via radiation, conduction, convection, or neutrino
loss, and 3.) thermonuclear reactions are the only form of energy production inside
the Sun [10] [11]. The last assumption is the one of interest due to its correlation toneutrino production.
The net thermonuclear reaction that takes place in the Sun’s core is the fusion offour protons and two electrons which forms a 4 He nucleus along with two electron
neutrinos and Q = 26.73 MeV of energy:
4p + 2e− → 4 He + 2νe + Q.

This happens via two groups of reactions: the proton-proton (pp) chain and theCarbon-Nitrogen-Oxygen (CNO) cycle.
The CNO cycle is the lesser contributor of the two, producing only 1.6% of theSun’s total output energy. This is due to the Sun’s core temperature being lower
than is required for the CNO chain to acquire dominance (this is around 1.8×107 K
whereas the core temperature is 1.5×107 K) [12]. By way of carbon, nitrogen and18oxygen catalysts, there are three reactions that produce neutrinos:
13

N →13 C + e+ + νe

15

O →15 N + e+ + νe

17

F →17 O + e+ + νe .

The upper boundaries of the emitted neutrino energies are 1.20 MeV, 1.73 MeV, and1.74 MeV respectively. With these low energies coupled with the low percentage of
total solar luminosity, the neutrino flux of the CNO cycle does not have a significant
impact on the study of solar neutrinos. Figure 2.1 shows the CNO cycle.

Figure 2.1: The CNO cycle. The neutrino-producing reactions are boxed.

The pp chain, making up for the remaining 98.4% of solar energy, can further be
broken down into four branches: pp-I, pp-II, pp-III, and hep. Of the processes leading19

up to the formation of 4 He in these branches, five produce neutrinos. In contrast
to the CNO cycle neutrinos, those from the pp chain are the major contributors tosolar neutrino detection on Earth. The pp chain’s neutrino producing reactions are
named pp, 7 Be, pep, 8 B, and hep and are listed below.
p+p →

2

H + e+ + νe ,Be + e− →7

Li + νe ,p + e− + p →2

H + νe ,

8

B →

8

Be + e+ + νe ,He + p →4

He + e+ + νe .

73with neutrino energies <0.42, 0.86, 1.44, ≤14.06, and ≤18.77 MeV respectively.
Figure 2.2 shows the pp chain.
In total, eight reactions (including the CNO cycle) give rise to the predicted
neutrino fluxes of the BP2004 SSM as summarized in Table 2.1. The large errors in
the CNO and 8 B fluxes come from the uncertainties in chemical composition and is
20% for 8 B and around 35% for CNO. Another large uncertainty is in the 3 He→4 Hereaction rate and is 7.5% for 8 B and 8.0% for 7 Be [9].
Table 2.1: Table of the BP2004 SSM predicted flux values for each solar neutrino
source. All values are in units of cm−2 s−1 .
sourceBP2004pp5.94(1 ± 0.01) × 1010
pep
1.40(1 ± 0.02) × 108
hep
5.94(1 ± 0.16) × 103
7Be
5.94(1 ± 0.12) × 109
8B
5.94(1 ± 0.23) × 106
13N
5.94(1+0.37) × 108−0.35
15O
5.94(1+0.43) × 108−0.39
17F
5.94(1 ± 0.44) × 106

20

Figure 2.2: The pp chain. The energy percent contributions for the different branches
are labeled. The neutrino-producing reactions are boxed.

21

2.2

T S N P

2.2.1

B

A problem arose in 1968 when the Homestake experiment detected a neutrino fluxof about one third of that of the SSM. Of course, investigations began into thedetection efficiencies and SSM assumptions (or lack of) but ultimately yielded noadequate explanation for the deficit. The newer Kamiokande experiment begancollecting data and in 1989 reported a flux slightly over half of the expected SSM
value. In 1992, the GALLEX collaboration said they also observed a flux 65% of the
SSM (closely related GNO and SAGE also reported similar deficits). After muchstruggle to bring the SSM into agreement with experiment, an inconsistency uncovered in the temperature dependence of the solar neutrino flux and the current threeobserved values pointed to the very likely possibility that fundamental particle
physics needed to be revised and not the SSM.
The Homestake experiment detected neutrinos from a 0.814 MeV threshold. This
includes 7 Be, pep and large portions of the 8 B and hep fluxes. Kamiokande detected
only the upper tails of 8 B and hep from a 7.5 MeV threshold. Given that the 8 B
neutrino flux dependence is heavily dependent on the Sun’s core temperature (as
25
11
Tcore
[13]), and less dependent for 7 Be (Tcore), a slight temperature adjustment in

the SSM could result in agreement with Kamiokande’s 8 B measurement. However,the 7 Be flux becomes not as strongly suppressed as Homestake would suggest with
their greater flux deficit. On the other hand, the pep and pp flux are inversely−1
proportional to core temperature (∼ Tcore) but pp-sensitive GALLEX indicates flux

loss, not flux gain. Perhaps the neutrinos themselves have an energy dependent
property.

22

2.2.2T O H

The solution to the solar neutrino problem came after considering the possibilityof neutrino flavor change. As stated earlier, the theory of neutrinos oscillating
between flavor states was proposed by Maki, Nakagawa, and Sakata in 1952 andlater Pontecorvo in 1967. If the solar neutrino data fit the oscillation hypothesis, itwould mean solar neutrinos were endowed with finite mass.
A simple example of solar neutrino oscillations is the transition of νe producedin the core of the Sun to νµ or ντ during its flight to the Earth. Since this takes
place in the vacuum of space, the survival probability of Equation 1.23 with L =149.6 × 106 km (1 au, the Earth-Sun distance) can be used. Additionally, if L issubstituted with



t
,
L = (1 au) × 1 −  cos 2π
T

(2.1)

where  = 0.0167 is the eccentricity of the Earth’s orbit and T = 1 year, a seasonalpattern could be observed in the rates throughout the year. This possible scenario
is known as the vacuum (VAC) solution.
Another possibility also exists when considering matter effects. The electron
density at the Sun’s core is Ne,core ' 150 g/cm−3 and Equations 1.32 and 1.33 areused in 1.23 to calculate the survival probability. But the electron density outside of
the core is not constant and instead follows a roughly inverse exponential trend (seeFigure 2.3). It is then possible for Ne,Sun (r) (where r is the ratio of the distance fromthe core to the solar radius and making Ne,Sun (r = 0) = Ne,core and N(r = 1)e,Sun = 0)
to equal Ne,crit , the resonance condition (MSW effect) mentioned in Section 1.3.3.
The neutrino mass eigenstates in matter are νM,1 and νM,2 and correspond toenergy eigenvalues given in 1.31. By applying the leptonic mixing matrix of 1.21
(with the substituted effective mass splitting and mixing angle) to the matter mass23

3

e density in mol/cm

10

2

10
1
-1

-

10
10

-2

0.1

0.2

0.3

0.4

0.5

0.6

0.7
0.8
0.9
1
distance/sun radiusFigure 2.3: The Sun’s electron density profile as a function of distance from the coreto the surface (defined as r in the text).

eigenstates, it can be seen thatνe = νM,1 cos θM + νM,2 sin θM ,(2.2)

νµ,τ = −νM,1 sin θM + νM,2 cos θM .

(2.3)

At the large density Ne,core in the core, the effective mixing angle is pushed to π/2,
making core-born νe mostly νM,2 . If the neutrino propagates adiabatically to thesurface of the Sun, it will stay in state νM,2 . But as Ne,Sun (r) decreases, so will theeffective mixing angle and thus increase the contribution of νµ,τ in νM,2 . At theresonance Ne,Sun (r) = Ne,crit , the two flavor states are maximally mixed, and at thesurface where N(r = 1)e,Sun = 0, νM,2 has become the vacuum state ν2 . The neutrino
does not oscillate after emergence from the Sun. If the neutrino propagates throughthe Sun non-adiabatically, it can possibly jump into the νM,1 state at the resonanceand emerge as ν1 . Figure 2.4 shows the vacuum eigenstates (expressed as m2 ) as afunction of solar electron density.
There exists the increased possibility of the ν2 mixture converting back into νeand is a consequence of interactions with Earth’s matter. In this case, an experimentsensitive to solar neutrinos with the capabilities of realtime detection might see a

24

Figure 2.4: The journey of νe from the core to the surface of the Sun. Ne,crit is thepoint of resonance.

slightly higher nighttime rate when compared to the daytime rate. As with theSun, the Earth’s density as a function of radius is not constant but is at values lessthan an order of magnitude of the Sun’s. There is no resonance inside the Earth.
The combinations of (vacuum) mixing and mass-squared difference in the regime
where matter effects are important is the plane typically above ∆m2 > 10−9 eV2 . Regions on the plane allowed by solar neutrino data are known as the small mixing
angle (SMA), the large mixing angle (LMA) and the low solutions1 . All four solutions are shown in Figure 2.5.
The low (LOW) solution refers to the small ∆m2 values for matter oscillations and the relatively
low possibility of it being the solution when considering SMA and LMA.
125

Figure 2.5: The naming conventions of possible combinations of the mixing parameters as allowed by solar neutrino data.

26

2.3

S N E

2.3.1

H

The radiochemical experiment Homestake (1967-1994), located in the South Dakota
Homestake gold mine at 1478 m (4200 meters water equivalent2 ) depth, employed
the reaction 37 Cl + νe → 37 Ar + e− by filling a tank with 615 tons of C2 Cl4 (tetrachloroethylene) with a 24.23% concentration of the 37 Cl isotope and letting it sit
for around 30 days to capture solar neutrinos. Argon (which contained an admixture of the 37 Ar isotope after exposure) was then removed from the tank bypurging it with helium gas for 20 hours. Residual tetrachloroethylene and otheractive gases were removed and the argon was extracted (95% efficient). The 37 Ar
isotopes were counted by the emission of Auger electrons (81.5% at a total energyof 2.823 keV) and X-rays (8.7%) when 37 Ar decayed back into 37 Cl. It was a total ratemeasurement unable to distinguish direction or energy of the captured neutrinos.

2.3.2

K

Kamiokande (1987-1995, solar measurement) was the first solar neutrino experiment after Homestake and was started 20 years later. The original purpose of
Kamiokande was the detection of proton decay to test certain Grand Unified Theories. It was upgraded in 1987 to facilitate the collection of solar neutrinos. The
detector was a cylindrical tank located 1000 m (2700 mwe) below ground in the
Kamioka mine in Japan and was filled with 3000 tons of pure water surrounded by
948 photomultiplier tubes (PMT) of 50 cm diameter. The PMTs collected Cherenkov
light from electrons recoiling from elastic scattering with solar neutrinos. The reaction νe + e− → νe + e− was the dominate process but Kamiokande was also sensitive
2

Meters water equivalent (mwe) is the corresponding distance in water that would provide equalshielding.

27to νµ,τ + e− → νµ,τ + e− but at a much lower rate and it could not be discriminated.
Kamiokande was a realtime experiment which could discern the direction andenergy of recoil electrons from neutrino scattering, but limited statistics hid the appearance of possible spectral distortions which are a telltale sign of the parametersthat govern neutrino oscillation.

2.3.3

GALLEX/GNO & SAGE

Like Homestake, GALLEX (May 1991-June 1997) was a radiochemical experiment.
It detected solar neutrinos through the reaction 71 Ga + νe →71

Ge + e− which

has a threshold of Eν =0.233 MeV. This made the detection of all solar neutrino
sources possible. Located in the underground national laboratory of Gran Sassoin Italy, GALLEX had 101 tons of liquid gallium chloride solution (GaCl3 -HCl)under an overhead of 3300 mwe. After exposures lasting about 30 days, 71 Ge
was chemically extracted and proportional counters waited for its decay back into
71

Ga. Improvements to the extraction process gave birth to GNO (May 1998 - April

2003), a very similar experiment using the same detector as GALLEX. In total,
GALLEX/GNO performed 123 extraction runs.
SAGE (January 1990-December 2001) was a joint Soviet-American Gallium Experiment (later Russian-American Gallium Experiment) which employed the same
reaction as GALLEX/GNO. It was located at the Baksan Neutrino Observatory 2000
m (4700 mwe) under the Caucasus Mountains dividing Asia and Europe. Its volume was a total 50 tons shielded by 60 cm low-radiation concrete and a 6 mm steel
shell. By the end of the experiment, SAGE had conducted nearly 100 successful
solar neutrino measurements.

28

2.3.4S-K

Super-Kamiokande (SK, April 1996-present) was initiated in June, 1996 and is the
successor of the Kamiokande experiment employing the same Cherenkov radiation
detection technique. In the solar neutrino analysis, the larger volume of SK gavea statistical advantage over Kamiokande allowing discrimination of the neutrinoenergy spectrum shape. With a shape identified, it would then be possible to distinguish a favorable solution out of the possible four set forth by oscillation theory andearlier solar experimental data. This came to fruition with the publication statingthat the SMA and vacuum solutions were disfavored by not observing distortionsin the spectrum shape [14]. Later, after further exploiting SK’s strength of improved
statistics, it was announced that the LMA solution was favored, and with the combination of all solar neutrino experimental results, LMA was the unique solution
to the solar neutrino problem at very high confidence [15]. The SK experiment iscurrently divided into three stages: SK-I, SK-II, and SK-III.

2.3.5

S N OThe Sudbury Neutrino Observatory (SNO, 1999-2006) was the first experimentable to measure the total solar neutrino rate and compare it to the rate of νe that
did not undergo oscillation. Located 2092 m (6010 mwe) below the surface in the
Creighton mine in Sudbury, Canada, SNO was a water Cherenkov detector thatused heavy water (D2 O) as its target. The neutrino rates from the 8 B neutrino flux
were measured using the following charged current (CC), neutral current (NC),29and elastic scattering (ES) processes:
CC : νe + D→

p + p + e− ,(2.4)

NC : νx + D→

n + p + νx ,(2.5)

ES : νx + e−→

νx + e− ,(2.6)

where x = e, µ, τ.
By subtracting the νe CC rate from the total NC rate, SNO could show that
two-thirds of the neutrinos from the Sun were changing into flavors not sensitive
to the CC reaction in the detector. Likewise, the ES-minus-CC weighted difference
also pointed to neutrino flavor change and was in good agreement with the NCminus-CC rate. The NC rate confirmed experimentally the theoretical SSM modelon neutrino emission for the 8 B flux.
SNO was in three phases: Phase I using pure D2 O [16], Phase II using NaCl dissolved in the heavy water (the salt phase) [17], and Phase III with the salt removed
and an array of 300 3 He proportional counter tubes added. Each successive phase
was an effort to improve upon the measurement of the NC total rate by increasing
both the neutrino capture cross section and collection efficiency. At the time of thiswriting, Phase III results have not been published.

2.3.6

S  C S N DThis section presents a comparison in graphical and tabular form of the SSM predicted flux values and those measured by experiment.

30

Figure 2.6: The predicted BP2004 SSM fluxes as a function of neutrino energy. The
thresholds of the solar neutrino experiments described earlier are bracketed above.
Taken from [19]

31

32

Table 2.2: For each experiment described in Section 2.3, the neutrino detection reaction, neutrino energy threshold, measured
flux, and percentage of the BP2004 SSM are listed. In the flux column, the first error given is statistical, the second issystematic. In the BP2004 column, the statistical and systematic errors have been added in quadrature. Data are takenfrom [6] [16] [17] [18].
experiment
reactionthreshold flux (∗ SNU, ∗∗ × 106 cm−2 s−1 ) % of BP04
Homestakeνe +37 Cl → e− +37 Ar
0.814
2.56 ± 0.16 ± 0.16∗
30.1 ± 2.7∗
GALLEX/GNO
69.3 ± 4.1 ± 3.6
52.9 ± 4.2
νe +71 Ga → e− +71 Ge
0.233+5.3 +3.7∗
SAGE
70.8−5.2 −3.2
54.0 ± 4.0∗∗
Kamiokande
6.7
2.80
±
0.19
±
0.33
48.4 ± 6.6
νx + e− → νx + e−
∗∗
SK-I
4.7
2.35 ± 0.02 ± 0.08
40.6 ± 1.4
+0.06
−
∗∗
νe + D → p + p + e
6.9
1.76−0.05 ± 0.09
30.4 ± 1.9+0.46∗∗SNO - D2 O
νx + D → n + p + νx
2.224
5.09+0.44
87.9± 11.1−0.43 −0.43
0.24
−
−
∗∗
νx + e → νx + e
5.7
2.390.23 ± 0.12
41.3 ± 4.7+0.08∗∗−
νe + D → p + p + e
6.9
1.68 ± 0.06−0.09
29.0 ± 1.7
SNO - NaClνx + D → n + p + νx
2.224
4.94 ± 0.210.38∗∗
85.3 ± 7.5
−0.34
−
−
∗∗
νx + e → νx + e
5.7
2.35 ± 0.22 ± 0.15
40.6 ± 4.6

1.0+0.09−0.08

pp, pep

GALLEX/GNO

0.290±0.017
SAGE

SK

H2 O

Cl

0.540
±0.0400.529
0.484 ±0.042±0.066
Kamiokande

0.406±0.014

Homestake

0.301±0.027Ga
7Be

1.0 ± 0.23 0.853±0.0751.0 ± 0.23

8

B

D2 O
CNO

SNO

1.0 ± 0.23

SNO

1.0 ± 0.21

D2 O

experiment

Figure 2.7: The predicted SSM flux values of solar neutrinos compared with theirexperimentally measured values. All values are ratios to the BP2004 SSM. The
figure is based on [20].

33C 3
T S-K-II D
Particle physics today is at the stage where theory outpaces experiment. To confirm or exclude the myriad of hypotheses by experimental means has often required increasingly higher energies and larger detectors. As a result, collaborations of physicists form and grow, each contributing a share of responsibility todevelop and maintain what possibly could be a mammoth scientific instrument.
Super-Kamiokande was borne out of the desire to improve the already successful
Kamiokande measurement. Sixteen times larger than its predecessor, the SK experiment has operated almost continually since June, 1996 and is presently in its third
stage. The current chapter aims to present the construction and detection methodsof the second stage detector, Super-Kamiokande-II1 .

3.1E OThe Super-Kamiokande experiment utilizes a large water Cherenkov imaging detector located in a zinc mine in Hida City (formally Kamioka Town), Gifu Prefecture,
Japan. The mine is owned and operated by Kamioka Mining and Smelting Com1The Super-Kamiokande-II detector was the result of a partial recovery from a November 12th,
2001 accident which destroyed over half of the PMTs in SK-I (details on the SK-I detector canbe found in [21]). Reconstruction was finished in September 2002 but with 19% photo-cathodecoverage compared with SK-I’s 40%.
The cause of the accident was determined to be an imploding PMT located at the bottom of thedetector. This created a propagating shock-wave in water. The SK-I tank at that time was fillingwith water after a recent electronics upgrade and was approximately two-thirds full. The resulting
shock-wave destroyed 6,777 out of the 11,146 PMTs, virtually all PMTs below the water line.
To help ensure such an accident would not occur again, the PMTs were encased in blast shields
consisting of a fiber-reinforced plastic shell with a clear acrylic cover over the PMT photo-sensitivearea. After SK-II completed in October 2005, full restoration of the PMTs took place and SK-III turnedon in July, 2006. SK-III currently has 11,129 PMTs and are all encase in blast shields. Information istaken from [22][23]34

pany. The detector is 1,000 meters (2,700 mwe) under the summit of Mt. Ikeno.
The coordinates are 36◦ 25’33” N, 137◦ 18’37” E and is 371.8 m above sea level. It
is approximately 250 km from Japan’s capital of Tokyo. Figure 3.1 shows SK’s
location in Japan.

Figure 3.1: The SK experimental site’s location in Japan.

The detector is the largest experiment currently under task at the Kamioka
Observatory, part of the University of Tokyo’s Institute for Cosmic Ray Research(ICRR). SK and other experiments at the underground observatory take advantageof the reduced cosmic ray muon rate (2.2 Hz) which is a background in manymeasurements. The types of analyses performed at SK are atmospheric neutrino,
solar neutrino, supernova neutrinos (realtime and relic), proton decay, and othertopics.
The experimental site is accessed by a 1.5 km horizontal tunnel that is traveledby reduced emission, non-spark ignition diesel vehicles. The SK site is an enclosed
area consisting of a control room with data collecting and monitoring computers
and a dome area immediately above the SK tank. Electronics huts which house35the equipment to power and read the information of the light collecting PMTs
are located in this area as well as various calibration tools and equipment. An
electron linear accelerator (LINAC) is housed in a room to the side of the domearea. Figure 3.2 shows the SK site and tank.

Figure 3.2: Layout of the SK experiment site in the Kamioka mine.

The SK collaboration of researchers and engineers numbers about 140 and comefrom five countries: Japan, US, Korea, China, and Poland. The SK experiment islargely funded by Japan’s Ministry of Education, Culture, Sports, Science and Technology and in part by the U.S. Department of Energy and the U.S. National Science
Foundation. Additional funds come from the Korean Research Foundation, theKorean Ministry of Science and Technology, and the National Science Foundationof China.

36

3.2

D M

3.2.1

N I

Solar neutrino detection at SK is achieved by the collection of Cherenkov light
generated by recoil electrons in water scattered by incoming solar neutrinos. The
scattering is elastic, meaning the neutrino and electron are retained throughout theinteraction. SK is sensitive to all three types of neutrinos:
νe + e− → νe + e− ,νµ + e− → νµ + e− ,ντ + e− → ντ + e− .
All processes are neutral current processes but only the νe can interact with theelectron via charged current as well. This implies a larger cross section for νe andmakes it possible to measure solar νe disappearance since contamination from νµ,τinteractions is small. The total cross sections for 10 MeV incident neutrinos are


−44
2


 8.96 × 10 cm for νe
σtot = .
 1.57 × 10−44 cm2 for νµ,τ(3.1)

The scattering is highly forward making the recoil electron’s direction is more
uncertain by multiple scattering rather than the variance of kinematic angles.

3.2.2C R

Cherenkov radiation is light generated when a charged particle travels in a mediumof refractive index n faster than the speed of light c in that medium (v > c/n). For arecoil electron to emit Cherenkov light in water, its energy must be greater or equal
to the Cherenkov threshold energy of 0.767 MeV.
37

The differential number of photons generated per unit wavelength dλ per unit
distance dx the electron2 travels is


2πα
1
2πα
d2 N
= 2 1−=
sin2 θC ,dxdλ
λ(n(λ)β)2λ2(3.2)

where β = v/c and α is the fine structure constant. Here, the refractive index n is afunction of wavelength. The total number of photons is found by integrating overtravel distance and all wavelengths. The right side of 3.2 has an angle known as the
Cherenkov opening angle and describes the direction the photons travel relative
to the path of the electron. The resulting radiation forms a cone with opening 2θCaround the electron trajectory. The opening angle can alternatively be writtencos θC =

c
,
n(λ)v

(3.3)

and can be easily seen to vary as a function of electron velocity. For a recoil electronin water, θC ∼ 42◦ .

3.3

D

3.3.1

T

The SK detector is a stainless steel cylindrical tank 41.4 m tall and 39.3 m in diameter.
It is in a rock-hewn cavity with concrete filling the narrow area between the tankand rock. Two meters from the inside wall is a 1 m wide trussed support structurethat climbs the entire height of the detector. There is also a trussed ”floor” and”ceiling” 2 meters from the bottom and top of the tank. This support structure
essentially divides the detector into two sections: inner detector (ID) and outer2

This discussion is valid for all charged particles. Electrons are used as an example to keep incontext with SK’s solar neutrino detection method.

38

detector (OD). Its purpose is to have mounted on it the photomultiplier tubes(PMT) that are responsible for light collection. Also, opaque sheets are attached tothe structure to function as an optical division between the ID and the OD.

3.3.2

I DThe ID consists of the space inside the support structure. On the structure aremounted 5182 50 cm PMTs in a checker board pattern (see Figure 3.3). Each PMT
is encased in a fiber-reinforced plastic shell capped with a clear acrylic dome overthe PMT’s photo-sensitive surface. The purpose of the shield is to protect against
propagating shock-waves that might occur if a PMT should succumb to fatigue
and implode under the pressure of water and vacuum. At normal incidence, theacrylic domes have a transparency better than 98% at 400 nm in light wavelength.
At 300 nm, it is about 86%. The PMTs are sensitive in the range of 300-600 nm. To
reduce reflected light in the ID, black polyethylene sheets are placed on all areas ofthe support structure where no PMT is mounted. The total PMT coverage of the
SK-II detector is 19%.

3.3.3

O DThe OD consists of the space between the support structure and the stainless steel
wall. It acts as a veto detector for incoming muons and a water shield from neutrons
and gamma rays emitted from the surrounding rock. On the support structure aremounted 1885 20 cm PMTs in a grid pattern. The OD PMTs are not encased in blast
shields. White Tyvec R sheets are placed on all areas where no PMT is mounted toincrease light collection from light reflection.

39

Figure 3.3: A view of the inner detector from the bottom.

40

3.3.4

P TThe ID is equipped with Hamamatsu R3600 PMTs with a photo-sensitive surface(the photocathode) 50 cm in diameter (Figure 3.4. They were originally developed
for the Kamiokande experiment. The bulbs of the PMTs are hand blown fromborosilicate glass to a thickness of about 5 mm. The inner surface of the glass iscoated with Bialkali (Sb-K-Cs) photocathode and is matched so its sensitivity tolight (quantum efficiency, or QE) coincides with the peak of the Cherenkov light
spectrum. The QE is about 21% at the peak between wavelengths 360 nm and 400
nm (see Figure 3.5). The 11 stage ”Venetian blind” dynode structure multiplies theejected electron (photoelectron, or p.e.) from the photo-cathode to create a gain of
107 electrons through an applied voltage of 1700 V to 2000 V. This signal can now
be read out via a 70 m coaxial cable that winds up the support structure into thedome area and into one of four electronic huts.
The OD PMTs are Hamamatsu R1408 PMTs with a photo-cathode of 20 cm indiameter. An acrylic wavelength shifting plate of dimensions 60×60×60 cm3 anddoped with 50 mg/l of bis-MSB is attached to the bulbs of the OD PMTs. The plates
serve the function of absorbing light in the ultraviolet and shifting it to blue-green
to match the peak sensitivity of the PMTs. This improves the collection efficiency
by a factor of 1.5.

3.4

P S

3.4.1

W

Removing impurities from the tank water is important to increase the water transparency which directly affects the collection of Cherenkov light from events. Someof the impurities may also be radioactive (222 Rn) and contribute to unwanted back-41

~ 70000

cable length
φ 116

φ 254 10φ 82 2glass multi-seal

photosensitive area > φ 460< φ 520

cable

water proof structure( 610 20 )~ 720
Figure 3.4: Diagram of an ID PMT.

42

(mm)

Figure 3.5: The quantum efficiency of the ID PMTs as a function of wavelength andthe Cherenkov spectrum in pure water.

ground events. The SK tank water is originally drawn from two streams in the
Kamioka mine that are sourced by the natural seepage of rain and snow melt
through the rock. It is then pumped into a water purification system that removes
contaminants before placement in the tank. This water is then recirculated at a rateof about 35 tons per hour. The purification process is listed below and the systemsetup is shown in Figure 3.6.
1 µ  
A series of filters removes small particle in the water.

1  
The heat exchanger functions to cool the water to reduce the growth of bacteria
and PMT dark noise rate.

43

C 
The polisher removes Na+ , Cl− , Ca+2 , and other heavy ions.

R-   This step dissolves reduced-radon air into the water to increase the efficiency ofthe vacuum degasifier.

R  
An additional filter to further remove contaminants.

V 
The degasifier removes radon gas and oxygen dissolved in the water. The efficiencyof extraction of radon is about 96%.

2  
The second heat exchanger removes heat in the water mainly coming from theprior filtering processes.

U 
The ultra filter further removes contaminants down to sizes of 10 nm in diameter.

M 
A second degasifier, it removes radon gas with an 83% efficiency.
After filtering, the water’s radon concentration is reduced from about 2 mBq/m3 to
0.4±0.2 mBq/m3 . The water is then introduced into the SK tank from the bottom.

44

VACUUM
DEGASIFIERCARTRIDGE
POLISHERPUMPULTRA MEMBRANEFILTER DEGASIFIERUVSTERILIZERFILTER(1µm Nom.)HEAT
EXCHANGER

PUMP
HEAT
EXCHANGER

PUMPREVERSEOSMOSISBUFFERTANK
PUMP

RN-LESS-AIR

RN-LESS-AIR
DISSOLVE TANKREVERSEOSMOSISSK TANK

SUPER-KAMIOKANDE WATER PURIFICATION SYSTEM

Figure 3.6: Schematic of the SK water purification system.

3.4.2

A

The ambient air in the mine is heavily concentrated with radon emanated fromthe surrounding rock. In the summer months, radon levels are around 2000-3000
Bq/cm3 while in the winter months they are around 100-300 Bq/m3 . This seasonal
dependence is shown in Figure 3.7. Due to the air outside the mine being at a nearconstant of 10-30 Bq/m3 during the year, the outside air is pumped into the SK
dome area. This is accomplished by a housing of air pumps, filters, and a cooler
located outside the mine that draws fresh air and channels it into the SK dome area
via an air duct at 50 m3 per minute. As a result, radon concentrated mine air isdisplaced and the radon levels are typically 20-30 mBq/m3 .
Reduced-radon air is also pumped into the ∼ 60 cm air gap between the water’ssurface and the top of the stainless-steel tank roof. Radon is further reduced byan air purification system (shown in Figure 3.8) that reduces the air to less than 3
mBq/m3 . This system consists of a compressor (to 7.0 8.5 atm), 0.3 µm filter, buffertank, air drier (to improve efficiency of radon removal and to remove CO2 ), carbon
45

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

radon level [Bq/m^3]

2500

2000

1500

1000

500

0
01/01/00

03/01/00

05/01/00

07/01/00

09/01/00

11/01/00

01/01/01

Figure 3.7: Radon levels (in Bq/m3 ) at the SK site as a function of time. The solidline is the level in the mine. The dotted line is the level in the air-purged control
room.

column (to remove radon gas) and finally two filters of 0.1 and 0.01 µm mesh.
COMPRESSORCARBONCOLUMNBUFFERTANK

AIR FILTER(0.3mm)AIR DRIERCARBONCOLUMNHEATEXCHANGER

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

COOLEDCHARCOAL(-40 oC)

SUPER-KAMIOKANDE AIR PURIFICATION SYSTEM

Figure 3.8: Schematic of the air purification system at SK.

46

3.5

D A S

3.5.1

ID DAQ

The first task of ID data acquisition (DAQ) system is the collection of PMT signals3 .
This is done by Analog Timing Modules (ATM). They are electronic boards thatreceive signals from PMTs via 12 channels (one PMT per channel). Each channel
has a charge dynamic range of 550 pC at 0.2 pC (∼0.1 p.e.) resolution and a time
dynamic range of 1.2 µs at 0.3 ns resolution. Once a PMT receives a hit, it sends its
charge information to the ATM channel where it is amplified 100 times and thensplit into four signals. One signal is sent to PMTSUM which is an analog sum of
the 12 channels used for the Flash ADC DAQ. More information on Flash ADC canbe found elsewhere [25]. Another signal is sent to a discriminator to compare thevoltage with a preset threshold. The two other signals are sent to two Charge to
Analog Converters (QAC). The presence of two QACs (and two TACs, explainedin the next paragraph) is to reduce electronics dead time for quick succession of
events such as a muon followed by its decay electron or supernova neutrino bursts.
The signal sent to the discriminator is measured against a threshold of 0.25 p.e. If
it exceeds this threshold, a HITSUM signal is generated with a 15 mV pulse heightand 200 ns width. This represents one hit PMT and is sent to the global triggeringsystem for collection with other ID PMT hits. At the same time of the generation of
the HITSUM signal, the QACs are instructed to open a 400 ns wide gate to store thecharge information. Likewise, Time to Analog Converters (TAC) are instructed toopen a gate to later be closed by the global trigger. There are two TACs per channel
to reduce electronics dead time.
Once all the PMT hits are assembled and pass a predefined threshold, the global
trigger will send out signals to all the ATMs to save the charge and timing in3

Information on the DAQ was retrieved primarily from [24].

47formation they have collected. At this time, the TAC closes its gate marking theend of all hits. The QAC and TAC information is then digitized by the Analog to
Digital Converter (ADC) and sent to the First In First Out (FIFO) internal memoryof the ATM for storage. If the total of PMT hits do not exceed the threshold, theglobal trigger does not send out signals to the ATMs and the TAC gate closes at its
maximum width of 1.2 µs. All information in the TACs and QACs are discarded.
Figure 3.9 shows a schematic of the analog input of one ATM channel.
to PMTSUMHITOUTDISCRI.
CURRENT SPLITTER&AMPLIFIER

PMT INPUT

W:200ns

to HITSUM

ONE-SHOT HIT
W:300ns

SELF GATESTART/STOP

TAC-A

TAC-A

CLEAR
GATE

QAC-A

QAC-A

to ADCSTART/STOP

TAC-B

TAC-B

CLEAR
GATE

THRESHOLD

QAC-B

QAC-B

DELAY

GATE
TRIGGERPED_START

Figure 3.9: The analog input of one ATM channel.

Each ATM board is housed in a TRISTAN KEK Online (TKO) crate (20 boards
maximum) along with GO/NoGo (GONG) and Super Control Header (SCH) modules. The GONG module serves to distribute the global trigger signal to all 20 ATM
boards while the SCH module collects the digitized ATM FIFO data to send to theSuper Memory Partner (SMP) module. There is one SMP module for every TKO
crate and are housed in Versa Module Europe (VME) crates (six per VME crate).
The SMP modules then relay the data to computer servers finally to be collectedby an online host computer. The data from all hit PMTs is assembled into eventswith various information relating to charge, time, number of hit PMTs, location of48

hit PMTs, etc. The events are then sent to the offline system. Figure 3.10 shows theorganization of the SK DAQ.
In total, there are 12 channels per ATM board and 20 ATM boards per TKO
crate. 12 TKO crates, 2 VME crates with 6 SMP modules, and 2 computer servers
are housed in one electronics hut. There are 4 electronics hut and one central hutthat houses the online host computer and global trigger hardware. These 5 huts
are in the dome area immediately above the SK tank. Since there are 5182 PMTs in
SK-II, each ATM has an average of 5.4 live channels out of a possible 12.

3.5.2

OD DAQThe OD DAQ serves to record hits in the OD that come before and after ID global
trigger events. The OD PMT signals are sent to Charge to Time Converters (QTC)
where an OD HITSUM is generated (0.25 p.e.) threshold. If the OD trigger, which
serves the same purpose as the ID global trigger, issues a signal, the OD HITSUM
data is digitized and saved to a TDC module. The TDC can store up to 8 QTC
pulses at a 0.5 ns resolution. Its timing dynamic range is 16 µs with 10 µs before
and 6 µs after the ID global trigger. The data is saved only if an ID global trigger
was issued. The OD information is then sent to the online computer and mergedwith the event information of the ID.

3.5.3

T S

The HITSUM signal generated by the ATM boards are collected at the global
triggering system in the central hut. This triggering system adds all HITSUMs tocreate a total HITSUM. If the total HITSUM exceeds 110 mV (roughly 7 hits), thena global trigger is issued.
There are 3 main levels to the triggering system. They are the Super Low Energy49

20-inch PMT
SCH
ATM
x 20
x 240

ATM
GONG

20-inch PMTSCHATMinterface
Ultra Sparc

x 20
x 240

SMP

ATM

SMP

GONG

online CPU(slave)

SMP
SMP
SMP
SMP

x6

Super Memory Partner

VME
20-inch PMT
SCHUltra Sparc

ATM

online CPU(slave)

x 20
x 240

Ultra Sparc

ATM

online CPU(host)

Analog Timing Module

GONG

online CPU(slave)
TKO

FDDI

Ultra Sparc

Ultra sparc

online CPU(slave)
Ultra Sparc

online CPU(slave)
Ultra Sparc

FDDI

online CPU(slave)
Ultra Sparc

online CPU(slave)

online CPU(slave)interface
Ultra Sparc
SMP
SMP

Ultra Sparc

online CPU(slave)

SMP
SMP
SMP
SMP

Super Memory Partnerinterface

VME
VME

interrupt reg.
TRG

20-inch PMT
SCH
TRIGGERATM

TRIGGER

x 20
x 240

PROCESSORHIT INFORMATIONATM
Analog Timing Module

GONG

TKO

PMT x 11200

ATM x ~1000

SMP x 48

online CPU(slave) x 9

Figure 3.10: The layout of the ID DAQ.

50

(SLE) trigger, the Low Energy (LE) trigger, and the High Energy (HE) trigger. An
event is said to be an SLE, LE, or HE event if the total H ITSUM exceeds 110 mV, 152
mV, or 180 mV respectively. A fourth, independent trigger monitoring OD PMThits registers an event if there are more than 20 OD hits.
The SLE triggered events are mostly radioactive background from the ID walls
and PMTs and gamma rays from the outer rock wall. To save limited storage
space, these events need to be filtered. This is accomplish by a fast online vertex
reconstruction algorithm. Originally SK-I’s standard vertex reconstruction tool, it
was modified and optimized for SK-II to use online (see Section 4.1.1 for details
on the fitter). Those events that reconstruct outside the fiducial volume (definedas 2 meters from the ID wall) are rejected. All others are recorded for later offline
analysis. The SLE trigger and the vertex reconstruction algorithm are collectively
known as the Intelligent Trigger (IT). The reduction in data is around a factor of 6.

3.6W T M

3.6.1

L L I

To measure the effects of light scattering and absorption in water, SK-II has an
automated system of 8 laser light injectors tunable to different wavelengths: 337
nm, 371 nm, 400 nm, and 420 nm. The injection points are diagramed in Figure 3.11.
Light is injected at regular intervals and the total charge Q is collected on the
opposing wall. This is direct, unimpeded light. Reflected light is collected bycounting the number of hit PMTs on all other surrounding walls. The ratio of
hit-PMT/Qbottom is then plotted as a function of time and is used to adjust (tune) a
quantified model of intensity given by
I(x) = I0 ·1
· exp(−x/Latten ),
x2
51

(3.4)

where I0 is the initial intensity and x is the distance from the light source. Latten is
the attenuation length and given as
Latten =λ4αabs + αRayleigh + αMie

.

(3.5)

The α terms are the tunable coefficients and are adjusted to match the ratio-timing
distributions of laser data and MC. The αRayleigh and αMie coefficients correspondto Rayleigh and Mie scattering respectively. The αabs coefficient corresponds toabsorption and is tuned with LINAC data instead (see Section 4.7.2 for a descriptionof absorption). The SK-II values are αRayleigh = 1.7 × 108 and αMie = 5.0 × 106 . An
example of the laser timing distribution is shown in Figure 3.12. The first broad
peak in the timing distribution is due to light scattered from water. The second,smaller peak is due to reflection off the acrylic blast shield, PMT, and black sheet.
The attenuation coefficients are plotted in Figure 3.13 with their λ−4 dependence.

3.6.2

T V

The water quality changes with time and needs to be accounted for during event energy reconstruction. Decay electrons from cosmic ray muons provide an abundantand constant source to measure the SK water’s transparency length periodically.
The charge collected from a stopping muon’s decay electron is fitted to the function
Qi = Q0 ·f (θi )
· exp(−x/Latten ),
x2

(3.6)

where, xi is the distance from the light source to the i-th PMT, Q0 is a constant,and fi (θ) is the i-th PMT acceptance at angle θ (Figure 3.14). Since the Cherenkov
light cannot be isolated in frequency, Latten is measured for the entire (averaged)
spectrum. Latten as a function of time during the SK-II data taking period is shown52

Top new

Top old

Barrel 1(1232.25 cm)
Barrel 2(595.95 cm)
Barrel 3(-40.35 cm)
Barrel 4(-605.95 cm)
Barrel 5(-1242.25 cm)
BottomFigure 3.11: The positions of the 8 laser light injectors in the SK-II tank. ”Top old”
refers to the original SK-I light injector which is still used in SK-II.

53

Number of
hit PMTs

REAL DATA
MC

Number of
hit PMTs

Total Charge

Radius= 10 mRadius= 10 mTotal Charge

Figure 3.12: An example of laser light injection from the top (upper left) and barrel(lower left). Barrel injection uses a 10 m radius around the injector to collect
both charge and hit-PMT count. The right figure shows the timing distribution of
hit-PMT/Q and its agreement with MC.

54scattering-absorption coefficient (1/m)

1

10

10

10

10

-1

-2

-3

-4

200

250

300

350

400

450

500

550

600

650

700

wavelength (nm)Figure 3.13: The solid line is the total scattering and absorption coefficient describing light-loss and scattering in water for SK-II MC. The bold dotted line is Rayleighscattering only. The light dotted line is Mie scattering only.

in Figure 3.15

3.7

C

3.7.1

PMT C

Three types of calibration are performed on the PMTs: relative gain, absolute gain,
and timing. They are to insure that all PMTs give a uniform response to reduce
systematic bias. The information obtained by calibration data are used to adjusteach individual PMT.

55

Figure 3.14: The left plot shows the PMT acceptance as a function of incident angle.
The right angle shows the incident angle on the PMT.

56

water transparency (m)

120

115

110

105

100

95

90

85

80
2500 2600 2700 2800 2900 3000 3100 3200 3300 3400
elapsed day

Figure 3.15: The water transparency as a function of time and measure by the decay
electron from cosmic ray muons.

R GThe relative gain of a PMT is defined as
Gi =

Qi
li
· l2i · exp
Q0 × f (θ)
L

(3.7)

where Qi is the collected charge, Q0 is a constant normalization factor, li is the
distance from a light source to the PMT, L is the water attenuation length, and f (θ)
is the PMT acceptance function dependent on incident angle θ (Figure 3.14). The
light source used in calibration is a scintillating ball exposed to a xenon lamp and isdesigned to emit light uniformly in the tank at a peak wavelength of 440 nm. Since
the gain is related to its applied high voltage as G α V a (a is a constant), the voltage
is tuned until all PMTs obtain a uniform collected gain. Xenon lamp calibration
data is taken at several positions in the tank and for several high voltage values.
The relative gain spread is realized at 7%. The remaining difference is compensated57

for in the event reconstruction software. Figure 3.16 shows the setup of the xenon
lamp calibration system. Figure 3.17 shows the relative gain for all PMTs.
UV filter ND filter
Xe Flash Lamp

Photo
Diode

SK TANK

Photo
Diode

Optical fiber

ADC

Scintilator Ball
PMT
Scintilator

Monitor20inchPMT

Trigger

Figure 3.16: The xenon lamp calibration system. The ultraviolet (UV) filter acts
to pass only light in the higher spectrum, the neutral density filter (ND) adjusts
intensity, and the scintillator ball (BBOT and MgO powder) emits light at 440 nm
in all directions.

A GThe intensity of a PMT signal is quoted in photoelectrons (p.e.) during analysis but
it is charge in pico-Coulombs that is read from the PMTs. Therefore, a conversion
factor needs to be determined at the single p.e. level to render meaningful physicalquantities. A Ni-Cf source is used for the absolute gain calibration. The source
apparatus is shown in Figure 3.18. 252 Cf undergoes spontaneous fission and its
emitted fast neutrons are thermalized in water. The neutrons are then captured
on Ni wire which then emits gamma rays around 9 MeV in energy. The gamma
rays, now outside the source apparatus, Compton scatter with electrons, producing
Cherenkov light. The energy is low enough to assume each PMT is hit with a single
photon. A charge distribution is made (Figure 3.19) to find what mean charge
58

Number of PMTs in Each Bin

1400

1200

σgain = 7.0%
1000

800

600

400

200

0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Relative PMT Gain

Figure 3.17: The relative gain as determined by xenon lamp calibration. The spread
is 7%.

corresponds to these one p.e. hits. It is determined that 2.055 pC is equivalent to
one p.e. and is used as a conversion factor.

T C
The relative timing of all PMTs must be adjusted so that it is possible to reconstruct
event vertices and directions accurately. PMTs can display relative timing offsets
depending on the amount of charge a PMT collects. The charge dependence ismainly due to ”slewing,” an effect in which greater charge hits are triggered earlier
than those with less charge. A fast pulsing light source is used to correct for this.
A N2 laser with pulse width ∼ 3 ns and attched to a dye laser module fires intenselight of 384 nm wavelength into two channels: one going to a monitoring systemand the other to a light-diffuser ball lowered into the SK tank (Figure 3.20). The
light is diffused without significantly affecting the time spread. The light intensity
is varied to correspond with a PMT signal of one p.e. to several hundred p.e. and
the PMTs are read. The distribution of timing and charge is plotted in Figure 3.21.
59

30mm
130mm

125mm

10mm
200mm

50mm

20mm

Nickel wire + Water

Figure 3.18: A cross section of the Ni-Cf source apparatus that is lowered into the
SK tank.

1

-1

10

-2

10

-2

0

2

4

1 p.e. distribution

6

8

10

Charge (pC)

Figure 3.19: The charge distribution for single photoelectrons as measured by the
Ni-Cf calibration source.

60

The errors on the measurements are the timing resolutions of the PMTs and are
typically ∼ 3 ns. The data are then used to counter-offset the PMTs to match one
time regardless of the differing amounts of charge they receive.
ND filter
Trigger

PMT

Dye

N2 LaserDiffuser Ball

LUDOX

Diffuser Tip(TiO2)

Super-Kamiokande Detector

Figure 3.20: The N2 laser setup for the PMT timing calibration.

3.7.2

E C

To better determine the oscillation parameters tan2 θ and ∆m2 , the SK solar analysis
looks at the shape of the recoil electron energy spectrum. The presence and natureof distortions in the spectrum points to these parameters. It is therefore vital to calibrate the energy scale and energy resolution to a high degree or accuracy. SK uses
two calibration sources exclusively employed for the solar neutrino measurement:
monochromatic electrons and 16 N decay products.

LINAC C
To better understand the recoil electron energies from scattered solar neutrinos, SK
first measures the energies of electrons from a controlled source: an electron linear
accelerator (LINAC) [26]. The LINAC used is a Mitsubishi ML-15MIII LINAC
which was originally manufactured for medical purposes. After purchasing, in-

61

980
970

Linear Scale

Log Scale960

T (nsec)

950
940
930
920
910
900
890
880
1

5

10

100

Q (p.e.)Figure 3.21: The timing-charge map (TQ-map) of all ID PMTs. The circles represent
the timing resolutions (∼3 ns).

stalling at the detector site, and performing modifications (which included reduction of the electron gun current to decrease the number of accelerated electrons),
the first successful regular operation of the SK LINAC was in September 1997. The
SK-II LINAC data taking was in March, 2004.
The LINAC accelerates electron ”bunches” using 2 µs-wide microwave pulses.
The number of electrons in a bunch (in the order of 106 ) is too high for calibration
purposes and is collimated down to a few electrons per bunch. The beam leaves
the radiation-shielded room where the LINAC is housed through an evacuatedbeam pipe. To reach the SK tank, the beam must be steered and is done so by
three electromagnets (D1, D2, and D3, see Figure 3.22). The D1 magnet not only
bends the beam downward at a 15◦ angle but also acts as a momentum selector by
deflecting electrons with momenta other than the momentum desired. The beam
passes additional collimators before being bent 90◦ into the SK tank. The end of the62

TOWER FOR INSERTING BEAM PIPE
D2 MAGNET

D3 MAGNET

D1 MAGNET
LINAC

1300 cm
4

2

5

3

6

-12m

-4m

4200 cm

1+12m

BEAM PIPE
0m

-12m

Z

X
4000 cmY

Figure 3.22: The LINAC system at SK. The dotted lines show the fiducial volume
and the numbers indicate positions where LINAC data was taken in SK-II

63

beam pipe is closed with a 50 µm-thick titanium cap which allows electrons to passwithout much attenuation. Also at the beam pipe’s end is a LINAC trigger counter
which can be connected to the main SK trigger system to discriminate calibration
events. The electron in-tank energies taken for SK-II are 5.8, 6.8, 8.8, and 13.4 MeV.
The energy calibration proceeds by reproducing downward-traveling electronsof the LINAC energies taken in the MC detector simulation program4 . These andthe LINAC calibration data events are then reconstructed with the same methodsof the solar neutrino analysis. The MC and data energy distributions are fittedwith a Gaussian function and are compared. Any position dependent deviation
is minimized by adjusting parameters in the MC simulation. These include water
scattering, water absorption, PMT and black sheet reflection, and others. Once
this minimum has been achieved, the absolute gain of the MC PMTs is adjusted
to eliminate any difference in the averaged MC energy and the averaged LINAC
data energy. The top plot of Figure 3.23 shows the MC energy scale deviation from
LINAC data. Using the equation
18

1 X 2
σ,
σ =
18 i=1 i
2(3.8)

where σi are the deviations for the 6 positions over 6.8, 8.8, and 13.4 MeV, an energy
scale systematic uncertainty is assigned to 1.4%.
Uncertainty in energy resolution can also be estimated. Energy resolution is theone sigma value from the Gaussian fit and is compared with MC. An average valueof 2.5% deviation is obtained.
LINAC calibration data can also be used to estimate vertex and direction reconstruction uncertainties. They are discussed in their respective sections.
4

The LINAC energies are determined by aiming the beam into a Ge detector which has an energy
resolution of 1.92 keV. This is done after each successful beam injection into the tank.

64

(MC-LINAC)/LINAC

0.04
0.03
0.02
0.01
0
-0.01
-0.02
-0.03
-0.04

4

6

8

10

12

14

(MC-LINAC)/LINAC

Energy (MeV)

0.15
0.1
0.05
0

-0.05
-0.1
-0.15

4

6

8

10

12

14
Energy (MeV)

Figure 3.23: Deviation in energy scale (top) and energy resolution (bottom) between
LINAC MC and data.

65

DTG
The Deuterium-Tritium Generator (DTG) [27] is a neutron generator used to induce
the creation of 16 N in the SK tank. 16 N’s subsequent decay is then used as an energy
calibration source. See Figure 3.24 for a schematic of the DTG setup.
16N calibration(a)! 4He + n

(b)

(c)

! p + 16N

erator

E n=14.2 MeV
16

O(n,p)16 N

2m
n

n
n

n
n

n

16

N

~106 neutrons / pulse
16~1%ofofcreateFigure 3.24: The setup
the neutrons
DTG calibration
apparatus. N
The generator is lowered
into the tank (a).16Neutrons
are iscreatedwhich interact
with 16 O in the tank andN decay
precisely
known.
16
make N (b). The generator is then lifted 2 meters to allow the decay of 16 N to
6.129MeVγ+
proceed with minimal66.2%
photocathodeshielding by
the4.29MeV
generator (c).β,28.0% 10.419 MeV β, etc.
Data
taken
at various
positions.
The productionof thissource starts
with collisionof deuterium and tritium
Uniform direction complementary to
inside the source generator to create 4 He and a neutron. This is done in pulses with
Y.Takeuchi@WIN’05
14
LINAC calibration.
about 106 neutrons in each pulse. The neutrons then exit the generator into thewater and react with 16 O and release 16 N and protons. The 16 N decays with a halflife of 7.13 seconds with decay products of 6.13 MeV gamma rays and 4.29 MeV
beta particles (66.2% contribution) along with 10.42 MeV betas (28.0% contribution).
These events are reconstructed for energy and an absolute energy scale uncertainty
66

is found to be 1.2% and in good agreement with LINAC data and MC. Since the
DTG is relatively small and portable, data can be taken more often and in manytank positions to better track the time and position dependence of the energy scale.
Also, since 16 N decays isotropically, zenith angle dependence of the energy scale
can be measured and is found to be 0.5%. This is used as an uncertainty in the
day-night asymmetry value. Figure 3.25 shows the absolute and zenith energy

0.04
0.03

(1)

(2)

(3)

(4)

(5)

0.02
0.01
0
-0.01
-0.02

(a)

(b)

(c)

-0.03
-0.04

-12
0
12
-15
-12
0
12
15
-15
-12
0
12
15
-15
-12
0
12
15
-15
-12
0
12
15
-15
-12
0
12
15
-12
0
12
-12
0
12

(MC-SOURCE)/SOURCE

scales using DTG data and MC.

(MC-SOURCE)/SOURCESOURCE z-position (m)

0.015
0.01
0.005
0

-0.005-0.01-0.015-1 -0.8 -0.6 -0.4 -0.20

0.2 0.4 0.6 0.8

1cos(zenith angle)

Figure 3.25: 16 N energy scale deviation from MC. Representing the varying
times calibration data were taken, (1) Nov. 2003, (2) March 2004, (3) July
2004, (4) Nov. 2004, (5) Sep. 2005. (a), (b), and (c) represent the x positions
15.20 m, 10.96 m, −14.49 m respectively. All other calibration data were taken at
x = 0.35 m. The bottom plot shows the 16 N energy scale deviation from MC for 6
selected zenith angles of the detector (-1 is down).

67C 4
SK-II S E
SK-II’s high threshold of 7.0 MeV which corresponds to 8 B and hep neutrino detection is in an area of large, unwatned background. These include gamma rays,
nuclear spallation products, muon decays, radon decays, and electronic noise. To
successfully separate neutrino events from this background, highly efficient reconstruction techniques need to be employed. It is then necessary to remove those
events which are most unlike those expected from neutrino-electon scattering. Finally, the identification of the recoil events from solar neutrinos is carried out by
a comparison of simulated events based on theoretical models and Monte Carlo
random generation.

4.1

E R

4.1.1

V

The determination of event vertex in SK-II is done by an algorithm specificallydeveloped to address the reduced light collection capability of SK-II relative toSK-I. Since this algorithm marks a significant departure in the event reconstruction
methods of SK-I, it warrants a comparison.
The motivation for a new way to reconstruct event vertices comes from thereduced photo-cathode coverage in SK-II and the SK-I standard vertex fitter’s
inability to perform well at lower energies. As an example, the efficiency of the
SK-I standard fit significantly drops at energies below the SK-I analysis thresholdof 5.0 MeV. With 40% photocathode coverage, this corresponds roughly to 25 PMT

68

hits. At 19% coverage in SK-II, 25 hits translates to 8 MeV. The SK-I and the new
SK-II standard vertex fitters function as follows.
The timing residual in an event is defined as the time difference between a PMT’s
hit time ti and the emission time t0 (fitted to minimize all timing residuals) minus
the time it would take Cherenkov light to reach that PMT given the event’s vertex~
v in the tank:tresidual = (ti − t0 ) − |~v − ~hi |/c,

(4.1)

where ~hi is the vector location of the hit PMT and c is the group velocity of light inwater.
In SK-I, vertex reconstruction is accomplished by selecting a limited number of
hit PMTs from an event (to reduce bias from PMT dark noise and scattered light hits)
and calculating a goodness relation based on the timing residuals of those selectedhits and a candidate vertex ~v. A systematic grid search of candidate vertices isperformed until the goodness reaches a maximum value. After that, the vertex
position is fine-tuned to further maximize the goodness. The SK-I reconstruction
will not attempt a vertex fit for less than 10 hits.
In contrast, SK-II uses all hits from an event to form the timing residuals fordetermination of the vertex position. PMT dark noise is taken into account byconstructing a likelihood describing the shape of the timing residual distribution
from LINAC calibration data. This likelihood is then maximized from a vertex
search based not on a grid pattern but from a list of vertex candidates calculated
from PMT hit combinations of 4 hits each. The four-hit combinations each define
a unique vertex given their timing constraints. Any event with four hits or more isreconstructed.
SK-II also makes use of the SK-I goodness-grid search method in its online andinitial offline analysis for filtering background events. The final reconstruction, or
the standard fit based on the residual likelihood method, is the final determination69

of the vertex and can also be seen as a correction for any misreconstructed events
which survived the filtering process. Figure 4.1 shows the various vertex resolutions for the SK-II vertex reconstruction. The uncertainty of the measured solar
neutrino rate due to systematic shifts in vertex position is estimated to be 1.1%.
SK-II also utilizes a fast fit online reconstruction method for pre-filtering low
energy events. Details can be found elsewhere [18].
vertex resolution (cm)

300

250

200

150

100

50

0
5

standard fit
goodness fit
fast fit
67

8

9 10 11 12 13 14
electron total energy (MeV)

Figure 4.1: Vertex resolution (defined as 68.2% of reconstructed events which
reconstruct inside a sphere of radius σ from the correct vertex) of 8 B Monte Carloevents as a function of total recoil electron energy.

4.1.2

D

To recognize the Sun as the source of a neutrino scattering rate rising over a relatively flat background rate, event direction reconstruction is important. To dothis, SK-II uses a maximum likelihood method that scans directions which correspond with Cherenkov ring patterns. A function f (cos θdir ) describes distributionsof opening angles between a candidate particle direction d~ and the reconstructedvertex-to-hit PMT position (~v − ~hi ). It is made by MC and an example for 10 MeV
70

electrons is shown in Figure 4.2. The effects of electron multiple scattering and lightscattering in water can be seen by the broad peak around the 42◦ Cherenkov angle.
Opening angles θi between the reconstructed vertex ~v and the vector normal to the
PMTs inward-facing surface and weighted by the PMT acceptance function a(θi )are then summed as a product with f (cos θdir ) in a likelihood function

L(~
v) ≡

N30
X

log( f (cos θdir ))i ×icos θi,a(θi )(4.2)

where N30 is the number of hit PMTs with timing residuals within a 30 ns window
around tresidual = 0. The direction d~ is scanned at various levels of precision until
Equation 4.2 is maximized. An absolute angular resolution is then defined as the
maximum angular difference between 68% of the reconstructed and true event directions as determined by MC. It is plotted as a function of energy in Figure 4.3.
The systematic uncertainty in the angular resolution is 6.0% when comparing resolutions of data and MC. This results in a ±3.0% error on the total flux measurement.

4.1.3

E

To approximate the energies of solar neutrinos, the recoil electrons off which theyscatter is reconstructed and acts as a good equivalence to neutrino energy since therecoil electron mostly preserves the incident neutrino’s direction. Whereas vertex
and direction reconstruction rely on the location and pattern of Cherenkov lightreceived by PMTs, energy reconstruction is dependent on the intensity of that light.
Cherenkov photons from recoil electrons travel in the tank and are detectedby the 50 cm PMTs through generation of photoelectrons in the photocathode.
The number of Cherenkov photons are approximately proportional to the electronenergy and in turn proportional to the number of generated photoelectrons. But
given the poor charge resolution (∼ 50%) of the PMTs and the fact that electrons71relative probability

1

0.8

0.6

0.4

0.2

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

0.2 0.4 0.6 0.8 1
cosθdir

directional resolution (degree)

Figure 4.2: The likelihood function describing the cosine of opening angles between
candidate particle directions and reconstructed vertex-to-hit PMT position for 10
MeV electrons.

45
40
35
30
25
20
15

5

6

7

8

9
10
11
12
13
electron total energy (MeV)

Figure 4.3: Directional (angular) resolution of Monte Carlo events as a function ofrecoil electron total energy.

72below 20 MeV produce roughly one photon per hit PMT, the number of hit PMTs,
or Nhit , is counted instead.
To reduce the contamination of noise hits, Nhit is defined as the number of hits
whose timing residuals (Equation 4.1) fall within a 50 ns window1 . But this alone
is not adequate for energy calculation. Corrections must be made for detectorspecific dependencies including PMT occupancy, late hits, dark noise, dead PMTs,
photo-cathode coverage, water transparency, and PMT gain. These are manifested
as an ”effective” hit sum
Nhit


X


1
ri
Nall
×
× exp
× Gi (t) .
Neff =(Xi + tail − dark ) ×
Nlive S(θi , φi )
λeff
i=1

(4.3)

O
To estimate the effect of multiple photoelectrons in the i-th hit PMT, the occupancy
Xi is defined as− log (1−xi )xi < 1xi
Xi = 
,
 2.5
xi = 1(4.4)

where xi is the ratio of hit PMTs in a 3×3 area surrounding the i-th PMT over the
total number of live PMTs in the same area. The − log (1 − xi ) term is then theestimated number of photons per one PMT in that area and is determined fromPoisson statistics. The value when xi = 1 is assigned to 2.5. It should be mentionedthat in SK-II the usual number of PMTs in a 3×3 area is 5 (9 in SK-I) and is arranged
in a checkered pattern.
1

Introduced in the last section was N30 , the number of hits within a 30 ns window. The generalnotation in this work is the number of hits N subscripted by the width of the timing window in
nanoseconds. If Nhit is given without numerical scripting, 50 ns is assumed.

73

L HTo account for hits falling outside the 50 ns timing window, the term tail = (N100 −N50 )/N50 is added to the occupancy. Such hits could arise from Cherenkov photons
being scattered and reflected and thus arriving late to the PMT surface.

D N
Dark noise is subtracted from the occupancy with the term dark = (100 ns × Nlive ×
rdark )/N50 . Nlive is the number of functioning PMTs for a given run and rdark is themeasured dark rate for that run.

D PMTTo correct for non-functioning PMTs, the factor Nall /Nlive is applied. Nall is the total
number of PMTs and is 5182 in SK-II.

P-C C
The term 1/S(θi , φi ) is to account for the direction-dependent photocathode coverage. It acts as a scaling function on the average coverage of 19%.

W T
The water transparency is accounted for by ri /λeff where ri is the distance from thereconstructed vertex to the i-th hit PMT. λeff is the measured water transparency.
PMT G
The last factor, Gi (t), is the PMT gain correction. This is to adjust the relative gainof the PMTs at the single photoelectron level. The differences in gain arise from the74

various stock and fabrication dates of the PMTs.

After determining Neff , a corresponding value for an event’s energy must be
obtained. This is done by generating MC events at discrete input energies between
5 and 80 MeV, calculating their Neff values, and then interpolating the energyfunction. This function is then used to calculate an event’s total energy in MeV.
The energy values are checked with LINAC and 16 N calibration data and MC (see
LINAC and 16 N sections).
Since the corrections in Neff depend on the water transparency, the reconstructedenergy also varies slightly with changing water quality. See Figure 4.4 for Neff as afunction of time for a given water transparency. When calculating energy for data
events, the water transparency value as determined by decay electrons from cosmic
ray muons is used as an input parameter. However, for MC events, the change
in water transparency is not simulated due to its relative stability. A calculated,
constant value of 101 m is used for all MC events in the SK-II solar neutrino analysis.

4.2D R

The average LE trigger rate in SK-II is about 63 Hz. Therefore, each day 5.4×106
events above 8.0 MeV are expected in the SK-II tank. The vast majority of these
events are not recoil electron events from solar neutrinos (even considering allenergies, the BP2004 expected rate is 325.4 events per day). To separate the solar
neutrino events from unwanted events, or background, a series of reduction cutsmust be applied to the initial data set. These cuts are
1. Noise reduction, to reduce the contribution of triggered events originatingfrom PMTs and electronic components and to eliminate charged particles
coming from outside the detector. It also discards entire runs or subruns that75

Figure 4.4: Upper figure shows the time variation of the measured water transparency during SK-II (identical to Figure 3.15). Lower figure shows the stability of
the SK-II energy scale as a function of time.

are deemed inadequate for the solar neutrino analysis.
2. Spallation cut, to reduce beta-decay and gamma ray events arising fromspallation products of incoming muons.
3. Timing and hit pattern cut, to verify the trustworthiness of reconstructed
events in effort to further eliminate noise events that might have passed
previous cuts.
4. Gamma cut, to cut gamma-ray events originating from the ID wall.
The reduction cuts utilize the information obtained from the PMT timing, charge,
and event reconstruction and are applied to all LE-triggered events as well as those
events that survived the SLE filtering process (see Section 3.5.3).

76

4.2.1

N R

Noise reduction is a multistep process that relies on a variety of information. It isalso known as first reduction since it is the first effort to reduce background. The
steps and their descriptions follow.

P C
To remove high energy events such as cosmic ray muons and atmospheric neutrino
events, a total p.e. cut is applied that keeps events registering p.e.≤500.

R  A T
This is simply a removal of those events that might have an additional trigger incoincidence with an LE or SLE trigger. These are mainly calibration events andatmospheric analysis-related pedastal events.

C  T D  E
When two events follow closely in time, they may be a stopping muon and its
decay electron or other unwanted electronic noise such as signal reflection in the
PMT-ATM cable. To eliminate this background, when the event time differenceis less than 50 µs apart, they are removed. Figure 4.5 shows the time difference
between events in a representative run.

R  OD- E
A charged particle such as a cosmic ray muon may enter the ID and leave its
signature in the OD. Therefore, when the OD registers an event (p.e.≥20), it isremoved. Figure 4.6 shows the reduction of events after the OD trigger cut.

77events

10

5

10

4

10

3

10

2

10

1

2

3

4

5

6
7
8
9
10
log(time difference (ns))

eventsFigure 4.5: The time difference cut. Events that are less than 50 µs apart areremoved. The shaded region show the events which survive the cut.

10

4

10

3

10

2

0

10 20 30 40 50 60 70 80 90 100
number of OD hit PMTs

Figure 4.6: Events which have OD triggers are removed. The shaded region shows
the distribution of events after the OD trigger cut.

78

R  A E N
Ambient noise from electronic components come from noisy PMTs or from various
effects originating in the dome area above the tank. This noise typically produces
an absolute PMT charge no greater than 0.5 p.e. A ratio can be formed by defining
N as the total number of hit PMTs with |QPMT | < 0.5 and S as the total numberof PMTs regardless of charge. A cut is made for those events with N/S < 0.55.
Figure 4.7 shows the ratio before and after reduction.
The same noise events are often clustered on one ATM. If the number of hit
PMTs on one ATM is greater than 95% of the total PMT channels on that ATM, thenthe corresponding events are rejected. Figure 4.7 shows a typical distribution of

eventsthis percentage.
6

events

10
5
10
4
10
3
10
2
10
10
1
0
10

5

10

4

10

3

0

0.2

0.4

0.6

0.8

1
N/S ratio

0.2

0.4

0.6

0.8

1
ATM %

Figure 4.7: Top: Events with a ratio of low charge PMTs to total PMTs less than
0.55 are removed. The shaded area show the events that passed the cut. Bottom:
Events with a large number of hit PMTs on one ATM are removed.

79E  F PMTA flashing PMT is caused by an electrical discharge between dinodes and can becaused by water seeping into the PMT housing. This can severely affect a normallyuniform trigger rate in the detector and hinder event reconstruction. Fortunately,
flashing PMTs with a high charge can easily be identified and immediately turned
off. Unfortunately, those that are not readily identifiable can remain in the data set.
To identify a flashing PMT for removal, its neighboring PMTs are scrutinized forhigh charge. If the number of PMTs NPMT surrounding the PMT with the maximum
charge Qmax in an event exceeds the function




10
Qmax < 17.02





NPMT > −0.51 × Qmax + 18.75 17.02 ≥ Qmax < 32.59 ,






 2
32.59 ≥ Qmax

(4.5)

eventsz (cm)

then that event is rejected. Figure 4.8 shows the relation of Nmax and Qmax .

10

5

10

4

10

3

10

2

2000

1500

1000

500

0

10
-500

1
-1

10

-1000

0
2.5

5
N 7.5
10
PM
12.5
T
15
17.5
20
22.5

-1500

40
0 20

60

200
160180120140
100
Q MAX
80

-2000
0

500

1000

1500

2000

2500
2

3000
3
x 10

r (cm2)

Figure 4.8: Left: The events with a large number of hit PMTs surrounding the
maximum charge-hit PMT are rejected. The transparent bins are events that are
cut. Right: The z v.s r2 distribution of the SK tank. The cluster at high z value is a
flasher. The line represents the boundary of the tank.

80

R  F F
Failed fits refer to the inability of the vertex reconstruction algorithm to locate asuitable vertex for an event. At this stage and until the end of noise reduction,
the goodness-grid search algorithm of SK-I and optimized for SK-II, mentioned in
Section 4.1.1, is used to reconstruct events. Failure can arise from events having
less than the minimum 5 hits.
C B  G C
The vertex reconstruction assigns a value which gives a numerical gauge of thegoodness of the determined vertex. Those events reconstructed with a goodness
less than 0.4 are rejected.
F V C
A fiducial volume cut is applied using the reconstructed vertex of events. Those
events falling within 2 m of the ID wall are rejected. This is to eliminate noise comingfrom the wall (PMT support structure, PMT glass, black sheet, etc.). Figure 4.9
shows the z vs. r2 distribution of reconstructed events in the tank.
L E C
Finally, an energy cut removes all events (SLE and LE) below 6.5 MeV.

After the completion of all noise reduction cuts, various distributions of thedata (such as those presented in the above figures) are manually scanned for eachrun. The purpose is to identify anomalies that might have passed the automatedreduction process. These include low rate flashing PMTs, higher than normal or
erratic time dependent event rates, high occurrences of noise in a single run or
subrun, and others. When such data are observed, the run (or subrun, depending81z (cm)

2000

1500

1000

500

0

-500

-1000

-1500

-2000
0

500

1000

1500

2000

2500

3000
3
x 10

r2 (cm2)

Figure 4.9: The z v.s r2 distribution of the SK tank. The solid line represents theboundary of the tank. The dotted line is the fiducial boundary (2 m from the tankwall). All events outside the fiducial area are cut.

on the magnitude of contamination) they occupy is labeled as a bad run and is
discarded from the analysis. Short runs, typically under an hour, are discarded as
well.
The total systematic error due to differences in reduction efficiencies between
data MC is estimated to be ±1.0% on the total flux measurement.

4.3

S C

When cosmic ray muons travel in the SK tank, they break apart 16 O nuclei andcreate unstable nuclei known as spallation products. These products soon decay
releasing electrons, positrons, or gamma-rays with energies largely between 7 and20 MeV. This obvious background to the solar neutrino signal must be reduced.
This is accomplished by taking advantage of the correlation spallation products
have with the initial muon.

82

4.3.1

M T R

To identify muon events, PMT charge is scrutinized: if the total charge in the tankexceeds 3000 p.e. and at least one PMT has more than 100 p.e., then the event ismarked as an incoming muon. To reconstruct its track, the position of the earliest
hit PMT with at least 2 neighboring hit PMTs (hit within 5 ns of the first hit) is
identified as the entry point. The central point of all PMTs with more than 190 p.e.2
is identified as the exit point. The line from entry point to exit point is definedas the muon track. The quality of the track is measured by defining Lent as theminimum distance between each charge saturated PMT at the entry point and Lexit
as the maximum distance between each charge saturated PMT at the exit point.
The track is considered a good reconstruction if Lent > 300 cm and Lexit < 300 cm.
After the initial track reconstruction, the charge residual Qresidual of the muon
event is determined by
Qresidual = Qtotal − pL,(4.6)

where Qtotal is the total charge in the event, L is the reconstructed track length, andp = 11.4 p.e./cm and is the average number of photoelectrons ejected per unit track
length of the traveling muon. If a muon event has a large charge residual, the firstreconstruction may not be trusted. If Qresidual > 12, 000 p.e., it is refitted with asecond algorithm regardless of the quality of the first reconstruction.
The second reconstruction utilizes the same entry point as explain above but
searches for an exit point by following the Cherenkov wake left by the muon. A
goodness relation is defined as2 
1 ti −Ti σ2 exp − 2 1.5σi
i
,
P 1

P 1
gµ =

i σ2
i

2

A PMT with more than 190 p.e. is said to be charge saturated.

83

(4.7)

where σi is the time resolution of the i-th hit PMT and T is the time the muonentered the detector. The fitted entrance time ti is defined asti = Ti (~xi ) −

lµ (~
xexit )
c−lγ (~xexit )
c/n

(4.8)

where ~
xi is the location of the i-th hit PMT, Ti is the time of the i-th hit PMT,~
xexit is the exit point, lµ and lγ are the flight distance of the muon and Cherenkov
photons respectively, and n is the refractive index of water. Figure 4.10 shows thecomponents of the fitted entrance time equation. After searching candidate exit
points, the one that gives gµ > 0.85 is chosen.

4.3.2

S  S E

After the passing of a muon in the tank, the resulting nearby spallation products
decay with half lives up to 14 seconds. Therefore, the decay products’ spatial andtemporal proximity to the muon track and event time can be used as variables ina likelihood test. They are expressed as a difference of the event vertex and timeto that of a preceding muon (∆L and ∆T respectively). A third variable, the muon
event’s residual charge Oresidual (Equation 4.6), is also used in the likelihood andserves as an indicator of the muon’s lost energy and hence possibility of creating
spallation products. When compared to solar neutrino events, spallation eventsare expected to be spatially closer to a preceding muon track (small ∆L), temporally
closer to a muon’s event time (small ∆T), and greater residual in the tank (large
Qresidual ). Each candidate solar neutrino event (those that survived the various
noise cuts) are tested with the 200 preceding muon events. Those events with alikelihood greater than 0.98 are rejected. It is possible to use the likelihood even
if the muon track failed to be reconstructed. In this case, ∆T and Oresidual are usedand events with a likelihood greater than 0.92 are rejected. Since this cut inevitably

84

Figure 4.10: Relation of the variables in the muon track reconstruction.

85

results in signal loss, a dead time is calculated with a randomly collected sample ofevents not correlated with muons. The effect this has on the final flux measurement
is estimated to be 0.4%.

4.4

T & H P C

After noise reduction and the spallation cut, many background events remain due
to mis-reconstruction inside the fiducial volume. The 2-dimensional timing-hit
pattern cut is designed to remove those events whose reconstruction should not
be trusted. An optimized hit PMT timing goodness is defined (Equation 4.9) bycomparing two timing residual Gaussian distributions, one with a width of σ = 5
ns to encompass selected hits and the other with a ω = 60 ns width characteristicof the PMT timing resolution for a single photo-electron:


gt (~
v) =

Σe

τ (~
v)−t

τ (~
v)−t0

− 21 ( i ω 0 )+( i σv)−t0 2
1 τi (~)
ωΣe− 2 (

2
)

.

(4.9)

The effective hit time is defined as τi (~v) = ti − |~v − ~hi |/c which is just the timing
residual tresidual of Equation 4.1 with added t0 . The sums are over all hits.
The hit pattern goodness allows us to identify Cherenkov events by theirazimuthal-symmetric ring pattern from a reconstructed vertex and direction. Allothers are labeled mis-reconstructed or non-Cherenkov events. A goodness function gp (~v) is defined for all directional events.
A cut on the goodness values is made in tandem using a hyperbolic radius of
g2t − g2p > 0.25 and rejecting all other events. Figure 4.11 shows this background
reduction cut on data and 8 B Monte Carlo in the 7.0-7.5 MeV bin. When the cut isapplied to LINAC data and MC, a total flux systematic error of ±3.0% is obtained.

86

0.5

0.5

direction (pattern) goodness

0.5

data

0.45

0.4
0.4

0.4

0.35

0.35

0.3
0.3

0.3

0.25

0.25

0.2
0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05

00
0

0.1

0.2

0.3

0.4

0.5
0.5

0.6

0.7

0

-3

10-3

10

-4

10-4

10

0.1
0

MC

0.45

0

0.8
0

0.1

0.2

0.3

0.4

0.5
0.5

0.6

0.7

0.8

-5

10-5

10

vertex (timing) goodness

Figure 4.11: PMT timing and hit pattern cut. Data (left) show an excess of misreconstructed and non-Chrerenkov events to the upper-left of the diagonal cut line.
Approximately 78% (8%) of data (MC) events between 7.0-7.5 MeV are rejected by
the cut. The color scale is to show the relative (normalized) number of events. The
axes are squared goodness.

87

4.5

G- C

Although a 2 m fiducial volume cut is applied to reject events that are coming fromthe ID wall, similar events might remain inside the fiducial volume. To eliminate
them, an additional gamma-ray cut is applied using the energy, direction, andvertex based on the standard fitter. With this information, events with direction
inward from the wall are rejected with the following criteria:
7.0 MeV < E ≤ 7.5 MeV

anddwall < 11 m

7.5 MeV < E ≤ 8.0 MeV

anddwall < 10 m

8.0 MeV < E ≤ 8.5 MeV

anddwall < 8 m

8.5 MeV < E ≤ 9.0 MeV

anddwall < 7 m

9.0 MeV < E ≤ 10.0 MeV

anddwall < 5 m

10 MeV < E ≤ 20 MeV

anddwall < 4 m,

where E is the reconstructed event energy and dwall is the distance from the wallin the direction of the reconstructed event (Figure 4.12). This further reduces
the number of events around the areas of large z or r2 as seen in Figure 4.9. The
uncertainty on the total flux measurement is estimated as ±1.0% for the gamma-ray
cut.

4.6

F D SThe SK-II data period lasted from December 10th, 2002 to October 6th, 2005. The
raw data sample before reduction contains 1.5×1010 events. This includes all events
before trigger cuts and low energy cuts. The first noise reduction (with bad runcut) greatly reduces this total to 6.1×106 events. Figure 4.13 shows graphically the88Figure 4.12: dwall is defined as the distance from the wall in the direction of thereconstructed event.

contributions of succeeding cuts and Figure 4.14 shows the efficiencies on MC of
those cuts. The final solar sample event count is 117,696. Still, the majority ofevents come from background that were not removed. The next step is to employ
Monte Carlo event simulations to separate the remaining background and solar
signal.

4.7

M C E S

To identify which recoil electron events are coming from scattered solar neutrinos, acomputer generated model is used for comparison with the reduced SK-II data set.
This model, based on theoretical rates, kinematical distributions and cross sections,and a detector response simulation, is also valuable for determining reduction
efficiencies and, later, oscillation spectra.

89Events/day/0.5MeV

10

4

10

3

10

2

After noise reduction
After spallation cut
After timing-hit pattern cut
After gamma cut

10
1
-1

10
-2

10

10

15

20
Energy (MeV)

Efficiency

Figure 4.13: Summary of the data reduction steps.

1

0.8

0.6

0.4
After noise reduction
After spallation cut
After timing-hit pattern cut
After gamma cut

0.2

0

4

6

8

10

12

14

Energy (MeV)

Figure 4.14: Summary of the efficiencies of the data reduction steps on MC.

90

4.7.1

N R  I

The first half of the SK-II Monte Carlo event simulation involves stating, statistically,
what recoil electron signal is expected due to scattering with neutrinos coming fromthe Sun. This can be divided into three steps3 , each determined separately:
1. The rate of neutrinos emitting from the Sun
2. The probability a neutrino of energy Eν will scatter with an electron
3. The resulting energy and direction of the recoil electron
The first step comes from the measured 8 B neutrino spectrum of Ortiz [28] andthe total flux prediction of the BP2004 Standard Solar Model (SSM) [9]. Using
these, it is possible to state the total rate of solar neutrinos arriving at earth. But
since at SK-II solar neutrinos are only detectable indirectly by scattering withelectrons, a scattering rate must be included in the equation. Taken from [29], thedifferential cross section dσ(Eν )/dE describes the probability a neutrino of energy Eν
pwill scatter off an electron with total energy E = p + m2 . The neutrino spectrum
and differential cross section is shown in Figure 4.15. Now it is possible to define
a total rate of electrons scattered off solar neutrinos:
Z ∞Z ∞
R8 B = ΦSSM Ne

dEν dEφ(Eν )00dσ(Eν ),
dE

(4.10)

where φ(Eν ) is the neutrino spectrum. The ΦSSM factor is the BP2004 8 B flux prediction and is 5.79 × 106 cm−2 s−1 . Ne is equal to 1.086 × 1034 and is the numberof electrons in the 32.4 kton volume of the SK-II inner detector. By summingover all electron and neutrino energies, the total expected rate in the SK-II tank is325.4 events per day. The errors on the neutrino spectrum and cross section come3

When considering neutrino oscillations, a fourth step will be interjected.

91

from [30] and [31]. Their effect on the total flux measurement is ±1.9% and ±0.5%

0.14
0.12
0.1
0.08

events per day (32.4 ktons)

differential. c.s. (10E-19)

emission probability

respectively.
0.1

0.08

0.06

0.06
0.04

10

2

10
1
-1

10
-2

10
-3

10

0.04

-4

10

0.02

0.02

-5

10
0
0

2.5

5

7.5

10 12.5 15 17.5 20
neutrino energy (MeV)

0
0

5

10

15
20
electron total energy

0

2.5

5

7.5

10 12.5 15 17.5 20
electron total energy

Figure 4.15: Left: The black solid line is the 8 B neutrino spectrum from [28] and thered dotted line is the hep neutrino spectrum. Both are normalized to unity. Center:
The black solid line is the νe − e elastic scattering differential cross section in unitsof cm2 MeV−1 × 10−42 . The red dotted line is the elastic scattering differential crosssection for νµ,τ − e. Right: The black solid line is the expect number of events per
day as a function of electron energy coming from scattering with 8 B neutrinos. The
red dotted line is for scattering with hep neutrinos.

Since neutrinos from the hep chain are also in the energy range which SKII is sensitive, hep neutrinos are included in the simulation. The rate for hep isconstructed the same way as the 8 B rate in Equation 4.10 but substituting φ(Eν ) forthe hep neutrino spectrum φhep (Eν ). To calculate the total rate of hep neutrinos in
the SK-II tank, the BP2004 SSM value of ΦSSM,hep = 7.88 × 103 cm−2 s−1 is used andcomes to 0.64 events per day.
The third step in determining an expected recoil electron signal in SK-II is theelectron’s energy and direction. By the construction of Equation 4.10, the energy
information is already included. The integral over all electron energies is dropped
and R8 B becomes r8 B (E), a function of the recoil electron energy. The rate spectrum isshown in Figure 4.15. The direction of the scattered electron begins by recognizing
the incident neutrino originates from the Sun for whose direction is constantly
known. Therefore, the incident neutrino’s path is along the same vector from the92

Sun to the detector. The scattered electron’s direction is finally determined by thekinematics of the elastic collision and its scattering angle with respect to the initial
neutrino direction. It is given by
1 + Emν,cos θ = q
2m
1 + ET(4.11)

where m is the mass of the electron and ET is the kinetic energy of the recoil electron(disregarding mass). In the azimuth, scattering is symmetric.

4.7.2

D R

Having obtained a model describing the rate of recoil electrons from 8 B and hep
neutrinos, it can now be used as input information for the SK-II detector simulation
computer routines collectively called skdetsim for SK detector simulation. Theseroutines simulate the physical processes of energetic electrons in water and, ingeneral, the scattering and absorption of photons produced by these processes in
the SK-II tank.
The skdetsim program is based on CERN’s GEANT 3.23 package which allows
simulation of electromagnetic processes in the range of 10 keV to 10 TeV. After
instructing the relevant GEANT routines to simulate electrons in water with thespectra plotted in Figure 4.15 with recoil direction 4.11 from the Sun-detector vector,
the simulation package tracks the electrons and models the physical reactions that
could appear during its journey. Such reactions include Cherenkov radiation, multiple scattering, ionization loss, gamma-ray production, bremsstrahlung radiation,
pair creation, and Compton scattering.
The skdetsim program’s next role is to simulate the scattering and absorptionof generated photons in the SK-II tank. To describe light scattering, the laser
data obtain in the water transparency measurement (Section 3.6.1) is used. For93

absorption in the longer wavelengths above 350 nm, a third party independentmeasurement [32] of pure water using an Integrating Chamber Absorption Meter(ICAM) is used. For shorter wavelengths (λ < 350 nm), the MC energy output in thesimulated tank is highly dependent on the absorption coefficient and is carefully
tuned with LINAC calibration data. The most suitable coefficient is chosen byminimizing the differences in reconstructed LINAC data and MC energies. This
minimum was found to be a 1.4% difference for 13.4 MeV and 8.8 MeV eventsusing no absorption below 350 nm. See Section 3.6.1 for a discussion of the watertransparency model with its tuned coefficient values. The absorption coefficient isscattering-absorption coefficient (1/m)

shown in Figure 4.16

1

10

10

10

10

-1

-2

-3

-4

200

250

300

350

400

450

500

550

600

650

700

wavelength (nm)Figure 4.16: The solid line is the total scattering and absorption coefficient describing light-loss and scattering in water for SK-II MC. The dotted line is thecontribution of absorption only.

Having traversed the simulated water, the GEANT photons have the possibility
to reach the blast shield-encased PMTs4 . The acrylic front shield introduces another
4

The other possibility is light may travel to the inner detector black sheet. Although the black
sheet is meant to absorb photons, scattering can occur and is modeled after test data taken in the
SK-II tank.

94

medium for which scattering and absorption could occur. Extensive studies wereconducted to determine the overall transparency of the acrylic shields; at normal
incidence transparency is better than 98% above 400 nm in wavelength and is about
86% at 300 nm.
Other light-interacting properties of the PMTs are also simulated using observed
or calculated values such as reflection off the PMT glass and absorption in thephotocathode. Once the photon is ready for conversion into a photoelectron,
PMT timing resolution, PMT collection efficiency, and PMT quantum efficiency are
adjusted appropriately in skdetsim.
Finally, a trigger simulator is applied to the MC-simulated data. The output
is identical in form to that of real data but consists solely of signals generated
from simulated recoil electrons off solar neutrinos. In addition, the original momenta of these events are preserved and are used to gauge the performance of thereconstruction tools and reduction cuts.

4.7.3

E R F

Although not utilized for the extraction of SK-II solar neutrino data, an analytical
function of the detector’s energy resolution can be constructed with previously
determined information. This function can be used to apply the SK-II energy
response to theoretical spectra for comparison with SK-II data.
The MC events used for the Neff -to-energy conversion function are this timereconstructed for energy E0 . The distribution of E0 follows a Gaussian shape. They
are then fitted to determine their one sigma values and the fitting error on thosevalues (δσ ). Energy resolution over energy ((σ ± δσ )/E0 ) is plotted as a function ofE0 and is again fitted to obtain the function (in MeV)√
σE (E) = 0.0536 + 0.5200 E + 0.0458E,
95(4.12)

where E is the true electron energy. The Gaussian-fitted E0 distributions as well as

Number of events

Equation 4.12 are shown in Figure 4.17.

Mom=5MeV

2000

7MeV
9MeV

11MeV
13MeV

15MeV

22MeV

1000

E resolution/E

18MeV

0
0
0.4

26MeV

30MeV

10

20
30
Reconstructed energy (MeV)

10

20
30
electron total energy (MeV)

0.3
0.2
0.1
0

Figure 4.17: Top: The reconstructed energy distributions of MC events and their
Gaussian fits. Bottom: The energy resolution of the reconstructed distributions(black markers) and the function that best describes its shape (blue line, Equation 4.12)

This is used in a Gaussian probability density function
(E0 − E)2
R(E, E ) = √
exp −
.
2σ2E
2πσE
0

1

(4.13)

Equation 4.13 can now be used with Equation 4.10 to calculate the expected 8 B rate96

in SK-II as a function of measured energy E0 :Z ∞Z ∞
r8 B (E ) = ΦSSM Ne
0dEν dEφ(Eν )00dσ(Eν )
R(E, E0 ).
dE(4.14)The hep rate is determined in similar fashion by substituting ΦSSM and φ(Eν ) with
their hep counterparts. Figure 4.18 shows the calculated rates of 4.14 in the standard
SK-II binning and compared to the rates as determined by skdetsim and event
reconstruction. The maximum difference in binned rates between both methods is

Events/kton/year

no larger than 7.5%.

10

2

10

1

10

15

20
Energy(MeV)

Figure 4.18: The expected recoil energy spectrum of the BP2004 SSM as it would
appear in the SK-II tank. The black lines are the expected rates as determined by
the skdetsim detector response program. The blue lines are the expected rates as
determined by the energy resolution function.

97

4.8

R

4.8.1

S E

Since the recoil electrons from solar neutrino scattering are strongly correlated withthe vector from the detector to the Sun, information on the Sun’s position and the
expected dependence the number of recoil events has on this direction can be used
to extract the data signal.
This is done by an extended maximum likelihood fit with data and MC distributions. The likelihood function is
Nbin
Nbin Yni XYL = exp
Bi + S
Bi · bi (cos θi j ) + S · Yi · si (cos θi j ,i=1(4.15)

i=1 j=1

where Nbin = 17 and is the number of bins between 7.0 MeV and 20.0 MeV andni is the number of observed events in the i-th energy bin. Bi is a free parameterthat represent the number of background events and S · Yi is the number of signal
events again for the i-th energy bin. The free parameter S is the total number ofsolar neutrino events and Yi is the expected fraction5 of that total for each bin. The
solar angle dependent distributions bi (cos θSun ) and si (cos θSun ) (θSun ≡ θi j , shownin Figure 4.19) describe the expected shapes of the background and recoil electron
signal respectively.
The background shape bi (cos θSun ) has a non-flat distribution when viewed asa function of solar angle θSun . This is due to the effective coverage of backgroundsources seen from various vantage points in a cylindrical detector. To estimate
this background, bi (cos θSun ) is extracted from the data by fitting the zenith angle
and azimuthal event distributions with an eighth degree polynomial. The solar
neutrino events are those defined as cos θSun > 0.7 and are weighted to reduce5

As expected from the Ortiz [28] spectrum.

98

SK

θSun

Sun
Figure 4.19: The solar angle θSun .

99their contribution so the fitted function sees no solar peak. The background is thenre-expressed as a function of θSun .
The signal shape si (cos θij ) is obtained from MC and incorporates the effects ofelectron multiple scattering and the energy resolution of the detector. This causesa smearing of the peak in the direction of the Sun (cos θSun = 1.0) and is dependent

probability density

on energy. Figure 4.20 shows this distribution for 4 bins in energy.
0.25

0.2

20 MeV
12 MeV
10 MeV

0.15

8 MeV

0.1

0.05

0
0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
cosθsunFigure 4.20: The expected cos θSun distributions for 4 energies and obtained fromMC.

The values Bi and S are then chosen so the likelihood has a maximum value. Fora live time of 791 days of SK-II data from 7.0 to 20.0 MeV, the extracted number of+152.9signal events is 7212.8−150.9(stat.) +483.3(sys.). This corresponding 8 B flux is−461.66−2−1(2.38 ± 0.05(stat.)+0.16−0.15 (sys.)) × 10 cm sec .

The cos θSun distribution of data and the extracted signal and background can be100

seen in Figure 4.21. Figure 4.22 shows the distribution divided into energy bins ofEvent/day/kton/bin

the recoil electron spectrum.

0.2

0.15

0.1

0.05

0
-1

-0.5

0

0.5

1
cosθsunFigure 4.21: The cos θSun distribution of data showing a strong correlation with thesolar direction. The solid histogram is signal shape obtained by the likelihood fit.
The dotted line is the expected background contribution.

4.8.2

D-N  S V

The day and night fluxes can also be extracted from the SK-II final data set byrecognizing those events which happened when the Sun was above the local horizon and when it was below. Unlike the total flux, the day and night fluxes arequoted using a threshold of 7.5 MeV due to low signal to noise ratio and increased
deviation from a non-flat background for the 7.0-7.5 MeV bin in the solar direction101

7.0- 7.5MeV

7.5- 8.0MeV

8.0- 8.5MeV

8.5- 9.0MeV

9.0- 9.5MeV

9.5-10.0MeV

10.0-10.5MeV

10.5-11.0MeV

11.0-11.5MeV

11.5-12.0MeV

12.0-12.5MeV

12.5-13.0MeV

13.0-13.5MeV

13.5-14.0MeV

14.0-15.0MeV

15.0-16.0MeV

16.0-20.0MeV

Figure 4.22: The cos θSun distribution of data for each energy bin in the recoil
electron spectrum.

102

after the data set is divided. Their values areΦday = (2.31 ± 0.07(stat.) ± 0.15(sys.)) × 106 cm−2 sec−1 ,Φnight = (2.46 ± 0.07(stat.) ± 0.16(sys.)) × 106 cm−2 sec−1 .
To quantify the possible indication of regeneration back into νe as mentioned in
Section 2.2.2, an asymmetry value between the day and night fluxes can be definedas A = (Φday − Φnight )/( 21 (Φday + Φnight )). The SK-II day-night difference yields
A = −0.063 ± 0.042(stat.) ± 0.037(sys.).
Unfortunately, while the night flux reflects an increase in the νe rate over the
day flux, the combined statistical and systematic errors make the measurement
consistent with zero.
The total flux variation as a function of time, or seasonal variation, for both SK-I
and SK-II solar data is shown in Figure 4.23. Each bin represents 1.5 months and isseen to follow a sinusoidal trend consistent with the expected 1/r2 flux variations
due to the eccentricity of the earth’s orbit around the sun. SK-II has excellent
agreement with SK-I data, thus showing the continuation of the SK solar neutrino
measurement through two phases of the detector.

F C  S A
With the completion of SK-II, the solar neutrino flux measurement of the SuperKamiokande experiment spans an interval of 9.5 years. This closely coincides withthe full period of solar cycle 23. To address any possible correlation of solar neutrino
flux with sun spot number, the SK-I and II flux time variation data are compiledin 1-year bins between 1996 and 2006. The SK-I data set (from 1996 to 2001) istaken from a 5.0 MeV threshold while SK-II is from 7.0 MeV. Errors are statistical
103

6

2

Flux (x10 /cm /s)

4

3

2

1

1998

2000

2002

2004 2006
YEAR

Figure 4.23: Time dependence of the solar neutrino flux. The black points are fromthe 1496-day SK-I data set at a threshold of 5.0 MeV. The blue points are fromthe 791-day SK-II data set at a threshold of 7.0 MeV. The black line represents the
expected 1/r2 flux variations due to the eccentricity of the earth’s orbit around thesun. Errors are statistical only. The absence of data points between SK-I and SK-II
indicates dead time while construction of SK-II was occurring.

only. From 1996 to the end of the SK-II phase in October 2005, the solar neutrino
flux is stable and shows no pattern of correlation with the minima and maximumof solar cycle 23. This is consistent with (and a continuation of) the Kamiokande
measurement and comparison with solar cycle 22 [33], albeit with a greater levelof precision for Super-Kamiokande.

4.8.3

E S

The recoil electron energy spectrum is obtained by dividing the total flux into 17
energy bins ranging from 7.0 to 20.0 MeV. The bin boundaries and flux values arelisted in Table II. Figure 4.25 shows the observed energy spectrum divided by the
expected spectrum without oscillation determined from the BP2004 SSM. The line
through the spectrum represents the average of all bins. This spectrum is valuable
for constraining the parameters governing solar neutrino flavor change.

104

Table 4.1: SK-II observed energy spectra expressed in units of event/kton/year.
The errors in the observed rates are statistical only. The 7.0-7.5 MeV energy bin
is excluded from the day-night analysis. Correction is made for the reduction
efficiencies in Figure 4.14. The expected rates neglecting oscillation are for the
BP2004 SSM flux values. θz is the angle between the z-axis of the detector and thevector from the Sun to the detector.
EnergyObserved rate
Expected r ate
8
ALL
DAY
NIGHT
B
hep
(MeV)
−1 ≤ cos θz ≤ 1 −1 ≤ cos θz ≤ 0 0 < cos θz ≤ 1
7.0 − 7.5
43.7+5.2−−
112.4 0.257
−5.1
+3.6
+5.1
+5.2
7.5 − 8.0
40.0−3.5
36.4−4.9
43.6−5.0
99.1
0.245
+3.5
+3.5
8.0 − 8.5
34.9+2.5
34.4
35.5
85.9
0.231
−2.4
−3.4
−3.4
+2.0
+2.8
+2.8
8.5 − 9.0
30.1−1.9
27.0−2.7
33.0−2.7
73.5
0.215
+2.3
+2.3
9.0 − 9.5
24.5+1.6
23.9
25.0
61.4
0.198
−2.2
−2.2
−1.6
+2.0
+2.0
+1.4
9.5 − 10.0
22.0−1.4
20.7−1.9
23.3−1.9
50.3
0.181
+1.2
+1.7
+1.7
16.6−1.1
15.4−1.6
17.6−1.6
40.7
0.163
10.0 − 10.5
+1.5
+1.5
10.5 − 11.0
13.9+1.0
13.5
14.2
32.1
0.145
−1.0
−1.4
−1.4
+0.9
+1.3
+1.2
11.0 − 11.5
10.3−0.8
11.3−1.2
9.4−1.1
25.3
0.129
+1.00
+1.03
8.06+0.71
7.11
8.96
19.51
0.113
11.5 − 12.0
−0.66
−0.90
−0.94
+0.62
+0.94
+0.86
12.0 − 12.5
6.28−0.58
6.82−0.84
5.79−0.77
14.67 0.098
+0.50
+0.73
+0.70
12.5 − 13.0
4.07−0.45
4.18−0.63
3.97−0.61
10.96 0.084
+0.62
+0.61
13.0 − 13.5
3.32+0.43
2.95
3.66
7.91
0.071
−0.38
−0.53
−0.53
+0.35
+0.57
+0.44
13.5 − 14.0
2.23−0.30
2.95−0.48
1.59−0.35
5.74
0.060
+0.39
+0.60
+0.53
2.77−0.35
2.99−0.51
2.58−0.45
6.90
0.091
14.0 − 15.0
+0.42
+0.45
1.75+0.30
1.37
2.08
3.41
0.063
15.0 − 16.0
−0.26
−0.32
−0.37
+0.27
+0.37
+0.40
16.0 − 20.0
1.37−0.22
1.11−0.28
1.60−0.31
2.52
0.089

105

3

150

2

100

1

50

0

0
1998 2000 2002 2004 2006
YEAR

6

Number of sunspots

200

2

Flux (x10 /cm /s)

4

Figure 4.24: Time variation of the solar neutrino flux overlaid with sun spot number
for solar cycle 23. Errors are statistical only. The SK-I and II 1-year binned solar
flux data gives an agreement of χ2 = 6.11 (52% c.l.) when compared to a straight
line.

4.9

S  S E

A listing of all assigned systematic errors is given in Table 4.2. The second column
is the error when applied to the total flux measurement and the third column(day-night) is the error applied to the day-night difference.
Table 4.3 is the list of energy correlated systematic errors (energy scale, energy
resolution, 8 B spectrum) assigned to each bin in the recoil electron energy spectrum.

106

Data/SSMBP2004

0.8

0.6

0.4

0.2

5

10

15
20
Energy(MeV)

Figure 4.25: Ratio of observed and expected energy spectra. The purple lines
represent a ±1 sigma level of the energy correlated systematic errors. The blackline represents the SK-I 1496-day average and shows agreement with SK-II.

107

Table 4.2: SK-II systematic error of each item in %. Numbers in parentheses are thevalues obtained from calibration data before application to the neutrino flux.
flux
day-night
Energy scale (absolute ±1.4%) +4.2 − 3.9
Energy scale (relative ±0.5%)
±1.5
±0.3
Energy resolution (2.5 %)8
B spectrum±1.9
Trigger efficiency±0.5
st
1 reduction±1.0
nd
2 reduction±3.0
Spallation dead time
±0.4
Gamma cut
±1.0
Vertex shift±1.1
Non-flat background
±0.4
±3.4
±3.0
Angular resolution (6.0%)
Cross section±0.5
Live time
±0.1
±0.1
Total
+6.7 − 6.4±3.7

Table 4.3: SK-II energy correlated systematic errors for each energy bin in %.
8
EnergyB Shape
Energy Scale Energy Res.
7.0 − 7.5
+0.47, −0.54+1.07, −1.02+0.42, −0.41
7.5 − 8.0
+0.52, −0.65+1.71, −1.54+0.36, −0.36+0.64, −0.77+2.34, −2.08+0.27, −0.28
8.0 − 8.5
8.5 − 9.0
+0.82, −0.89+2.96, −2.62+0.15, −0.17
9.0 − 9.5
+1.05, −1.03+3.57, −3.180, −0.02
9.5 − 10.0 +1.30, −1.181+4.17, −3.75+0.15, −0.18+4.77, −4.32+0.36, −0.37
10.0 − 10.5 +1.58, −1.35
10.5 − 11.0 +1.88, −1.53+5.36, −4.90+0.60, −0.57
11.0 − 11.5 +2.19, −1.73+5.95, −5.48+0.87, −0.80
11.5 − 12.0 +2.51, −1.95+6.55, −6.06+1.18, −1.04+7.16, −6.63+1.52, −1.30
12.0 − 12.5 +2.84, −2.19
12.5 − 13.0 +3.18, −2.44+7.78, −7.20+1.89, −1.57
13.0 − 13.5 +3.53, −2.71+8.43, −7.76+2.30, −1.87
13.5 − 14.0 +3.89, −2.98+9.09, −8.31+2.74, −2.20
14.0 − 15.0 +4.44, −3.41+10.15, −9.12 +3.47, −2.76
15.0 − 16.0 +5.17, −3.98 +11.68, −10.16 +4.59, −3.68
16.0 − 20.0 +6.46, −5.19 +15.95, −12.83 +8.18, −7.40

108C 5O A
Since the announcement by Super-Kamiokande of the discovery of neutrino oscillation in 1998, tremendous progress has been made in neutrino physics. Throughthe efforts of Super-Kamiokande, SNO and KamLAND, the solar neutrino problem
appears to be understood, for the first time since its discovery over 30 years ago.
There is now strong evidence that neutrinos have mass and their flavor states mix
with each other. The evidence exists from atmospheric neutrinos, solar neutrinos,
reactor experiments, and long baseline oscillation experiments.
By the methods presented in the previous chapter, solar neutrinos events in
the initial SK-II data set are extracted and the corresponding flux determined.
Although the results from SK-II does not significantly change the experimentally
determined precision of the solar neutrino mixing angle and mass difference, it does
give overwhelmingly consistent results which further solidifies the hypothesis ofoscillating neutrinos. Since this is the second phase of the Super-Kamiokande
experiment, the SK-II oscillation analysis is very similar to that performed earlier.
It is the goal of this chapter to present this analysis and its results in detail.

5.1

A   SK-II χ2

The comparison of observed data rates and SSM rates is done by fitting their energy
spectra in a χ2 analysis. The finite spectra boundaries were chosen based on SK-II’senergy threshold and the limited statistics and energy resolution in the high energy
region. The binning of the spectra is the same as presented in Table 4.1. To begin,109

a simplified version1 of the χ2 is given asχ2 (β, η) =


2
17
X
di − βbi (tan2 θ, ∆m2 ) − ηhi (tan2 θ, ∆m2 )
σ2i

i=1

.

(5.1)

The di is the data spectrum term with statistical error σi . The β and η are freeparameters scaling the SSM-derived 8 B (bi ) and hep (hi ) rates, respectively, allowing
for arbitrary total neutrino rates to be chosen to minimize the χ2 . The SSM rate ismodified for a number of neutrino oscillation parameter sets (tan2 θ, ∆m2 ), each ofwhich has χ2 -minimizing values of β and η. The final step will be to find whichset (or sets) best favors the data spectrum by utilizing the distinctive signaturesdifferent parameter sets produce.
To further clarify, the terms in equation 5.1, di , bi , and hi are in fact ratios of the
SK-II data and oscillated SSM MC rates over the unoscillated SSM MC rate. This isshown as
di =
bi =

Di
B i + Hi
Bosci

B i + Hi
Hiosc
.
hi =
B i + Hi

(5.2)
(5.3)
(5.4)

The di equation is the spectrum shown in Figure 4.25 and Di is listed in Table 4.1
as the observed total rate. For MC, whereas Bi and Hi are the expected rates inTable 4.1, Boscand Hiosc must be determined from neutrino survival probabilities
iand are hence known as oscillated rate predictions.
1

Additional considerations relating to systematic errors as well as time variation effects will be
discussed later.

1105.1.1O R PThe SSM MC rate was obtained in the previous chapter to show how a SSM solar
neutrino flux would appear in SK-II and to compare with actual observation. ThisMC rate is transformed by including the probability of electron neutrino survival.
The electron neutrino survival probability is given by Pνe (Eν ) and incorporated into
Equation 4.14:
Z ∞Z ∞
r(E) = ΦSSM Ne00 dσνedσνµ,τ
Pνe +(1 − Pνe ) .
dEν dEe φ(Eν )R(Ee , E)
dEe
dEe

(5.5)

The dEν and dEe integration is taken over all true neutrino and recoil electronenergies. E is the recoil electron energy in the SK-II tank. The no-oscillation limit
occurs when when Pνe (Eν ) = 1. For other probabilities, the σνµ,τ cross section must
be considered due to νµ and ντ interacting in SK-II (albeit with a much lower crosssection than νe ).
The explicit form of Pνe (Eν ) depends on whether or not matter effects play asignificant role in an electron neutrino’s journey from the core of the Sun to the
SK-II tank. Since this analysis considers neutrinos of mass differences equal to andgreater than 10−9 eV2 . Vacuum oscillations can be ignored. Oscillation must thenadjust for matter effects prompting P(Eν ) to have a matter effective probability (seeSection 1.3.3).
However, the effective mass squared difference and mixing angle assume constant mass densities. From the profiles in Figure 5.2, it is seen that densities in the
Sun and Earth vary greatly with the core-surface distance. This must be taken into
account and is done so numerically by adjusting the value A to correspond to thecorrect solar density at r. Since it is assumed that the density in the Sun varies only
radially and is otherwise isotropic when viewed from the neutrino-producing core,
the probability of flavor change does not vary with time. The path through the111

Earth is accounted for similarly but is slightly more complicated due to the Earthaxis rotation. The path of the neutrino through the Earth to SK transverses different
chords within the Earth depending on the time of day. A continuous model of the
myriad of different paths a neutrino might take is not developed for this analysis.
Instead, a system of binning based on the SK-II detector’s zenith angle with the
Sun is adopted. 1000 bins are utilized for propagation through the Earth (nightwith 0 < Θz < 1) and 1 bin is used when Earth matter effects do not occur (day with
−1 < Θz < 0). P(Eν ) becomes P(Eν , cos Θz ) and can be determined based on which
bin the neutrino would be traveling.
Since Di represents the total day and night combined data rate for the i-th energybin, Boscand Hiosc need to be expressed as a 24-hour rate as well. Taking the newlyidefined P(Eν , z), where z = cos Θz , and placing it in the oscillated rate prediction ofequation 5.5, Boscand Hiosc can be obtained for SK-II observed energy E:i
B (E) =osc1001
X
τ(z j )
j=1

τtot

8B

· ri (E, z j ),

H

osc

(E) =

1001
X
τ(z j )
j=1

τtot

hep

· ri (E, z j ),(5.6)

where τ(z j ) equals the live time in the jth zenith angle bin and is obtained from LE
and SLE trigger live time data. Figure 5.1 shows the 1000-bin night live time. τtot is
the total live time of the SK-II experiment (791 days). The 8 B and hep superscripts
on the un-normalized rates distinguish which SSM flux φ(Eν ) is used to computethe rates. The oscillated SSM rates in Equation 5.6 are then binned for the SK-II
energy spectrum by
Bosc
=
i

Z Ehigh,i

Hiosc =

oscdE · B (E),
Elow,i

Z Ehigh,idE · Hosc (E)Elow,ifor the i-th energy bin with boundaries Elow,i and Ehigh,i .

112

(5.7)

Oscillated MC/MC

Days

1
0.8
0.6
0.4

7-7.5 MeV

0.48

7.5-8 MeV
8-8.5 MeV
8.5-9 MeV
9-9.5 MeV
9.5-10 MeV
10-10.5 MeV
10.5-11 MeV
11-11.5 MeV
11.5-12 MeV
12-12.5 MeV
12.5-13 MeV
13-13.5 MeV
13.5-14 MeV

0.47

0.46

0.2
0

0.49

0

0.2

0.4

0.6

0.8
1
cos(ΘZ)

0.45

-1

-0.5

14-15 MeV
15-16 MeV
16-20 MeV

0

0.5

1

-0.5

0

0.5

1

cos(ΘZ)e density in mol/cm

3

Figure 5.1: Left: The number of days in the 791-day sample of SK-II that occur ineach of the 1000 bins of nighttime solar zenith angle. Right: The predicted rates foreach of the 1001 (added one for day) bins of solar zenith angle.
2

10
10
1

-1

-

10
-2

density in g/cm3

10

14
12
10
8
6
4
2
0

0.1

0.2

Core
0

0.2

0.3

0.4

0.5

0.6

Mantle

0.7
0.8
0.9
1
distance/sun radius

Mantle

0.4
0.6
0.8
0
distance/earth radius

0.2

0.4

Core
0.6

0.8
1
cos ΘzFigure 5.2: Top: The electron number density of the Sun as a function of distanceto the surface. Bottom left: The mass density of the Earth as a function of distanceto the surface. Bottom right: The minimum (dashed horizontal line), the average(solid line), and the maximum (upper dashed line) mass density a neutrino passes
while traveling through zenith angle Θz .

113

5.1.2

SK-II S E T

The χ2 of equation 5.1 only includes statistical errors. Therefore, it is necessary to
add consideration for systematic uncertainties. These come in two types: energy
correlated and energy uncorrelated.

E C S E
The energy correlated (EC) systematic uncertainties consist of errors on the energy
scale, energy resolution, and the theoretical 8 B neutrino spectrum (the assignmentof these errors is given in Chapter 4). Since these uncertainties directly impact
the shape of the SK-II energy spectrum, the χ2 must reflect these possible changes
when comparing Monte Carlo to data.
The assigned EC systematic uncertainties in Table 4.3 are applied to the spectrum
as multiplicative energy shape factors to reflect an actual one sigma shift. However,since the data may better agree with a lesser or greater shift from the one sigma
assigned value, the factors are varied until the χ2 gives a minimum value.
Written out explicitly, the energy shape factors are
fx (Ei , δx ) = (1 + δx ∗ ±x (Ei ))−1
where x denotes the type of EC error (B for 8 B spectrum shape, S for energy scale,and R for energy resolution), δx is a unit-less parameter that is used to vary theassigned error, and ±x (Ei ) is the assigned percent error from Table 4.3. When δx
is less than 0, the absolute value of the negative error |− | is used. These form a
combined energy shape factor
f (Ei , δB , δS , δR ) = fB (Ei , δB ) × fS (Ei , δS ) × fR (Ei , δR ),114

which is used to modify the predicted combined rate ratio as
β · bi + η · hi × f (Ei , δB , δS , δR ).

(5.8)

The χ2 becomesχ2 (β, η, δB , δS , δR ) =


2
~
17
X
di − (βbi + ηhi ) × f (Ei , δ)
σ2i

i=1

+ (δB )2 + (δS )2 + (δR )2 .

(5.9)

The δx factors are constrained to 0 to assure sensible sigma-level values when theχ2 is minimized. The σi remain statistical uncertainties only. See Figure 5.3 for theOscillated/Unoscillated

effect the energy shape factors have on the predicted rate (Equation 5.8).

0.55
0.5
0.45
0.4
0.35
0.3

10

15

20
Energy(MeV)

Figure 5.3: The predicted oscillated-unoscillated spectrum ratio (black line) andshifts due to the estimated energy correlated systematic errors (colored lines). Red
is 8 B shape, blue is energy scale, green is energy resolution, and the dotted line isall 3 errors combined. The oscillated rate reflects that of a typical LMA solution−(tan2 θ = 0.52, ∆m2 = 6.3 × 10 5 eV2 ). All shifts are one sigma.

The f energy shape factor is essentially a correction to reflect possible uncertainties in the data spectrum di . However, rather than modify the data according to the115assigned EC systematic errors, the predicted combined rate is inversely modified
thereby retaining the χ2 formulation and treating the δx s as minimizing parameters.
The minimization itself is difficult to perform analytically in full so the β and η parameters are calculated analytically at each point of a three-tiered iteration of the δxparameters. This is done coarsely at first (each δx is independently incremented by
0.5 sigma) and then repeated in a smaller, minimally localized area using a simplex~
search. The minimum χ2 is then identified for a unique combination of β, η, and δ.
E U S E
The energy uncorrelated (EU) systematic uncertainties are those which are notreflected by explicit energy related errors. Mainly, they are comprised of uncertainties in event reconstruction, extraction, and simulation and must be accounted
for in the oscillation analysis.
Table 4.2 list the EU systematic errors for the total flux measurement in SKII. A complete treatment of these systematics would include not only an error
estimate for the total flux but also an estimate for each individual energy bin as
is done with the EC systematic uncertainties. For use in the χ2 , they would thenbe combined in quadrature with the energy spectrum’s statistical uncertainties
and σi in Equation 5.9 would refer to the combined statistical-systematic errorsof the SK-II spectral bins. This is the method applied to the SK-I analysis but isomitted for SK-II. This is due to the EU errors having negligible contributions whencompared to the large uncertainty in the SK-II energy scale (an EC error which is
notably small for SK-I). In the observed spectrum, the energy scale systematic error
increases with increasing energy leaving SK-II greatly affected not only due to thelarge 1.4% sigma total value (SK-I: 0.64%), but also due to the analysis thresholdof 7.0 MeV (SK-I: 5.0 MeV). Below 7.0 MeV, the energy scale error would not havesuch a dominating effect and thus EU systematics would need to be included. But
116

in SK-II, energies below 7.0 MeV are not analyzed.
However, the total flux EU systematic uncertainty is applied. This is due to the8

SK-II spectrum constraining the combined rate (βφ B + ηφhep ) in the χ2 . To accountfor the total flux EU systematic error, it is helpful to start by Taylor expanding2
equation 5.1 (the energy shape factors will be added again shortly) to second orderaround the minimum, βm and ηm . Written in terms of the curvature matrix, it
becomes3χ (β, η) = χ2m (βm , ηm ) +2!
!Tβ − βmβ − βmC0,η − ηmη − ηm

(5.10)

where the curvature matrix C0 is defined as
 2
b
17 
X
 i2
 σi
C0 =

 bi ·hii=1 2bi ·hi
σ2i
h2i

σi

σ2i




 .



It is known in this formulation that the curvature matrix is proportional to theinverse of the errors of the fitting parameters β and η. Therefore it is possible toscale the curvature matrix by αsys to allow the addition of the total EU systematic
error on the total flux scale factors. αsys is defined asαsys =andσ20
σ20 + σ2sys

,

where

χ2α (β, η) = χ2m (βm , ηm ) +

σ20 =

17
X
1
2
σstat,ii=1!T!
β − βmβ − βmαsys · C0.
η − ηmη − ηm

(5.11)

It can be seen in the above equation that the minimum does not change (the second
2

The Taylor series of a quadratic function is exact and not an approximation due to the third andhigher derivatives vanishing.
3
Since setting the partial first derivatives of χ2 with respect to β and η to zero is required to
calculate the parameters’ minimizing values, the first order term in the Taylor expansion, wheresuch first derivatives would appear, is omitted.

117

order term vanishes at β = βm ) but β and η are allowed a greater range that directlyreflects the increased error due to the total flux systematic uncertainty. Adding theenergy shape factors, the total χ2 for the SK-II energy spectrum shape is χ2spec (β, η) = Min χ2α β, η, δB , δS , δR + (δB )2 + (δS )2 + (δR )2 .

(5.12)

This treatment can also be found in [34].

5.1.3

T-V A

So far, the χ2 comparison of data and MC has been confined to the energy spectrum
shape. While this is sufficient to exclude some parameter set candidates, it is
excluding the phenomenon of matter effects which can provide a more powerfultool when investigating flavor change.
The likelihood function that was used to extract the solar signal from background(Equation 4.15) can be modified with a scaling function that reflects the deviationfrom the average number of events arising from time variations. Using a suitable
substitution for the zenith angle to reflect its time dependence, t ≡ Θz , the oscillated
rate predictions of Section 5.1.1 can be used to form the ratio ri (t j )/rave
where t j isithe time that corresponds to the j-th event and the denominator is the average overall zenith bins. This then scales the probability density function of solar neutrino
events si, j giving a likelihood of

L =e−Σi (Bi +S)

Nbin Yni Yri (t j ) Bi · bi j + S · Yi · si j · ave .
ri
i=1 j=1(5.13)

With a likelihood function describing solar neutrinos with a 24-hour period time
variation, it is possible to test the increase in likelihood when compared to neutrinos
with the time variation averaged (ri /rave= 1, the original likelihood). This is donei118

by a likelihood ratio test and then casting it in terms of a change in χ2 values:




L
−2 log= −2 log L − log Lave = χ2tv − χ2ave = ∆χ2tv
Lave
Before combining with the SK-II spectrum χ2spec , it is noted that the oscillated
rate predictions are altered by the flux and energy shape factors coming from the~ in χ2 andχ2spec fit. This results in the fitted rate predictions (βbi + ηhi ) × f (Ei , δ)
spec
8
~ in the likelihood differing only by the exclusion of energy(βr B (t j )+ηrhep (t j ))× f (Ei , δ)binning and a sum of all zenith angle bins in the time-variation rate predictions.
The full SK-II χ2 becomes:
χ2SK-II = χ2spec + ∆χ2tv .

5.2(5.14)

R

In presenting the SK-II χ2 fit results, two initial distinctions will be made: the recoil
electron spectrum fit of equation 5.12 and the time variation fit of equation 5.14. The
purpose for differentiating the fits is to show how the recoil spectrum shape alone
can favor certain oscillation parameters. This is purely the effect of the energy
dependence on neutrino survival probability and is heavily dependent on the
mixing angle and mass-squared difference. With the time variation fit, disfavoredregions will be seen based on Earth’s matter effects prompting νe regeneration in
the Earth.
To remind the reader, the χ2 fits in the equations referenced above have noconstraints on the β and η parameters. After minimizing these parameters for a
specified oscillation parameter set, tan2 θ and ∆m2 are systematically incremented
in value until 90,601 sets have been minimized. The χ2 maps are therefore twodimensional depending on tan2 θ and ∆m2 and contours are plotted for minimum
χ2 values corresponding up to a 95% confidence level (c.l.).
119

Lastly, each contour χ2 plot contains three fitted maps: SK-II (violet), SK-I (thinblack line) and SK-I and SK-II combined fit (light blue). In the SK-Combined fit,
two separate χ2 ’s are minimized together sharing the flux factors β and η. Also,
the 8 B spectrum shape systematic error is common between the two. All othersystematics, data, and MCs are not correlated.

5.2.1

SK-II S & T-V FIn the spectrum fit, SK gives exclusion areas where the oscillation parameters are
not favored at 95% c.l. (Figure 5.4, left side). While the significance of the SK-II
exclusion is smaller than that of SK-I, it does follow the same trend and can beseen on the right side of Figure 5.4 when SK-II’s exclusion probability is loweredto the 68.3% c.l. The loss of sensitivity is due to both the combined effects of larger
statistical errors, larger EC errors, and the absence of the four low-energy bins
between 5.0 and 7.0 MeV. This is evident in the values of the χ2 itself where lessdata and larger denominators bring the SK-II value down relative to SK-I. At theirrespective minimum χ2 values, SK-II is almost 17 units less than SK-I.
SK-II, while excluding fewer oscillation candidates, does offer slight increases
in exclusion power in the SK-Combined fit, most notably along tan2 θ at ∆m2 ≈5 × 10−4 eV2 and around tan2 θ ≈ 0.1 and ∆m2 ≈ 10−6 eV2 .
By adding the time variation term ∆χ2tv to the χ2 , greater exclusion is obtained.
Also, the addition of SK-II data to SK-I continues to show improvement. Contours
at 95% c.l. are shown in Figure 5.5. These results, along with the spectrum fit, areindependent of third party data.
The SK-II flux factors β and η have minimizing values that correspond to 0.37 ×
hep

ΦSSM and 50.78 × ΦSSM for the 8 B and hep fluxes respectively. The large valueof hep is due to η compensating the low β and for the higher 16-20 MeV bin in
the Data/SSM spectrum. The energy shape factors give best fit sigma scalings of
120

-3

2
2

∆m in eV2

∆m in eV

2

-3

10

-4

10

-5

-4

10

-5

10

10

-6

-6

10

10

-7

-7

10

10

-8

-8

10

10
νe→νµ/τ (95%C.L.)

-9

10

10

-4

10

-3

10

-2

10

νe→νµ/τ (68.3%C.L.)

-9

-1

10

1

10

2

10
10
2
tan (Θ)

-4

10

-3

10

-2

10

-1

10

1

2

10
10
2
tan (Θ)Figure 5.4: The minimum χ2 map for various oscillation parameters tan2 θ and ∆m2 .
The left side shows at 95% c.l. SK-II (violet), SK-I (black line) and SK-Combined(light blue) exclusion areas. The right side shows SK-II only at 68.3% c.l. to illustrate
the consistency, albeit less sensitivity, of the SK-II exclusion when compared to SK-I
at 95% c.l.

121

4.1 × 10−3 , −6.3 × 10−2 , and 5.7 × 10−2 for the 8 B shape, energy scale, and energy
resolution systematic errors.

-3

2
2

∆m in eV2

∆m in eV

2

-3

10

-4

10

-5

-4

10

-5

10

10

-6

-6

10

10

-7

-7

10

10

-8

-8

10

10
νe→νµ/τ (95%C.L.)

-9

10

10

-4

10

-3

10

-2

10

νe→νµ/τ (68.3%C.L.)

-9

-1

10

1

10

2

10
10
2
tan (Θ)

-4

10

-3

10

-2

10

-1

10

1

2

10
10
2
tan (Θ)Figure 5.5: The spectrum and time variation fit of SK data. The contours are SK-II(violet), SK-I (black line) and SK-Combined (light blue) and are at 95% c.l. (leftside) and SK-II only at 68.3% c.l. (right side)

5.2.2

SK-II R C F

To render regions in the χ2 plane where data favors various oscillation scenarios,
it is necessary to constrain the total rate of 8 B neutrinos during fitting. Previously
when plotting for excluded regions, the 8 B rate factor β was free to take any (positive) value in order to minimize the difference between data and MC oscillated
predictions. For the rate constrained fit, the rate factor β is tied to the total rate of
all active flavors of neutrinos measured via neutral current (NC) interactions on
Earth .
122

First, to constrain β, a term of (φcon. − β)2 /σ2φ is added to the χ2 of equation 5.14.
The value of φcon. and its accompanying error is the NC value coming from themeasurement of SNO [17]. It is again divided by the BP2004 flux value and is
0.85 ± 0.07.
Figure 5.6 shows contours for both the SSM and NC constrained fits. It favors
the LMA and LOW regions in the SK-I and SK-Combined contours. This is dueto the relative flatness of the LMA and LOW spectral distortions. SK-II alone is,again, lacking the sensitivity of SK-I.
It is important to mention that the comparison between the SK-I and SKCombined fits. Unlike with the spectral and day-night exclusion fits, the rate
constrained χ2 of SK-Combined does not offer greater precision despite the inclusion of SK-II data. The LMA and low regions essentially remain unchanged.
The SK-Combined best fit parameters are tan2 θ = 0.55 and ∆m2 = 6.6 × 10−5 eV2 .
This is in close agreement with the SK-I result [18] of tan2 θ = 0.52 and ∆m2 =6.3 × 10−5 eV2 . The fitted 8 B flux is 0.84 × ΦSSM .

5.3

C  O S E

5.3.1

S N O

SNO’s published flux and day-night asymmetry values are used to further constrain the allowed oscillation parameter region. The data include the NC and CC
flux measurements from both the 306-day D2 O phase [16] and the 391-day salt
phase [17]. Their values are listed in Table 2.2. The SNO analysis presented here isa total rate comparison of data and prediction and does not include any informationof the SNO spectra.
MC oscillated rate predictions that reflect the SNO energy resolution and uniquescattering target, deuterium, are needed for comparison with the data. The oscil123

2∆m in eV

2

-3

10

-4

10

-5

10

-6

10

-7

10

-8

10

νe→νµ/τ (95%C.L.)

-9

10

-4

10

-3

10

-2

10

-1

10

1

2

10
10
2
tan (Θ)Figure 5.6: The spectrum and time variation fit of SK data with the 8 B flux constrained to the SNO Phase II NC measurement. The contours are SK-II (violet),SK-I (black line) and SK-Combined (light blue) and are at 95% c.l.

124lated rate predictions are formed in the same manner as SK-II (Equation 5.5) but
with the substitution of the neutrino-electron differential cross section with theneutrino-deuteron differential cross section. Also, the resolution function R(Ee , E)
is given analytically for SNO and isR(Te , T) = √(Te − T)2
exp −
,
2σ2T
2πσT
1

(5.15)

where a the change from total energy E to kinetic energy T is taken into accountwhen integrating with the differential cross section. The value σT differs between
Phase I and Phase II and isp
σT (Te ) = −0.0684 + 0.331 Te + 0.0425Tep
σT (Te ) = −0.131 + 0.383 Te + 0.03731Te

Phase I,
Phase II.

The rate predictions are analogous to the rates in 5.7 but are integrated over allenergies to form a total rate with no spectral information. They are then divided bytheir unoscillated counterparts and are labels BCC and HCC . They are differentiatedfrom the SK-II spectrum predictions bi and hi by the uppercase lettering. The diratio has as its SNO equivalent value
DCC = φCC /φSSM ,where φSSM is the Standard Solar Model flux of 5.79×106 cm−2 s−1 and φCC is thevalue listed in Table 2.2. No MC predictions are made to compare with the NC flux
since the ratio DNC = φNC /φSSM is directly reflected by the flux factor β.
With these terms, a SNO CC-NC χ2 can be formed using the flux factors β and125η for minimization. Written in matrix form,→−→−
χ2SNO (β, η) = φ T V −1 φ(5.16)

where
DCC − βBCC − ηHCC→−
φ=
DNC − βb − ηh

!



2 
 σ2
 CC σct  .
and V −1 =  σ2 σ2 ct
NC

The constants b and h are the SSM ratios of 8 B and hep fluxes over the total flux andare b = 0.9986 and h = 0.0014. The σct matrix element is the cross term correlationof the CC and NC fluxes and can be found from
σ2r = σ2CC









∂r 2
∂r 2
∂r
∂r
2
2
+ σNC
+ 2σct,∂φCC∂φNC∂φCC ∂φNC(5.17)

where r ± σr is the published CC/NC ratio with uncertainties and σCC and σNC are
the published uncertainties for the CC and NC fluxes.
The final SNO χ2 includes martix terms for both phases of SNO. Also, the Phase
I day-night asymmetry (7.0 ± 5.1%) is added as a constraint. The SNO χ2 isχ2SNO (β, η) = χ2SNO,I (β, η) + χ2SNO,II (β, η) +(ADNCC − ADNpred (β, η))2
σ2ADN

,

(5.18)and is added to the SK-I and SK-II χ2 . The resulting contour is shown in Figure 5.7.
It shows that with the addition of the CC fluxes and day-night asymmetry of SNO,
the LOW solution is no longer favored at 95% c.l. and only LMA remains.

5.3.2

R E & KLAND

The Homestake, GALLEX/GNO, and SAGE [36] rates can also be added to the χ2 toform a ”global” contour encompassing 5 different solar neutrino experiments. The
126

2∆m in eV

2

-3

10

-4

10

-5

10

-6

10

-7

10

-8

10

νe→νµ/τ (95%C.L.)

-9

10

-4

10

-3

10

-2

10

-1

10

1

2

10
10
2
tan (Θ)Figure 5.7: The SK-I, SK-II, and SNO combined contour at 95% c.l.

127

radiochemical rates are constrained to the best fit β and η factors of the SK-SNO fit.
The parameter set that corresponds to a minimum χ2 is found to be tan2 θ = 0.40
and ∆m2 = 6.03 × 10−5 eV2 and is in the LMA region of solutions.
The KamLAND [37] experiment, a liquid scintillator experiment measuring theinverse beta decay process ν̄e + p → e+ + n, was initially constructed to verify
the solar measurements’ favored LMA solution. Detecting antineutrinos fromsurrounding nuclear reactors in Japan, KamLAND is able to measure ∆m2 withgreatly improved precision. The best fit contour is shown overlaid with the global
solar contour in Figure 5.8 and does confirm the LMA as the most likely parameterregion for quantifying νe → νµ,τ oscillation.

5.4

C

The conclusion of Super-Kamiokande-II has brought measurements consistent withprevious solar experiments. The 8 B flux in SK-II is (2.38 ± 0.05(stat.)+0.16(sys.)) ×−0.15
106 cm−2 sec−1 and a day-night asymmetry value was observed to be −0.063 ±0.042(stat.) ± 0.037(sys.) which is consistent with zero. SK-II has brought the total
SK time-dependent flux measurement to a length of 9.5 years and this measurement
is compared with solar activity in solar cycle 23 resulting in no strong correlation.
In the combined SK-I and SK-II global oscillation analysis, the best fit is foundto favor the LMA region at tan2 θ = 0.40 and ∆m2 = 6.03 × 10−5 eV2 , in excellent
agreement with previous solar neutrino oscillation measurements. Although noadditional precision in the measurements was gained, reconstruction tools andreduction cuts that were developed to compensate for SK-II’s loss of photocathodecoverage are now being used to lower the threshold of the SK-III analysis. SK-II, in
short, was the catalyst for improved research techniques at the Super-Kamiokande
experiment.

128

2
2

∆m in eV

2 -4
x10

1

0

νe→νµ/τ (95%C.L.)
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 5.8: The SK-I, SK-II, SNO, and radiochemical rate combined contour at 95%
c.l. The green overlay is KamLAND.

129

B
[1] Raymond Davis’s autobiography located at
http://nobelprize.org/nobel prizes/physics/laureates/2002/davisautobio.html.
[2] J.N. Bahcall, arXiv:hep-ph/9711358v1[3] B. Kayser in Neutrino Mass, eds. G. Altarelli and K. Winter (Springer-Verlag,
Germany, 2003), p. 5.
[4] B. Kayser, talk at NuFact Summer School, Tokyo Metropolitan University,
Japan, July 22, 2004.
[5] P. Langacker, Phys Rep. 72, 185 (1981).
[6] C. Giunti and C.W. Kim, Fundamentals of Neutrino Physics and Astrophysics,
Oxford University Press, 2007.
[7] S.P. Mikheyev and A.Y. Smirnov, Sov. J. Nucl. Phys. 42 (1985) 913;
L. Wolfenstein, Phys. Rev. D17 (1978) 2369.
[8] J.N. Bahcall, arXiv:astro-ph/0010346v1.
[9] J.N. Bahcall and M.H. Pinsonneault, Phys. Rev. Lett. 92 (2004) 121301.
[10] Y. Koshio, Study of Solar Neutrinos at Super Kamiokande, Ph.D. thesis, Universityof Tokyo (1998).
[11] K. France, review of standard solar models located at
http://www.pha.jhu.edu/ france/PAPERS/solmodel.pdf.
[12] M. Fukugita and T. Yanagida, Physics of Neutrinos and Applications to Astrophysics, Springer-Verlag, 2003.
[13] J.N. Bahcall and Andrew Ulmer, Phys. Rev. D53 (1996) 4202.
[14] S. Fukuda et al., Phys. Rev. Lett. 86 (2001) 5651-5655.
[15] S. Fukuda et al., Phys. Rev. Lett. B539 (2002) 179-187.
[16] B. Aharmim et al., arXiv:nucl-ex/0610020v1.
[17] S.N.Ahmed et al., Phys. Rev. Lett. 92 (2004) 181301.
[18] J. Hosaka et al., Phys. Rev. D73 (2006) 112001.
[19] J.H. Bahcall and C. Pena-Garay, arXiv:hep-ph/0404061v1.
[20] J.H. Bahcall, arXiv:hep-ph/0412068v1
130

[21] S. Fukuda et al., Nucl. Instr. Meth. A501 (2003) 418.
[22] First report on the Super-Kamiokande accident at
http://www-sk.icrr.u-tokyo.ac.jp/cause-committee/index-e.html/[23] Second report on the Super-Kamiokande accident at
http://nngroup.physics.sunysb.edu/.
[24] T. Kato, Tau Neutrino Appearance via Neutrino Oscillations in Atmospheric Neutrinos, Ph.D. thesis, Stony Brook University (2007).
[25] H. Nishihama, Development of Flash ADC Modules for Super-Kamiokande, Masters Thesis, Tokyo Institute of Technology (2002).
[26] M.Nakahata, et al. Nucl. Instr. Meth. A421 (1999) 113.
[27] E.Blaufuss, et al. Nucl. Instr. Meth. A458 (2001) 636.
[28] C.E. Ortiz et al., Phys. Rev. Lett. 85 (2000) 2909.
[29] J.N. Bahcall, M. Kamionkowski, and A. Sirlin, Phys. Rev. D51 (1995) 6146-6158.
[30] J.N.Bahcall et al., Phys. Rev. C54 (1996) 411.
[31] J.N. Bahcall, M.H. Pinsonneault, and G.J. Wasserburg, Rev. Mod. Phys. 67(1995) 781.
[32] R.M.Pope and E.S.Fry, Appl. Opt. 36, 8710-8723 (1997).
[33] Y. Fukuda, et al., Phys. Rev. Lett. 77 (1996) 1683.
[34] M.B.Smy, arXiv:hep-ex/0202020.
[35] Q.R.Ahmed et al., Phys. Rev. Lett. 89 (2002) 011301; Q.R.Ahmed et al., Phys.
Rev. Lett. 89 (2002) 011302.
[36] B.T.Cleveland et al., Astrophys. J. 496 (1998) 505; J.N.Abdurashitov et al., Phys.
Rev. C60 (1999) 055801; M.Altmann et al., Phys. Lett. B490 (2000) 16; alsopresented at Neutrino 2004 by C.Cattadori.
[37] T.Araki et al., Phys. Rev. Lett. 94 (2005) 081801.

131