UNIVERSITY OF CALIFORNIA,
IRVINEFirst Direct Evidence for Matter Enhanced Neutrino Oscillation, Using Super-Kamiokande
Solar Neutrino Data
DISSERTATION

submitted in partial satisfaction of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
in Physics

byAndrew Lee Renshaw

Dissertation Committee:
Professor Henry W. Sobel, Chair
Professor Steven Barwick
Professor Jonas Schultz2013c 2013 Andrew Lee Renshaw

Dedication

To my amazing wife Lindsey, and precious daughters Annabelle and Zoey, whom I love sovery much. They stood behind me, through the thick and the thin, and supported me inevery way they could. Here’s to closing one chapter of our lives, and to the opening of anew one. And to my mom and dad. I could not have accomplished so much without theirsupport, and not just financially, but in the form of continuous love and encouragement.
They made it possible for me to dedicate all of my working time to Physics. And to theloving support given to me from my brother, sister and grandmothers. All of their help andsupport also eased the grueling process of gradschool.

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Table of Contents
Page
List of Figuresvii

List of Tablesxv

Acknowledgments

xvii

Curriculum Vitae

xviii

Abstract of the Dissertationxx

1 Introduction to Neutrinos
1.1 History of the Neutrino . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Neutrinos in the Standard Model of Particle Physics . . . . . . . . . . . . . .
1.3 Neutrino Production and Oscillation . . . . . . . . . . . . . . . . . . . . . .
1.4 Neutrino Flavour Oscillation in Vacuum . . . . . . . . . . . . . . . . . . . .
1.5 Neutrino Flavour Oscillation in Matter . . . . . . . . . . . . . . . . . . . . .

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2 Solar Neutrinos
2.1 The Standard Solar Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 The Solar Neutrino Problem . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Solar Neutrino Flavour Oscillation . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Solar Neutrino Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 The Homestake Experiment . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 The Kamiokande Experiment . . . . . . . . . . . . . . . . . . . . . .
2.4.3 The Gallium Experiments . . . . . . . . . . . . . . . . . . . . . . . .
2.4.4 The Super-Kamiokande Experiment . . . . . . . . . . . . . . . . . . .
2.4.5 The Sudbury Neutrino Observatory Experiment . . . . . . . . . . . .
2.4.6 The Borexino Experiment . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Current Neutrino Oscillation Parameter Results . . . . . . . . . . . . . . . .
2.5.1 Results from Solar Neutrino Experiments . . . . . . . . . . . . . . . .
2.5.2 The KamLAND Experiment . . . . . . . . . . . . . . . . . . . . . . .
2.5.3 Other Neutrino Oscillation Mixing Parameters . . . . . . . . . . . . .
2.6 Solar Neutrino Measurements in this Thesis . . . . . . . . . . . . . . . . . .

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3 The Super-Kamiokande Experiment
3.1 Experimental Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Solar Neutrino Detection Inside Super-Kamiokande . . . . . . . . . . . . . .
3.3 The Super-Kamiokande Detector . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Detector Tank and Volumes . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Water Purification System . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Radon-Reduced Air Systems . . . . . . . . . . . . . . . . . . . . . . .

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3.3.4 Photomultiplier Tubes . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.5 Data Acquisition System . . . . . . . . . . . . . . . . . . . . . . . . .
Phases of Super-Kamiokande . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Super-Kamiokande Detector Calibration
4.1 PMT Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Absolute Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Relative Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.3 Quantum Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.4 Timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Water Transparency Measurements . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Measurement by Laser Light Data . . . . . . . . . . . . . . . . . . . .
4.2.2 Time Dependence by Decay-e Energy Measurements . . . . . . . . .
4.2.3 Position Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Absolute Energy Scale Determination . . . . . . . . . . . . . . . . . . . . . .
4.3.1 LINAC Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 DT Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.3 Energy Scale Systematic Uncertainty . . . . . . . . . . . . . . . . . .

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3.4

5 Event Reconstruction and Simulation
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5.1 Vertex Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2 Direction Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.3 Energy Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.4 Event Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6 Data Reduction
106
6.1 Run Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.2 Pre-Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.3 Solar Neutrino Analysis Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.3.1 Total Number of Hit PMTs . . . . . . . . . . . . . . . . . . . . . . . 110
6.3.2 Calibration Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.3.3 Flasher Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.3.4 Time Difference Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.3.5 OD Event Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.3.6 Event Quality Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.3.7 Spallation Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.3.8 Hit Pattern Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.3.9 External Event Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.3.10 16 N Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.3.11 Small Hit Cluster Event Cut . . . . . . . . . . . . . . . . . . . . . . . 121
6.3.12 Tight Fiducial Volume Cut . . . . . . . . . . . . . . . . . . . . . . . . 122
6.4 Summary of the Reduction Steps . . . . . . . . . . . . . . . . . . . . . . . . 124
6.5 Multiple Scattering Goodness . . . . . . . . . . . . . . . . . . . . . . . . . . 126

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7 Solar Neutrino Flux Analysis
128
7.1 Signal Extraction Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
7.2 Systematic Uncertainty on the Extracted Solar Neutrino Flux . . . . . . . . 132
7.2.1 Energy Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
7.2.2 8 B Neutrino Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 134
7.2.3 Signal Extraction Method . . . . . . . . . . . . . . . . . . . . . . . . 135
7.2.4 Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.2.5 Angular Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
7.2.6 Trigger Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.2.7 Background Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.2.8 Reduction Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.2.9 Summary of the Systematic Uncertainty on the Extracted Flux . . . 142
7.3 Extracted Solar Neutrino Flux . . . . . . . . . . . . . . . . . . . . . . . . . . 143
8 Solar Neutrino Spectrum Analysis
149
8.1 Solar Neutrino Spectrum Extraction . . . . . . . . . . . . . . . . . . . . . . 149
8.2 Systematic Uncertainty of the Solar Neutrino Energy Spectrum . . . . . . . 152
8.2.1 Energy Bin Correlated Systematic Uncertainties . . . . . . . . . . . . 152
8.2.2 Energy Bin Uncorrelated Systematic Uncertainties . . . . . . . . . . . 153
8.3 Solar Neutrino Spectrum Results . . . . . . . . . . . . . . . . . . . . . . . . 154
8.4 SK-I/II/III/IV Combined Spectrum Analysis . . . . . . . . . . . . . . . . . . 156
8.5 SK+SNO Spectrum Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
9 Solar Neutrino Day/Night Flux Asymmetry Analysis
169
9.1 The Solar Neutrino Day/Night Flux Asymmetry . . . . . . . . . . . . . . . . 169
9.2 The Solar Neutrino Day/Night Amplitude Fit . . . . . . . . . . . . . . . . . 170
9.3 Solar Neutrino Day/Night Flux Asymmetry Systematic Uncertainty . . . . . 176
9.3.1 Energy Scale Systematic Uncertainty . . . . . . . . . . . . . . . . . . 176
9.3.2 Energy Resolution Function Systematic Uncertainty . . . . . . . . . . 177
9.3.3 Background Shape Systematic Uncertainty . . . . . . . . . . . . . . . 177
9.3.4 External Event Cut Systematic Uncertainty . . . . . . . . . . . . . . 178
9.3.5 Earth Model Systematic Uncertainty . . . . . . . . . . . . . . . . . . 179
9.3.6 Summary of the Systematic Uncertainty . . . . . . . . . . . . . . . . 180
9.4 Solar Neutrino Day/Night Flux Asymmetry Results . . . . . . . . . . . . . . 181
9.4.1 Straight Day/Night Flux Asymmetry Results . . . . . . . . . . . . . 181
9.4.2 Day/Night Amplitude Fit Results . . . . . . . . . . . . . . . . . . . . 184
9.5 SK+SNO Day/Night Asymmetry Results . . . . . . . . . . . . . . . . . . . . 187
10 Solar Neutrino Oscillation Analysis
189
10.1 Spectrum Shape Oscillation Analysis . . . . . . . . . . . . . . . . . . . . . . 189
10.2 Solar Neutrino Flux Time Variation Oscillation Analysis . . . . . . . . . . . 191
10.3 SK Oscillation Analysis Results . . . . . . . . . . . . . . . . . . . . . . . . . 192
10.4 SK+SNO Oscillation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 193
10.5 Solar Neutrino Global Oscillation Analysis . . . . . . . . . . . . . . . . . . . 195v

11 Discussion
199
11.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
11.2 Future Outlook for Solar Neutrino Measurements . . . . . . . . . . . . . . . 203
11.2.1 Future Solar Neutrino Measurements at Super-Kamiokande . . . . . . 203
11.2.2 Other Current and Future Solar Neutrino Experiments . . . . . . . . 206
11.3 Closing Words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
Bibliography

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List of Figures
Page
2.1

Schematic view of the pp chain. The boxed reactions are neutrino producing
reactions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Schematic view of CNO cycle reactions. The boxed reactions are neutrino
producing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 pp chain solar neutrino energy spectra predicted by BP2004 SSM [6]. . . . .
2.4 The Sun’s electron density profile as a function of the ratio of the distancefrom the core to the solar radius. . . . . . . . . . . . . . . . . . . . . . . . .
2.5 The vacuum eigenstates in matter as a function of electron density. The point
where the dotted lines cross give the point of the MSW resonance. . . . . . .
2.6 The resulting production rate of 37 Ar within the Homestake Cl detector [13].
The errors shown for individual measurements are statistical errors only, while
the errors shown for the cumulative result is the combination of the statisticaland systematic uncertainties in quadrature. . . . . . . . . . . . . . . . . . . .
2.7 Recoil electron energy spectrum in Kamiokande, shown as the ratio of themeasured flux to the SSM expectation [14]. The hatched area shows thesystematic uncertainty. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8 The solar neutrino flux measured at Kamiokande during the daytime andnighttime [14]. The nighttime is also sub-divided into 5 regions. δsun shows
the angle between the direction from the Sun and the detector z-axis. . . . .
2.9 SAGE results for each year [16]. The shaded band is the combined best-fit andits uncertainty for all years. The error bars shown for individual measurements
are statistical uncertainties only. . . . . . . . . . . . . . . . . . . . . . . . . .
2.10 The combined Gallex and GNO result [18]. The error bars are statisticalerrors only. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.11 Borexino MC simulation based fit results in the units of counts/day/100 ton,spanning the energy range of 270 to 1600 keV [22]. . . . . . . . . . . . . . .
2.12 Locations of possible solutions for the solar neutrino oscillation parameters(tan2 θ12 vs. ∆m221 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.13 Reactor ν̄e energy spectrum obtained by the KamLAND experiment, alongwith the best-fit oscillation parameters case [26]. . . . . . . . . . . . . . . . .
2.14 Best-fit and allowed neutrino oscillation parameters coming from MINOS andSK [28]. Recent results from the Tokai to Kamioka (T2K) experiment are also
shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.15 Current solar neutrino experimental constraints on the low energy solar neutrino νe survival probability (Pee ). The gray band shows the MSW-LMA
predicted Pee with the 1 σ range estimated by [33]. . . . . . . . . . . . . . .
2.16 Measured day/night flux asymmetry, with statistical (blue) and systematic(red) uncertainties, for SK-I, II and III. . . . . . . . . . . . . . . . . . . . . .
3.1

The SK experimental site location within Japan. . . . . . . . . . . . . . . . .
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3.2
3.3

Layout of the SK experimental site [19]. . . . . . . . . . . . . . . . . . . . .
Differential cross section of (νe -e) (solid) and (νµ,τ -e) (dashed) elastic scattering in the case of 10 MeV incident neutrino energy. . . . . . . . . . . . . . .
3.4 Cross section of (νe -e) (solid) and (νµ,τ -e) (dashed) elastic scattering as afunction of incident neutrino energy. . . . . . . . . . . . . . . . . . . . . . .
3.5 The upper figure shows the neutrino energy spectrum measured by Winter[35]. Lower plot shows the experimental uncertainty of the Winter spectrum(black band) and the ratio between the Winter and Ortiz [36] spectra and its±1 σ experimental uncertainty (dashed line). . . . . . . . . . . . . . . . . . .
3.6 The expected number of events per day as a function of recoil electron totalenergy, coming from the elastic scattering of 8 B (hep) neutrinos with electronsin black (red). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7 Definition of the Cartesian coordinate system used in SK. . . . . . . . . . . .
3.8 Picture of the SK ID taken from the bottom of the detector. . . . . . . . . .
3.9 Schematic view of one unit of the PMT support structure [19]. The thicknessof the structure is about 55 cm. Each unit holds twelve 50 cm PMTs for the
inner detector and two 20 cm PMTs for the outer detector. . . . . . . . . . .
3.10 Water injection and removal points and rates within the SK tank. . . . . . .
3.11 Schematic view of the SK water purification system. . . . . . . . . . . . . . .
3.12 Time dependence of the supply water temperature. In November of 2009 the
new temperature control system was installed. . . . . . . . . . . . . . . . . .
3.13 Radon levels (in Bq/m3 ) throughout the year at the SK site[19]. The solidline is the level in the mine. The dotted line is the level in the air-purged
experimental areas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.14 Schematic of the air purification system at SK [19]. . . . . . . . . . . . . . .
3.15 Diagram of an ID PMT [19]. . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.16 The quantum efficiency of the ID PMTs as a function of wavelength, alongwith the Cherenkov spectrum in pure water. . . . . . . . . . . . . . . . . . .
3.17 One pe signal of the ID PMTs. . . . . . . . . . . . . . . . . . . . . . . . . .
3.18 Layout of the data acquisition system of SK-I, II and III. . . . . . . . . . . .
3.19 Schematic view of the input and output channels of the ATM. . . . . . . . .
3.20 Overview of the global trigger generation during SK-I, II and III. . . . . . .
3.21 Block diagram of the signal processing in the QBEE module. . . . . . . . . .
4.1

4.2

4.3

The Xe lamp calibration system. The UV filter acts to only pass light aroundthe UV region of the spectrum. The neutral density (ND) filter adjusts theintensity and the scintillator ball (BBOT and MgO powder) emits light at 440
nm in all directions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Output charge distribution of 1 pe hits for SK-III (black) and SK-IV (red).
The SK-IV distribution is multiplied by 0.844 so that the peak amplitude of
the SK-IV distribution matches the peak amplitude of SK-III. . . . . . . . .
Relative PMT gains measured using the laser calibration data. The 1 σ spreadis ∼6%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.4

Position dependence of the PMT hit rate for Ni calibration data (red) andMC simulated events (blue) [38]. The upper, bottom left and bottom rightfigures show the hitrate for the barrel, top and bottom PMTs, respectively.
The horizontal axis of the upper figure is the z (cm) position of the PMTs.
For the lower figures, the horizontal axis is the square of the distance fromthe center (cm2 ) of the detector. . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Visual explanation of the time-walk effect. Higher charged signals will trigger
the threshold earlier than lower charged signals. . . . . . . . . . . . . . . . .
4.6 Schematic view of the USHO laser calibration system [38]. . . . . . . . . . .
4.7 Single PMT timing vs. charge distribution obtained by the USHO laser calibration. The horizontal axis shows Qbin defined in Equation 4.7. Black, redand blue points refer to the three QBEE charge channel ranges defined in
Section 3.3.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8 Schematic view of the TQ smearing method. Each bin has its own binningeffect, so each bin is expanded to a Gaussian distribution. The standard
deviation of the Gaussian is defined for each Qbin value. After smearing thetiming distribution an asymmetric Gaussian fit is made from the lower timing
region to the higher timing region. . . . . . . . . . . . . . . . . . . . . . . . .
4.9 Timing vs. charge distribution after making TQ map corrections. . . . . . .
4.10 Definition of the symmetric and asymmetric scattering in the MC simulation.
4.11 Schematic view of the water coefficients measurement by laser light injection.
In SK-IV, a N2 laser was used for 337 nm measurements and laser diodes forthe other wavelength measurements. . . . . . . . . . . . . . . . . . . . . . .
4.12 Hits/Qtot distributions coming from the laser calibration for data (black) andMC simulation (red) [38]. The horizontal axis shows time (nsec) and the vertical axis shows hits/Qtot . The first and second peaks correspond to photons
scattering in the water and those which are reflected off the bottom PMTs
and black sheet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.13 Measured and fit water transparency coefficients from April, 2009 [38]. The
black line shows the fit result of the total water transparency, red is absorption,
blue is symmetric scattering and pink is asymmetric scattering. . . . . . . .
4.14 (Top) Time variation of the W T as measured by decay-e’s. (Bottom) Timevariation of the reconstructed mean energy of cosmic-ray muon decay-e’s before (after) the W T correction has been applied in black (red). A W T of 90
m is assumed before the W T correction, hence the reason for the vertical shift
before and after correction. After correction, the mean energy is stable within±0.5%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.15 Fractional change in the absorption coefficient, relative to the coefficient when
the absolute energy scale calibration was done, vs. µ decay-e measured watertransparency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.16 Hitrate of data (black) and MC simulated events (β=0 for green and β=0.016(best-fit) for red). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.17 Data vs. MC simulation χ2 for various β parameters. The minimum χ2 value
determines the β value to be used in the MC simulation of solar neutrinos. .

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4.18 Diagram of the LINAC calibration system and the injection points of themonoenergetic electrons into the SK tank [40]. . . . . . . . . . . . . . . . . .
4.19 Linearity of the Ge detector. The horizontal axis is the output from the Ge
detector (counts) and the vertical axis shows the energy of the calibration
gamma-ray source (MeV). Deviation of the output of the Ge detector fromthe fitted line (%) is shown on the right. For the solar neutrino energy region,
the difference is within 0.1%. . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.20 Neff distribution for data (MC simulation) in black (red), taken during theAugust, 2010 LINAC calibration. 13.2 (6.5) MeV is shown on the left (right).
4.21 Schematic view of the DT generator system. . . . . . . . . . . . . . . . . . .
4.22 Direction dependence of the energy scale, from the DT calibration. . . . . . .
4.23 Position dependence of the energy scale, from the DT calibration. . . . . . .
5.1

5.2

5.3

5.4
5.5
5.6
5.7
5.8

5.9
6.1
6.2
6.3

Probability density function of the timing residuals used for the vertex reconstruction maximum likelihood fit. The second and third peaks around 30 and
100 nsec are caused by PMT after pulses. . . . . . . . . . . . . . . . . . . . .
Vertex resolution for SK-I, III and IV shown by the dotted (blue), dashed(red) and solid (black) lines, respectively. The SK-III vertex resolution improvement over SK-I comes from using an improved vertex reconstruction.
The slightly improved timing resolution and agreement between data and MC
is responsible for the further improvement in SK-IV. . . . . . . . . . . . . . .
Vertex shift of the Ni calibration events. The start of the arrow gives the true
Ni-Cf source position and the direction indicates the averaged reconstructed
direction at that given position. The length of the arrow indicates the magnitude of the vertex shift. To make the vertex shifts easier to see the lengthis scaled by a factor of 20. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The energy dependence of the angular likelihood used in the direction reconstruction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The incident angle dependence of the PMT acceptance. . . . . . . . . . . . .
Definition of angles θi and φi for the i-th PMT. . . . . . . . . . . . . . . . .
The effective PMT photo-cathode coverage correction factor as a function ofθ and φ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(Top) Relationship between Neff and the reconstructed energy, shown withthe fit function described by Equation 5.3. (Bottom) Ratio between the MC
simulated points and the fit function. . . . . . . . . . . . . . . . . . . . . . .
PMT response for incident light. . . . . . . . . . . . . . . . . . . . . . . . . .

87

88
89
90
91
92

94

95

95
97
97
100
100

103
105

The top (bottom) shows an example of a good event (bad event) which has alower (higher) direction reconstruction goodness. . . . . . . . . . . . . . . . . 109
Correlation between gv2 (horizontal) and ga2 (vertical) for background (DTcalibration) events on the left (right). The line shows gv2 − ga2 = 0.25. . . . . 110
Cut criteria for flasher events. The horizontal axis shows the maximum chargeof a PMT in an event and the vertical axis shows the number of hits in thesurrounding 24 PMTs. The top figure shows a run which has many flasher
events, while the bottom figure shows a typical good run. . . . . . . . . . . . 112
x6.4

Spallation cut likelihood distribution for the spallation-like sample (black) andthe random sample (red). The horizontal axis shows log(Lspa ). . . . . . . . .
6.5 Position dependence of the dead time resulting from the spallation cut. The
horizontal axis of the top (bottom) figure is the detector z (r2 ) position in cm(cm2 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6 Definition of θPMT , fwall and deff . . . . . . . . . . . . . . . . . . . . . . . . . .
6.7 Lpattern for data (black) and solar neutrino MC simulated events (red), for
6.5-8.0 MeV (left), 8.0-12.0 MeV (middle) and >12 MeV (right). . . . . . . .
6.8 deff distributions for data (black) and MC simulated solar neutrino events(red), for 5.5-8.0 MeV (left) and 8.0-8.5 MeV (right). The black line gives thevalue used for the external event cut in each energy region. . . . . . . . . . .
6.9 deff distributons for lower energy events (4.5-5.0 MeV) for each of the threedifferent values of pwall , for data (black) and MC simulated solar neutrino
events (red). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.10 Vertex distribution of 4.0-4.5 MeV (recoil electron kinetic energy) (top left),
4.5-5.0 MeV (top right) and 5.0-5.5 MeV (bottom) events in the SK-IV final
data sample. The units are events/day/bin. The tight fiducial volume cut
made in each bin is shown by the red curve (lines). . . . . . . . . . . . . . .
6.11 Event rate of the final data sample for SK-III (black) and SK-IV (red). . . .
6.12 SK-IV energy spectrum after each of the main reduction steps. The filledsquares (triangles) correspond to the sample after the pre-reduction (spallation) cut. The stars give the spectrum after the external event cut, and the
SK-IV final data sample is shown by the open circles. The SK-III final data
sample is shown by the black circles. . . . . . . . . . . . . . . . . . . . . . .
6.13 Selection efficiency for the solar neutrino MC simulation events after each of
the main reduction steps. The open stars give the trigger efficiency. The filledsquares (triangles) correspond to the reduction step after the pre-reduction(spallation) cut. The filled stars give the efficiency after the external event
cut and the SK-IV final efficiency is given by the open circles. The SK-III
final selection efficiency is shown by the dotted lines. . . . . . . . . . . . . .
6.14 Multiple scattering goodness (MSG) for 4.6 MeV (8.4 MeV) LINAC data inblack (red) and MC in blue (cyan), normalized by number of events. . . . . .
7.1
7.2
7.3
7.4
7.5
7.6
7.7

Definition of the angles θsun and θz . . . . . . . . . . . . . . . . . . . . . . . .
SK-IV final data sample cos θsun distribution for 4.0-19.5 MeV events. . . . .
Cosθsun probability density function of the background (bij ) for three different
energy bins. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cosθsun probability density function of the solar neutrino signal (sij ) for threedifferent energy bins. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Energy resolution function taken from monoenergetic electron MC simulated
events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Vertex shift systematic error on the flux in each energy bin. The energy
dependence below 5.0 MeV comes from the tight fiducial volume cut. . . . .
The difference of the event quality cut efficiency between data and MC simulated events, obtained by the LINAC calibration. . . . . . . . . . . . . . . .
xi

116

117
118
119

120

120

123
124

125

125
127
129
129
131
131
134
138
139

7.8
7.9

Time variation of the dead-time resulting from the spallation cut. . . . . . .
The difference of the hit pattern cut efficiency between LINAC data and MC
simulated events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.10 The method to artificially shift the reconstructed vertex and direction according to the measured vertex shift and angular resolution. The direction isshifted within the plane of the true and reconstructed directions. . . . . . . .
7.11 The difference of the small hit cluster cut efficiency between data and MC
simulated DT events for energy bins 4.0-4.5, 4.5-5.0 and 5.0-5.5 MeV. . . . .
7.12 Solar angle distribution of 4.0-19.5 MeV events. Black point shows the observed data while the solid (dashed) red line shows the fit signal plus background (background). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.13 Measured flux of all SK phases individually. The solid line gives the combined
SK flux average, while the dashed lines give the combined statistical plus
systematic error on the flux average. All phases are consistent with eachother within the smallest value of the statistical error (SK-IV value of 1%). .
7.14 Combined solar angle distribution for SK-I-IV, a total livetime of 4,141 days.
8.1

8.2

8.3

8.4

8.5

8.6
8.7

8.8

SK-IV final data sample cosθsun distributions for various energy regions. Red
shows all events in each energy region, while the green, blue and cyan show
events for MSG betwen 0-0.35, 0.35-0.45 and 0.45-1, respectively. . . . . . . .
Ratio of the statistical errors of the extracted number of events when using
MSG sub-samples and when not (i.e. w/o MSG). The red and black points
are for the positive and negative errors, respectively. . . . . . . . . . . . . . .
Energy bin correlated systematic uncertainties. The solid, dotted and dasheddistributions correspond to the energy bin correlated systematic uncertainties of the 8 B spectrum shape, absolute energy scale and energy resolution,
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
MSG scaling functions applied to the solar neutrino MC simulated events toestimate the systematic uncertainty on the energy spectrum. The blue, blackand red lines correspond to 15.8, 8.36 and 4.59 MeV LINAC data over MC. .
Cosθsun distributions for the energy bins which use MSG sub-samples (3.5-7.5
MeV) (MSG bins go low to high in going left to right). Black points show thedata, the blue and red histograms show the best-fit to the background andsignal plus background, respectively. . . . . . . . . . . . . . . . . . . . . . .
Cosθsun distributions for energy bins not utilizing MSG sub-samples. . . . . .
SK-IV energy spectrum, using MSG below 7.5 MeV. The horizontal dashedline gives the SK-IV total flux average (2.37 × 106 cm−2 sec−1 ). Error bars
shown are statistical plus energy bin uncorrelated systematic uncertainties.
SK-I, II, III and IV recoil electron spectra. The green (blue) shape is the best
fit to SK data using the global solar (solar+KamLAND) best-fit oscillation
parameters. The orange (black) line is the best fit to SK data with a generalparabolic (cubic) Pee survival probability. . . . . . . . . . . . . . . . . . . . .

xii

140
141

142
143

144

146
148

150

151

152

154

157
158

159

160

8.9

νe survival probability Pee based on the solar global best-fit parameters (thicksolid black) and the solar+KamLAND best-fit parameters (thick solid blue).
The solid red (cyan) line is the best exponential approximation to the thicksolid black (blue) line. The dashed (dotted) orange line is the best quadratic(cubic) approximation to the thick solid black. The dashed (dotted) green
line is the best quadratic (cubic) approximation to the thick solid blue line. .
8.10 Allowed areas of the shape parameters (e1 and e2 ) of an exponential fit to thesurvival probability Pee of SK data. The parameter set corresponding to thesolar neutrino (solar+KamLAND) data best fit is indicated by the dark green(blue) star. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.11 Allowed survival probability 1 σ band from SK data. The red (blue) area isbased on an exponential (quadratic) fit, the green band is based on a cubic fit.
The 8 B flux is constrained by the NC measurement from SNO. The absolutevalue of the 8 B flux doesn’t affect much to the shape constraint, just the
average value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.12 SK-I+II+III+IV recoil electron spectrum. The green (blue) shape is the best
fit to SK data using the global solar (solar+KamLAND) best-fit oscillation
parameters. The orange (black) line is the best-fit to SK data with a generalparabolic (cubic) Pee survival probability. . . . . . . . . . . . . . . . . . . . .
8.13 Allowed areas of the shape parameters c1 vs. c2 (c1 vs. c0 ) of a quadratic fit
to the survival probability Pee of SK (green) and SNO (blue) data on the left
(right). Also shown is a combined fit (red). The parameter set corresponding
to the solar neutrino (solar+KamLAND) data best-fit is indicated by the darkgreen (blue) star. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.14 Allowed survival probability 1 σ band from SK (green) and SNO (blue) data.
The red band gives the combination of SK and SNO. Also shown are predictions based on the best-fit parameters from the global solar (green) andsolar+KamLAND (blue) analyses. . . . . . . . . . . . . . . . . . . . . . . . .
9.1
9.2
9.3

9.4

The livetime in each cosθz (Cz ) bin (in days/bin). Cz bin = 1 and 1,000
correspond to cosθz = 0.001 and 1, respectively. . . . . . . . . . . . . . . . .
Predicted solar zenith angle time variation of the ratio of the oscillated rate
to the unoscillated rate (∆m221 =4.90×10−5 eV2 and sin2 θ12 =0.314). . . . . . .
The solar zenith angle dependence of the solar neutrino flux (4.49-19.49 MeV).
The day data is sub-divided into five bins, while the night data is divided into
six bins. Solar neutrinos in the last night bin pass through the Earth’s core.
The error bars show are statistical errors only. . . . . . . . . . . . . . . . . .
Data/MC simulation (unoscillated) dependence on the solar zenith angle forvarious energy regions (zenith angle and energy bins defined in Table 9.2).
The MC simulation assumes a 8 B flux of 5.25×106 cm−2 sec−1 . Overlaid in red(blue) are predictions when using the global solar (solar+KamLAND) best-fit
oscillation parameters and the assumed neutrino fluxes fit to best describe thedata. The error bars shown are statistical uncertainties only. . . . . . . . .

xiii

162

164

165

166

168

168
171
175

182

183

9.5

9.6

9.7

9.8

Measured day/night asymmetry (fitted day/night amplitude times the expected day/night asymmetry (red)) vs. ∆m221 , for sin2 θ12 =0.314. The light(dark) gray band give the 1 σ statistical (statistical plus systematic) uncertainty. Overlaid are the allowed 1 σ ranges from solar neutrino data (greenband) and KamLAND (blue band). . . . . . . . . . . . . . . . . . . . . . . .
Dependence of the day/night asymmetry on sin2 θ12 for ∆m221 =4.90×10−5 eV2 .
The light (dark) gray band shows the 1 σ statistical (statistical plus systematic) uncertainty. The red line shows the expected day/night asymmetry. . .
SK combined day/night asymmetry vs. energy (measured day/night amplitude times the expected asymmetry (red)). The error bars shown are statistical uncertainties only. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
SK+SNO measured day/night asymmetry (fitted day/night amplitude times
the expected day/night asymmetry (red)) vs. ∆m221 . The light (dark) grayband gives the SK plus SNO (SK) 1 σ statistical plus systematic uncertainty.
Overlaid are the allowed ranges from solar neutrino data (green band) andKamLAND (blue band). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

185

186

186

188

10.1 Excluded contours of ∆m221 vs. tan2 θ12 from the SK-I/II/III/IV spectra(green) and day/night data (blue) at the 1, 3 and 5 σ confidence levels (5sigma is filled). The unfilled region gives the 1 and 2 σ allowed regions. The
2
 2
13 −0.0242. . . . 191
red contours are the combined fit. θ13 is constrained by sin θ0.0026
10.2 Excluded and allowed contours of ∆m221 vs. tan2 θ12 from the SK-I/II/III/IV
spectra and day/night data (blue). The shaded blue region gives the excluded
regions (1-5 σ) and the unfilled regions are the 1 and 2 σ allowed regions.
Green shows the same data, but now with the 8 Bflux constraint
 included, atthe 1-5 σ confidence levels. θ13 is constrained bysin2 θ13 −0.0242
0.00262. . . . . . . 194

10.3 Allowed contours of ∆m221 vs. tan2 θ12 from the SK-I/II/III/IV spectral andday/night data using the 8 B flux constraint (green), and the SNO results(blue), at the 1-5 σ confidence levels. The SK and SNO combined result is 2
2sin θ13 −0.0242
shown in red. θ13 is constrained by. . . . . . . . . . . . . . . 195
0.002610.4 Allowed contours of ∆m221 vs. sin2 θ12 from solar neutrino data (green) at
1-5 σ and KamLAND data (blue) at the 1-3 σ confidence levels. Also shown
are the combined results in red. For comparison, the almost identical resultsof the SK+SNO combined fit
 are shown by the dashed dotted lines. θ13 is2
sin2 θ13 −0.0242.
0.0026
Allowed contours of sin2 θ13 vs.

constrained by

10.5. . . . . . . . . . . . . . . . . . . . . . . . . 198

sin2 θ12 from solar neutrino data (green) at
1-5 σ, and KamLAND data (blue) at the 1-3 σ confidence levels. Also shown
are the combined results in red. . . . . . . . . . . . . . . . . . . . . . . . . . 198

xiv

List of Tables
Page
2.1

Table of the BP2004 SSM predicted flux values for each solar neutrino source. 15

3.1

Trigger thresholds in each SK data taking period. *The SLE threshold was
lowered time to time during SK-I, the final value is shown here. . . . . . . .
Observable charge range and resolution for each of the three QBEE input
channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Summary of the current software trigger types, hit thresholds, events widths
and even rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Summary of the key features of each phase of SK. The livetime of SK-IV isfrom September, 2008 to the end of 2012 (used in this thesis). . . . . . . . .

3.2
3.3
3.4
4.1

4.2

The absolute gain for each SK phase. This factor converts the output chargeof the PMTs (pC) to the number of pe’s. The absolute gain factor becomeslarger with time, but new PMTs were installed just before SK-III and have aslightly lower gain than those used during SK-II. . . . . . . . . . . . . . . . .
Characteristics of the USHO laser. . . . . . . . . . . . . . . . . . . . . . . .

60
61
63
64

67
73

5.1

Angular resolution difference between LINAC data and MC for SK-I, III and
IV. The energy refers to the electron in tank kinetic energy. . . . . . . . . . 101

6.1

Reduction efficiencies after each of the pre-reduction steps. The data set is
reduced to ∼1 % of its original size by the pre-reduction. . . . . . . . . . . . 109
Calibration sources considered for the calibration source cut. The water temperature sensor is considered after run 33027 (September 4th , 2007). . . . . . 111
List of possible spallation products in SK. . . . . . . . . . . . . . . . . . . . 114

6.2
6.3
7.1
7.2
7.3
7.4
8.1
8.2

8.3
8.4

SK-IV energy resolution difference between LINAC data and MC simulation.
The energy refers to the electron in tank kinetic energy. . . . . . . . . . . . .
Angular resolution difference between LINAC data and MC for SK-I, III and
IV. The energy refers to the electron in tank kinetic energy. . . . . . . . . .
Summary of the systematic uncertainty on the total flux for each SK phase.
SK measured solar neutrino flux by phase. . . . . . . . . . . . . . . . . . . .
Energy bin uncorrelated systematic uncertainties on the spectrum shape. . .
The observed event rates in each energy bin at 1 AU. The errors are statisticalerrors only. The selection efficiencies are corrected and the expected event
rates are for an unoscillated flux of 5.25×106 cm−2 sec−1 . . . . . . . . . . . .
Best approximations of the MSW resonance curves when using exponential
and polynomial parameterizations of Pee . . . . . . . . . . . . . . . . . . . . .
Elastic scattering rate ratios and energy uncorrelated uncertainties for each
SK phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xv

132
136
144
147
153

155
162
163

8.5

8.6
9.1
9.2

9.3

SK measured exponential and polynomial fit coefficients and correlations.
Also given are SNO’s quadratic fit coefficients (slightly different than the
published values since the day/night asymmetry is not fit), as well as SK and
SNO combined measured quadratic fit coefficients and correlations. . . . . . 163
Spectrum fit χ2 comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
Day/night asymmetry systematic uncertainties by SK phase. The total isfound by adding the contributions for each phase in quadrature. . . . . . . . 180
The observed zenith angle dependence of event rates in each energy region at
1 AU. Errors are statistical uncertainties only. The reduction efficiencies arecorrected and the expected event rates are for a flux of 5.25×106 cm−2 sec−1 . 181
Day/night asymmetry for each SK phase, coming from separate day and night
rate measurements (left column) and the amplitude fit (middle column), withthe expected day/night asymmetry in the right column. The uncertainties
shown are statistical and systematic. The final row gives the combinationof all four phases, for both methods. The entire middle and right columns
assume ∆m221 =5.90×10−5 eV2 , sin2 θ12 =0.314 and sin2 θ13 =0.025. . . . . . . . 183

xvi

Acknowledgments
I must first thank Dr. Michael Smy, my academic mentor. Our daily discussions, and thepush to further investigate problems at hand, to acquire a much deeper understanding of
the solutions, have turned me into the scientist I am today. I will forever be indebted forthe time and effort he has given me, and look forward to our continued friendship. Without
Michael, none of this work would have been possible.
I would like to thank Dr. Hank Sobel, for giving me the opportunity to come and work withsuch a great group of people. I am grateful for all of his support, in every needed area. It
was truly a pleasure and an honor to work under him. I also want to thank him for showing
me that there are still professors out there who know how to get their hands dirty. I will
forever remember our working days spent in Japan, in the snow.
Many many thanks to Dr. Masayuki Nakahata, the fearless leader of the SK low energy
analysis group, and a man truly devoted to good science. I am grateful to have had thepleasure of working under his guidance and supervision. I am also grateful for the many
opportunities he gave to me to extend my scientific abilities, including presenting my work
to the world.
I would also like give many thanks to Jeff Griskevich. He has given me insight into many
aspects of life which many graduate students do not encounter. The skills he taught me in
the machine shop, and while doing a wide range of jobs together, will stay with me for the
rest of my life. I am grateful to have met such a good friend.
I must thank Dr. Mark Vagins, whom readily took me on as a legitimate member of the
SK gadolinium research and development team, and pushed in many cases for me to makeappearances at conferences and meetings. I very much enjoyed the many, and sometimes
completely off topic, conversations we have had. Thanks for all your support and guidance
to the many aspects of what it takes to be a success scientist.
I need to thank to my committee members Dr. Jonas Schultz, Dr. Steven Barwick and
Dr. Mu Chun Chen. I appreciate the time they took out of their busy schedules, and alltheir constructive comments. I must thank everyone in the UCI SK group, for all the help
and support. They all made work (both at UCI and in Japan) a more enjoyable place tobe. Especially good luck to Pierce Weatherly, as he now untangles the workings of the low
energy analysis at SK, and thanks for all the help this last year.
I would like to thank the other members of the solar analysis team, including Dr. T.
Yokozawa, Dr. Y. Takeuchi, Dr. Y. Koshio, Dr. M. Ikeda, Dr. B. Yang and Y. Nakano.
This work is a product of many sleepless nights, by all of us. And of course many manythanks to all of the members of the SK low energy analysis group, and other members of
the SK experiment, both past and present. Finally, I would like to thank all my friends andcolleagues I met along the way, from UCI, Japan, and at conferences and meetings. Thanks
for all the hard work done, and fun times had together.
xvii

Curriculum Vitae
Andrew Lee Renshaw
RESEARCH INTERESTS
Particle physics, with focuses in neutrino physics and astrophysics and advanced detector
research and development. Hardware and software development for testing equipment anddata analysis.
EDUCATION
PhD, Physics
University of California, Irvine

2013
Irvine, CA

M.S., Physics
University of California, Irvine

2010
Irvine, CA

B.S., Physics
California State Polytechnic University, Pomona

2008
Pomona, CA

RESEARCH EXPERIENCE
Graduate Research Assistant
University of California, Irvine

2009–2013
Irvine, CA

TEACHING EXPERIENCE
Teaching Assistant
University of California, Irvine

2008–2009
Irvine, CA

REFEREED JOURNAL PUBLICATIONS
Solar Neutrino Results in Super-Kamiokande-III, The Super-Kamiokande Collaboration, Phys. Rev. D 83, 052010 (2011).
Supernova Relice Neutrino Search at Super-Kamiokande, The Super-Kamiokande
Collaboration, Phys. Rev. D 85, 052007 (2012).
Search for GUT Monopoles at Super-Kamiokande, The Super-Kamiokande Collaboration, Astro. Phys. 36 (2012) 131-136.
Search for Proton Decay via p→µ+ κ0 in Super-Kamiokande I, II, and III, The
Super-Kamiokande Collaboration, Phys. Rev. D 86, 012006 (2012).
xviii

Solar Neutrino Measurements in SK-IV, The Super-Kamiokande Collaboration, To
appear in Phys. Rev. D.
First Direct Indication of Matter Effects on Neutrino Oscillation, The SuperKamiokande Collaboration, To appear in Phys. Rev. Lett.
Irvine Device Evaluating Attenuation Length, A. Renshaw et al., To appear in Nucl.
Instrum. Meth.
REFEREED CONFERENCE PUBLICATIONS
Evaluating Gadolinium’s Action on Detector Systems
The XXV International Conference on Neutrino Physics and Astrophysics

June 2012

Research and Development for a Gadolinium Doped
Water Cherenkov Detector
Technology and Instrumentation in Particle Physics 2011

June 2011

GADZOOKS!
The XXIV International Conference on Neutrino Physics and Astrophysics

June 2010xix

Abstract of the Dissertation
First Direct Evidence for Matter Enhanced Neutrino Oscillation, Using Super-Kamiokande
Solar Neutrino Data
By
Andrew Lee Renshaw
Doctor of Philosophy in Physics
University of California, Irvine, 2013
Professor Henry W. Sobel, Chair

Super-Kamiokande-IV has proven to be the largest and most precise SK data set yet. The
data from this fourth phase of SK are combined with data from the previous three phases
to give an extracted solar neutrino flux between 4.0 and 19.5 MeV (recoil electron kinetic
energy) as (2.37±0.015(stat.)±0.04(syst.))×106 cm−2 sec−1 . The SK combined recoil electronenergy spectrum slightly favors a distorted shape over a flat shape. Comparing the day andnight solar neutrino interactions rates separately, SK measures the day/night asymmetry as(-4.2±1.2±0.8)%. A maximum likelihood fit to the amplitude of the expected solar zenith
angle variation of the neutrino-electron elastic scattering rate in SK, results in a day/night
asymmetry of (-3.2±1.1±0.5)%. These results give 2.8 and 2.7 σ significance for a non-zero
day/night asymmetry. This is not only the first evidence for the regeneration of electron typesolar neutrinos as they travel through Earth’s matter, but the first direct evidence for matterenhanced neutrino oscillation of any kind. SK solar neutrino data measures the solar mixing+1.8
2
angle as sin2 θ12 =0.341+0.029−0.025 , and the solar neutrino mass squared splitting as ∆m21 =4.8−0.9 ×

10−5 eV2 . When these results are combined with data from other solar neutrino experiments−5
and the KamLAND experiment, sin2 θ12 =0.304±0.013 and ∆m221 =7.45+0.20
eV2 give−0.19 × 10

the most precise measurements to date.

xx

Chapter 1
Introduction to Neutrinos
1.1 History of the Neutrino
The story of the neutrino begins in 1896, with the discovery of strange radiation comingfrom uranium salts by Kenri Becquerel, and the isolation of radium by Pierre and Marie
Curie. Within a few years Ernest Rutherford had discovered two distinct types of radiation,
which he coined alpha (α) and beta (β) radiation in 1899. A third type of radiation was foundby Paul Villard and termed gamma (γ) radiation in 1900. After the discovery of multiple
types of radiation, in 1902, the Curies were able to show that β radiation was nothing morethan electrons. Around the same time, Frederick Soddy and Ernest Rutherford showed thatthe three types of radiation were in fact coming from different types of radioactive decays.
By 1904, the three types of radiation had been defined as α, γ and β radiation, in which
either a helium 4 nucleus (4 He), a photon or an electron are emitted from a radioactive
nucleus, respectively.
Once the three types of radiation had been discovered, and were found to be distinctively
different, the race was on to characterize these types of radiation and the decays responsible
for their emission. Things got very interesting in 1911 when Lise Meitner and Otto Hahn
performed an experiment which showed that the energies of electrons emitted during β-decay
had a continuous spectrum, rather than the discrete spectrum predicted if the electron was
the only particle emitted. This lead to the obvious problem of having to account for thevariability of the energy of the electrons, while still taking into account the conservation of
momentum and angular momentum. In his most famous letter written in December of 1930,
1

Wolfgang Pauli suggested that in addition to the electrons and protons contained inside theatom, there are also extremely light neutral particles, which he called neutrons. He proposedthat these neutrons, which were also emitted during β-decay and accounted for the missing
energy and momentum, had simply not yet been discovered. In 1931 Enrico Fermi built upon
Pauli’s ideas of a yet undetected particle, and three years later published a very successfulmodel of β-decay in which neutrinos were produced. At this point Fermi coined the term
“neutrino”, which is Italian for “small neutral one”, thus giving birth to the neutrino itself.
In 1934 Hans Bethe and Rudolf Peierls showed that the cross section between neutrinos
and matter should be extremely small. They estimated that it should be billions of times
smaller than that for electrons and matter. This would allow a neutrino to travel throughthe entire Earth without deviation, making it extremely elusive to detect. In 1954 Ray
Davis Jr. attempted the first detection of neutrinos. He was trying to observe an increasein the rate of production of argon (Ar) when surrounding nuclear reactors were tuned from
off to on.

37

Ar is produced from the neutrino-chlorine (Cl) interaction, which will only

proceed if the neutrinos are truly neutrinos and not antineutrinos. Unfortunately for Davis,
it is really antineutrinos which are produced inside nuclear fission reactors, and so this firstattempt at detecting the neutrino gave a null result. It wasn’t until 1956 that Clyde Cowan
and Frederick Reines discovered the neutrino (actually the antineutrino), by observing theinverse beta decay reaction p + ν̄e → n + e+ occurring from antineutrinos produced in a
nuclear reactor. This proved the hypothesis set forth by Pauli many years before.
Then in 1962, in an experiment done by Leon Lederman, Melvin Schwartz and Jack
Steinberger, it was shown that the neutrinos released in the decay of pions to muons (π + →µ+ + νe + νµ ) were in fact distinguishable from the neutrinos coming from β-decays. These
new neutrinos were named muon neutrinos (νµ ), with those coming from β-decays being
called electron neutrinos (νe ). This brought rise to the idea that each charged lepton should
also have a chargeless neutrino partner, also a lepton. Along with this partnering of the2charged leptons with neutrinos came the idea of the conservation of lepton family number.
Leptons are assigned Le,µ =1 and antileptons are assigned Le,µ =-1, with the total lepton
family numbers Le and Lµ being conserved in an interaction. A third family of leptons
would soon appear with the discovery of the tau lepton in 1975 by Martin Perl, and theeventual discovery of the tau neutrino (ντ ) in 2000 by the DONUT experiment at Fermilab.
The different families of leptons are referred to as lepton flavours, and to this day there
remains only the electron (e), muon (µ) and tau (τ ) flavours within the standard model ofparticle physics.

1.2 Neutrinos in the Standard Model of Particle
Physics
The standard model (SM) of particle physics accepts the electromagnetically neutral neutrino as a fundamental lepton, along with the charged leptons; the electron, muon and tau.
This means they have h̄/2 spin angular momentum. As explained in the previous section,
each of the three types of neutrinos, and the corresponding charged lepton, make up a lepton family, where the total lepton family number is conserved across particle interactions.
According to the SM, these interactions can take place via two of the four types of fundamental forces (gravitational, electromagnetic, weak and strong). Neutrinos do not interact
strongly, and because they are electromagnetically neutral, they also do not interact via
the electromagnetic force. Although in the SM the neutrino is considered massless, various
experiments have proven that the neutrino must have a small, yet non-zero mass. Also, thecurrent framework of the standard model does allow for the introduction of massive neutrinos. This means that the neutrino interacts gravitationally. However, due to its extremely
small mass, any gravitation effects can be neglected. This leaves only the weak force forthe neutrino to communicate with the rest of reality. The neutrino can interact via both3

interaction types of the weak force, charged current (CC) and neutral current (NC) interactions. The CC (NC) interaction is mediated by the massive W ± (Z) boson. The large
masses introduce a large suppression factor into the cross section, making the neutrino a“ghostlike” particle with the ability to travel through extremely large quantities of matter,with no detectable interactions taking place. For this reason, the neutrino has proven to
be very difficult to study experimentally, usually requiring very large detectors employing
sophisticated detection techniques for observation.

1.3 Neutrino Production and Oscillation
Electron flavour neutrinos are generated in fusion nuclear reactions, such as those occurring in the Sun (solar neutrinos). Both electron flavour neutrinos and antineutrinos are
created in fission nuclear reactions, such as those occurring inside the Earth’s crust and mantle (geo neutrinos). Electron antineutrinos are also created in man made nuclear reactors(reactor neutrinos). Neutrinos of the electron and muon flavours are created when cosmic
rays interact with nuclei contained in Earth’s atmosphere. The cosmic-ray interactions withatmospheric nuclei produce showers of often unstable particles (charged π’s and κ’s), whose
decay products can include electron and muon flavoured neutrinos (atmospheric neutrinos).
Man made neutrino beams have also started to come into existence, in which protons are accelerated into a target, producing unstable products which decay into neutrinos (accelerator
neutrinos). However, regardless of the method of production of the neutrinos, the reactions
which produce the different types of neutrino flavours are known. This allows for predictionsof the various fluxes for the different types neutrinos, which can then be measured in orderto test fundamental theories.
The story is not quite so simple though. Various experiments observed fluxes far lessthan those predicted, bringing fundamental principles into question. In 1958, Pontecorvo
4had already come up with the first theory of neutrino oscillation [1], of the type particle ↔
antiparticle (ν ↔ ν̄). This was long before the discovery of multiple neutrino flavours. The
idea of the mixing of massive neutrinos was introduce in 1962 by Maki [2]. In 1969 Gribov
and Pontecorvo had developed a consistent phenomenological theory of neutrino mixing
and oscillation [3]. This is a purely quantum mechanical effect, resulting from the factthat the weak eigenstates, which are produced in weak decays, are different from the mass
eigenstates. The mass eignestates determine the propagation of particles in the vacuum.
The time dependent mass eigenstates are given as|νj (t, x)i = |νj (0)ie(−i(Ej t−~p·~x)) .

(1.1)

This means that two conditions are necessary for the appearance of neutrino oscillation;
lepton family quantum numbers are not strictly conserved (∆Lα 6= 0), and at least two of
the neutrinos must have different mass values. If all the neutrino masses are equal, mi = m,and hνβ |M |να i = mδαβ . This would make transitions of the type να ↔ νβ impossible. So
if the SM assumption of zero neutrino mass was correct, then there would be no neutrino
oscillation. The discovery of atmospheric neutrino flavour conversion by Super-Kamiokande(SK) in 1998 [4] provided the first evidence that neutrinos must have the ability to change
from one flavour to another, and therefore they must have a non-zero mass.

1.4 Neutrino Flavour Oscillation in Vacuum
In the standard neutrino mixing scheme, the neutrino flavour eigenstates are coherent
superpositions of the neutrino mass eigenstates, and vice versa [8]. This is the basic quantum
mechanical reason for the existence of the neutrino oscillation phenomenon. The flavour
eigenstates |να i (α = e, µ, τ ) can be expanded in terms of the neutrino mass eigenstates |νi i5

(i = 1, 2, 3) as|να i = U|νi i,(1.2)

where U is a unitary matrix, often called the Pontecorvo-Maki-Nakagawa-Sakata (PMNS or
just MNS) mixing matrix. Because of the tight constraint on the number of active neutrino
flavours being three (coming from the measurement of the Z boson total width [9]), the 3×3
matrix U will contain only 4 independent parameters. It is often parameterized as




−iδ

0
0   C13
0 S13 e
 1



U=
0
1
0
 0 C23 S23  −S13 eiδ 0
C13
0 −S23 C23  C12 S12 0   −S
,
C
0
12
12




0
0 1

(1.3)

where Cij ≡cosθij and Sij ≡sinθij denote the mixing angles. δ is a CP violating phase, andwill be non-zero if CP violation occurs for neutrinos.
A neutrino will be produced weakly at time t = 0, in a specific flavour state α, made upof the super position of the three mass eigenstates. At some time t later, the probability offinding the neutrino in a given flavour eigenstate β will be

Pνα →νβ = |hνβ |να (t)i|2 .

(1.4)

Then using Equation 1.2 to translate to the mass eigenstates, the fact that t = v/L 'c/L = 1/L (in the current units) and the assumption that the neutrinos are highly relativisticp(pν  mi so that Ei = 2 p2ν + m2i ≈ E + m2i /(2E)), Equation 1.4 can be written asPνα →νβ =

3
X

m2
i

2

U∗βi e−iEt e−i 2E L Uαi .

(1.5)

i=1

L is the distance from the neutrino source to the point of observation and E is the neutrinoenergy.

6

Recent measurements of the mixing angle θ13 (sin2 θ13 ≈0.025) imply that there is very
little mixing between the third mass eignestate and the electron flavour weak eigenstate.
For this reason, the amount of electron to tau neutrino flavour conversion will be very small.
It is then convenient to simplify the picture and reduce the number of neutrinos from three
to two, reducing the number of free parameters in the matrix U to one. U is then written
as





 cos θ sin θ 
U=.
− sin θ cos θ(1.6)

θ in this matrix is ≈ θ12 from Equation 1.3. Using this approximation, Equation 1.5 becomes22

Pνe →νx = sin 2θ12 sin



L
∆m221

4E


,

(1.7)

for two neutrinos. α has been replaced with e (electron neutrino) and β with x (µ or τ
neutrino). The probability that a neutrino created as an electron neutrino will be observed
as an electron neutrino, called the survival probability, is


2 L
.
Pνe →νe = 1 − Pνe →νx = 1 − sin 2θ12 sin ∆m21
4E
2

2

(1.8)

The oscillation parameter ∆m221 = |m21 − m22 | is called the solar mass squared splitting(difference). The factor involving the angle θ12 describes the amount of “mixing” there is
between states, and so the oscillation parameter θ12 is called the solar mixing angle. The
mass squared splitting ∆m221 in the second sin term and the neutrino energy E give thefrequency of the flavour change.
Equation 1.8 makes it easy to see that if the masses of the neutrinos were all the same(including zero), then the oscillation would in fact completely turn off. Then there would
be no vacuum transitions from one flavour to another. It is also easy to see the oscillatory
nature of the survival probability, as the expectation to find one flavour of neutrino will7

rise and fall. Unfortunately, when measuring neutrino oscillation, only the mass squared
parameter can be tested, and so no handle on the true mass of any of the mass eigenstates
is available. The sensitivity in the value of ∆m221 can be seen by∆m221 L
∼ 1.
2E

(1.9)

The ability to detect solar neutrino oscillation at all also depends on L/E and ∆m221 . If wemake the substitution L0 = 4πE/∆m221 , Equation 1.8 becomes2

2

Pνα →να = 1 − sin 2θ12 sin




L
π
.
L0

(1.10)

It is now seen that if L  L0 , the oscillation period becomes large and the second sin
factor goes to zero, making solar neutrino oscillation detection extremely difficult. However,
if L  L0 , the frequency will be too high and only the mixing angle can be probed. For
the case of solar neutrinos, in which L  L0 , assuming only vacuum oscillation would resultin the ability to probe only the solar mixing angle θ12 . To probe the solar mass squared
splitting ∆m221 , experiments must take into account matter effects coming from both the
Sun and the Earth, described in the next section.

1.5 Neutrino Flavour Oscillation in MatterAs neutrinos transverse ordinary matter (made up of electrons, protons and neutrons),
they can interact with the ambient particles via the CC (NC) interaction through the exchange of charged (neutral) W (Z) bosons. This can lead to forward elastic scattering, inwhich an interaction potential V brings an added contribution to the Hamiltonian. This
acts like an index of refraction for the neutrinos (different flavours are affected differently).
This will affect electron favour neutrinos greater, as they will interact with ordinary matter8

through both the CC and NC interactions. Muon and tau flavour neutrinos will interact
via the NC interaction only. In order to form the potential V it is convenient to write the
Hamiltonian in the flavour eigenbasis. To do this the Schrödinger equationid|νi (t)i = H|νi (t)i = Ei |νi (t)i,dt(1.11)

where H is diagonal with eigenvalues Ei is expressed in the flavour basis by applying themixing matrix U. This givesidU|νi (t)i = UHU† U|νi (t)i = H(α) |να (t)i.
dt(1.12)The Hamiltonian has been rotated toH(α) = EI +m∆m2  − cos 2θ sin 2θ 
I+

.
4E
4E
sin 2θ cos 2θ

P(1.13)

It is now possible to add the matter dependent term V to H(α) . Because all flavoursof neutrinos will interact via Z boson exchange, while only electron flavour neutrinos willinteract via W boson exchange, V can be written as




√
1
 1 0 Vint = − √ GF Nn I + 2GF Ne 
,
2
0 0

(1.14)

where GF is the Fermi constant and Nn and and Ne are the neutron and electron number
densities, respectively. The identity matrix I terms in both V and H(α) can be dropped, as
this will contribute an additional phase to all of the flavours equally, thus not contributing

9

to the effects of neutrino flavour change. The Hamiltonian in matter is then written as
H0 = H(α) + Vint =







∆m2  − cos 2θ sin 2θ  √ 1 0 
 + 2GF Ne .
4E
sin 2θ cos 2θ0 0(1.15)

We can now diagonalize the Hamiltonian back into the mass eigenbasis, giving the new
eigenvalues∆m2 A ∆m2±
EM1,2 =
2E
4E

q(cos 2θ − A)2 + sin2 2θ.

(1.16)The effective mass squared splitting in matter is then written as∆m2M = ∆m2

q(cos 2θ − A)2 + sin2 2θ.

(1.17)√
A = 2 2EGF Ne /∆m2 is a unit-less parameter which describes the influence matter has on
neutrino oscillation. The case of pure vacuum oscillation is recovered if A=0. Likewise, one
can write the effective mixing angle in matter assin2 2θM =sin2 2θ
.
sin2 2θ + (cos 2θ − A)2(1.18)The vacuum mass squared splitting ∆m2 and mixing angle sin2 θ terms in Equation 1.8 canbe replaced with Equations 1.17 and 1.18 to give the survival probability of νe propagating
through matter of constant electron density Ne . In the situations of the Sun and the Earth,where electron densities can be large and highly variable, neutrino oscillation can be greatly
affected. For instance, when the electron density reaches the critical value Ne,crit which makes
A=cos2θ, although the mixing angle in vacuum could be small, propagation through matter
will promote the mass eigenstates to maximal mixing (θM = π/4). This effect, called theMSW effect, was first theorized by Mikheyev, Smirnov and Wolfenstein [11] and has direct
application to the case of solar neutrinos propagating through the Sun.

10

Chapter 2
Solar Neutrinos
2.1 The Standard Solar Model
The standard solar model (SSM) is the solar model which is constructed with the best
available physics and input data. It is required to fit the observed luminosity and radiusof the Sun at the present epoch, as well as the observed heavy element to hydrogen ratioat the surface of the Sun [5]. In the SSM, the Sun is assumed to be spherical and to have
evolved quasi-statically over a period of about 4.6 billion years (age of the Sun). One ofthe principle goals of the SSM is to accurately model the energy production within the coreof the Sun. This energy is released in the form of either neutrinos created in the core of
the Sun, or photons which leave the surface. Therefore, understanding the exact method ofenergy production can give theoretical predictions of the solar neutrino flux leaving the Sun.
These theoretical predictions can then be compared to experimental measurements to test
either the accuracy of the SSM, or the completeness of neutrino theory. The SSM consideredin the following is that of BP2004 [6], developed by the late John N. Bahcall and Marc H.
Pinsonneault in 2004.
The net thermonuclear reaction which takes places inside the core of the Sun is the fusionof four protons and two electrons into a 4 He nucleus, two electron neutrinos and an energy
release of Q = 26.73 MeV, in the form of photons. This process can be written as

4p + 2e− → 4 He + 2νe + Q.

11

(2.1)

With the extremely large number of reactions taking place inside the core of the Sun, it
can be seen that a large flux of electron flavour neutrinos is created. But this is not the entire
story, as Equation 2.1 only gives the net reaction. This reaction can actually proceed via
two different processes: the proton-proton (pp) chain, shown in Figure 2.1, and the CarbonNitrogen-Oxygen (CNO) cycle, shown in Figure 2.2. According to the SSM, 98.4% of thetotal solar luminosity is predicted to be generated by the pp chain, while the remaining 1.6%
is thought to be generated by the CNO cycle. The ratio of pp chain interactions to CNO
cycle interactions is directly related to the elemental makeup of the Sun. If there are moreheavy elements within the Sun, then there will be more CNO cycle energy production.
Within the pp chain, electron neutrinos are produced by five different reactions, eachbeing characterized by the reactants of the reaction, and having either unique or maximalenergies. The five reactions, which produce pp, 7 Be, pep, 8 B and hep neutrinos, and either
their monoenergetic or endpoint energies, are given as;

p + p → 2 H + e+ + νe
p + e− + p → 2 H + νe7

Be + e− → 7 Li + νe8
3

B → 8 Be∗ + e+ + νe

He + p → α + e+ + νe(pp, Eν <0.42 MeV),(2.2)(pep, Eν =1.442 MeV),(2.3)(7 Be, Eν =0.861, 0.383 MeV),(2.4)(8 B, Eν <14.06 MeV),(2.5)(hep, Eν <18.77MeV).

(2.6)

The CNO cycle also proceeds via multiple branches, with electron flavour neutrinos being

12

Figure 2.1: Schematic view of the pp chain. The boxed reactions are neutrino producing
reactions.

Figure 2.2: Schematic view of CNO cycle reactions. The boxed reactions are neutrino
producing.

13

Figure 2.3: pp chain solar neutrino energy spectra predicted by BP2004 SSM [6].

produced via the three reactions;13

N → 13 C + e+ + νe(Eν <1.27 MeV),

(2.7)

15

O → 15 N + e+ + νe(Eν <1.73 MeV),(2.8)(Eν <1.74 MeV).

(2.9)

17

F → 17 O + e+ + νe

Because of the low energies of the CNO cycle neutrinos and the low percentage of thetotal solar luminosity coming from the CNO cycle, the neutrino flux coming from the CNO
cycle does not currently play a significant role in the study of solar neutrinos. The flux
coming from the pp chain is the current focus of study. Figure 2.3 shows the generated solar
neutrino energy spectra, for pp chain neutrinos, predicted by the BP2004 SSM [6]. Becauseof the higher energy, wider range and having a much larger flux than the hep neutrinos, 8 B
neutrinos give the best opportunity for various solar neutrino measurements. The 8 B solar
neutrino flux predicted by BP2004 is 5.79(1±0.23)×106 cm−2 sec−1 . The other predicted
fluxes are listed in Table 2.1.

14

Table 2.1: Table of the BP2004 SSM predicted flux values for each solar neutrino source.
Source
pp
pep
7
Be
8
B
hep
13
N
15
O
17
F

BP2004 Flux (cm−2 sec−1 )5.94(1 ± 0.01) × 1010
1.40(1 ± 0.02) × 108
4.86(1 ± 0.12) × 109
5.79(1 ± 0.23) × 106
7.88(1 ± 0.16) × 103
85.71(1+0.37−0.35 ) × 10+0.43
85.03(1−0.39 ) × 10
5.91(1 ± 0.44) × 106

2.2 The Solar Neutrino Problem
Once John Bahcall had calculated the expected neutrino flux coming from the Sun, basedon his SSM, it was possible for an experiment to make a measurement and check the accuracy.
Ray Davis, Jr.’s Homestake experiment was the first to do this in the late 1960’s. However,the detected solar neutrino flux turned out to be about one third of that predicted by the
SSM. This lead to the immediate investigation of the detector and its efficiencies, along with
the SSM assumptions. Both of these avenues turned out to give no clear answer for the
deficit of the flux. In 1989, the newly built Kamiokande experiment reported a flux slightly
over half of that expected from the SSM. The deficit was further confirmed in 1992, when the
GALLEX collaboration released an observed flux of 65% the SSM expectation. The closely
realted GNO and SAGE experiments also reported similar deficits as GALLEX.
There was an immediate struggle to try and bring these experimental measurements into
agreement with the SSM, but this only lead to the realization that there was an inconsistency
in the temperature dependence of the solar neutrino flux. The observed flux values pointed
to the likely possibility that it was not the SSM which needed revision, but instead the fundamental particle physics. The Homestake experiment had the ability to detect neutrinos
with a 0.814 MeV threshold, giving it access to 7 Be, pep, 8 B and hep solar neutrinos. The
Kamiokande experiment detected only 8 B and hep neutrinos, with a threshold of 7 MeV.
15

25
The 8 B neutrino flux depends heavily on the Sun’s core temperature (Tcore), while the de11
pendence for the 7 Be flux is much smaller (Tcore) [7]. This means that a slight temperatureadjustment in the SSM could bring Kamiokande’s 8 B measurement into agreement with theSSM expectation, but the 7 Be flux would not be as strongly suppressed as the Homestake
experiment observed. Then there are the pep and pp neutrinos, which have fluxes inversely−1
proportional to the Sun’s core temperature (∼ Tcore). The GALLEX experiment, which wassensitive to pp neutrinos, also observed a flux deficit and not an excess. It appeared as if
the neutrinos themselves have some sort of energy dependent property. Could it be neutrino
flavour oscillation?

2.3 Solar Neutrino Flavour Oscillation
The simplest example of solar neutrino flavour oscillation begins with the creation ofpurely νe ’s within the core of the Sun, and making a two neutrino flavour approximation(νe ↔ νx = νµ,τ ). During propagation from their production location within the Sun to theirdetection point on Earth, vacuum oscillation will cause the νe to change flavour into bothνµ,τ , in accordance with Equation 1.7 (using L=149.6×106 km≡1 au, the average Earth-Sun
distance). L is further modified to include the time dependence coming from the eccentricity
of Earth’s orbit around the sun as



t
.
L = (1 au) × 1 −  cos 2π
T(2.10)

Here =0.0167 is the eccentricity of Earth’s orbit and T =1 year. Without this time variationof the Earth-Sun distance included, a seasonal variation of the solar neutrino interaction rate
is possible to observe throughout the course of a year. This comes from the fact that thesolar neutrino flux will go like 1/r2 , regardless of what neutrino oscillation will do. Whenthe eccentricity is taken into account, any time variartion of the neutrino flux throughout
16

3

e density in mol/cm

10

2

10
1

-

10
10

-1
-2

0.1

0.2

0.3

0.4

0.5

0.6

0.7
0.8
0.9
1
distance/sun radiusFigure 2.4: The Sun’s electron density profile as a function of the ratio of the distance fromthe core to the solar radius.

the course of the year may be the effect of neutrino oscillation in vacuum. This is known as
the vacuum (VAC) solution.
Other scenarios can be expected when taking matter effects into account, while still
making the two flavour approximation. Inside the core of the Sun, the constant electron
density is Ne,core =150 g/cm3 . Using this, the effective mass squared splitting (Equation 1.17)
and the effective mixing angle (Equation 1.18) inside Equation 1.8, will yield the survival
probability of νe . Outside the core of the Sun, neutrino oscillation starts to get interesting, as
the electron density begins to follow an approximate inverse exponential function Ne,Sun (r),see Figure 2.4. Here r is the ratio of the distance from the core to the solar radius. For
reference Ne,Sun (r = 0) = Ne,core and Ne,Sun (r = 1) = 0. Because of the large range ofelectron densities surpassed by the neutrino, it is possible for Ne,Sun (r) = Necrit , which is thecondition necessary for a resonance flavour conversion to occur (MSW effect). This leads tomaximal conversion of the νe into the second mass eigenstate, still dependent on the effective
mixing angle, of course.
We can apply the mixing matrix U to the mass eigenstates in matter while substituting
in the effective mass squared splitting and the effective mixing angle, to get the flavour

17eginstatesνe = νM1 cos θM + νM2 sin θM ,(2.11)

νx = −νM1 sin θM + νM2 cos θM .

(2.12)

Inside the core of the Sun, where the electron density is large, the effective mixing angle isdriven to π/2, making νe born in the core of the sun mostly νM2 . The neutrino will stay in thestate νM2 if it propagates to the surface of the Sun adiabatically. For oscillation parameters
within the LMA region, neutrinos will propagate adiabatically through the Sun. As it travels
to the surface Ne,Sun (r) decreases, and so will the effective mixing angle. This will lead to anincreasing contribution of νx in the νM2 state. At the point where the resonance condition
is met Ne,Sun (r) = Necrit , the two flavour states are maximally mixed. When the neutrino
finally emerges from the surface of the Sun (Ne,Sun (r = 1) = 0), the νM2 state has become
the vacuum state ν2 . At this point the neutrino will not oscillate again until it reaches morematter. Figure 2.5 shows the vacuum eigenstates as a function of electron density, with
the intersection point of the dotted lines giving the point of critical electron density. If theneutrino were to propagate through the Sun non-adiabatically, it can jump into the νM1 state
and then emerge from the surface of the Sun in the ν1 state. This work will assume thatneutrinos travel through the Sun adiabatically.
Because the MSW effect depends not only on the value of the mass squared splitting,but also on the neutrino energy, the survival probability of the generated νe is also energy
dependent. For certain values of the mass squared splitting, the energy dependence ofthe νe survival probability can lead to a distortion of the observed energy spectrum. For
values of the mass squared splitting and mixing angle currently preferred by solar neutrino
data, the MSW resonance in the Sun will occur at ∼3 MeV, highly distorting the observed
solar neutrino spectrum by inducing an “upturn” when going from high to low energies.

18

Figure 2.5: The vacuum eigenstates in matter as a function of electron density. The point
where the dotted lines cross give the point of the MSW resonance.

The observation of this “upturn” in the solar neutrino energy spectrum would be a direct
signature of MSW type solar neutrino oscillations.
When solar neutrinos arrive at Earth in a pure ν2 state, the matter of the Earth can thenlead to an increase in the probability of the neutrinos converting back into νe . It is thenpossible for a solar neutrino experiment with realtime detection capabilities to observe an
asymmetry in the νe interaction rates during the daytime rD (when the Sun is above thehorizon) and the nighttime rN (when the Sun is below the horizon). This would be seen asa slightly higher interaction rate during the nighttime, and is known as “the solar neutrino
day/night effect”. Although the Earth’s electron density changes with the distance from thecore of the Earth, it is always at values less than an order of magnitude of the Sun’s, andhence there is no resonance condition within the Earth. Observation of a larger nighttime
interaction rate than that of the daytime would also be a direct signature of solar neutrino
oscillations.

192.4 Solar Neutrino Experiments
2.4.1 The Homestake Experiment
As previously explained, the Homestake experiment was the first to observe solar neutrinos, accomplished around 1968 [12], with a detector located 1,478 meters underground (4,200
meters water equivalent1 ) in the Homestake Gold Mine, in South Dakota. The experiment
ran from 1967 until 1994, and consisted of a 380 m3 tank of perchloroethylene (a common
dry-cleaning fluid) placed underground in order to shield from cosmic-ray backgrounds. Solar neutrinos interacted with the chlorine (Cl) atoms inside the fluid, producing argon (Ar)
atoms by the reaction
37

Cl + νe → 37 Ar + e− .

(2.13)The energy threshold of this interaction is 0.814 MeV, so the Homestake experiment
was sensitive to 8 B, 7 Be, pep and CNO neutrinos. The experiment was left to sit for 1-3
month periods, to collect solar neutrinos, after which time the generated 37 Ar atoms wereextracted by purging the tank with He gas for 20 hours. The collection efficiency of the37

Ar atoms was 95%. The 37 Ar atoms were then counted by the emission of Auger electrons(81.5% at a total energy of 2.823 keV) and x-rays (8.7%), as they decayed back into 37 Cl.
Figure 2.6 shows the results of the production rate of the 37 Ar atoms [13]. The averagerate is 2.56±0.16(stat.)±0.16(syst.) SNU2 , which corresponds to approximately one third
the predicted rate of 9.30+1.2−1.4 SNU. This would come to be known as “the solar neutrino
problem”.
The Homestake experiment achieved the first detection of solar neutrinos, but was only
able to make a rate measurement. It could not distinguish the direction, energy or time of1
2

Meters water equivalent (mwe) is the corresponding distance in water that would provide equal shielding
SNU(Solar Neutrino Unit) is the number of interactions per 1036 target atoms per second20Figure 2.6: The resulting production rate of 37 Ar within the Homestake Cl detector [13]. The
errors shown for individual measurements are statistical errors only, while the errors shownfor the cumulative result is the combination of the statistical and systematic uncertainties
in quadrature.

interaction of the neutrinos. It is also important to point out that the reaction in Equation 2.13 only allows for the detection of electron flavour neutrinos. This means that if
neutrino flavour conversion was indeed occurring, the Homestake experiment would observea rate less than that predicted by the SSM, because it is not sensitive to the other flavours.
So, while the Homestake experiment was capable of discovering the solar neutrino problem,
further experiments would be needed to solve the riddle.

2.4.2 The Kamiokande Experiment
The Kamioka Nucleon Decay Experiment (Kamiokande) was the first solar neutrino experiment to come after the Homestake experiment (20 years later), and took data from 1987
until 1995 [14]. Originally it was constructed with the purpose of detecting proton decay,
to test certain Grand Unified Theories (GUTs). After observation of neutrinos coming fromSupernova 1987a [15], it was upgraded to facilitate the observation of solar neutrinos. The
detector consisted of a cylindrical stainless steel tank located 1,000 m (2,700 mwe) below

21

ground in the Kamioka mine in Japan. It was filled with 3,000 tons of pure water, viewed by
948 50 cm diameter photomultiplier tubes (PMTs), providing 20% photo-cathode coverage.
The fiducial volume for the solar neutrino measurement was 680 tons of water.
The PMTs collected Cherenkov light coming from recoil electrons, resulting from theelastic scattering of solar neutrinos ν+e− → ν+e− . This reaction is dominated by the process
νe +e− → νe +e− , which is ∼6 times more likely to occur than the process νµ,τ +e− → νµ,τ +e− .
This is because the νe process will proceed via both the neutral and charged weak currents,
while the νµ,τ process will proceed only via the neutral weak current. However, these twodifferent processes cannot be separately identified inside a water Cherenkov detector. The
lower energy threshold of the Kamiokande detector was 7 MeV, making it sensitive to 8 B
and hep neutrinos.
One of the main advantages of the realtime Kamiokande experiment was its ability to determine the timing of events. It could also distinguish the direction of the incoming neutrinos,
due to the highly forward scattering of the (νe -e) elastic scattering and the directionality of
the Cherenkov light. For the first time a clear solar neutrino signal coming from the Sun
could be identified. The total effective livetime of the Kamiokande experiment was 2,079
days, with an observed 8 B solar neutrino flux of Φ8 B,obs =(2.80±0.19(stat.)±0.33(syst.)) ×106
cm−2 sec−1 . This observed value is 0.423±0.029±0.050 times the predicted value coming fromthe SSM, and once again confirmed the solar neutrino flux deficit, although slightly less so.
This could still be explained with solar neutrino flavour conversion. However, this is because
the Kamiokande experiment was sensitive to all flavours of solar neutrinos, allowing for alarger observed to expected ratio of solar neutrinos.
The Kamiokande experiment also had the ability to do energy reconstruction, and hence
could search for distortions in the recoil electron energy spectrum, coming from neutrino
oscillation. Figure 2.7 shows the observed recoil electron energy spectrum, with no deviation
found beyond the uncertainties. Kamiokande also searched for a difference between the22

Figure 2.7: Recoil electron energy spectrum in Kamiokande, shown as the ratio of the measured flux to the SSM expectation [14]. The hatched area shows the systematic uncertainty.

daytime and nighttime solar neutrino flux. Figure 2.8 shows the daytime and nighttime
solar neutrino fluxes. Because of the limited statistics, no difference during the daytime andnighttime was found, within the uncertainties. So once again, the solar neutrino problem
could be confirmed but not solved, with neutrino flavour conversion still a likely candidate
for the solution.

2.4.3 The Gallium Experiments
The Soviet-American Gallium Experiment (SAGE) in Russia [16] and the Gallex/Gallium
Neutrino Observatory (GNO) expermients in Italy [17][18] used a gallium (Ga) target for
detecting neutrinos via their absorption on Ga. The reaction isνe +71 Ga → e− +71 Ge,(2.14)

which has a threshold energy of 0.233 MeV. This was the main advantage of the Ga experiments, as this low energy threshold makes it possible to detect pp solar neutrinos. It is23

Figure 2.8: The solar neutrino flux measured at Kamiokande during the daytime and nighttime [14]. The nighttime is also sub-divided into 5 regions. δsun shows the angle between the
direction from the Sun and the detector z-axis.

important to measure the pp neutrinos, because they are generated by the most fundamental
solar neutrino production reaction and the flux practically depends only on the solar luminosity. This makes the ambiguity in the calculated rate much smaller than for any of theother types of solar neutrinos. The predicted rate for any Ga experiment is given as 137+8−7
SNU.
The SAGE experiment ran from 1990 to 1992 and was located at the Baksan Neutrino
Observatory in Russia, at a depth of 4,715 mwe. It used 55 tons of liquid metal Ga as thetarget, housed in seven chemical reactors heated to 30◦ C. This was so the Ga metal would
remain molten. The detector was left to expose for about 3 weeks, at which time the resulting
Ge atoms were then extracted and counted by a proportional counter observing the Auger
electrons and x-rays emitted by the 71 Ge electron capture. Figure 2.9 shows the SAGE results+2.6
for each year, with the total combined observed rate given as (65.4+3.1−3.0 (stat.)−2.8 (syst.)) SNU.

This amounts to a ratio of the observed rate to the expected rate of 0.504±0.073+0.036−0.051 .
The Gallex experiment began observation in 1991 and ran until 1997. It was then replaced24Figure 2.9: SAGE results for each year [16]. The shaded band is the combined best-fit andits uncertainty for all years. The error bars shown for individual measurements are statisticaluncertainties only.

by GNO, which ran from 1998 until 2003. The Gallex/GNO detector was located in the Gran
Sasso Underground Laboratory in Italy, at a dept of 3,300 mwe. It used a 100 ton target
of GaCL3 -HCl solution, which contained 30.3 tons of Ga. In a similar fashion to the SAGE
procedure, the detector was left to expose for a period of 4-6 weeks, after which time the
Ge atoms were extracted and counted. Figure 2.10 shows the combined Gallex/GNO result,with the combined rate of Gallex and GNO found to be (69.3±4.1(stat.)±3.6(syst.)) SNU,
which is 0.556±0.047±0.036 the expected rate. The combined average rate coming from
SAGE, Gallex and GNO is calculated as (66.1±3.1) SNU. This is about half the predicatedvalue, again confirming the solar neutrino problem, but providing no answers.

2.4.4 The Super-Kamiokande Experiment
The Super-Kamiokande (SK) experiment [19] is the world’s largest water Cherenkov detector. Weighing in with 50,000 tons of ultra-pure water and viewed by over 11,000 50 cm
PMTs, it is the successor of the Kamiokande experiment, taking data from 1996 and stillrunning. SK is divided into 4 different phases. SK-I, II and III have already been completed25

Figure 2.10: The combined Gallex and GNO result [18]. The error bars are statistical errors
only.

and analyzed, while SK-IV is currently in operation. The extreme size difference between
Kamiokande and SK gives SK a tremendous statistical advantage. This allows for solar
neutrino spectral shape analysis, as well as the possibility to search for the solar neutrino
day/night effect. SK and SNO (described in the next section) were the first experiments
to give indication that the solar neutrino problem could be solved by a theory of neutrino
flavour conversion. SK will be descibed in great detail in the coming chapters, and hence isonly mentioned here.

2.4.5 The Sudbury Neutrino Observatory Experiment
The Sudbury Neutrino Observatory (SNO) experiment [20] was a Cherenkov detector
utilizing 1,000 tons of heavy water (D2 O), running from 1999 to 2006. It was located 2,092m (6,010 mwe) below the surface in the Creighton mine in Sudbury, Canada. The SNO

26

detector was able to observe 8 B neutrinos via three different reactions;νx + d → νx + p + n(Neutral Current (NC)),(2.15)

νe + d → e− + p + p(Charged Current (CC)),(2.16)

νx + e− → νx + e−

(Elastic Scattering (ES)),(2.17)

where x = e,µ,τ . The characteristics of each of the reactions are;• Neutral Current (NC):
Equally sensitive to all three active neutrino flavours. This reaction can determine
the total 8 B neutrino flux (Φ8 B ) independently of any specific active neutrino flavour
oscillation hypothesis. SNO used three different methods for observing the neutron
coming from this reaction, marking the three different phases of SNO.
• Charged Current (CC):Only sensitive to νe and observed via the Cherenkov light coming from the resulting e− .
Comparing this rate to the NC reaction rate, it is possible to determine the neutrino
survival probability as a function of energy.
• Elastic Scattering (ES):Sensitive to all neutrino flavours, although the cross section for νe is approximately six
times larger than that for νµ,τ .

The SNO detector operated in three different phases, each characterized by the method of
detection of the neutron coming from the NC interaction (Equation 2.15). The three phasesare;• Phase 1 : n + d → t + γ (6.25 MeV) (D2 O),
27

• Phase 2 : n +35 Cl →36 Cl + γ (8.6 MeV) (Salt),• Phase 3 : n +3 He → p + t + 0.76 MeV (NCD array).

The first phase utilized the capture of the neutron on a deuteron inside the heavy water,
releasing the 6.25 MeV gamma, easily observable inside the SNO detector. In the second
phase, more than 2 tons of salt was added to the detector, giving a neutron capture efficiency
on 35 Cl of ∼83%. The neutron capture on Cl releases a gamma cascade peaked around 8
MeV in energy. In the third and final phase of SNO, neutral current detectors (NCDs),
made up of helium three (3 He) proportional counter tubes, were hung in a grid within theD2 O. 3 He has a very large cross section for the capture of thermal neutrons, producing anenergetic proton-triton pair ultimately resulting in an electrical pulse in the counter wire.
SNO measured the 8 B solar neutrino flux through the NC interaction rate as
+0.36
6
−2
−1
(5.54+0.33(although recent SNO results have revised this−0.31 (stat.)−0.34 (syst.)) ×10 cm sec

number). By subtracting the νe ES rate (measured by SK) from the total NC rate, SNO
and SK showed that two-thirds of the electron flavour neutrinos produced in the Sun werechanging into other flavours. Likewise, the ES rate minus the CC rate weighted difference
also pointed to neutrino flavour change, and was in good agreement with the NC rate minus
the CC rate. The NC rate experimentally confirmed the theoretical SSM model of neutrino
emission for the 8 B flux. Along with the SK measurement of the ES rate, it was shownfor the first time that the solution to the solar neutrino problem was indeed solar neutrino
flavour conversion. Experiments would now set out to determine the exact mechanism of
this solar neutrino flavour converion, most likely in the form of neutrino oscillation. The
various SNO solar neutrino results will be discussed in detail in Sections 2.5.1, 8.4, 8.5 and
10.4.

28

Figure 2.11: Borexino MC simulation based fit results in the units of counts/day/100 ton,spanning the energy range of 270 to 1600 keV [22].

2.4.6 The Borexino Experiment
The Borexino experiment [22, 24] is a liquid scintillator experiment located in the Laboratori Nazionali del Gran Sasso (LNGS). It is placed 3,800 mwe underground and has beenrunning since 2007. The detector is filled with ∼278 tons of organic liquid scintillator, andis able to observe the sub-MeV 7 Be solar neutrinos by means of neutrino-electron elastic
scattering. The 7 Be monoenergetic neutrinos have energy of 863 keV, and are predicted by
all SSMs to be the second most frequently produced neutrino species (after the primary pp
neutrinos) in the Sun. The flux is about 1,000 times that of 8 B neutrinos. Because they aremonoenergetic, the theoretical predictions of neutrino oscillation are more unique and much
simpler to model.
Borexino uses a Monte-Carlo (MC) simulation based fit over the energy range of 270 to1600 keV, as shown in Figure 2.11, to extract the 7 Be flux. A fit done over the energy
range of 1 to 1.5 MeV is used to extract the pep and CNO fluxes. The Borexino experiment29

obtained the 7 Be, pep and CNO neutrino fluxes as

7

Be rate (862 keV line):

46.0 ± 1.5(stat.)+1.5−1.6 (syst.) counts/day/100 ton→ 2.78 ± 0.13 × 109 cm−2 sec−1 ,pep rate:

3.1 ± 0.6(stat.) ± 0.3(syst.) counts/day/100 ton→ 1.6 ± 0.3 × 108 cm−2 sec−1 ,CNO rate:(2.18)(2.19)< 7.9 counts/day/100 ton→ < 7.7 × 108 cm−2 sec−1 (95% C.L.).

(2.20)The 7 Be rate is 0.51±0.07 that of the predicated rate coming from the SSM. The CNO flux
is consistent with both the low and high metallicity SSMs, giving no sensitivity to distinguishbetween the two. Further solar neutrino oscillation analysis results from Borexino will be
discussed in the next section.

2.5 Current Neutrino Oscillation Parameter Results
2.5.1 Results from Solar Neutrino Experiments
Based on current solar neutrino experimental data, various types of neutrino flavour
oscillation analyses can be carried out. These include analyses of the total rate, spectral
shape and time variation measurements. The most immediate and obvious approach attrying to determine the solar neutrino oscillation parameters ∆m221 and θ12 (more commonly
expressed as sin2 θ12 or tan2 θ12 ) is to compare the observed rate before and after taking
oscillations into account. The regions within the plane of oscillation parameters allowed by
the Homestake, Gallium and Kamiokande rate measurements are separately distributed at
the small mixing angle (SMA), the large mixing angle (LMA) and the low mass difference30

Figure 2.12: Locations of possible solutions for the solar neutrino oscillation parameters(tan2 θ12 vs. ∆m221 ).

(LOW) solutions. The location of these three regions, along with the location of the vacuum
solution, are shown in the plane of oscillation parameters in Figure 2.12. The region of theplane above ∆m221 > 10−9 eV2 is where matter oscillation effects are important.
SNO is able to express the power of their solar neutrino measurements through an oscillation analysis. The SNO collaboration evaluates the observed interaction rates and theratios between them, as well as the shape of the νe survival probability as a function ofenergy and its day/night difference. SNO by itself cannot distinguish between the LMA and
LOW solutions [20]. Because Borexino has only carried out an oscillation analysis using themonoenergetic 7 Be neutrinos, they do not have the ability to look at spectral shape information. However, it is much easier for them to look at the rate difference, both between theobserved and expected rates, as well as the separate rates during the day and night times.
At the low energies of the 7 Be neutrinos, Borexino would expect a large day/night effect(∼12%) for oscillation parameters within the LOW solution, which it fails to observe. For
this reason Borexino excludes the LOW solution at the 8.5 σ level [23].
31The previous results from the first three phases of the SK experiment used the observed
spectral shape and time variation measurements to pick out oscillation parameter solutions.
By constraining the 8 B neutrino flux to the NC current measurement of SNO and the hep
neutrino flux to the SSM prediction, SK is able to select the LMA solution over the LOW and
SMA solutions, at more than the 3 σ confidence level. With the inclusion of all other solar
neutrino data into a global solar neutrino data oscillation analysis (solar global analysis), the−5best-fit oscillation parameters within the LMA solution are found to be ∆m221 =6.2+1.1−1.0 × 10

eV2 and sin2 θ12 =0.31±0.01 [25].

2.5.2 The KamLAND Experiment
The Kamioka Liquid Scintillator Antineutrino Detector (KamLAND) experiment [26] waslocated inside the Kamioka mine, in the same experimental hall which housed Kamiokande.
The experiment started data taking in 2002 and observed the electron antineutrino ν̄e flux
coming from all the nuclear reactors within Japan (and also from Korea) until 2007. ν̄e willinteract inside the detector most commonly through the inverse beta reaction ν̄e +p → e+ +n.
The positron emits prompt scintillation light, while light from a 2.2 MeV gamma-ray isemitted when the neutron is captured by a proton. The KamLAND experiment was able todetermine a precise value for the neutrino oscillation parameter ∆m221 , and place stringent
limits on sin2 θ12 , by directly observing a distortion in the reactor ν̄e energy spectrum. The
observed reactor energy spectrum from 2002 to 2007 is shown in Figure 2.13. KamLAND−5
selects the best-fit solar neutrino oscillation parameters as ∆m221 =7.49+0.21
eV2 and−0.19 × 10

sin2 θ12 =0.309+0.40−0.29 . Under the assumption of CPT invariance, neutrinos and antineutrinos
should oscillate with the same mixing parameters. A combination of all current solar neutrino
data with the KamLAND reactor antineutrino data [25] (solar+KamLAND) then yields the32

Figure 2.13: Reactor ν̄e energy spectrum obtained by the KamLAND experiment, along withthe best-fit oscillation parameters case [26].

best-fit solar neutrino oscillation parameters as∆m221 = (7.6 ± 0.2) × 10−5 eV2(2.21)

sin2 θ12 = 0.31 ± 0.01.

(2.22)

2.5.3 Other Neutrino Oscillation Mixing Parameters
While the two flavour neutrino approximation describes well solar neutrino oscillation, it
is already a well known fact that there are indeed three flavours of neutrinos. This means
there are still the four other neutrino oscillation parameters which are not directly measured
by solar neutrino measurements. The atmospheric mass squared splitting term ∆m232 ismeasured precisely by the Main Injector Neutrino Oscillation Search (MINOS) experiment as−3|2.32|+0.12
eV2 [27], using both atmospheric neutrinos and accelerator muon neutrinos−0.08 × 10

and antineutrinos. MINOS and other experiments do not have the power to determine the33

Figure 2.14: Best-fit and allowed neutrino oscillation parameters coming from MINOS andSK [28]. Recent results from the Tokai to Kamioka (T2K) experiment are also shown.

sign of the parameter at this time. The atmospheric mixing angle θ23 is measured by SK
atmospheric neutrino data as sin2 θ23 =0.5+0.83−0.93 , with the exact quadrant of the mixing angle
still unknown [28]. Figure 2.14 shows the ∆m232 vs. sin2 θ23 plane of allowed oscillation
parameters coming from the MINOS and SK measurements [27]. It is important to notethat these mixing parameters have a negligible effect on the oscillation of solar neutrinos.
The two remaining neutrino oscillation parameters θ13 and δ have been two of the main
focuses of neutrino physics over the past few years. However, these parameters play little
to no role in the oscillation of solar neutrinos. Although there have been weak limits placedon the value of δ, it still remains unmeasured, while θ13 has seen tremendous success injust the past couple of years. First with the verification of a non-zero θ13 value by theT2K experiment [29], and then with very significant measurements being made by the shortbaseline reactor antineutrino experiments Double Chooz [30], Daya Bay [31] and the Reactor
Experiment for Neutrino Oscillation (RENO) [32]. Taking the average of the three shortbaseline experiments gives the value of sin2 (θ13 )=0.025±0.003 [10]. At this value of θ13 , theeffect on solar neutrino oscillation is quite small.

34

2.6 Solar Neutrino Measurements in this Thesis
Direct observation of matter enhanced solar neutrino oscillation effects, using the SK
detector, is the main goal of this thesis. The solar neutrino problem was confirmed by all
experiments, and finally resolved with the measurements of SK and SNO showing that the
produced solar νe ’s were in fact changing flavour into νµ,τ ’s. But just showing that neutrinoschange flavour does not demand that the governing process be neutrino oscillation, and more
specifically MSW oscillation. The current best description to all solar neutrino data points
to LMA oscillations parameters, at better than 5 σ. It would be very beneficial to thecommunity if one experiment could give smoking gun proof of MSW oscillation.
To prove that MSW oscillation is in fact the reason for the solar neutrino flavour conversion, two methods are available. First, as stated in Section 2.3, the MSW effect in the
Sun will cause a characteristic “upturn” in the survival probability of νe , as seen by the grayband in Figure 2.15. Although there is a step up in survival probability when going fromthe higher energy measurements to those at lower energy, the current measurements do notcover the range where the “upturn” is expected, and so it has yet to be seen. However, using
the SK detector, this “upturn” can be searched for by looking for the same “upturn” in therecoil electron energy spectrum. SK has worked to lower the energy threshold and increase
the precision of the measurement so this could be done.
Second, due to matter enhanced oscillation effects coming form the Earth, there is an
expected increase in the νe interaction rate when comparing the nighttime rate to the daytime rate. However, this day/night effect will depend on the exact values of the oscillation
parameters, as well as the detector position on Earth. This effect is expected to be about
3% at SK, for the observable energy range and oscillation parameters chosen from the global
solar analysis. SK measures the day/night asymmetry as the difference between the day flux

35Figure 2.15: Current solar neutrino experimental constraints on the low energy solar neutrino
νe survival probability (Pee ). The gray band shows the MSW-LMA predicted Pee with the 1
σ range estimated by [33].

ΦD and the night flux ΦN , over the average flux, given byΦD − ΦN.
(ΦD + ΦN )2

ADN = 1

(2.23)

Figure 2.16 shows the observed day/night asymmetry from the first three phases of SK,
which show consistency, but no clear sign of a non-zero day/night asymmetry.
This thesis will analyze the solar neutrino spectral data from the SK-IV phase, as well
as a reanalysis of solar neutrino spectral data from the SK-II and SK-III phases. The
observed recoil electron energy spectra are then combined with the spectrum of SK-I to doa full SK combined analysis of the solar neutrino recoil electron energy spectrum shape.
The SK spectral shape analysis will then be combined with the spectral shape analysis of
SNO. A search for the solar neutrino day/night effect using an unbinned extended maximum
likelihood fit to the solar zenith angle variation of the solar neutrino interaction rate is also
carried out for all phases. The results from all four phases are combined into one result for
the entire SK data set. The spectral and time variation data is then used to carry out an36

Figure 2.16: Measured day/night flux asymmetry, with statistical (blue) and systematic(red) uncertainties, for SK-I, II and III.

oscillation analysis, which combines all phases of SK together. The SK data is also combined
into one analysis with the SNO data, which is then combined with data from other solar
neutrino experiments to give a global solar neutrino analysis. The global analysis is then
combined with the KamLAND experiment, to attain the world’s best measurement of thesolar neutrino oscillation parameters.

37

Chapter 3
The Super-Kamiokande Experiment
3.1 Experimental Overview
The Super-Kamiokande (SK) experiment [19] is a large ring imaging Cherenkov detector,
located 1,000 m (2,700 mwe) underground, inside an old zinc mine in Mount Ikenoyama
in Kamioka, Japan. The last three letters in Kamiokande now stand for both “Nucleon
Decay Experiment”, as well as “Neutrino Detection Experiment”. The coordinates of thedetector are 36◦ 25’33” N, 137◦ 18’37” E and it sits at an elevation of 371.8 m above sea
level. Figure 3.1 shows the location of SK within Japan, with Tokyo shown for reference.
SK is the largest of many underground experiments at the Kamioka Observatory, whichis part of the University of Tokyo’s Institute for Cosmic Ray Research (ICRR). The ∼130
collaborators on the experiment include researchers and engineers from Japan, The U.S.,
Korea, China, Poland, Canada and Spain. As a result of being deep underground, SK takes
advantage of the low cosmic-ray muon rate (2.2 Hz), which is a common background in manyof the different analyses taking place. SK has performed analyses on atmospheric, solar andaccelerator neutrinos, as well as searches for supernova neutrinos (from bursts as well as
relics), proton decay and other exotic particles. SK is largely funded by Japan’s Ministry of
Education, Culture, Sports, Science and Technology and in part by the U.S. Department ofEnergy and the U.S. National Science Foundation. The Research Foundation of Korea, theKorean Ministry of Science and Technology, the National Science Foundation of China andthe Spanish Ministry of Science and Innovation provide addition funding.
Figure 3.2 shows the experimental layout of the SK experiment, along with a perspective
38

Figure 3.1: The SK experimental site location within Japan.

Figure 3.2: Layout of the SK experimental site [19].

39view of its placement inside the mountain peak. The SK site is accessed via a 1.5 km
horizontal tunnel by certified, non-spark, diesel mine vehicles. The entire experimental areaof SK is made up of four separate sections. The main cavity of SK houses the 41.4 m talland 39.3 m diameter cylindrical stainless steel tank. On top of the tank there is a set offour electronics huts, which house all of the equipment necessary to power and read out the
PMTs. Also housed on top of the SK tank are the necessary tools needed for calibration of
the detector. Directly adjacent to the main cavity is a small set of connected rooms, whichhouse the computers for controlling the detector, as well as the network devices necessary
to export the data out of the mine. On the opposite side of the SK tank from the control
room sits the linear electron accelerator (LINAC) room. This houses the LINAC necessary
for the energy scale calibration at the lowest energies. The final experimental space is thewater system room, which is outside of the main SK experimental area across the mine road.

3.2 Solar Neutrino Detection Inside Super-Kamiokande
In SK, solar neutrinos are detected via neutrino-electron (ν-e) elastic scattering, given by
the equation
ν + e− → ν + e− .

(3.1)

All three flavours of neutrinos will feel this interaction, with the cross section described by
G2 me
Te 2
me Te
dσ
= F [A0 + B0 (1 −) + C0 2 ].
dTe
2π
Eν
Eν

(3.2)

GF , me , Eν and Te are the Fermi coupling constant, the electron mass, the incident neutrinoenergy and the kinetic energy of the recoil electron, respectively. The factor

40

G2F me2π

is equalto 4.31 × 10−45 cm2 /MeV, and the parameters A0 , B0 and C0 are define as
A0 = (gV + gA )2 ,(3.3)

B0 = (gV − gA )2 ,(3.4)

C0 = (gA2 − gV2 ),

(3.5)

1
gV = 2 sin2 θW + ,2
1
gV = 2 sin2 θW − ,21
21
gA = −
2

gA = +for νe ,(3.6)

for νµ,τ .

(3.7)

θW is the Weinberg angle and is given by sin2 θW = 0.2317 [34].
Figure 3.3 shows the differential cross section of (ν-e) elastic scattering for an incident
neutrino energy of 10 MeV, for both νe (solid) and νµ,τ (dashed), with the horizontal axis
giving the kinetic energy of the recoil electron (Te ). To calculate the differential cross sections
seen in Figure 3.3, radiative corrections of the one-loop electroweak, QCD and QED processes
have been considered. This reduces the relative probability of observing recoil electrons by∼4%, for the energy region considered in SK. It is also important to note that the crosssection of (νe -e) scattering is ∼6 times larger than that of (νµ,τ -e) scattering, because νewill elastically scatter via both the neutral and charged weak currents, while the νµ,τ will
elastically scatter via the neutral current only.
The total cross section of (ν-e) elastic scattering can be calculated as a function of incident
neutrino energy by integrating Equation 3.2, for recoil electron energies between 0 and Tmax .
Tmax is limited by the incident neutrino energy and the mass of the electron by

Tmax =Eν
me .
1 + 2Eν(3.8)

Figure 3.4 shows the total elastic scattering cross section for the case of νe (solid) andνµ,τ (dashed). The total cross section for νe is also ∼6 times larger than that for νµ,τ , due
41dσ/dTe (x10-462

cm /MeV)

100
75
50
25
0

0

2.5
5
7.5
10
Recoil Electron Kinetic Energy(MeV)

Figure 3.3: Differential cross section of (νe -e) (solid) and (νµ,τ -e) (dashed) elastic scatteringin the case of 10 MeV incident neutrino energy.

to the inclusion of the charged current interaction for νe .
Because SK observes (ν-e) elastic scattering by observing the recoil electron only, andbecause the energy of the recoil electron can be anything between 0 and Tmax , the 8 B solar
neutrino energy spectrum must be converted to an energy spectrum of the observed recoil
electron. This energy spectrum is described by
Z Eν,maxφ(Eν )F (Te )dTe =0dσ
dEν dTe ,
dTe

(3.9)

where φ(Eν ) is the 8 B solar neutrino spectrum shown in Figure 3.5. For the shape of
the 8 B solar neutrino spectrum, the calculation based on the α-spectrum measurement of
8

B → 8 Be∗ + e+ + νe , 8 Be∗ → 2α by Winter [35] has been adopted. For comparison, thebottom panel of Figure 3.5 shows the experimental uncertainty of the Winter spectrum(black band) and the ratio between the Winter spectrum and the Ortiz spectrum [36]. The
resulting recoil electron energy spectrum observed in SK is shown in Figure 3.6.

42

Figure 3.4: Cross section of (νe -e) (solid) and (νµ,τ -e) (dashed) elastic scattering as a functionof incident neutrino energy.

Figure 3.5: The upper figure shows the neutrino energy spectrum measured by Winter [35].
Lower plot shows the experimental uncertainty of the Winter spectrum (black band) andthe ratio between the Winter and Ortiz [36] spectra and its ±1 σ experimental uncertainty(dashed line).

43

events per day (32.4 ktons)

10

2

10
1
10
10
10
10
10

-1
-2
-3
-4
-5

0

2.5

5

7.5

10 12.5 15 17.5 20
electron total energy

Figure 3.6: The expected number of events per day as a function of recoil electron totalenergy, coming from the elastic scattering of 8 B (hep) neutrinos with electrons in black(red).

For the case of (ν-e) elastic scattering, the angle between the incoming neutrino and therecoil electron is given by
e
1+ m
Eν
.
cos θ = pe
1 + 2mTe(3.10)

The angular distribution of the recoil electron is then given by
Z Eν,max
F (θ)dθ =φ(Eν )
0dσ dTedEν dθ.
dTe dθ

(3.11)For the energy range of 8 B solar neutrinos, the angular distribution described by (3.11)
will be highly peaked at zero. This means the recoil electron more or less follows the directionof the incoming neutrino (the reconstructed direction will depend on the amount of multiple
Coulomb scattering). SK uses this angular distribution as a handle for statistically selectingout the solar neutrino signal from the background. The energy and angular distributions

44

of the recoil electrons, described by Equations 3.9 and 3.11, are used as inputs to the MC
simulation of 8 B and hep solar neutrinos (see Section 5.4).
Recoil electrons coming from the elastic scattering interaction are detected in SK bymeans of their emitted Cherenkov radiation. The radiation is detected using many PMTs.
Cherenkov radiation is light which is 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). The energy
threshold for an electron to emit Cherenkov radiation in water (n=1.33) is 0.767 MeV,
much below the energy threshold of SK. For any charged particle, it is possible to calculate
the number of Cherenkov photons generated per unit wavelength dλ per unit distance thecharged particle travels dx using


2πα
1
2πα
d2 N
= 2 1−= 2 sin2 θC .
2dxdλ
λ(n(λ)β)λ(3.12)

β = v/c, α is the fine structure constant and the refractive index n is now wavelength
dependent n(λ).
The total number of produced photons can then be calculated by integrating over the
charged particle travel distance and all wavelengths. The right side of Equation 3.12 substitutes in the parameter θC , which is known as the Cherenkov opening angle and describes the
direction the emitted photons travel relative to the direction of the charged particle. The
resulting Cherenkov radiation will then form a cone with an opening angle of 2θC aroundthe charged particle’s trajectory. The opening angle can be written ascos θC =

c
,
n(λ)v

(3.13)

where it is easily seen to vary with the charged particle’s velocity. For a recoil electron
coming from the elastic scattering of a solar neutrino in SK (and in general in water), theopening angle of the Cherenkov radiation cone will be θC ≈ 42◦ .
45

3.3 The Super-Kamiokande Detector
3.3.1 Detector Tank and Volumes
The SK 41.4 m (39.3 m) in height (diameter) cylindrical detector tank is constructed outof stainless steel and consists of a total volume of 50,000 m3 . It is filled with ultrapure water,
and utilizes the Cartesian coordinate system described in Figure 3.7. It sits inside a large
excavated cavity which is used as a structure for the tank (see Figure 3.2). After excavation
the cavity was plastered with 40-50 cm thick, reinforced concrete on the side walls and thefloor and then laid with 3-4.5 mm thick stainless steel plates. The top of the tank is a trussed
style ceiling, also covered by stainless steel plates. To eliminate effects on the PMTs comingfrom the geo-magnetic field, 26 sets of Helmholtz coils are arranged in the tank. Above
the tank the cavity was excavated in a dome shape high above the tank top, and the rockcovered with “Minegaurd”. This is a polyurethane material made by the Canadian company
Urylon, and blocks radioactive radon gas emitting from the rock. Radon gas is the mostserious background for the solar neutrino analysis. The tank is also sealed tightly to keepradon gas from entering the tank. The dome area gives ample space above the tank top forthe placement of PMT cables, electronics huts, calibration equipments and general storage.
Two meters from the inside tank wall sits a wide, trussed support structure which runs
the entire height of the tank. The main purpose of this structure is to house the PMTs used
to collect the Cherenkov light and to divide the detector into two distinct volumes, the innerdetector (ID) and the outer detector (OD). The ID is the 32,000 m3 main inner portionof the tank, which is viewed by ∼11,100 50 cm PMTs (the number varies with each SK
phase). Figure 3.8 shows a picture of the ID of SK. The 2 m OD region is viewed by 1,885
20 cm PMTs and is used as an active veto for identifying radioactive background comingfrom the rock, and to tag cosmic-ray muons entering the detector. In order to increase
the light detection efficiency in the OD, the walls of the OD are lined with white tyvek
46

5-FGVGEVQT\$CTTGN

6QR

TZ
[XGTVGZFKTGEVKQP

\

[ Z
Z

[

φ

$QVVQO

Figure 3.7: Definition of the Cartesian coordinate system used in SK.

Figure 3.8: Picture of the SK ID taken from the bottom of the detector.

47

sheet possessing a reflectivity above 80%. Behind the PMTs of the ID, black polyethylene
terephthalate sheets (black sheets) are hung, and along with the white tyvek sheets in the
OD, optically separates the ID from the OD. Figure 3.9 shows a schematic views of the
PMT support structure and the placement of the PMTs, black sheets and tyvek sheets on
the structure.

3.3.2 Water Purification System
The SK tank water is originally drawn from two streams inside the Kamioka mine, which
are sourced by the natural seepage of rain and snow melt through the mountain rock. Before
filling the tank, the water is pumped through a water purification system to remove contaminants and make it ultrapure. It is important for the water in SK to be as pure as possible,
as the large light travel distances causes the water transparency to have a large impact on
the collection of the Cherenkov light. It is also important to remove the contaminants asthey may be radioactive (such as radon 222 Rn) and can contribute to unwanted background
events occurring throughout the detector volume.
Once the purified water has filled the tank, it is continuously circulated through the water
purification system with a flow rate of 60 tons/hour (1 ton=1 m3 of water), in order to keep
the water as pure as possible. The water injection and removal points and rates within the
SK tank are shown in Figure 3.10, and have been tuned to reduce convection within thetank. Convection within the tank can cause radioactive elements such as Rn to enter into
the fiducial volume from the support structure and the PMT glass and cases, contaminating
the final data sample. A schematic view of the SK water purification system can be seen in
Figure 3.11. The main components are as follows:• 1 µm Filters: Series of mesh filters which remove dust (∼1µm) and small particles.

48

Figure 3.9: Schematic view of one unit of the PMT support structure [19]. The thicknessof the structure is about 55 cm. Each unit holds twelve 50 cm PMTs for the inner detector
and two 20 cm PMTs for the outer detector.

49

Figure 3.10: Water injection and removal points and rates within the SK tank.

Figure 3.11: Schematic view of the SK water purification system.

50

Figure 3.12: Time dependence of the supply water temperature. In November of 2009 the
new temperature control system was installed.

• Heat Exchanger (HE): The water temperature rises due to the heat generated bypumps, PMTs, the surrounding rock and so on. The temperature of the supply water is controlled with an accuracy of 0.01◦ C, to further suppress the convection insidethe detector. This is done with the use of a new temperature control system, installed in November, 2009. Figure 3.12 shows the time dependence of the supply watertemperature and the large effect the new control system has on making the supply
water temperature more stable. The processed water temperature is about 13.5◦ C,
low enough to suppress bacteria growth.
• Ion Exchanger: Resin for the high molecular, three dimensional network structure
within the ion exchange group. It is able to remove ions such as Na+ , Cl− , Ca+2 andothers.
• UV Sterilizer: Uses ultraviolet radiation to kill bacteria.
• Rn-free-air Dissolving System: Dissolves radon free air into the water to increase theefficiency of the Rn removal of the vacuum degasifer.
• Reverse Osmosis (RO) filter: Further removal of contaminants heavier than 1,000
molecular weight. The RO is newly added during the SK-III phase.
• Vacuum Degasifier: Removes gas components dissolved in the water. The main purpose51

is to remove Rn gas, which it does by ∼96%. It is also able to remove ∼99% of oxygen.
• Ultra Filter (UF): Further removal of small particles down to the size of 10 nm.
• Membrane Degasifier: Removes radon gas with ∼83% efficiency. After going throughthe entire purification system, the water’s Rn concentration is reduced to less than
0.4±0.2 mBq/m3 .

3.3.3 Radon-Reduced Air Systems
The ambient air in the mine is rich with Rn gas, which is continuously emitted from thesurrounding rock. Because of the different air flow patterns within the mine, air blows out of
the mine during the summer and into the mine during the winter. Because of the changing
air-flow patterns, the concentration of Rn gas inside the mine is ∼1500 Bq/m3 during thesummer time and ∼30 Bq/m3 during the winter time. This seasonal dependence is shownin Figure 3.13. The level during the summer time is not only high enough that it can cause
problems with analyses, but it is also a health hazard. However, air outside of the mine has
a constant Rn concentration of 10-30 Bq/m3 , throughout the entire year. For this reason a
special air circulation system consisting of a large air blower, air filters and a heat exchanger,
are housed outside of the mine. It is used to force the fresh air from outside the mine into
the SK experimental area, at a rate of 70 m3 per minute. As a result, the Rn concentrated
mine air is displaced from the experimental area and the Rn level is typically kept at 20-30
mBq/m3 .
Rn-reduced air is also pumped into the ∼60 cm air gap between the top of the stainlesssteel detector tank and the water’s surface, as Rn within this space can dissolve into thewater. The Rn concentration in the air is further reduced by a special air purification system(shown in Figure 3.14), which reduces the concentration to less than 3 mBq/m3 . The system
consists of an air compressor (compresses to 7.0-8.5 atm), a 0.3 µm air filter, a buffer tank, an52

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 3.13: Radon levels (in Bq/m3 ) throughout the year at the SK site[19]. The solid lineis the level in the mine. The dotted line is the level in the air-purged experimental areas.

air drier (to improve the efficiency of the Rn removal and to remove CO2 ), carbon columns(to remove Rn gas) and two 0.1 and 0.01 µm mesh air filters.
COMPRESSORCARBONCOLUMNBUFFERTANK

AIR FILTER(0.3mm)AIR DRIERCARBONCOLUMNHEATEXCHANGER

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

COOLEDCHARCOAL(-40 oC)

SUPER-KAMIOKANDE AIR PURIFICATION SYSTEM

Figure 3.14: Schematic of the air purification system at SK [19].

3.3.4 Photomultiplier Tubes
The ID is lined with more than ∼11,100 Hamamatsu R3600 PMTs, each with a photosensitive surface (the photocathode) 50 cm in diameter, covering ∼40% of the ID wall. These
53

PMTs were originally developed by Hamamatsu Photonics K.K. in cooperation with members of the Kamoikande experiment, although those used in SK have been further improved
over those used in Kamiokande. The size and shape of the PMTs are shown in Figure 3.15.
The bulbs of the PMTs are hand blown from borosilicate glass to a thickness of about 5 mm.
The photo-cathode is made of bialkali (Sb-K-Cs) and is sensitive to wavelengths between
300 and 600 nm, matching well the spectrum of Cherenkov light in water. The quantum
efficiency (QE) of the PMTs is shown in Figure 3.16, with a peak value of ∼21% between
360 and 400 nm.

~ 70000

cable length
φ 116

φ 82 2glass multi-seal

φ 254 10

photosensitive area > φ 460< φ 520

cable

water proof structure( 610 20 )

~ 720

(mm)

Figure 3.15: Diagram of an ID PMT [19].

A Venetian-blind type dynode is used because of the large photosensitive area of the PMT.
The large photosensitive area does have the downfall that the transit time becomes longer(about 50 nsec at the 1 photo electron (pe) light level), leading to a larger spread in the transit
times. For this reason, and because this type of PMT also has a smaller collection efficiency
for secondary electrons compared to others, the bleeder chain was optimized in order toachieve good timing response and collection efficiency. As a result of this optimization, an
11-stage voltage divider (with ratios of voltage division 8:3:1...:1) was used, and three typesof focusing mesh-plates between the cathode and the first dynode were adopted. This dynode
structure multiplies the ejected pe from the photo-cathode to create a gain of 107 electrons,54

Figure 3.16: The quantum efficiency of the ID PMTs as a function of wavelength, along with
the Cherenkov spectrum in pure water.

through an applied voltage of 1,700 to 2,000 V. The high voltage system for the PMTs
consists of a distributor (A933K), a controller (SY527) and an interface module (V288), allproduced by CAEN Co. The value of the high voltage is set so each PMT has the same gain,
is monitored every 10 minutes for every channel, and is continuously re-calibrated.
The resulting average collection efficiency is more than 70%, with a transit time spreadof about 2.2 nsec. Figure 3.17 shows the clear PMT signal coming from a one pe hit. It
is important in the solar neutrino analysis for the PMTs to have the ability to see 1 pe,and to have good timing resolution. This is because the number of photons arriving at a
PMT is generally one and the timing resolution directly affects the vertex reconstruction.
As previously mentioned, compensation coils line the SK tank, reducing the geo-magnetic
field to less than 100 mG at every PMT position, and giving the PMTs uniform response.
The average dark rate of the PMTs is ∼4.7 kHz, leading to ∼3 accidental hits (due to darknoise) within any 50 nsec time window. This dark noise rate is taken into consideration in

55

Figure 3.17: One pe signal of the ID PMTs.

the MC simulation program. The PMTs are read out via 70 m coaxial cables which wind up
the support structure to the top of the tank and into one of the four electronics huts, where
the data acquisition system (DAQ) is housed.
The OD PMTs are Hamamatsu R1408 PMTs with a photo-cathode 20 cm in diameter.
To further increase the detection efficiency of the OD PMTs, an acrylic wavelength shifting
plate of dimensions 60 cm×60 cm is placed around the bulb of each of the PMTs. The plates
are doped with 50 mg/l of bis-MSB and serve to absorb light in the ultraviolet region andisotropically re-emit about 55% of the photons with a longer wavelength, better matching
the QE of the PMTs. This improves the collection efficiency by a factor of 1.5.

56

3.3.5 Data Acquisition System
Original Data Acquisition System

The original layout of the SK data acquisition system is shown in Figure 3.18. The basic
unit of the front-end electronics, called the analog timing modules (ATM), was developed
based on Tristan KEK Online (TKO) standards. The integrated charge and timing information are digitized by an analog to digital converter (ADC) housed on board the ATM.
One ATM processed the signals coming from 12 PMTs, and so about 1,000 ATMs were used
in total. Figure 3.19 shows the schematic view of the analog input of one ATM channel.
Input signals were amplified 100 times and divided into four signals by the hybrid integrated
circuit (IC).
One of the four divided input signals was sent through a discriminator with a threshold
set to 100 mV, corresponding to ∼0.32 pe. If this threshold was exceeded, two pulses weregenerated. One of these pulses had a width of 200 nsec and a heigth of -15 mV. ThisHITSUM signal was given a 200 nsec width because event durations inside SK last up to 200
nsec (the time it takes light to cross the longest distance in the detector). The pulses of the
input 12 PMT channels were then summed and output as a signal called the HITSUM. The
HITSUM signals from the ATMs were eventually summed in the central electronics hut andif a threshold was exceeded, a global trigger was generated. Figure 3.20 shows the overview
of the global trigger generation.
Originally there were three levels of the global trigger, called the high energy trigger (HE),the low energy trigger (LE) and the super low energy trigger (SLE). These triggers were
determined by setting different threshold values for the summed HITSUM. The thresholdfor each global trigger type, for each phase of SK (phases described in Section 3.4) using theATMs, is given in Table 3.1. The second of the two pulses had a 900 nsec width and -15 mV

57

Analog sum of 12PMT signals
20-inch PMT

:

:
Sparc

KPVGTHCEGSCH(NCUJ#&%ATM

8ch(12x8=96PMT)/board

online CPU(slave)

x 15
x 20
(NCUJ#&%

x 240

ATM

8/'

8ch x15 x8 =960ch

GONG
20-inch PMTSCHATMinterface
Sparc

x 20
x 240

SMP

ATM

online CPU(slave)

SMP

GONG

SMP
SMP
SMP

Fast Ethernet

SMP

x6

Super Memory Partner

VME
20-inch PMT
SCH

Sparc

ATM

online CPU(slave)

x 20
x 240

Sparc

ATM
GONG

online CPU(host)

Analog Timing Module

online CPU(slave)
TKO
Fast Ethernet

Sparc

Sparc

online CPU(slave)Sparc

online CPU(slave)

Fast Ethernet

Sparc

online CPU(slave)Sparc

online CPU(slave)

online CPU(slave)interface
Sparc
SMP

Sparc

online CPU(slave)

SMP
SMP
SMP
SMP
SMP

interface

Super Memory Partner

VME
VME

interrupt reg.
TRG

20-inch PMT
SCH
TRIGGERATM

TRIGGERx 20

PROCESSORHIT INFORMATION

x 240(HITSUM)

ATM
GONG

Analog Timing ModuleTKO

PMT x 11129

ATM x ~1000

SMP x 48

online CPU(slave) x 17

FADC x 120

Figure 3.18: Layout of the data acquisition system of SK-I, II and III.

58

to PMTSUM
HITOUTDISCRI.

CURRENT SPLITTER&AMPLIFIER

PMT INPUT

W:200ns

to HITSUM

ONE-SHOT HIT
W:300ns

SELF GATESTART/STOP

TAC-A

TAC-A

CLEAR
GATE

QAC-A

QAC-A

to ADCSTART/STOP

TAC-B

TAC-B

CLEAR
GATE

THRESHOLD

QAC-B

QAC-B

DELAY

GATE
TRIGGERPED_START

Figure 3.19: Schematic view of the input and output channels of the ATM.

Figure 3.20: Overview of the global trigger generation during SK-I, II and III.

59

height, called the selfgate signal and was used as the trigger to start the charge to analog
converter (QAC) and time to analog converter (TAC) integration.
Table 3.1: Trigger thresholds in each SK data taking period. *The SLE threshold was
lowered time to time during SK-I, the final value is shown here.

SLE
LE
HE

SK-I
186 mV*
320 mV
340 mV

SK-II
SK-III
110 mV 186 mV
152 mV 302 mV
180 mV 320 mV

The remaining two divided signals were sent to a QAC, which integrated the charge for
400 nsec, and to a TAC to generate the hit timing. If a global trigger was generated within
1.3 µsec, the integrated QAC and TAC information was digitized by an ADC. The final of
the four signals was output as a PMTSUM signal, corresponding to the sum of the input
PMT signals, for each ATM board, and was run into a flash ADC (FADC) that recorded
the waveform information.

New Data Acquisition System

To achieve stable observation for the next 10∼20 years, and to improve the sensitivity ofthe detector, new front-end electronics were installed in September, 2008. The existing ATMs
were replaced with QTC-based electronics with ethernet readout modules (QBEEs). The
QBEEs are housed in the same TKO crates used to house the ATMs, but the readout scheme
is completely changed. One QBEE module processes signals from 24 PMTs. Figure 3.21
shows the block diagram of signal processing in the QBEEs. The essential components on
the QBEE module for the analog signal processing and digitization are the charge-to-time
converter (QTC) ASIC [37] and the multi-hit time-to-digital (TDC), respectively.
The QTC detects the PMT signals and drives output timing signals whose width repre-

60

Figure 3.21: Block diagram of the signal processing in the QBEE module.
Table 3.2: Observable charge range and resolution for each of the three QBEE input channels.
Gain Channel
Small
MediumHigh

Observable Charge Range
0∼51 pC
0∼357 pC
0∼2500 pC (∼1250 pe)

Charge Resolution
0.1 pC/count (∼0.05 p.e./count)
0.7 pC/count (∼0.35 p.e./count)
4.9 pC/count (∼2.5 p.e./count)sents the integrated charge of the PMT signal. To satisfy the physics requirements of keeping
good charge resolution at the few pe level, and of setting a larger dynamic range comparedwith the ATMs, the QTC has three input channels called the small, medium and high gains.
These gains are set to 1, 1/7 and 1/49. The observable charge range and resolution of each
input channel summarized in Table 3.2. The overall charge dynamic range is 0.2 to 2,500
pC, which is about five times wider than that of the ATMs. The charge integration gate is
set to 400 nsec.
The output signal from the QTC is sent to a multi-hit TDC, which measures the width61

of the QTC output pulse by detecting the leading and trailing edges of the pulse. A field
programmable gate array (FPGA) reads and processes the data stored in the TDC. The
width information is digitized into the charge information and the arrival timing information
is digitized into the timing information. The digitized signal is sent to the front-end PCs via
ethernet. The charge and timing resolutions of the QBEEs, at the one pe level, are 10% and
0.3 nsec, respectively. Both are better than the intrinsic resolutions of the ID PMTs. Finally
the QBEEs achieve high-speed signal processing by combining the pipelined components of
the QTC, TDC and FPGA into one signal. The front-end PCs sort the received data by time
and then send it to the merger PCs. At the merger PC stage, a software trigger program is
run to extract events from the hits data, from all the PMTs. The triggered event data arecollected by the organizer PC and then finally recorded to disk.
Because of the achievement of high-speed signal digitization, every hit of every PMT iscollected at the merger PCs. At this point the merger PCs apply the software trigger bysearching for events which have an Nhit,200ns (number of hits within a 200 nsec sliding time
window) above a set threshold. The threshold values for Nhit,200ns , for each of the trigger
types now used, are given in Table 3.3, along with the event widths and rates of each of
the triggers. The key feature of the software trigger is that it is possible to select an event
width dependent upon the trigger type, whereas previously with the ATMs, the event width
was always set to 1.3 µsec. Now the LE, HE and SHE triggers have a wider event widthof 40 µsec, better suited for detecting possible pre-activity (i.e. prompt gamma-rays) andpost-activity (i.e. muon decay-e’s). The event width of the SLE trigger is is set to 1.5 µsec
because of its high rate due to radioactive backgrounds. The software trigger has made it
possible to tag external triggers for calibration events and is also much easier to model in theMC simulation, reducing the systematic uncertainty associated with the trigger threshold.

62

Table 3.3: Summary of the current software trigger types, hit thresholds, events widths andeven rates.
Trigger Type
SLE
LE
HE
SHE
SHE

Hit Threshold (hits/200 nsec)
34
47
50
70 (before Sep.2011)
58 (after Sep.2011)

Event Width (µsec)
1.5 (-0.5∼+1.0)
40 (-5∼+35)
40 (-5∼+35)
40 (-5∼+35)
40 (-5∼+35)

Trigger Rate
3.0-3.4 kHz∼40 Hz∼10 Hz3.4 Phases of Super-Kamiokande
The SK detector began observation in April, 1996 with 11,146 ID PMTs and 1,885 OD
PMTs. After five years of continuous running, observation was stopped in order to replace
bad PMTs which had high dark rates and to do general detector maintenance. This first 5
years of data taking is known as the SK-I phase. After the replacement work was finished,refilling of the tank with pure water was started in September, 2001. In November of thesame year, when the height of the water in the tank was ∼30 m, one of the bottom ID PMTs
imploded, releasing an energetic shock wave which triggered a chain reaction. The result
was 6,777 ID and 1,100 OD PMTs being destroyed. After cleaning the inside of the SK tank,the surviving ID PMTs were redistributed throughout the detector and new OD PMTs were
installed where those had been damaged. At this time the ID PMTs were all enclosed inprotective cases made of fibre reinforced plastic (FRP) and acrylic, in order to avoid further
chain reactions. The SK-II phase started data taking in October, 2002 with 5,182 ID and1,885 OD PMTs, and ran for three years.
During the running of the SK-II phase, Hamamatsu produced ∼6,000 50 cm PMTs, inorder to replace those which had been destroyed in the chain reaction. During the timefrom October, 2005 to July, 2006, work to mount the newly produced PMTs was done.
This period is called the full reconstruction. The newly produced PMTs were also enclosed
in the protective cases at this time, matching those still inside the detector. After full63

Table 3.4: Summary of the key features of each phase of SK. The livetime of SK-IV is from
September, 2008 to the end of 2012 (used in this thesis).
SK Phase
SK-I
SK-II
Start
Apr. 1996 Oct. 2002
End
Apr. 2001 Oct. 2005
Solar ν Analysis Livetime 1496 days 791 days
Number of ID PMTs
11,146
5,182
Number of OD PMTs
1,885
1,885
Photo-Cathode Coverage
40%
19%
Front-End Electronics
ATM
ATM
Trigger Type
Hardware Hardware

SK-III
SK-IV
Jul. 2005 Sep. 2008
Aug. 2008 Current
548 days 1306 days
11,12911,129
1,885
1,885
40%
40%
ATM
QBEE
Hardware Software

reconstruction was finished, the SK-III phase data taking began in July, 2006 with 11,129
ID PMTs. SK-III ran for three years before being stopped in August, 2008 for the upgradeof the front-end electrons. The front-end electronics and DAQ replacement was finished in
September, 2008, and data taking resumed once again as the SK-IV phase. SK-IV continues
to run today. A summary of the key features of each of the SK phases is given in Table 3.4.

64Chapter 4
Super-Kamiokande Detector
Calibration
4.1 PMT Calibration
4.1.1 Absolute Gain
The details of the SK-IV detector calibration are described in [38]. There are two factors
which are used to describe the PMT charge response, the quantum efficiency (QE) and thegain. The QE is defined as the probability of emitting a pe when a photon hits the PMT
photo-cathode. The gain Ai is defined as the amplification factor of each resulting pe by the
PMT dynode and can be separated into two parts, the absolute gain and the relative gain.
The absolute gain is a common gain factor applied to all PMTs and gives the conversion fromthe output charge in pC (value readout from PMT) to the number of pe (used in analyses
for energy measurement). The absolute gain depends on the HV supplied to the PMT, andso initial HV values were determined for each PMT at the start of SK-III. This was doneusing the PMT charge responses Qi,obs to a xenon (Xe) light source. The Xe light source
calibration system is shown in Figure 4.1. Qi,obs was determined as

Qi,obs = Ni,photon × QEi × Ai ,(4.1)

where i is the PMT number index and Ni,photon is the number of photons which hit thephoto-cathode in the i-th PMT. The goal for Qi,obs was to have better than a ∼1% spread

65

Figure 4.1: The Xe lamp calibration system. The UV filter acts to only pass light aroundthe UV region of the spectrum. The neutral density (ND) filter adjusts the intensity andthe scintillator ball (BBOT and MgO powder) emits light at 440 nm in all directions.

and to successfully determine the HV to produce uniform charge output with an RMS valueless than 1.3%.
After determination of the HV for each PMT, the absolute gain was calibrated using anickel-californium (Ni-Cf) source (Ni calibration). 252 Cf decays via α-decay (∼96.9%) andspontaneous fission (∼3.1%) with a multiplicity of 3.8 neutrons, with a half life of 2.65 years.
The emitted neutrons thermalize and are then captured by the Ni through the reactionnNi(n, γ)n+1 Ni, releasing high energy gamma-rays. The emitted gamma-rays are isotropic

and have energies which depend on the isotope of Ni which captures the neutron. The range
of energies is from 6.1 MeV to 9.0 MeV (67.9% at 9.0 MeV). This gives a nice stable source
for weak Cherenkov light (at the one pe level), emitted uniformly, and is physically smallenough to make data taking at different detector positions much easier.
Using the light emitted from the Ni source, the output charge distribution of the 1 pe hits
is obtained, and shown in Figure 4.2 for both SK-III and SK-IV. The amplitude of SK-IV
66

Figure 4.2: Output charge distribution of 1 pe hits for SK-III (black) and SK-IV (red).
The SK-IV distribution is multiplied by 0.844 so that the peak amplitude of the SK-IV
distribution matches the peak amplitude of SK-III.
Table 4.1: The absolute gain for each SK phase. This factor converts the output charge of
the PMTs (pC) to the number of pe’s. The absolute gain factor becomes larger with time,
but new PMTs were installed just before SK-III and have a slightly lower gain than thoseused during SK-II.
Phase
SK-I
SK-II
SK-III
SK-IV

Absolute Gain
2.055
2.297
2.243
2.645

has been multiplied by 0.844 in order to match the peak amplitude in SK-III (for the sakeof comparison). The higher peak value in SK-IV comes from the PMT gain increasing by
2∼3% each year, and the different performance of the charge digitization in the new frontend electronics. The change in the 1 pe distribution shape between SK-III and SK-IV in thevery low charge range is caused by the difference of the front-end electronics performanceand the difference in charge discrimination. From the peak value in Figure 4.2, the absolute
gain for SK-IV is set as 2.645 pC/pe. Table 4.1 shows the absolute gain used in each SK
phase.
67

4.1.2 Relative Gain
The relative gain adjusts for relative differences in the gain between the PMTs and insuresthat all PMTs give a uniform response, minimizing systematic effects. This is measured with
two sets of laser calibration data, one with high light output (∼50 pe/PMT) and one with low
light output (1 pe/PMT). The charge detected by the i-th PMT Qi,obs in the high intensity
data follows
Qi,obs = Ii,high × QEi × Gi ,(4.2)

where Ii,high is the light intensity seen by the i-th PMT. QEi is the PMT quantum efficiency
and Gi is the relative gain for the i-th PMT. For the low intensity data the number of hit
counts over many laser fires are counted and defined as Ni,hit . Taking the ratio between
Qi,obs and Ni,hit gives the relative gain in the i-th PMT as
Qi,obs= Gi .
Ni,hit(4.3)

This method of measuring the relative gain removes any effects coming from things likewater transparency and geomagnetic field variations. The distribution of the relative gainis shown in Figure 4.3, with a 1 σ spread of ∼6%. Both the absolute gain and the relative
gain are used to convert the output charge from the PMT to the number of pe for dataprocessing.

4.1.3 Quantum Efficiency
As previously stated, the PMT QE is defined as the probability of a photo-electron being
produced when a photon hits the photo-cathode. Every PMT may have a different QE, andso it is necessary to make a measurement of the individual PMT QEs. This is done using

68

Number of PMTs per bin

1600
1400
1200
1000
800
600
400
200
0

0.6 0.7 0.8 0.91

1.1 1.2 1.3 1.4
Relative PMT gain

Figure 4.3: Relative PMT gains measured using the laser calibration data. The 1 σ spreadis ∼6%.

the Ni calibration source. The hit rate of each PMT is used, while the PMTs are separated
into groups based on their position in the detector. PMTs 1-7,650 are placed in the barrel ofthe detector, PMTs 7,651-9,398 are in the top of the detector and the remaining PMTs up
to number 11,146 are placed in the bottom. The hit rate in each PMT includes dark noiseand radioactive background. To remove this from the data, two timing windows within thetime of flight from the vertex subtracted timing distribution is made. The hits within these
two windows are called either the “on-time hits” or the “off-time hits”. The “on-time” hits
fall inside a 300 nsec window around the timing peak, while the “off-time” hits fall inside
the 300 nsec window which starts 150 nsec after the “on-time” window. The number ofhits within the “off-time” window is subtracted from the number of hits in the “on-time”
window, giving the true hit rate hi . In order to take into account the position dependenceof the PMTs, a geometrical factor is applied ashi,corr =hi × Ri2,a(θi )

69(4.4)

where Ri is the distance from the source position to the i-th PMT and a(θi ) is the acceptanceof the PMT. The acceptance of the PMTs is obtained from the geometry of the PMTs andis desribed asa(θi ) = 0.205 + 0.524 cos θi + 0.390 cos2 θi − 0.132 cos3 θi .

(4.5)

The remaining step is to calculate the hitrate hri by normalizing hi,corr by the total numberof hits in the detector divided by the number of PMTS,hri =hi,corr.
(Σhi,corr /NPMT )(4.6)

For SK-III and IV, NPMT is equal to 11,129. Even after the above dark noise and geometrical corrections are applied, there is a remaining position dependence in hri . This isespecially true for the case of photon reflections by neighboring PMTs or photon scatteringin the water. Fortunately these types of effects can be model by MC simulation. The data
and MC simulation hitrates for the top, bottom and barrel regions of the detector are shownin Figure 4.4. The remaining ratio between the data and the MC simulation hitrates isdefined as the QE table. The data used for this calibration was taken when the water waswell mixed, and so the position dependence of the water transparency was assumed to be
negligible. It is also assumed to be uniform in the MC simulation for this calibration.

4.1.4 Timing
The PMT timing information is very important for the reconstruction of the vertex position. Ideally, the PMTs would all have the same timing response. Due to slight differences in
the length of the PMT signal cables and varying responses of the response of the electronics,
the timing responses can differ. There is also an effect due to time-walking, which comes70Figure 4.4: Position dependence of the PMT hit rate for Ni calibration data (red) and MC
simulated events (blue) [38]. The upper, bottom left and bottom right figures show thehitrate for the barrel, top and bottom PMTs, respectively. The horizontal axis of the upper
figure is the z (cm) position of the PMTs. For the lower figures, the horizontal axis is thesquare of the distance from the center (cm2 ) of the detector.

71

Figure 4.5: Visual explanation of the time-walk effect. Higher charged signals will trigger
the threshold earlier than lower charged signals.

from the fact that if the amount of charge in a PMT is high, the signal will exceed the TDC
discriminator threshold earlier than if the amount of charge was lower. A visual explanationof this effect is given in Figure 4.5. These differences in the timing response of the PMTs iscalibrated out using a timing-charge relation table called the TQ map. The TQ map gives
the relationship between the timing and the observed charge of each PMT.
To calibrate the TQ map, a laser light source is used because of its fast rise time and easeof light intensity control. At the beginning of SK-IV, data was taken using the USHO (Usho
Optical Systems Co.) laser calibration system, shown in Figure 4.6. The characteristics ofthe laser are shown in Table 4.2. Although the pulse width of this laser is very short, thelong term stability is not so good, and so other laser light sources are used for the long term
stability check of the TQ map.
To make the TQ map, first the timing distribution for each PMT is made, after subtracting
the time of flight from the source position to each PMT. Figure 4.7 shows an example ofone of the PMT timing distributions vs. the observed charge. Here the charge value Qobs is72

Table 4.2: Characteristics of the USHO laser.
USHO N2 Laser
Wavelength337.1 nmMax. Rate
20 Hz
Pulse Width (FWHM)
0.3 nsec
USHO Dye Module
Wavelength
396 nmMax. Rate
20 Hz
Pulse Width (FWHM)
0.2 nsec
Spectrum Bandwidth (FWHM) 0.5 nm

0.CUGT

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

5-+&

8CTKCDNGHKNVGT&KHHWUGTDCNN
Figure 4.6: Schematic view of the USHO laser calibration system [38].

73Figure 4.7: Single PMT timing vs. charge distribution obtained by the USHO laser calibration. The horizontal axis shows Qbin defined in Equation 4.7. Black, red and blue points
refer to the three QBEE charge channel ranges defined in Section 3.3.5.

divided into 180 bins, with the following sizes:∆Qbin ≡


 0.2 pC,

for 1 ≤ Qbin ≤ 50 (0 pC ≤ Q ≤ 10 pC)

Qbin −1Qbin 10 50− 10 50 ,for 51 ≤ Qbin ≤ 180 (10 pC ≤ Q ≤ 3981 pC).

(4.7)

Unfortunately there is a binning effect when making the time vs. charge distributions, andso it is beneficial to smooth the timing histograms. A smearing method is used to do this.
Figure 4.8 showing how the smearing method works. First a Gaussian expansion is made foreach bin and the standard deviation found. Next an asymmetric Gaussian fit from tlow tothigh is made across multiple bins. tlow and thigh are defined as the values of the timing whichgive 0.05% of the peak value. The asymmetric Gaussian fit results are then used as a timing
resolution for each observed charge. These are input into the MC simulation and measured
as ∼3 nsec, at the 1 pe level. After getting the maximum value for each charge bin, we apply
a 3rd order polynomial fit to the smeared timing vs. charge distributions in the low charge
region, a 5th order polynomial fit to the middle charge region and a 7th order polynomial
fit to the high charge region. These three regions are the three input channels of the QBEE

74

Figure 4.8: Schematic view of the TQ smearing method. Each bin has its own binning effect,so each bin is expanded to a Gaussian distribution. The standard deviation of the Gaussian isdefined for each Qbin value. After smearing the timing distribution an asymmetric Gaussian
fit is made from the lower timing region to the higher timing region.

and were defined in Section 3.3.5. An example of the timing vs. charge distribution, after
the TQ map correction has been made, is shown in Figure 4.9. The distribution for each
PMT is used to calculate the observed charge of each PMT, given its observed timing of thehit.

4.2 Water Transparency Measurements
4.2.1 Measurement by Laser Light Data
The coefficients normally used to characterize the water transparency are the wavelength
dependent absorption αAbs , Rayleigh scattering αRay and Mie scattering αMie coefficients.
However, in SK, the scattering coefficients of the MC simulation are taken instead as thesymmetric scattering coefficient αsym (Rayleigh plus symmetric Mie scattering) and the asymmetric scattering coefficient αasy (forward Mie scattering). These are defined in Figure 4.10.
Each of the coefficients are independently measured by either an N2 laser (337 nm) or one
75

Figure 4.9: Timing vs. charge distribution after making TQ map corrections.

Figure 4.10: Definition of the symmetric and asymmetric scattering in the MC simulation.

of the laser diodes (LD) (375 nm, 405 nm, 445 nm and 473 nm).
Figure 4.11 shows a schematic view of the laser light calibration system used to measure
the water transparency coefficients. The method works by firing the lasers every one minuteand using the observed hit patterns in different regions of the detector. The method to
obtain the coefficients is as follows:• Make the scattered hits/Qtot distributions (hits/Qtot hereafter) for each detector region(Top, B1,..., B5, Bottom) defined in Figure 4.11.
• Generate sets of MC simulated laser light injection using varying water coefficients,
then make the same hits/Qtot distributions. Figure 4.12 shows the hits/Qtot distribu76

Figure 4.11: Schematic view of the water coefficients measurement by laser light injection.
In SK-IV, a N2 laser was used for 337 nm measurements and laser diodes for the otherwavelength measurements.

tions of data (black) and MC simulation (red).
• Compare each MC simulated set with data using a χ2 defined as

χ2 = Σ

(DAT A − M C)2
.
2
2
σDAT
A + σM C(4.8)• Find the minimum value of the χ2 for each water transparency coefficient.

After finding the best set of water transparency coefficients for each wavelength, they are
fit using continuous functions, taking the discrete wavelength measurements to a continuous

77Figure 4.12: Hits/Qtot distributions coming from the laser calibration for data (black) andMC simulation (red) [38]. The horizontal axis shows time (nsec) and the vertical axis shows
hits/Qtot . The first and second peaks correspond to photons scattering in the water andthose which are reflected off the bottom PMTs and black sheet.

78

distribution. The fit functions for each of the water transparency coefficients are given as:1r2
αsym (λ) = r1 × 4 1.0 + 2 ,λλ
4.623 × 106
2
αasy (λ) = m1 × 1.0 +× (λ − 3992.4) ,λ410.94 7
2.691×10
λ a1 ×+ a1 × 1.164 500.0
λ4αabs (λ) = a × a2 + PF(λ)
1
λ4

(4.9)
(4.10)
forλ ≤ 463.918,(4.11)
forλ ≥ 463.918.

r1 , r2 , m1 and a1 are multiplicative factors which adjust the function to match the watertransparency, and λ is of course the wavelength of the Cherenkov light. The additional factor
PF(λ) in the absorption coefficient function, above 463.9 nm, comes from the measurement
by Pope and Fry [39]. Figure 4.13 shows the observed water transparency coefficients foreach wavelength, along with the fitted functions, using data taken from April, 2009. The
total water transparency W T is then defined by

WT =1(αabs + αsym + αasym ).

(4.12)

As it turns out, W T does not strongly depend on the scattering coefficients, but only
on the absorption coefficient. For this reason, the scattering coefficients are taken as theconstant values shown in Figure 4.13, for the entire SK-IV period, while the absorption
coefficient changes based on the measured W T using muon decay electrons (explained in thenext section). The values of r1 , r2 , m1 and a1 are taken as the constant values of
r1 = 8.514 × 108 ,(4.13)

r2 = 1.138 × 106 ,(4.14)

m1 = 1.002 × 10−4 ,(4.15)

a1 = 0.7142.

(4.16)

79

-1

10

-2

10

-3

10

-4

10

300

325

350

375

400

425

450

475

500

Figure 4.13: Measured and fit water transparency coefficients from April, 2009 [38]. The
black line shows the fit result of the total water transparency, red is absorption, blue issymmetric scattering and pink is asymmetric scattering.

The value of a2 is varied with time.

4.2.2 Time Dependence by Decay-e Energy Measurements
Because SK sits at a given depth below the surface, the flux and spectrum of cosmic-ray
muons entering the detector are well known. These muons can decay in flight to an electron(decay-e), a νµ and a ν̄e . The energy spectrum of the electrons is well known. To check thestability of the water transparency, the mean energy of the decay-e spectrum is measured asa function of time. The selection criteria for the decay-e events is:• The time difference between the parent muon event and the decay-e candidate event80

is 3.0 µsec ≤ ∆t ≤ 8.0 µsec.
• The reconstructed vertex of the decay-e candidate event is within the 22.5 kton fiducial
volume, i.e. greater than 2 m from the PMT structure.
• The distance between the stopping point of the cosmic-ray muon and the decay-e
candidate event is within 250 cm.
• The number of hit PMTs within the 1.3 µsec time window selects 120 ≤ Nhits ≤ 600.

The event rate of decay-e’s is about 1,500 events per day. Using these selected events,
PMT hits are divided by the distance from the reconstructed decay-e vertex position toeach PMT. The water transparency is found by fitting the number of PMT hits vs. thedistance distribution, taking the azimuthal symmetry of the Cherenkov light distribution
into account. To further remove the effects of noise hits and indirect light, only PMT hits
giving a cone opening between 32◦ and 52◦ , and with a residual timing within 50 nsec, areused. The amount each of the hit PMTs contributes to the energy is the same as for theenergy reconstruction (see Section 5.3). The measured water transparency for each given
day is calculated using +/- one week of data, and can be seen in the top panel of Figure 4.14.
This value is used as an input parameter to the MC simulation. The bottom panel shows
the decay-e mean energy before and after the water transparency corrections have been
applied. After the corrections are made, the decay-e mean energy is stable within ±0.5%.
The relationship between the measurement of the water transparency coming from decaye’s and the absorption coefficient is shown in Figure 4.15, with the vertical axis taken as
the ratio of the absorption coefficient to that measured in August, 2010 (when the absoluteenergy scale was established).

81

WT (m)Energy (MeV)

140
130
120
110
42
40
38
36

2009

2010

2011

2012
Year

Fractional Change in Abs. Coeff.

Figure 4.14: (Top) Time variation of the W T as measured by decay-e’s. (Bottom) Timevariation of the reconstructed mean energy of cosmic-ray muon decay-e’s before (after) theW T correction has been applied in black (red). A W T of 90 m is assumed before theW T correction, hence the reason for the vertical shift before and after correction. Aftercorrection, the mean energy is stable within ±0.5%.

1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2

11000

12000

13000

14000 15000
WT (cm)Figure 4.15: Fractional change in the absorption coefficient, relative to the coefficient when
the absolute energy scale calibration was done, vs. µ decay-e measured water transparency.

82

4.2.3 Position Dependence
It is known that there is a position dependence of the water transparency, especially inthe z direction. This is mainly due to supplying water from the bottom of the detector andextracting it from the top. Since the beginning of the SK-III phase this position dependence
has been continuously monitored by the Xe calibration source (every one minute) and also by
the Ni calibration source (once per month). To define the amount of change in the hitrate asa function of the z position, a parameter called the “top-bottom asymmetry” T B is definedasTB =< hrtop > − < hrbottom >.
< hrbarrel >(4.17)For the Ni calibration source, < hrtop >, < hrbottom > and < hrbarrel > are the mean hit
rates in the top, bottom and barrel regions of the detector, respectively. The situation is
also similar for the Xe calibration source, but with the mean hit rate replaced by the mean
charge.
At the time of the PMT QE measurement, by the Ni calibration, the water transparency
was uniform. This resulted from a relatively larger amount of water convection within thetank. At this time the hit rate difference between the top and bottom of the tank was
almost zero (-0.5%). When the water circulation is more stable, from the measured valuesof the T B, there is a 3∼5% level of z-dependence of the water transparency, which canaccount for the ±0.5% systematic uncertainty of the directional dependence of the energy
reconstruction in the SK-I solar neutrino analysis. This is because the MC simulation used in
SK-I did not take into account position dependence of the water transparency, but assumed
it to be constant throughout the detector volume. Now, to take into account the hit rate
difference in the top and bottom of the detector, a new parameter β was introduced into
the MC simulation during SK-III. This parameter introduces a z-position dependence to the83absorption coefficient in the form

αabs (λ, z) =
 αabs (λ)(1 + β × z),

for z ≥ −1100cm,

 αabs (λ)(1 − β × 1100),for z < −1100cm.

(4.18)

To tune the β parameter for use in the MC simulation, the hitrate analyses from the Ni
and Xe calibrations are used. The monthly Ni calibration sets β, while the more continuous
Xe calibration checks the time dependence. In principle, these hitrate analyses are used tomeasure the QE. So, in order to cancel the effects of the QE for each PMT, Equation 4.4,
which calculates the corrected number of hits in a given PMT, is modified by normalizingout the QE of the hit PMT,
hi,corr = hi ×

Ri2.
a(θi )/QEi(4.19)

After the hitrate table is generated for both data and MC simulation (using β between 0
and 0.016 (1/m)), the χ2 between data and the MC events for each β value is determined.
The selected β parameter is taken as the one which minimizes the χ2 . Figure 4.16 shows
an example of the hitrate comparison for data and MC events, for two values of the βparameter. Figure 4.17 shows the χ2 dependence on β, with β=0 meaning no z-dependenceof the absorption coefficient, and a larger β giving a larger z-dependence of the absorption.
It was found that the T B value and the β parameter are correlated according to

β = {−0.163T B 2 × 3.676T B} × 10−3 .

84(4.20)

Figure 4.16: Hitrate of data (black) and MC simulated events (β=0 for green and β=0.016(best-fit) for red).

85

Figure 4.17: Data vs. MC simulation χ2 for various β parameters. The minimum χ2 value
determines the β value to be used in the MC simulation of solar neutrinos.

4.3 Absolute Energy Scale Determination
4.3.1 LINAC Calibration
Precise determination of the absolute energy scale is the most important calibration for
low energy data analysis. This includes the exact determination of the correlation between
the observed amount of Cherenkov light coming from a charged particle and the energy ofthat particle (e in the case of solar neutrino interactions), and also the determination andcorrection for any position, direction or time dependence of the energy scale. To determine
the absolute energy scale for the low energy analyses at SK, a modified Mitsubishi ML15MIII electron linear accelerator (LINAC), originally used for medical purposes, is installed
above the SK tank (see [40] for exact details). The LINAC has been modified to make it
more useful in calibrating single electron events, including a special electron gun, steering

86

TOWER FOR INSERTING BEAM PIPE
D2 MAGNET

D3 MAGNET

D1 MAGNET
LINAC

1300 cm
1

4

2

5

3

6

4200 cm+12m

BEAM PIPE
0m

Z

-12m
-12m

-4m

X
4000 cmY

Figure 4.18: Diagram of the LINAC calibration system and the injection points of themonoenergetic electrons into the SK tank [40].

magnets and collimators to reduce the number of electrons injected into the SK tank to
0.1 per bunch. The monenergetic electrons with kinetic energies between 4.6 and 15.8 MeV
are injected into the tank at three different heights (z=-12, 0 and 12 m) for two different
positions ((x, y)=(-12,-0.7) m and (-4,-0.7) m), in the downward going direction. LINAC
calibration data was taken at these various positions, for various energies, twice during SKIV. Because the water transparency was highly variable during the LINAC calibration donein August, 2009, LINAC calibration data from August, 2010 was used to set the absoluteenergy scale of SK-IV. A systematic uncertainty on the absolute energy scale coming fromthe water transparency is assigned as 0.2%.
The position of the LINAC with respect to the SK tank, and the various injection pointsof electrons inside the SK tank, are shown in Figure 4.18. To precisely determine the energyof the LINAC electron beam, the D3 magnet which bends the electrons from the horizontal
direction down into the tank is switched off and the beam is dumped into a germanium(Ge) detector. The Ge detector has very good energy resolution (1.92 keV at 1.33 MeV)

87

Deviation (%)

Energy of γ-ray (MeV)

10
9
8
7

1
0.8
0.6
0.4

6

0.2

5

0

4

-0.2

3

-0.4

2

-0.6

1

-0.8

0

0

250

500

-1

750 1000 1250 1500 1750 2000

Channel

0

1

2

3

4

5

6

7

8

9

10

Energy of γ-ray (MeV)

Figure 4.19: Linearity of the Ge detector. The horizontal axis is the output from the Ge
detector (counts) and the vertical axis shows the energy of the calibration gamma-ray source(MeV). Deviation of the output of the Ge detector from the fitted line (%) is shown on theright. For the solar neutrino energy region, the difference is within 0.1%.
and output linearity for the given range of electron energies used in the LINAC calibration.
Figure 4.19 shows the output charge of the Ge detector as a function of gamma-ray energy,
for several different gamma-ray emitting sources, between 0.662 MeV (for a 137 Cs source)
and 9.000 MeV (from the Ni(n,γ)Ni reaction). The deviation from the true energy is also
shown. For the few MeV region, most important to the solar neutrino analysis, the differenceof the output of the Ge detector count to the fitted line is within 0.1%, and contributes a
total of 0.21% to the systematic uncertainty of the energy scale in SK-IV.
Once the true beam energy is known, it can be related to the effective number of PMT
hits Neff (see Section 5.3) seen inside the SK tank, using MC simulation. The LINAC-tuned
MC simulation is used to generate uniformly distributed events at several fixed energies,
with the mean value of Neff being obtained for each energy by fitting the distribution witha Gaussian function, as seen in Figure 4.20. The correlation between the fit Neff and thetrue beam energy is then fit with a 4th order polynomial function, thus setting the absolute88Figure 4.20: Neff distribution for data (MC simulation) in black (red), taken during theAugust, 2010 LINAC calibration. 13.2 (6.5) MeV is shown on the left (right).
energy scale of the detector. The accuracy of this method of determining the absolute energy
scale is better than 1% for all LINAC energies at all injection positions, as determined fromcomparison of the LINAC data to MC simulation.

4.3.2 DT Calibration
The deuterium-tritium neutron generator (DT generator) is a calibration device whichemits a neutron n through the reaction3

H + 2 H → 4 He + n.

(4.21)The energy of the generated neutron is 14.2 MeV, which is large enough to generate 16 N inwater, via the reaction
16

O + n → 16 N + p.

(4.22)

The 16 N decays by gamma-ray and/or β emission with a half-life of 7.13 seconds. The
89

Figure 4.21: Schematic view of the DT generator system.

main decay modes of the 16 N is to produce a 6.1 MeV gamma and a 4.3 MeV β (66%) ora 10.41 MeV β (28%). Figure 4.21 shows a schematic view of the DT generator calibration
system. First the crane is used to lower the DT generator into position inside the SK
tank. The DT generator is then fired, generating a cloud of neutrons at the tip of thedevice, which then generate 16 N in the water. Immediately after firing, a short veto time(∼few sec) is applied giving the crane time to retract the DT generator from the cloud(otherwise it could severely shadow light and impact the reconstruction). After the shortveto time (and because the half-life of 16 N is 7.13 sec), SK begins to collect the thousands of16

N decay events. The reconstructed mean energy of this data is then compared against MC

simulation, with the difference in comparison giving an estimation of the difference in various
aspects of the energy scale (including position, direction and time). The DT calibration has
a few main advantages over the LINAC system, including its ease of data taking, higher
variability of position deployment and the fact that the decay modes of the 16 N give off
isotropic light, lending to the directional dependence determination of the energy scale. The
DT calibration is done once every three months to help determine the time dependence ofthe energy scale. Figure 4.22 shows the directional dependence of the energy scale comingfrom DT calibration data and MC simulation (less than 0.1%), along with an exponential90(MC-Data)/Data (%)

0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
cosθzSK

Figure 4.22: Direction dependence of the energy scale, from the DT calibration.

function with a horizontal offset fit (used to estimate further systematics). Figure 4.23 gives
the position dependence of the calibration (0.44%).

4.3.3 Energy Scale Systematic Uncertainty
The systematic uncertainty on the absolute energy scale comes from a few different
sources. The uncertainty of the energy scale coming from the determination of the LINAC
electron beam energy by use of the Ge detector is set as 0.21%, while the variation in thewater transparency during the LINAC calibration can account for 0.2% variation in the energy scale. From the comparison of DT calibration data and MC simulation, the directional
and position dependences of the energy scale are found to be 0.1% and 0.44%, respectively.
By taking the sum in quadrature of all four of these contributions, the total systematic uncertainty on the energy scale is found to be 0.54%, which is very comparable to the 0.53%
found in SK-III [25].

91(MC-Data)/Data (%)1.5
1(x,y)=(-12,-0.7)m(x,y)=(-0.35,12)m(x,y)=(-4,-0.7)m(x,y)=(0.35,-0.7)m(x,y)=(0.35,-12)m(x,y)=(11,-0.7)m

0.5
0

-0.5
-1
-1.5
-15

-10

-5

0

5

10

15
z (m)

Figure 4.23: Position dependence of the energy scale, from the DT calibration.

92

Chapter 5
Event Reconstruction and
Simulation
5.1 Vertex Reconstruction
Low energy charged particles, such as electrons emitted from the interaction of solar
neutrinos in SK, will only travel ∼10 cm in water. Because of the large size of SK, and its
poor vertex resolution (inherent to the water Cherenkov detection technique), the resulting
electron is treated as a point source at the vertex position of the interaction. To reconstruct
the vertex position, a maximum likelihood fit to the timing residuals of the Cherenkov signal
is used, implemented inside a special fitting program called BONSAI (Branch Optimization
Navigating Successive Annealing Iterations) [42]. The timing residual of each PMT hit isthe difference of the PMT hit time t minus the time of flight ttof to the PMT from the vertex
position, minus the time of the interaction t0 . The likelihood function is described byL(~x, t0 ) =Nhit
Xlog(P (t − ttof − t0 )).

(5.1)

i=1

P (t − ttof − t0 ) is the probability density function (PDF) of the timing residuals for a single
pe signal, shown in Figure 5.1. ~x is the testing vertex. The PDF is obtained from the LINAC
calibration, along with the vertex resolution. The vertex resolution is taken as the distancein which 68% of the reconstructed LINAC vertices agree with the true vertex position. The
vertex resolutions for SK-I, III and IV are shown in Figure 5.2. The major improvementfrom SK-I to SK-III comes from the improvement of the vertex reconstruction program. The

93

.KMGNKJQQF

661(6=PUGE?
Figure 5.1: Probability density function of the timing residuals used for the vertex reconstruction maximum likelihood fit. The second and third peaks around 30 and 100 nsec arecaused by PMT after pulses.

slight improvement from SK-III to SK-IV comes from a sharpening of the timing residuals
peak and a better data to MC simulation comparison, as a result of the front-end electronics
upgrade. The vertex resolution in SK-IV is ∼60 cm at 8 MeV.
During the SK-II and III phases, there was a bias in the reconstruction of the vertex,
known as the vertex shift. Using the Ni calibration source, the vertex shift was checked atseveral positions inside the SK tank. Figure 5.3 shows the vertex shift in SK-IV, as measured
by the Ni calibration events. This shift is much less than that observed during the SK-II and
III phases. This is an effect of the improvement of the understanding of the timing residuals,
coming from the front-end electronics upgrade.

94

Vertex Resolution (cm)

250
200
150
100
50
0

5

7.5
10
12.5
15
True Electron Energy (MeV)z (m)Figure 5.2: Vertex resolution for SK-I, III and IV shown by the dotted (blue), dashed (red)and solid (black) lines, respectively. The SK-III vertex resolution improvement over SK-I
comes from using an improved vertex reconstruction. The slightly improved timing resolution
and agreement between data and MC is responsible for the further improvement in SK-IV.

10 cm

10

0

-10

0

5

10

15

r (m)

Figure 5.3: Vertex shift of the Ni calibration events. The start of the arrow gives the true
Ni-Cf source position and the direction indicates the averaged reconstructed direction atthat given position. The length of the arrow indicates the magnitude of the vertex shift. To
make the vertex shifts easier to see the length is scaled by a factor of 20.

95

5.2 Direction Reconstruction
Because of the distinct ring like angular distribution of Cherenkov photons emitted when
an electron travels through water, it is possible to reconstruct the direction of the electronby means of a maximum likelihood function. This function is described by

~ ≡
L(d)

N20
X

log(f (cos θi,E )) ×icos θi.
a(θi )(5.2)

N20 is the number of PMT hits within a 20 nsec window around t − ttof − t0 =0. f (cos θi,E ) is
the expected distribution of opening angles between the electron direction d~ and the vector
from the reconstructed vertex to the hit PMT. f (cos θi,E ) is obtained from monoenergetic
electron MC simulation, and as seen from Figure 5.4, the likelihood peaks at the characteristic
42◦ . The reason for the distribution expanding to a wide range comes from the electron
direction changing due to multiple Coulomb scattering. At this point it is also possible tosee the energy dependence of the multiple Coulomb scattering, where lower energy electrons
will incur more multiple scattering and thus have a wider and more scattered distribution of
PMT hits (see Section 6.5). The second factor in the likelihood corrects for PMT acceptance,
where θi is the opening angle between the vector from the reconstructed vertex to the i-th hit
PMT and the vector normal to the PMT surface. Figure 5.5 gives the PMT acceptance of
Cherenkov light as a function of the incident angle. The best-fit direction is found by scanning
Equation 5.2 over various directions, until the likelihood is maximized. The direction which
maximizes the likelihood is chosen as the best-fit direction. Using this method in SK-IV
gives an angular resolution of the reconstructed direction of ∼25◦ , for 10 MeV electrons.
This is defined as the maximum angular difference between 68% of the reconstructed andtrue event directions, as determined from MC simulation.

96

cos theta

1

1

0.8

0.9

0.6

0.8

0.4

0.7

0.2

0.6

0

0.5

-0.2

0.4

-0.4

0.3

-0.6

0.2

-0.8

0.1

-1

4

6

8

10

12

14

16

kinetic electron energy(MeV)
Figure 5.4: The energy dependence of the angular likelihood used in the direction reconstruction.

Figure 5.5: The incident angle dependence of the PMT acceptance.

975.3 Energy Reconstruction
When solar neutrinos elastically scatter with electrons in SK, it is possible to use the recoil
electron energy spectrum to infer the incoming neutrino energy spectrum. It is important toreconstruct the energy as precisely as possible, as changes in the flux and distortions in theenergy spectrum can come from various energy dependent sources. As the charged electron
travels through water and emits Cherenkov light, the amount of light being emitted will
directly depend on the energy of the electron. The Cherenkov photons will be approximately
proportional to the electron energy, which will in turn be proportional to the number ofgenerated pe’s in the hit PMTs. The proportionality here get rather complicated due tothings like poor charge resolution (∼50%) of the PMTs, angular dependences of the PMT
acceptance and the fact that for electrons with energies below 20 MeV, roughly only 1
pe is produced per hit PMT. For this reason, low energy reconstruction in SK is done bycalculating the effective number of hits Neff from the number of hit PMTs within a 50 nec
time window Nhit , and not by the total charge of the hit PMTs. The 50 nsec time window
helps to reduce contamination from noise hits. Neff is calculated asNhit 
Xri
1Nall
× e λeff × Gi (t) .
×
Neff =(Xi + tail − dark ) ×
N
S(θ
,
φ
)
alive
i
i
i

(5.3)

The various corrections applied to Nhit are as follows:

Occupancy
The occupancy correction is used to correct for the fraction of hit PMTs which observe
more than a single photon, occurring more often for events near the wall and of higher
energies. If a hit PMT detects multiple photons, it is likely that many surrounding
PMTs are also hit. To account for the effect of multiple pe’s for the i-th PMT, the98occupancy is define as
Xi =


1

 log 1−xi ,xixi < 1 3.0,

xi = 1,(5.4)

where xi is the fraction of hit PMTs in a 3×3 patch around the i-th PMT.
Late hits
Some of the Cherenkov photons can be scattered or reflected and thus will fall outsideof the 50 nsec time window. To account for these late hits, the late hits correction isapplied astail =

N100 − N50 − Nalive × Rdark × 50nsec
.
N50

(5.5)

N100 is the maximum number of hits in a 100 nsec timing window, Nalive is the numberof functioning PMTs and Rdark is the average dark noise rate, measure for each run
and is ∼4.7 kHz in SK-IV.
Dark noise
The correction applied due to hits coming from dark noise isdark =

Nalive × Rdark × 50nsec
.
N50

(5.6)

Dead PMTs
The factor Nall /Nalive accounts for the number of dead PMTs, where Nall is the total
number of PMTs and is equal to 11,129 in SK-IV.
Photo-cathode coverage
The correction for the directional dependence due to the PMT photo-cathode coverageis dependent on the direction of the light falling on the i-th PMT. The factor S(θi , φi )is the effective photo-cathode area of the i-th PMT, given the angles θi and φi definedin Figure 5.6. The effective photo-cathode area as a function of the two angles is shownin Figure 5.7.
99

Figure 5.6: Definition of angles θi and φi for the i-th PMT.

Figure 5.7: The effective PMT photo-cathode coverage correction factor as a function of θand φ.

100

Table 5.1: Angular resolution difference between LINAC data and MC for SK-I, III and IV.
The energy refers to the electron in tank kinetic energy.
Energy (MeV)
4.59
5.54
6.50
8.36
10.5
13.2
15.8

SK-I(%)
-1.64
-1.38
2.32
2.33
1.52
1.07
0.883

SK-III(%)
0.74
0.40
-0.27
0.39

SK-IV(%)
-0.01
0.51
2.05
1.40
0.63

Water transparency correction
This factor corrects for the variation in the water transparency, where ri is the distance
between the reconstructed vertex and the i-th hit PMT. λeff is the water transparency
measured by the mean energy of decay-e’s.
PMT gain
The final factor corrects for the time dependence of the relative i-th PMT gain, atthe single pe level. This is measured using laser light calibration data, as described in
Section 4.1.2.

After all corrections are made, the total electron energy is calculated as a function of Neff .
Figure 5.8 shows the relationship between Neff and the reconstructed energy, with the best
fit function described by2
3
4
Erec = 0.82 + 0.13Neff − 1.11 × 10−4 Neff
+ 1.25 × 10−6 Neff
− 3.42 × 10−9 Neff
.

(5.7)

This function is precisely calibrated using LINAC data and MC simulation. It gives anaccuracy of the absolute energy scale better than 1%, and is used for energies up to 25 MeV(Neff <189.8). Above 25 MeV the function is described by the first order polynomial fit

Erec = 25.00 + 0.138(Neff − 189.8).
101

(5.8)

An important point to make is that this method does assume that the particle being
reconstructed is a positron or an electron, and should not be applied to other particles suchas π’s and µ’s. The energy resolution, which is the maximum difference between 68% of thetrue and reconstructed MC simulation energies, is 14% at 10 MeV electron total energy.

5.4 Event Simulation
Solar neutrino MC simulated events consist of single electrons generated within a fully
simulated SK detector. There are several steps which must be taken to make this happen;
generate the solar neutrino fluxes and cross-sections, determine the recoil kinematics, track
the Cherenkov light in water and simulate the response of the PMTs and electronics. The
8

B solar neutrino spectrum calculated by Winter [35] and the hep solar neutrino spectrum

calculated by Bahcall [43] have been used as the input spectra to the MC simulation. The
associated systematic uncertainties from these flux calculations are incorporated later in theenergy-correlated systematic uncertainty of the recoil electron spectrum.
The simulated event times are chosen according to the livetime distribution of SK-IV,
so that the zenith distribution of simulated events matches that of the data. True solar
neutrino events will come from the Sun. The vector from the Sun to the SK detector isalways known and can be modeled precisely. The angular distribution of reconstructed
directions is taken from Equation 3.10. The recoil electron energy spectrum is calculated byintegrating the differential cross section between zero and Tmax , the maximum recoil electronenergy (limited by the incident neutrino energy). There is also an energy dependence of theratio of the differential cross section between νe ’s and νµ,τ ’s, due to the fact that only the νe ’s
elastically scatter via both the neutral and charged weak currents. This energy dependence
was accidentally omitted during the SK-III phase, but has since been re-introduced, and the
SK-III data reprocessed with the correct energy dependence of the differential cross section
102

Figure 5.8: (Top) Relationship between Neff and the reconstructed energy, shown with thefit function described by Equation 5.3. (Bottom) Ratio between the MC simulated points
and the fit function.

103

ratios. The resulting recoil electron energy spectrum is shown is Figure 3.6, with the rate ofelectrons scattered off solar neutrinos calculated from
Z ∞Z ∞dEν dEφ(Eν )R8 B = ΦSSM Ne00

dσ(Eν )
.
dE

(5.9)

φ(Eν ) is the input neutrino spectrum, ΦSSM is the factor which accounts for the predicted
total 8 B solar neutrino flux (5.25×106 cm−2 s−1 ) and Ne is the number of electrons in the
32.5 kton SK ID (1.086×1034 ). The total expected number of events per day occurring in
the SK-IV detector is 294.74.
After obtaining the input recoil electron energy spectrum for the MC simulation, thenext step is to simulate the detector response to the charged electron traveling the very
short distance in water. This include the tracking of the Cherenkov photons from the pointof production to the point of termination. The skdetsim program is based on the GEANT
3 simulation package, and includes the full geometry of the detector and all components.
GEANT is able to track charged particles in water, simulating any interactions during theirtravel. It is then skdetsim’s job to track the photons as they travel through the detector, andthen determine the outcome of the photons hitting the PMTs and black sheet. The propertiesof photons traveling through the water are measured by the calibrations techniques describedin Section 4.2. Next, skdetsim must decide whether a photon which hits a PMT in the MC
simulation will trigger that PMT. When the photons reach the surface of the PMT oneof three actions is taken; (1) absorbance of the photon by the photo-cathode (with somefraction of it causing pe production), (2) reflectance of the photon from the PMT surface or(3) transmittance of the photon through the PMT. Each of these processes has an incident
angle dependence, described in Figure 5.9(b). The final probability of a pe being generated
is defined as

P rob. = QE(λ) × Pobs (θi , λ) × CPMT × QTi ,104(5.10)

wherePobs (θi , λ) =A(θi , λ).
A(0, λ)(5.11)

Here QE(λ) is the incident wavelength λ dependence of the quantum efficiency of the PMTs,
measured by Hamamatsu and shown in Figure 3.16. A(θi ,λ) is the summed fraction of thepe generation (green) and the pe absorption (black) in Figure 5.9(b), which is measured
by the laser light source calibration. CPMT is a correction factor which is fined tuned by
the LINAC calibration and QTi is the quantum efficiency table measured and described in
Section 4.1.3. After the photoelectron has been determined to be generated, the output
charge of the PMT is simulated, and if it exceeds the threshold of the front-end electronics,
the photon is considered detected by the PMT. The timing resolution of the PMTs is also
taken into account in the MC simulation.

(a)

(b)

Figure 5.9: (a) Schematic view of the PMT response for incident light. (b) PMT response forincident 420 nm unpolarized light, depending on the incident angle. The black area shows
the fraction of photons which are absorbed but don’t generate photo-electrons, while thegreen area shows the fraction which generate a pe.

105

Chapter 6
Data Reduction
6.1 Run Selection
To break up the SK data taking stream and to reduce the large size of the data files, thebasic unit of SK data is broken up into “runs”. Each run has a maximum livetime of 24hours, with the run range used for analysis in SK-IV spanning from 61525 to 70784. The
quality of the data is checked within each run, with runs falling under the criteria of a badrun being thrown out. The criteria for selecting bad runs are as follows;• If the run time is less than 5 minutes there is not enough pedestal data and so it isdifficult to judge whether or not the run is bad, therefore these runs are rejected.

• If the run occurred within 15 minutes of the high voltage being turned back on thenit is possible for the PMTs to still be noisy, so these runs are rejected.

• If any hardware or software problems occurred the run is rejected.

• Any calibration and detector maintenance runs are rejected.

106

6.2 Pre-Reduction
Although there are only ∼295 solar neutrino events expected in the SK detector eachday (efficiencies not included), the number of background events expected each day is on
the order of 2×105 . If all of this data were stored to disk the total file size would be on
the order of a few hundred gigabytes (GB) per day! Because the majority of this data is
known to be background, it is necessary to reduce the data size by a factor of ∼1/100 using
a pre-reduction step utilizing loose versions of the analysis cuts. The pre-reduction includes;• The fiducial volume cut removes events which occur near the wall of the detector andhave a vertex which reconstructs within 2 m of the ID wall. The event rate is muchhigher near the walls of the ID, the source of which is known to be background events.

• Background events coming from the PMTs and ID wall may have a reconstructedvertex which falls within the fiducial volume. To remove these events a parameter
called deff is defined as the distance from the wall to the vertex in the direction of thereconstructed event, as shown in Figure 6.6. In the pre-reduction step, events withdeff <400 cm are cut. A more stringent version of this cut is applied later, for eventswith energies less than 7.5 MeV, and is known as the external event cut.

• If the reconstructed energy is less than 3.0 MeV, the event is rejected by the prereduction step.

• Some events occurring inside the fiducial volume turn out to be mis-reconstructed. It
is possible to define a pair of goodness parameters. One of which is the vertex goodness107

gv , described byPv )−t0 2
1 τi (~)
we− 2 (

gV = all hits Pv )−t0 2
1 τi (~)σ

e− 2 (

v −t0 ) 2
1 τi (~)
we− 2 (.

(6.1)

all hitsτ (~v ) is a PMT hit time after time of flight subtraction, t0 is the fit peak time of the τdistribution, w is the resolution of of the τ distribution and σ is the timing resolutionof the PMTs. This goodness is the degree of vertex reconstruction quality estimatedby the hit PMT timing information and ranges in values from 0-1, with 1 being thebest quality. The direction goodness ga is defined asgA =max{θi,uniform − θi,Data } − min{θi,uniform − θi,Data }
.
2π

(6.2)

θi,uniform is the azimuthal angle between the best fit direction and the vector from thevertex to the i-th hit PMT, assuming that the hit PMTs are uniformly distributed
along a Cherenkov cone. θi,Data is the same angle not assuming a uniform distributionof Cherenkov light. The value of ga also ranges from 0-1, but this time with good events
having a goodness closer to 0. An example of a good event (low direction goodness)
and a bad event (high direction goodness) is shown in Figure 6.1. Figure 6.2 shows thecorrelation between the vertex and direction reconstruction goodnesses squared, forbackground (MC simulated solar neutrino) events on the left (right). The line shownat gv2 − ga2 = 0.25 determines the cut criteria later on (called the event quality cut),
but for the pre-reduction step events with gv2 − ga2 < 0.1 are cut.

The number of events after each step of the pre-reduction, for an average 24 hour run,is shown in Table 6.1. It is seen here that the number of events is reduce by a factor of∼1/100. The pre-reduction inefficiency for the solar neutrino MC simulated events is shownin Figure 6.13, and is less than 0.5% for energies above 5.5 MeV.

108

Figure 6.1: The top (bottom) shows an example of a good event (bad event) which has alower (higher) direction reconstruction goodness.

Table 6.1: Reduction efficiencies after each of the pre-reduction steps. The data set is reduced
to ∼1 % of its original size by the pre-reduction.
Pre-reduction Step
InitialAfter Fiducial Volume Cut
External Event Cut
Lower Energy Cut
Event Quality Cut

Number of Events
197,524
28,161
14,864
6,872
2,057

109

Efficiency
100%
14.2%
7.5%
3.5%
1.0%

Figure 6.2: Correlation between gv2 (horizontal) and ga2 (vertical) for background (DT calibration) events on the left (right). The line shows gv2 − ga2 = 0.25.

6.3 Solar Neutrino Analysis Cuts
6.3.1 Total Number of Hit PMTs
If the total number of hit PMTs in one event Ntot exceeds 400 then the events is cut.
Ntot =400 corresponds to an energy of ∼60 MeV for an electron type event, and so making
this cut will eliminate most of the cosmic-ray muons.

6.3.2 Calibration Cut
Events which are triggered by one of the external calibration triggers are scheduled calibration events and are thus rejected. The auto-calibration events are automatically triggered
during normal runs and include laser events for water transparency measurements (done frequently), LED-bursts which are used as a test for the supernova burst detection system(every 3 weeks or so), and pedestal events which occur every 30 minutes. We also remove

110

Table 6.2: Calibration sources considered for the calibration source cut. The water temperature sensor is considered after run 33027 (September 4th , 2007).
SourceXenon
LED
TQ Diffuser Ball
Supernova Burst Test
Water Temp. Sensor

x (cm) y (cm) z (cm)
353
-70.7
0
35
-350
150
-177
-70.7
100
-35
353
100
-35
1200
-2000

spurious events which may come from the calibration sources themselves. Events having reconstructed vertices closer than 2 m to a calibration source, or closer than 1 m to the source
cable (all cables run along the z-axis from the top down to the source position) means theevent is removed. Table 6.2 lists the various calibration sources which are considered for thiscut.

6.3.3 Flasher Cut
It is possible for a PMT to arc discharge on the dynodes. When this happens other PMTs
may also pick up the signal, causing what is called a “flasher event”. In a typical flasher
event, one PMT will contain a large amount of charge, with many other surrounding PMTs
showing hits. Figure 6.3 shows the two-dimensional plot of the maximum charge of a PMT
and the number of hits in the surrounding 24 PMTs. Events having a maximum charge
greater than 50 pe and the number of hit surrounding PMTs greater than three are regarded
as flasher events and rejected. The top panel is a run contaminated with flasher events,
while the bottom panel shows a typical good run.

111

Figure 6.3: Cut criteria for flasher events. The horizontal axis shows the maximum chargeof a PMT in an event and the vertical axis shows the number of hits in the surrounding 24
PMTs. The top figure shows a run which has many flasher events, while the bottom figure
shows a typical good run.

6.3.4 Time Difference Cut
To remove decay-e’s coming from cosmic-ray muons, and events caused by “ringing”
noise following a very energetic muon, events occurring within 50 µsec of each other are
removed. During the SK-IV phase, untagged muons, typically not energetic enough to
produce Cherenkov radiation, are also taken into account when calculating the time difference
to the previous event.

6.3.5 OD Event Cut
Solar neutrino elastic scattering occurring in the fiducial volume of SK should result inan event in which all of the Cherenkov light is observed in the ID of the detector. If there
is a large amount of hits in the OD PMTs, then the event is the result of a charged particle

112

entering the detector from outside. If the OD has been triggered (total charge >∼20 pe)then the event is cut. In addition to this cut on the OD trigger, any event which has morethan 19 OD hits is cut, regardless of whether or not the OD was actually triggered.

6.3.6 Event Quality Cut
As described in Section 6.2, the quality of the vertex and direction reconstruction can be
described by the goodness parameters gv and ga . As shown in Figure 6.2, the correlation
between gv2 and ga2 is quite different for the background and MC simulated solar neutrino
events. The placement of the line in this figure is determined so that the significance (signal√
efficiency/ background) is maximized. As one would expect, the optimization line of thiscut changes with energy, and so the energy dependent cut values of the difference between
gv2 and ga2 are assigned as
g 2 − ga2 > 0.29, 3.5 ≤ E < 4.5 MeV v
gv2 − ga2 > 0.25, 4.5 ≤ E < 7.0 MeV g 2 − g 2 > 0.20, E ≥ 7.0 MeV.
a
v

(6.3)

6.3.7 Spallation Cut
When high energy cosmic-rays enter the SK tank it is possible for them to interact bybreaking up an oxygen nucleus, often resulting in the production of various radioactive
elements. The general format for this reaction isµ + 16 O → µ + X + · · · ,(6.4)

where X is a radioactive nucleus and the other particles produced can include things such
as n, p, π ± , etc, which are produced by hadronic cascades induced by muon interactions.
113

Table 6.3: List of possible spallation products in SK.
Isotope82 He

τ 1 (sec)
2
0.119

8
3 Li
8
3B
9
3 Li

0.838
0.77
0.178

9
6C
11
3 Li

0.127
0.0085

11
4 Be

13.8

Decay Mode
β−
β− n
β−
β+
β−
β− n
β+ n
β−
β− n
β−

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

0.02360.02020.01100.01740.00860.0138
2.449

β−
β−
β+
β−
β+
β−
β−

16
6 C
16
7 N

0.747
7.13

β− n
β−

Kinetic Energy (MeV)
9.67 + 0.98 (γ)
16%
∼13
13.9
13.6 (50.5%)(∼50%)
3∼15
16∼20 (∼50%)∼16 (∼50%)
11.51 (54.7%)
9.41 + 2.1 (γ) (31.4%)
11.71
13.37
16.32
13.44
13.2 16.7
14.55+6.09 (γ)
9.77 (36.8%)
4.47+5.30 (γ)∼410.42 (28.0%)
4.29+6.13 (γ) (66.2%)

These secondary particles can then also be captured by other oxygen nuclei, resulting in
the formation of some radioactive nuclei. At times, the produced radioactive elements canbe quite long lived isotopes, making the correlation between them and the parent muons
extremely hard to accomplish. The decays will often times involve a β and/or γ decay,
signals which can exactly mimic that of a solar neutrino interaction. A list of the possible
resulting radioactive isotopes in SK which decay via an emitted β or γ are given in Table 6.3,
along with their half-lives (ranging from 0.001 to 14 sec) and energy modes (4-20 MeV). Theseβ and γ decay events are called spallation events, and because their energy range and signals
match those of solar neutrinos, they can be a major source of background.
To identify such spallation events in a statistical way, a likelihood function is used. The

114

spallation cut likelihood is defined as

Lspa = f (∆T ) × f (∆L) × f (Qres ).

(6.5)∆T is defined as the time difference between the spallation candidate event and the preceding muon event, while ∆L is defined as the distance from the reconstructed vertex of thespallation candidate event to the reconstructed track of the preceding muon event. Qres is
the residual charge of the preceding muon event and is defined as the difference between
the observed charge in the PMTs Qobs and the distance the preceding muon traveled (incm) times the expected observed charge per cm from a muon in water (∼11.4 pe/cm). If
the muon really does cause spallation, it will have more charge than expected, due to thecreation of hadronic cascades.
The likelihood function f (∆T ) is generated by considering the lifetime of the produced
radioactive elements, while the likelihood functions f (∆L) and f (Qobs ) are generated using
two real data samples, a spallation like sample taken from data and a random sample.
The spallation-like sample is composed of events with ∆T <0.1 sec and E >8 MeV The
random sample is composed of events with E <5 MeV and are assigned random vertices anddirections. This is because most of the low energy events are produced by radioactive sourcesat the ID wall and not by spallation. Figure 6.4 shows the likelihood distribution for the
spallation-like sample (black) and for the random sample (red). The value of the spallation
likelihood taken as the cut value is determined so that the dead time of the random sample is20%. The cut point in SK-IV is determined to be log(Lspa )=4.52. The position dependenceof the resulting dead time is also taken into account when the expected number of solar
neutrino events is calculated in the MC simulation, and is shown in Figure 6.5.

115

Figure 6.4: Spallation cut likelihood distribution for the spallation-like sample (black) andthe random sample (red). The horizontal axis shows log(Lspa ).

6.3.8 Hit Pattern Cut
When the spallation products resulting from cosmic-ray muon interactions decay via multiple γ and/or β rays, the pattern of the Cherenkov light will not be a clear ring. For thisreason it is possible to distinguish a single electron event from an event involving more than
one γ and/or β, by use of a likelihood function based on the pattern of hit PMTs. The hit
pattern likelihood is defined as
N

50
1 X
Lpattern (E, ~v ) =log(Pi (E, cos θPMT , fwall )),
N50 i

(6.6)

where Pi (E, cos θPMT , fwall ) is the pdf of the hit pattern, made by single electron MC simulated events and N50 is the number of hits within a 50 nsec time window. E is the energy ofthe electron, cos θPMT is the cosine of the angle between the reconstructed direction and thevector from the reconstructed vertex to each hit PMT. fwall is a parameter defined as theeffective distance between the wall and the reconstructed vertex. A schematic view showing
the definition of both θPMT and fwall is shown in Figure 6.6. Figure 6.7 shows the hit pattern

116

Figure 6.5: Position dependence of the dead time resulting from the spallation cut. The
horizontal axis of the top (bottom) figure is the detector z (r2 ) position in cm (cm2 ).

117

Figure 6.6: Definition of θPMT , fwall and deff .

likelihood for both data (black) and MC (red) simulated solar neutrino events, for threedifferent energy regions. The energy dependent cut on the hit pattern likelihood is thendefined as
P > −1.88 6.5 ≤ E < 8.0 MeV,P > −1.86 8.0 ≤ E < 12.0 MeV,
 P > −1.95 E ≥ 12.0 MeV.

(6.7)

Below 6.5 MeV it is difficult to distinguish a single electron event from an event withmultiple particles, due to the low number of hit PMTs. For this reason the hit pattern
likelihood cut is applied for events with E >6.5 MeV only.

118

Figure 6.7: Lpattern for data (black) and solar neutrino MC simulated events (red), for 6.5-8.0
MeV (left), 8.0-12.0 MeV (middle) and >12 MeV (right).

6.3.9 External Event Cut
External gamma-rays and radioactivity coming from the PMTs and other detector components can be reconstructed inside the fiducial volume. These events will mostly have areconstructed direction pointing toward the center of the detector, or “inward”. To rejectbackground of this sort, the distance to the wall from the vertex, in the direction opposite
to the reconstructed direction, is defined as deff (see Figure 6.6). Figure 6.8 shows the deff
distribution for 5.5-8.0 MeV and 8.0-8.5 MeV data (black) and solar neutrino MC simulated
events (red). Events are then rejected if deff > 650 cm,

6.5 ≤ E < 8.0 MeV,
 deff > 400 cm,

E ≥ 8.0 MeV.

(6.8)

Because the radioactivity around the edge of the ID is not uniformly distributed, a tightfiducial volume cut is necessary for E <5.0 MeV (see Section 6.3.12). For the determinationof the external event cut for E <5.5 MeV, events which are reconstructed in the central
region of the detector (z >-10 m and r2 <13 m2 ) are used. The cut further depends upon avariable called pwall , which is defined as either “top”, “bottom” or “barrel” (see Figure 3.7),

119Figure 6.8: deff distributions for data (black) and MC simulated solar neutrino events (red),for 5.5-8.0 MeV (left) and 8.0-8.5 MeV (right). The black line gives the value used for theexternal event cut in each energy region.

Figure 6.9: deff distributons for lower energy events (4.5-5.0 MeV) for each of the threedifferent values of pwall , for data (black) and MC simulated solar neutrino events (red).

and is the first detector surface which is hit when tracing back along the incoming neutrino
direction from the vertex. Figure 6.9 shows the deff distributions for events with E between
4.5 and 5.0 MeV total energy, for the three different values of pwall . Based on pwall and deff ,the external event cut criteria for events with E between 4.0 and 5.5 MeV isdeff > 1000 cmdeff > 1200 cm deff > 1300 cm(pwall is top),(pwall is barrel),(pwall is bottom).

120

(6.9)

6.3.10
16

16

N Cut

N can be produced inside the SK detector when low energy cosmic-ray muons arecaptured by 16 O. The 16 N will then decay via gamma-rays and/or β’s, with a half-life of
7.13 sec. This is a similar situation as that which occurs during the DT calibration, when16

N is produced by means of neutron absorption on the 16 O. As in the DT calibration themain decay mode of the 16 N is a 6.1 MeV gamma-ray and a 4.3 MeV β (66%) or a 10.41
MeV β (28%). In order to reject these events, the correlation between stopping muons in
the detector and the remaining candidate events are checked. To first select only captured
muons from the stopping muon sample, stopping muons which did not accompany an event
within 100 µsec are selected. The remaining candidate events whose reconstructed vertex
is within 335 cm from the stopping point of a muon, or the time difference between thecandidate event and a stopping muon is between 100 µsec and 30 sec, are rejected. The
dead-time resulting from this 16 N cut is estimated to be 0.53%, from accidental coincidence
between the selected stopping muons and the random sample events.

6.3.11 Small Hit Cluster Event Cut
The small hit cluster event cut targets radioactive background events in the PMT enclosures or glass, which coincides with an up-fluctuation of PMT dark noise. Only eventswith a reconstructed r2 larger than 180 m2 (155 m2 ), a reconstructed z smaller than -7.5 m(-3 m) or a reconstructed z larger than 13 m, for reconstructed energies less than 6.0 MeV(5.5MeV) are subject to this cut. To characterize small hit clusters, we select PMT hits
whose times (corrected by the travel time of the light signal assuming a straight line pathfrom the vertex to the PMT) are coincident within 20 nsec, and then find the smallest circle
around any of the selected PMTs that encloses at least 20% of all selected PMTs. The radiusof this circle is then multiplied by the ratio of PMT hits coincident within 20 nsec (without

121

time-of-flight correction) over Neff . Solar neutrino events near the edge of the fiducial volume
have a bigger radius×hit ratio (see also section III.C in [25], Figures 17 and 18) than theradioactive background. Events with a radius×hit ratio less than 75 cm are rejected if they
have r2 >155 m2 (r2 >120 m2 ), z <-7.5 m (z <-3 m or z >13 m) and energy between 5.0
and 5.5 MeV (4.0 and 5.0 MeV).

6.3.12 Tight Fiducial Volume Cut
Even after subjecting the SK data sample to the above cuts, there still remains a large
amount of low energy background near the bottom and edge of the barrel. It is important
for the vertex distribution to be as uniform as possible within the fiducial volume. Thisis so the background shapes will be uniform and the systematic uncertainties associatedwith them will be as small as possible. Figure 6.10 shows the vertex distribution for thethree lowest energy bins in the solar neutrino analysis (4.0-4.5, 4.5-5.0 and 5.0-5.5 MeV√
total recoil electron energy). Based on a significance calculation (signal/ background) andan estimation of the systematic effects, it was chosen to make no tight fiducial volume cut
above 5.5 MeV, but for energies between 4.0 and 5.5 MeV, the tight fiducial volume cut
shown by the red lines is made. For the energy bin of 5.0-5.5 MeV, events are kept if theyfall within the volume contained by r2 <180 m2 and z >-7.5 m (red lines). For events withenergies between 4.0 and 5.0 MeV the tight fiducial volume cut is described by the red curve
and the function(x2 + y 2 ) + (150∗ |z − 4.25|4 ) ≤ 150.
4
11.75

(6.10)

By making this tight fiducial volume cut, the fiducial volume used in solar neutrino
analysis between 4.0 and 5.0 MeV is reduced to 8.8 ktons, while between 5.0 and 5.5 MeV
it is reduced to 13.3 ktons. Above 5.5 MeV the total fiducial volume of 22.5 ktons is used.

122z (m)

z (m)

0.3

15

0.25

10

0.3

15

0.25

10

5

0.2

5

0.2

0

0.15

0

0.15

-5

0.1

-5

0.1

-10

50

100

150
200
rr-z distributionr^2 (m^2)

z (m)

-15
0

0.05

0

-10
-15
0

0.05

50

100

200
r^2 (m^2)

0

0.3

15
10

0.25

5

0.2

0

0.15

-5

0.1

-10

0.05

-15
0

150

50

100

0

150
200
r^2 (m^2)
Figure 6.10: Vertex distribution of 4.0-4.5 MeV (recoil electron kinetic energy) (top left),
4.5-5.0 MeV (top right) and 5.0-5.5 MeV (bottom) events in the SK-IV final data sample.
The units are events/day/bin. The tight fiducial volume cut made in each bin is shown by
the red curve (lines).

123Figure 6.11: Event rate of the final data sample for SK-III (black) and SK-IV (red).

6.4 Summary of the Reduction Steps
Figure 6.11 shows the time variation of the event rate in the final data sample for SK-III(black) and SK-IV (red). With improvements of the water circulation system and cut criteria,
a continuous lower background level is achieved during SK-IV. Figure 6.12 shows the energy
spectrum of the SK-IV data sample after each of the main reduction steps. The final data
sample is shown by the open circles (for comparison the SK-III final data sample is shownby the dotted lines). Figure 6.13 shows the selection efficiency of the MC simulated solar
neutrino events after each of the main reduction steps, with the final selection efficiency of
SK-IV shown by the open circles (dotted lines show SK-III final reduction efficiency). Above
6.5 MeV kinetic recoil electron energy, the selection efficiency of the SK-IV final data sample
is just about the same level as SK-III, while for 5.5-6.5 MeV the SK-IV selection efficiency isbetter than SK-III, caused by making a looser external event cut and removing the cluster
hit event cut at these energies. Below 5.5 MeV the SK-III final sample selection efficiency isbetter than that of SK-IV, caused by the tightening of the event quality cut for 5.0-5.5 MeV
and making a tighter fiducial volume cut for 4.0-5.0 MeV.

124

4

10

3

10

2

Events/day/kt/0.5MeV

10

10
1

10
10
10

-1
-2
-3

4

6
8
10
12
14
Recoil Electron Kinetic Energy (MeV)

Efficiency

Figure 6.12: SK-IV energy spectrum after each of the main reduction steps. The filledsquares (triangles) correspond to the sample after the pre-reduction (spallation) cut. The
stars give the spectrum after the external event cut, and the SK-IV final data sample isshown by the open circles. The SK-III final data sample is shown by the black circles.

1

0.75
0.5
0.25
0

4

6
8
10
12
14
Recoil Electron Kinetic Energy (MeV)

Figure 6.13: Selection efficiency for the solar neutrino MC simulation events after each of themain reduction steps. The open stars give the trigger efficiency. The filled squares (triangles)
correspond to the reduction step after the pre-reduction (spallation) cut. The filled stars
give the efficiency after the external event cut and the SK-IV final efficiency is given by theopen circles. The SK-III final selection efficiency is shown by the dotted lines.

1256.5 Multiple Scattering Goodness
Even at the low energies of the 8 B solar neutrinos, it is possible to use the PMT hit patternof the resulting Cherenkov cones to reconstruct the multiple Coulomb scattering of theelectrons. Very low energy electrons will incur more multiple scattering than higher energy
electrons and thus have a more isotropic PMT hit pattern. Radioactive background events
such as 214 Bi β decays generally have less energy than 8 B solar neutrinos. To characterize thishit pattern anisotropy, a “direction fit goodness is used. This goodness, called the multiple
scattering goodness (MSG), is constructed by the following method:• Select pairs of hit PMTs which occur within a 20 nsec sliding time window, after timeof-flight subtraction. The vector ~ri gives the direction from the vertex position to thei-th PMT.
• For each pair of hit PMTs, cones with opening angles of 42◦ are projected around eachof the ~ri ’s, and the number of intersection lines between the cones is counted. This
number will be either 0, 1 or 2.
• For pairs of PMTs which give intersection lines from the projection of the cones, thevectors from the vertex along each of the intersection lines are taken as “unit vectors”~ui .
• Clusters of ~ui ’s are found which are within 50◦ of a central ~ui . The unit vectors arethen summed along the direction of the central ~ui . There may be multiple clusters,
and hence multiple vector sums. The vector sum with the largest magnitude is kept
as the best fit direction d~best .
• d~best is then used at the central ~ui , and the above step of summing unit vectors alongd~best is repeated. This serves to maximize as well as center the clusters of unit vectors.
After a few iterations the longest vector sum is taken as the final best fit direction
d~best .
126

0.04
0.03
0.02
0.01
0

0

0.2

0.4
0.6
0.8
1
Multiple Scattering Goodness

Figure 6.14: Multiple scattering goodness (MSG) for 4.6 MeV (8.4 MeV) LINAC data inblack (red) and MC in blue (cyan), normalized by number of events.

• The MSG is defined as the length of the best fit direction vector sum, divided by the
total number of unit vectors which would result using all hit PMTs within the 20 nsec
time window.

The above method will give a value between 0 and 1. Events incurring more multiple
scattering will have a lower MSG (closer to zero), and events which do not multiple scatteras much will have a higher MSG (closer to 1). Figure 6.14 shows the comparison of the MSG
for 4.6 MeV (8.4 MeV) LINAC data and MC in black and blue (red and cyan). As expected,
higher energy electrons have a larger mean MSG. There is no cut made to the final data
sample based on the MSG. When extracting the number of solar neutrino events in eachenergy bin, the lowest energy bins are further separated into bins of MSG. This is done tofurther separate the background from the solar neutrino signal (see Section 8.1).

127Chapter 7
Solar Neutrino Flux Analysis
7.1 Signal Extraction Method
The main handle for extracting the solar neutrino signal in SK comes from Equation 3.11,
which gives a distribution of events highly peaked in the forward direction. This means recoil
electrons coming from the elastic scattering of solar neutrinos will point back to the direction
from which the neutrinos came from, in this case the direction of the Sun. For this reason it isconvenient to define an angle in SK called θsun , which is the angle between the reconstructed
event direction and the vector pointing from the Sun’s position (at the time of the event) tothe vertex (defined in Figure 7.1). In the case of surviving background events, we expect thereconstructed directions to be completely random in direction and have no correlation to thesolar position. Therefore, if we plot the distribution of cosθsun for events in SK, there should
be an excess of solar neutrino events on top of the constant background. The excess of solar
neutrinos should be strongly peaked at cosθsun =1. Figure 7.2 shows the distribution of the
SK-IV final sample events with energies between 4.0 and 19.5 MeV recoil electron kinetic
energy. As expected there is a large peak at cosθsun =1, coming from the detection of solar
neutrino elastic scattering events. The constant offset from 0 gives the flat distribution of
background events, which is uncorrelated with the solar position.
In order to precisely determine the number of solar neutrino elastic scattering eventsoccurring in SK, an extended maximum likelihood fit to the cosθsun distribution is used, forrecoil electron energies 4.0-19.5 MeV. This is the same method which is used in all phases of128

Event/bin

Figure 7.1: Definition of the angles θsun and θz .

7000

6000

5000

4000

3000

2000

1000

0
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1
cosθsunFigure 7.2: SK-IV final data sample cos θsun distribution for 4.0-19.5 MeV events.

129

SK, and has been described extensively in [45, 46, 25]. The likelihood function is defined as

L=e

N
ni
bin Y
Y
P
−( i Bi +S)(Bi · bij + S · Yi · sij ).

(7.1)

i=1 j=1

Nbin is the number of energy bins the final data sample.
In SK-IV Nbin =22 energy bins, 19 bins of 0.5 MeV from 4.0 to 13.5 MeV, two 1 MeV
energy bins between 13.5 and 15.5 MeV and one energy bin spanning from 15.5 to 19.5 MeV(recoil electron kinetic energy). ni is the number of observed events in the i-th energy bin,while Yi is the fraction of signal events expected in the i-th energy bin, calculated basedsunon MC simulated solar neutrino events. The background weights bij = βi (cos θij) and thesun
signal weights sij = σi (cos θij) are calculated from the expected shapes of the backgroundand the solar neutrino signal, respectively (these are the pdf’s). The background shapes
βi are based on the detector zenith and azimuthal angular distribution of real data, which
are fit by polynomial functions and then converted from these coordinates to the cosθsun
distribution. The background shapes, seen in Figure 7.3, are not necessarily completely flat,
due to the cylindrical shape of the detector inducing preferred detector zenith and azimuthal
directions of background events. The signal shapes σi shown in Figure 7.4 are obtained
from the MC simulated solar neutrino events. Finally the free parameters Bi and S are thenumber of background events in the i-th energy bin and the total number of signal events inall energy bins combined. These are the values obtained by maximization of the likelihood
function.

130

probaility

0.52
0.515
0.51
0.505
0.5
0.495
0.49

4.5MeV
0.485

8.0MeV
0.48

10.0MeV
0.475
0.47

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

cosθsunProbability

Figure 7.3: Cosθsun probability density function of the background (bij ) for three different
energy bins.

0.3

4.5MeV

0.25

8.0MeV

0.2

10.0MeV
0.15

0.1

0.05

0
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1
cosθ

Figure 7.4: Cosθsun probability density function of the solar neutrino signal (sij ) for threedifferent energy bins.

131

Table 7.1: SK-IV energy resolution difference between LINAC data and MC simulation. Theenergy refers to the electron in tank kinetic energy.
Energy (MeV)
4.38
6.28
8.16
12.93
15.58

∆σ(Eres ) (%)
-2.53
0.90
-0.95
1.03
0.76

7.2 Systematic Uncertainty on the Extracted Solar
Neutrino Flux
7.2.1 Energy Scale
The systematic uncertainty on the total flux due to the energy scale is estimated bytaking into account the systematic uncertainty of the absolute energy scale (0.54%). Firstthe energy scale is shifted by either +(-)0.54%, shifting the energies of the MC simulated
solar neutrino events. This produces new signal fractions Yiscale+ or Yiscale- to be used in thesignal extraction likelihood. The ±1.2% difference in the extracted number of signal events,
which is the results of using either Yi or Yiscale+,scale- , is taken as the systematic uncertainty
on the total flux coming form the energy scale. This large improvement from the 2.2%
estimated in SK-III is the result of the inclusion of the total livetime for all low energy bins(for SK-III data below 6.0 MeV only had about half the SK-III livetime). Also the higher
selection efficiency of SK-IV between 5.0 and 6.0 MeV and the extending of the low energy
threshold down to 4.0 MeV in SK-IV (from 4.5 MeV in SK-III) further lessens the impactof the energy scale and resolution uncertainties on the total flux determination.
The systematic uncertainty on the total flux coming from the uncertainty on the energy
resolution is estimated by taking into account the difference of the energy resolution between

132

LINAC data and MC simulated events, shown in Table 7.1. The difference in the energy
resolution between LINAC data and MC simulated events ∆σ(Eres ) is taken as
2.53% (E < 4.77 MeV),∆σ(Eres ) =1.0% (E > 6.75 MeV), interpolation (4.77 ≥ E ≥ 6.75 MeV).

(7.2)

The predicted observed electron energy spectrum is then calculated by distorting thetheoretically expected electron energy spectrum by the detector resolution, giving
Z EmaxF(Eobs ) =F (Ee )R(Ee , Eobs )dEe .

(7.3)

0

Ee is the total energy of the recoil electron, F (Ee ) is the originally predicted recoil electron
spectrum, calculated in Equation 3.9, and R(Ee , Eobs ) is the detector response function. The
detector response function takes into account the energy resolution σ(Ee ) and is describedby1e
R(Ee , Eobs ) = p2πσ(Ee )2



(E −E
)2
− e obs2
σ(Ee ).

(7.4)

The top panel of Figure 7.5 shows the energy distributions obtained from various MC
simulations containing single electron events of different monoenergetic energies. The black
points in the bottom panel show the standard deviation of the top Gaussian fits, while thered line gives the fit result of the energy resolution. The resulting energy resolution as afunction of recoil electron total energy is then given asσ(Ee ) =p1(−0.123 + 0.376 Ee + 0.0349Ee ).
Ee

(7.5)

The idea is then to replace σ(Ee ) in Equation 7.4 with either σ + (Ee ) = σ(Ee )(1+∆σ(Eres ))or σ − (Ee ) = σ(Ee )(1 − ∆σ(Eres )), obtaining the shifted expected observed recoil electron
133

9000
8000
7000
6000
5000
4000
3000
2000

Energy Resolution

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

5

10

15

20

25

5

10

15

20

25

30

35

40

30
35
40
Total Energy (MeV)

Figure 7.5: Energy resolution function taken from monoenergetic electron MC simulated
events.

spectra F + (Eobs ) and F − (Eobs ). Finally the ratios of F + (Eobs ) or F − (Eobs ) to F(Eobs ) are
obtained and used as weighting factors for the energy bin signal fraction Yi . This producesEthe energy resolution uncertainty shifted energy bin signal fractions Yi res+ and YiEres − . Using
these shifted energy bin signal fractions in the signal extraction method results in a ±0.15%
shift in the total extracted signal. This is taken as the systematic uncertainty on the total
flux coming from the energy resolution.

7.2.2

8

B Neutrino Spectrum

The uncertainty in the 8 B solar neutrino spectrum shape is estimated in [35] and includes
both experimental and theoretical uncertainties. To estimate the effect on the extractionof the total solar neutrino flux coming from the uncertainty on the 8 B spectrum shape, the134solar neutrino spectrum φ(Eν ) which is used to produce the MC simulated events is distorted
according to the spectrum shape uncertainties. By doing this, the expected observed recoil
88electron spectrum shifts from F(Eobs ) to either F B+ or F B− . These new expected recoil
8

electron spectra are then used to produce the shifted energy bin signal fractions Yi B+ and
8

Yi B− , which in turn replace the original Yi in the signal extraction likelihood. The difference
8B(+ or −)of ±0.35% in the extracted total flux when using Yi

compared to Yi is taken as

the systematic uncertainty on the total flux coming from the uncertainty on the 8 B solar
neutrino spectrum.

7.2.3 Signal Extraction Method
The systematic uncertainty coming from the signal extraction method is estimated byapplying the signal extraction to dummy data, in which the true number of signal andbackground events is known. The resulting difference in the extracted number of signal
events and the true number of signal events is taken at the uncertainty on the total flux.
From this method the uncertainty on the total flux coming from the signal extraction method
is estimated as ±0.7%.

7.2.4 Cross Section
The uncertainties associated with the solar neutrino interaction cross section are estimatedin [43], in which the radiative corrections parameters for (ν-e) elastic scattering have 1 σtheoretical uncertainties. The errors on ρ and κ come mainly from the lack of precise
knowledge of the top quark and higgs masses and quantum chromodynamic processes. These
parameters are shifted by the ±1 σ values and new expected observed recoil electron spectra
are created. These spectra are again used to determine the shifted energy bin signal fractions,which are in turn used when extracting the total flux. The ±0.5% change in the extracted135

Table 7.2: Angular resolution difference between LINAC data and MC for SK-I, III and IV.
The energy refers to the electron in tank kinetic energy.
Energy (MeV)
4.38
5.33
6.28
8.16
10.26
12.93
15.58

SK-I(%)
-1.64
-1.38
2.32
2.33
1.52
1.07
0.883

SK-III(%)
0.74
0.40
-0.27
0.39

SK-IV(%)
-0.01
0.51
2.05
1.40
0.63

total flux is taken as the estimated systematic uncertainty on the total flux, coming fromthe solar neutrino elastic scattering cross section uncertainty.

7.2.5 Angular Resolution
The angular resolution is obtained from LINAC data and MC simulated events, and isdefined as the angle which includes 68% of the events in the angular difference distribution
from the LINAC beam direction. The systematic uncertainty on the angular resolution is
obtained from the difference of the angular resolution between the LINAC data and MC
simulated events. This difference is shown in Table 7.2, and is comparable to SK-I and III.
To estimate the systematic uncertainty on the total flux new signal shapes sang+and sangare
ij
ij
made by shifting the reconstructed directions of the MC simulated solar neutrino events by
the uncertainty in the angular resolution. These new signal shapes are used when extracting
the total flux, and the ±0.36% change in the extracted flux is taken as the systematic
uncertainty coming from the angular resolution.

1367.2.6 Trigger Efficiency
The trigger efficiency depends on the vertex position, water transparency, number of andwhich of the PMTs are hit and various other systematic effects. To estimate the effect of thetrigger efficiency on the extracted total flux, Ni calibration data and MC simulated eventswere compared. For 3.5-4.0 MeV (4.0-4.5 MeV) the difference in trigger efficiency between
data and the MC simulated events is 4.5% (0.37%). Above 4.5 MeV the trigger efficiency is
100% and so its contribution to the flux uncertainty is negligible. These differences in thetrigger efficiency are applied as a change in the extracted number of events and the obtained
systematic uncertainty on the total flux coming is ±0.1%.

7.2.7 Background Shapesun
As described in Section 7.1, the predicted background shapes βi (cos θij) are generatedusing the detector zenith and azimuthal angle polynomial fits to the data sample. Because of
the background directional dependence coming from the cylindrical shape of the detector, thebackground distribution in φ will not be flat, but instead have an asymmetric shape. For thisreason the background distribution in φ is originally fit using a non-flat shape. To estimate
the uncertainty on the total flux coming from the background shape, the φ distribution isforced to fit with a flat shape, while still taking into account any non-flatness in the zenithsunangle distribution. This will result in a slight change of the βi (cos θij) shapes. These newshapes are used while extracting the total flux, and the 0.1% change taken as the systematic
uncertainty on the total flux coming from the background shape.

137

Vertex Shift Uncertainty (%)

0.5
0.4
0.3
0.2
0.1
0

4

6
8
10
12
14
Recoil Electron Kinetic Energy (MeV)

Figure 7.6: Vertex shift systematic error on the flux in each energy bin. The energy dependence below 5.0 MeV comes from the tight fiducial volume cut.

7.2.8 Reduction Steps
Fiducial Volume Cut

One of the main contributions to the systematic uncertainty on the extracted solar neutrino flux comes from events which are reconstructed close to the fiducial volume edge, andmay either be included or excluded by the fiducial volume cut depending on whether or not
the vertex shift is taken into account. To estimate the systematic uncertainty of the total
rate of solar neutrinos, the reconstructed vertex positions of the MC simulated solar neutrino events are artificially shifted following the arrows in Figure 5.3. The number of events
passing the fiducial volume cut before and after the shift are compared. Figure 7.6 showsthe energy dependence of the vertex shift, with the large jump at 5.0 MeV coming from thetight fiducial volume cut. The systematic uncertainty on the total number of extracted solar
neutrino events is estimated to be ±0.1%.

138

Figure 7.7: The difference of the event quality cut efficiency between data and MC simulated
events, obtained by the LINAC calibration.

Event Quality Cut

LINAC calibration data and MC simulated events are used to estimate the effect the event
quality cut has on the total flux. The difference of the parameters gv2 − ga2 between LINAC
data and the corresponding MC simulated events is shown in Figure 7.7, and is always lessthan 1%. This translates to an estimated uncertainty on the total flux of ±0.3%.

Spallation Cut

To estimate the systematic uncertainty on the flux coming from the spallation cut, thedifference in the observed dead-time between a random data sample and MC simulated
events (taking into account the position dependence of the dead-time, see Section 6.3.7) is
considered. This difference will also have a time dependence to it, as shown in Figure 7.8.
The resulting spallation cut associated systematic uncertainty on the total flux is ±0.2%.
139Figure 7.8: Time variation of the dead-time resulting from the spallation cut.

Hit Pattern Cut

The effect on the total flux coming from the hit pattern cut is also estimated by comparing
the difference of the cut efficiency between LINAC data and the corresponding MC simulated
events. This comparison is shown in Figure 7.9, again with the difference between data andMC simulated events being less than 1% everywhere. The hit pattern cut systematic error
on the total flux is then calculated to be ±0.3%.

External Event Cut

Because the value of deff is dependent on both the vertex position and the reconstructed
electron direction, it is possible for both the vertex shift and the angular resolution to effectthe cut efficiency of the external event cut. To estimate the total flux uncertainty coming
140

Figure 7.9: The difference of the hit pattern cut efficiency between LINAC data and MC
simulated events.

from the external event cut, the vertex and direction of events are shifted according to
Figure 7.10. The change in the selection efficiency is taken as the systematic error of theexternal event cut. The difference in the energy bin signal fractions are then applied while
maximizing the signal extraction likelihood, with the ±0.2% difference in the extracted total
flux taken as the systematic uncertainty on the total flux.

Small Hit Cluster Event Cut

The systematic uncertainty on the total flux coming from the small hit cluster event
cut is estimated by the difference of its efficiency between DT data and the corresponding
MC simulated events. DT events taken within the region of the small hit cluster event cut
are used. Figure 7.11 shows the obtained small hit cluster event cut distribution for DT
data (black) and MC simulated events (red). The difference of the small hit cluster event

141

5JKHVGF
FKTGEVKQP
1TKIKPCN
FKTGEVKQP

F GHH
UJKHV
5JKHVGFRQUKVKQP
1TKIKPCNRQUKVKQP

Figure 7.10: The method to artificially shift the reconstructed vertex and direction according
to the measured vertex shift and angular resolution. The direction is shifted within the planeof the true and reconstructed directions.

cut efficiency for 5.0-5.5 MeV, 4.5-5.0 MeV and 4.0-4.5 MeV is 0.44%, 0.64% and 1.1%,
respectively. The systematic uncertainty on the total flux is estimated to be 0.3%.

7.2.9 Summary of the Systematic Uncertainty on the Extracted Flux
The systematic uncertainty on the total flux is summarized in Table 7.3, for all phases.
The combined systematic uncertainty is calculated as sum in quadrature of all components,
and is found to be 1.7% in SK-IV. This is by far the best value seen throughout the four phasesof SK, much improved over the 2.2% seen in SK-III. The largest contributions to the reductionin the uncertainty comes from improvements in the uncertainties arising from the energycorrelated uncertainties (energy scale and resolution), the vertex shift, trigger efficiency
and the angular resolution. The reason for the improvement in the total flux uncertainty
coming from the energy-correlated components is largely because SK-IV data taking was donewith the lowest energy threshold (4.0 MeV) and with all energy bins acquiring the entire
phase livetime. The majority of the improvements coming from the energy-uncorrelated
components can be contributed to the replacement of the front-end electronics, which has

142

Figure 7.11: The difference of the small hit cluster cut efficiency between data and MC
simulated DT events for energy bins 4.0-4.5, 4.5-5.0 and 5.0-5.5 MeV.

led to an improvement of the timing residuals and the better matching of the timing residuals
between data MC simulated events.

7.3 Extracted Solar Neutrino Flux
The results of this and the following sections will be published in [44]. After applying
the extended maximum likelihood fit method to extract the signal and background components from the cosθsun distribution of the SK-IV final data sample, it is known how manybackground and solar neutrino events were observed in the detector. The black points in
Figure 8.6 show the cosθsun distribution of the SK-IV final data sample, including eventswith energies between 4.0 and 19.5 MeV, and broken up into 80 bins of cosθsun . Also shownby the solid (dashed) red distribution in this figure is the best fit result of the background(signal plus background) component coming from the extended maximum likelihood fit. The
total number of extracted solar neutrino elastic scattering events with recoil electron kinetic
energies 4.0-19.5 MeV, during SK-IV, was found to be24, 405+244−242 (stat.) ± 414(syst.) events.

143

(7.6)

Table 7.3: Summary of the systematic uncertainty on the total flux for each SK phase.

Event/day/kton/bin

SK-I SK-II
Threshold in MeVkin
4.49
6.49
Trigger Efficiency
0.4% 0.5%
Angular Resolution
1.2% 3.0%
Reconstruction Goodness
1.0% 3.0%
Hit Pattern
0.8%−
Small Hit Cluster−−
External Event Cut
0.5% 1.0%
Vertex Shift
1.3% 1.1%
Second Vertex Fit
1.3% 1.0%
Spallation Cut
0.2% 0.4%
Background Shape
0.1% 0.4%
Signal Extraction
0.7% 0.7%
Cross Section
0.5% 0.5%
Livetime
0.1% 0.1%
Multiple Scattering Goodness−
0.4%
Subtotal
2.8% 4.8%+1.7+4.2
Energy Scale−1.6 %−3.9 %
Energy Resolution
0.3% 0.3%+1.1
8
B Spectrum
1.9%−1.0 %
+3.5
+6.7
Total
−3.2 %−6.4 %

SK-III
3.99
0.5%
0.7%
0.4%
0.3%
0.5%
0.3%
0.5%
0.5%
0.2%
0.1%
0.7%
0.5%
0.1%
0.4%
1.4%
1.4%
0.2%
0.4%
2.2%

SK-IV
3.99
0.1%
0.4%
0.3%
0.3%
0.3%
0.2%
0.1%
−
0.2%
0.1%
0.7%
0.5%
0.1%
0.4%
1.2%
1.2%
0.15%
0.33%
1.7%

0.3

0.2

0.1

0
-1

-0.5

0

0.5

1
cosθsunFigure 7.12: Solar angle distribution of 4.0-19.5 MeV events. Black point shows the observed
data while the solid (dashed) red line shows the fit signal plus background (background).

144

This is by far the largest and most precise data set out of the four SK phases. This is
not because it has the largest livetime (SK-I still holds this title), but because the energy
threshold in SK-IV has been pushed to the lowest yet (4.0 MeV recoil electron kinetic energy)
while improving the understanding of the systematic effects in the detector. The number ofextracted solar neutrino events inside SK can be turned into an absolute flux measurement,
based on the number of expected solar neutrino interactions occurring inside the SK detector.
First, the expected number of solar neutrino elastic scattering interactions occurring within
the SK fiducial volume can be calculated according toZ Eνmax
Tlive φ(Eν )σtotal (Eν )Ne dEν .

(7.7)

0

Eν is the incident neutrino energy, φ(Eν ) is the energy dependent solar neutrino flux of8

B neutrinos and σtotal (Eν ) is the total energy dependent cross section for (ν-e) elastic

scattering. Ne is the total number of target electrons in the ID of SK (1.086×1034 ) and Tlive
is the SK-IV livetime (1306.3 days). The shape of the expected 8 B neutrino flux is thatgenerated based on the alpha spectrum measurements by Winter [35]. The total numberof events is normalized to the total 8 B solar neutrino flux measurement coming from theNC analysis of SNO data [21], which is 5.25×106 cm−2 sec−1 . Assuming no oscillation andmaking no analysis cuts (including energy), this leads to 383,739 expected solar neutrino
elastic scattering interactions in the SK ID, corresponding to a rate of 293.76 events per day.
But this is only the calculated number of expected interactions in SK, not the expected
number of observed solar neutrino events which SK will actually record. To determine theexact number of solar neutrino events SK should record, the MC simulated solar neutrino
events are generated with an extreme rate of 100 per minute (this is to give substantial
statistics), with the correct predicted spectrum and the vertex position set uniformly overthe volume of the ID. The energy and the direction of the recoil electrons are determined from
Equations 3.9 and 3.11. These MC simulated events are then processed through the same145

Figure 7.13: Measured flux of all SK phases individually. The solid line gives the combined
SK flux average, while the dashed lines give the combined statistical plus systematic error
on the flux average. All phases are consistent with each other within the smallest value ofthe statistical error (SK-IV value of 1%).

reduction chain as the SK data sample, and the surviving number of events (renormalized
to the SNO NC measurement of the solar neutrino 8 B flux) is taken as the expected numberof observed solar neutrino elastic scattering events in SK. This is found to be 54,403 events.
Then, assuming a pure νe flavour content (no oscillations), the ratio of the extracted numberof observed events to the expected number of observed events in SK-IV is 0.449. This
corresponds to a 8 B solar neutrino flux ofΦ8 B = (2.36 ± 0.02(stat.) ± 0.04(syst.)) × 106 cm−2 sec−1 .

(7.8)

As seen in Figure 7.13 and Table 7.4, the measured flux of each of the four SK phases is
consistent with the combined SK average (solid line), within the 1 σ statistical uncertainties
of each phase. The total uncertainty on the four phase flux average (dashed lines) assumes
the uncertainties are uncorrelated between phases, except that of the 8 B neutrino spectrum
prediction. This was removed from the uncertainty on the flux of each phase before the flux
146

Table 7.4: SK measured solar neutrino flux by phase.
Flux (×106 cm−2 sec−1 )SK-I
2.38±0.02±0.08
SK-II
2.41±0.05+0.16−0.15
SK-III
2.40±0.04±0.05
SK-IV
2.37±0.02±0.04
Combined2.37±0.15±0.04

average was made, and then put back into the total uncertainty on the average flux once, atthe end. The measured flux coming from the combination of all four phases of SK is 0.451
the expected flux, and is found to beΦ8 B = (2.37 ± 0.015(stat.) ± 0.04(syst.)) × 106 cm−2 sec−1 .

(7.9)

The χ2 to this average over the four SK phases is found to be 0.58/3 D.O.F, showing
consistency between the phases within the smallest statistical uncertainty. For illustrative
purposes, Figure 7.14 shows the solar angular distribution for the combined data of SK-I,
II, III and IV. The extracted total number of solar neutrino elastic scattering events during
the total SK livetime of 4,141 days is more than 60,000 events.

147

Events/bin

20000

15000

10000

5000

0
-1

-0.5

0

0.5

1
cosθsunFigure 7.14: Combined solar angle distribution for SK-I-IV, a total livetime of 4,141 days.

148

Chapter 8
Solar Neutrino Spectrum Analysis
8.1 Solar Neutrino Spectrum Extraction
As described in Section 7.1 (Equation 7.1), the solar neutrino signal of SK-IV is extracted
by an extended maximum likelihood fit to the cosθsun distribution of the data. The 8 B flux
analysis uses all 22 energy bins at once, fitting 22 background and one signal component,with the energy spectrum constrained by the Yi factors coming from the MC simulated
solar neutrino events. The solar neutrino energy spectrum is extracted by fitting one recoil
electron energy bin i at a time, with each Yi =1. This will extract the background and signal
for each of the energy bins, with no constraints on the spectrum shape. It was also possible
to extract the flux in the lowest energy bin (3.5-4.0 MeV recoil electron kinetic energy), andso it is included in the spectrum and oscillation analysis.
Because of the large amount of background present in the lowest energy bins, below 7.5
MeV each of the energy bins is further split into three sub-samples of MSG. The boundariesof the MSG sub-samples are set to MSG=0.35 and 0.45. The idea behind this comes fromthe fact that most of the lowest energy background comes from low energy β-decays. These
are most commonly coming from 214 Bi, which has an endpoint β-decay energy of 3.27 MeV.
These low energy β’s will make their way up into the solar neutrino data sample by meansof statistical fluctuations, coming in large part from the energy resolution of the detector.
Although these events may have reconstructed energies above 3.5 MeV, they will still multiple
scatter based on their true energy, and so should have a lower MSG than solar neutrino eventswith energies truly above 3.5 MeV. By dividing the lowest energy bins into sub-samples of149

Figure 8.1: SK-IV final data sample cosθsun distributions for various energy regions. Red
shows all events in each energy region, while the green, blue and cyan show events for MSG
betwen 0-0.35, 0.35-0.45 and 0.45-1, respectively.

MSG, it is possible to separate the background and signal components with different ratios,
into different MSG bins. More background is placed into the lower MSG bins, and moresignal is placed into the highest MSG bins. This effect can be seen for the SK-IV final data
sample in Figure 8.1, where for the lowest MSG bin the level of background is the highest,
and then as the MSG increases, the background level goes down and the solar neutrino peak
sharpens.
These three sub-samples are then fit simultaneously to a single signal and three independent background components. The signal fraction Yi in the i-th energy bin is changed tothe signal fraction in the g-th MSG and i-th energy bin, defined as Yig . The factors Yig aredetermined from the MC simulated solar neutrino events, just as the Yi ’s are, along with the
expected signal shapes σig . The background shapes βig are still taken from the SK-IV final
data sample. When the MSG sub-samples are used, the extended maximum likelihood fit
then extracts three background components (one for each MSG sub-sample) and one signal
components for each energy bin. The statistical precision of the extracted number of signal
events is increased by up 15% for energy bins utilizing MSG sub-samples (3.5-7.5 MeV).
This is seen by the comparison of the statistical errors when using and not using MSG sub-150

Figure 8.2: Ratio of the statistical errors of the extracted number of events when using MSG
sub-samples and when not (i.e. w/o MSG). The red and black points are for the positive
and negative errors, respectively.

samples (Figure 8.2). The additional systematic effect coming from the use of these MSG
sub-samples is small. As previously stated, the spectrum unconstrained signal extraction isdone using MSG sub-samples for energy bins below 7.5 MeV, while the signal extraction forenergy bins above 7.5 MeV does not use MSG sub-samples.

151

Energy Correlated Uncertainty

0.1

0

-0.1
5
10
15
Recoil Electron Kinetic Energy (MeV)

Figure 8.3: Energy bin correlated systematic uncertainties. The solid, dotted and dasheddistributions correspond to the energy bin correlated systematic uncertainties of the 8 B
spectrum shape, absolute energy scale and energy resolution, respectively.

8.2 Systematic Uncertainty of the Solar Neutrino
Energy Spectrum
8.2.1 Energy Bin Correlated Systematic Uncertainties
The uncertainty on the 8 B neutrino spectrum, as well as the uncertainty arising from the
SK-IV energy scale (±0.54%) and energy resolution (±2%), will shift all energy bins in acorrelated manner. The size of the correlations are calculated from the neutrino spectrum,
differential cross section, energy resolution function and the size of the systematic shifts. The
three systematic parameters (8 B neutrino spectrum shift, energy scale and energy resolution)
are varied individually, while running the signal extraction method for each energy bin.
The resulting energy-bin correlated systematic uncertainty coming from the 8 B neutrino
spectrum, the energy scale and the energy resolution are shown by the solid, dotted anddashed lines, respectively, in Figure 8.3. These systematic shifts are applied to the spectrum
when further oscillation analysis is carried out (see Chapter 10).

152

Table 8.1: Energy bin uncorrelated systematic uncertainties on the spectrum shape.
Energy (MeV)
Trigger Efficiency
Reconstruction Goodness
Hit Pattern
Small Hit Cluster
External Event Cut
Vertex Shift
Background Shape
Signal Extraction
Cross Section
MSG
Total

3.49-3.99
±4.5%
±0.8%
±1.1%
±0.1%
±0.5%
±2.9%
±2.1%
±0.2%
±0.4%
±5.9%

3.99-4.49±0.37%±0.8%±0.64%
±0.1%
±0.5%
±1.0%
±2.1%
±0.2%
±0.4%
±2.7%

4.49-4.99±0.8%±0.41%
±0.1%
±0.3%
±0.8%
±2.1%
±0.2%
±0.3%
±2.5%

4.99-5.49
±0.8%
±0.1%
±0.1%
±0.2%
±0.7%
±0.2%
±0.3%
±1.1%

5.49-5.99±0.64%
±0.1%
±0.1%
±0.1%
±0.7%
±0.2%
±0.3%
±1.0%

5.99-6.49±0.64%±0.64%
±0.1%
±0.1%
±0.1%
±0.7%
±0.2%
±1.7%
±2.1%

6.49-6.99±0.64%±0.64%
±0.1%
±0.1%
±0.1%
±0.7%
±0.2%
±1.7%
±2.1%

6.99-19.49±0.1%±0.28%
±0.1%
±0.1%
±0.1%
±0.7%
±0.2%
±0.8%

8.2.2 Energy Bin Uncorrelated Systematic Uncertainties
Multiple Scattering Goodness Systematic Uncertainty

Since we simultaneously fit sub-samples of MSG in the lowest recoil electron energy region,
any systematic shift in this goodness parameter of the data compared to that of the MC
simulated events would affect the measured rate in each energy bin. To estimate such effects
when using MSG sub-samples, the MSG distributions of LINAC data and MC simulated
events were compared, as seen in Figure 6.14. To bring the MC simulated events and data
into better agreement, the ratio of the data MSG distribution to the MC simulated events
MSG distribution is used as a scaling factor. The scaling functions used for three different
LINAC energies can be seen in Figure 8.4. These scaling functions are applied to the MC
simulated events in an energy dependent way, giving new signal shapes σig and shifting thesignal fractions in each MSG sub-sample Yig . The signal extraction method is then re-run
using these new shapes and signal fractions, and the change in the extracted number of
events in each energy bin taken as the systematic uncertainty. The systematic effect in eachenergy bin is summarized in Table 8.1, for SK-IV.
The various systematic uncertainties which plague the extraction of the total numberof signal events (4.0-19.5 MeV) also come into play when extracting the number of signal
events in each energy bin. The methods for estimating the various systematic effects are the
153

1.3
1.2
1.1
1
0.9
0.3

0.4
0.5
0.6
Multiple Scattering Goodness

Figure 8.4: MSG scaling functions applied to the solar neutrino MC simulated events toestimate the systematic uncertainty on the energy spectrum. The blue, black and red lines
correspond to 15.8, 8.36 and 4.59 MeV LINAC data over MC.

same as previously explained, except now implemented for one energy bin at a time. The
summary of the systematic uncertainties for each energy bin is shown in Table 8.1, with thetotal for each bin calculated as the quadrature sum of each of the components. The energy
bin uncorrelated total systematic uncertainty of energy bin i is added in quadrature to thestatistical uncertainty of energy bin i when carrying out further analysis.

8.3 Solar Neutrino Spectrum Results
The measured solar angle distributions for the lowest energy bins, which use MSG subsamples, are shown in Figure 8.5, for each of the MSG bins. The data is indicated by theblack points, while the best fit to the background (signal plus background) is shown by theblue (red) histogram. In Figure 8.6 the solar angle distributions are shown for energy bins
in which MSG sub-samples are not used (7.5-19.5 MeV). The extracted number of events
154Table 8.2: The observed event rates in each energy bin at 1 AU. The errors are statisticalerrors only. The selection efficiencies are corrected and the expected event rates are for an
unoscillated flux of 5.25×106 cm−2 sec−1
Energy(MeV)
3.49 − 3.99
3.99 − 4.49
4.49 − 4.99
4.99 − 5.49
5.49 − 5.99
5.99 − 6.49
6.49 − 6.99
6.99 − 7.49
7.49 − 7.99
7.99 − 8.49
8.49 − 8.99
8.99 − 9.49
9.49 − 9.99
9.99 − 10.49
10.49 − 10.99
10.99 − 11.49
11.49 − 11.99
11.99 − 12.49
12.49 − 12.99
12.99 − 13.49
13.49 − 14.49
14.49 − 15.49
15.49 − 19.49

Observed RateALLDAY
−1 ≤ cos θz ≤ 1 −1 ≤ cos θz ≤ 0
83.7+10.2
87.1+14.7
−10.0
−14.3
+5.9
78.7−5.8
63.9+8.4−8.0+5.4
83.9+3.9
76.7
−3.8
−5.2
+3.5
71.5+2.4
68.4
−2.4
−3.4
+1.8
60.5−1.8
58.9+2.6−2.5+2.4
55.3+1.7
50.4
−1.6
−2.3
+1.5
48.8−1.5
47.7+2.2−2.1+2.0
41.7+1.4
41.2
−1.3
−1.9
+1.8
37.8+1.2
38.6
−1.2
−1.8
+1.1
29.9−1.0
30.2+1.6−1.5+1.4
24.6+0.9
25.0
−0.9
−1.3
+0.8
18.7−0.8
17.9+1.1−1.1+1.0
14.6+0.7
15.7
−0.6
−1.0
+0.8
10.8+0.6
10.8
−0.5
−0.8
+0.50
+0.71
8.68−0.47
7.69−0.65
6.61+0.42
6.44+0.62
−0.40
−0.56
+0.47
4.38+0.34
3.75
−0.31
−0.42
+0.27
2.81−0.25
2.96+0.40
−0.35
+0.37
2.25+0.24
2.54
−0.22
−0.32
+0.20
1.47−0.17
1.49+0.30
−0.24
+0.32
1.83
1.68+0.20
−0.18
−0.27
+0.1840.518
0.520+0.125−0.100−0.096+0.166+0.096
0.297−0.106
0.228−0.067

155

NIGHT
0 < cos θz ≤ 1
80.5+14.2−13.8
92.2+8.5−8.1
90.6+5.7−5.5
74.3+3.3−3.2
62.0+2.5−2.5
59.9+2.4−2.3
49.8+2.1−2.1
42.2+1.9−1.8
37.1+1.7−1.6
29.6+1.5−1.4
24.2+1.3−1.2
19.4+1.1−1.1
13.6+0.9−0.9
10.8+0.8−0.7
9.58+0.73−0.67
6.76+0.60−0.54
4.97+0.50−0.45
2.66+0.38−0.33
1.98+0.32−0.28
1.45+0.28−0.23
1.53+0.27−0.23
0.526+0.186−0.1370.162+0.121−0.068

Expected Rate
8
B
hep
163.2
182.8
168.0
153.7
138.1
122.1
106.9
92.3
78.4
65.4
53.6
43.0
33.9
26.1
19.61
14.38
10.25
7.10
4.82
3.12
3.18
1.113
0.463

0.290
0.339
0.326
0.313
0.300
0.284
0.268
0.252
0.234
0.216
0.198
0.180
0.163
0.145
0.129
0.113
0.097
0.084
0.071
0.059
0.088
0.056
0.064

in each energy bin, as well as the expected number of events, are given in Table 8.2. The
resulting SK-IV solar neutrino spectrum, given in units of flux (cm−2 sec−1 ), is shown in
Figure 8.7, where below 7.5 MeV (indicated by the dashed vertical line) MSG sub-samples
are used, and above 7.5 MeV they are not used. The errors shown in this figure are thesum in quadrature of the statistical and energy bin uncorrelated systematic uncertainties.
The horizontal dashed line gives the extracted flux (for energies 4.0-19.5 MeV) of 2.37×106
cm−2 sec−1 (see Section 7.3). Further analysis of all four SK spectra (SK phases I-IV) will
be detailed in the next section.

8.4 SK-I/II/III/IV Combined Spectrum Analysis
Solar neutrino flavour oscillation above about 3 MeV are dominated by the solar MSW
resonance (see Section 2.3), while low energy solar neutrino flavour changes are due tovacuum oscillation. The MSW effect rests solely on standard weak interaction, so it is ratherinteresting to confront the expected resonance curve with data. Unfortunately, multiple
Coulomb scattering prevents the kinematic reconstruction of the neutrino energy in the (νe) elastic scattering interactions. However, the energy of the recoiling electron still provides alower limit of the neutrino’s energy. It is therefore possible to statistically infer the neutrino
spectrum from the recoil electron spectrum.
The νµ,τ differential cross section for elastic scattering off an electron is not just a factorof ∼6 smaller than that of νe , but it also has a softer energy dependence. This means
the observed recoil electron spectrum shape depends both on the flavour composition, andthe energy dependence of the composition of the solar neutrinos. Even a flat compositionof 33% νe and 67% νµ,τ will still distort the recoil electron spectrum, compared to one
which assumes 100% νe ’s. Additionally, according to the average day/night flux asymmetry
expected from the oscillation parameters sin2 θ12 =0.314 and ∆m221 =4.90×10−5 eV2 , and the156

Figure 8.5: Cosθsun distributions for the energy bins which use MSG sub-samples (3.5-7.5
MeV) (MSG bins go low to high in going left to right). Black points show the data, theblue and red histograms show the best-fit to the background and signal plus background,respectively.

157

0.02

0.025

Events/Day/Kton

0

0.02

7.5- 8.0MeV (kin)08.0- 8.5MeV (kin)08.5- 9.0MeV (kin)

0.02
0.01
0.01

0

9.0- 9.5MeV (kin)0

9.5-10.0MeV (kin)

0.01

0

10.0-10.5MeV (kin)

0.005
0.005

0

10.5-11.0MeV (kin)0

11.0-11.5MeV (kin)011.5-12.0MeV (kin)0.002
0.0025

Events/Day/Kton

0

0.002

12.0-12.5MeV (kin)012.5-13.0MeV (kin)

x 100

-3

13.0-13.5MeV (kin)

0.5
x 10

0.002

-3

0.5

0

13.5-14.0MeV (kin)014.0-15.5MeV (kin)015.5-19.5MeV (kin)Figure 8.6: Cosθsun distributions for energy bins not utilizing MSG sub-samples.

158

8

6

2

B Flux (10 /cm /s)

4

3

2

1
5
10
15
Recoil Electron Kinetic Energy (MeV)
Figure 8.7: SK-IV energy spectrum, using MSG below 7.5 MeV. The horizontal dashed line
gives the SK-IV total flux average (2.37 × 106 cm−2 sec−1 ). Error bars shown are statisticalplus energy bin uncorrelated systematic uncertainties.

rare hep neutrino interactions (which have a higher endpoint energy than 8 B neutrinos), thespectrum will be distorted. It is therefore necessary to take these effects into account when
making predictions and analyzing the SK spectra, in search of the MSW resonance upturn.
To analyze the 8 B solar neutrino spectrum, the spectra of SK-I, II, III and IV are fit simultaneously to their predictions (assuming different oscillation parameters or more generalfunctions), while varying the 8 B and hep neutrino fluxes within their uncertainties. The 8 B
flux is constrained to (5.25±0.20)×106 cm−2 sec−1 (motivated by SNO’s NC measurement/limits [20, 21]) and the hep flux to (8±16)×103 cm−2 sec−1 (from solar model calculations[47]). We have re-analyzed the SK-III spectrum below 7.5 MeV using the same method of
MSG sub-samples, for the same energy bins as in SK-IV. The entire SK-II spectrum has been
refit using the same MSG sub-samples, for all energy bins. SK-I data was not re-analyzed.
The four SK phase’s spectra are shown in Figure 8.8, in units of the measured flux divided
by the expected (unoscillated) flux. This spectrum of the ratio of the measured flux to thepredicted flux gives a measurement of the energy dependence of the νe flavour content, or in
159

Data/MC (unoscillated)

Data/MC (unoscillated)

SK I Spectrum

0.75
0.7

0.65

0.65
0.6

0.6

0.55

0.55

0.5

0.5

0.45

0.45

0.4
4

6

8

10

12

14

0.35

16
18
Ekin in MeVSK III Spectrum

10

12

14

16

18
Ekin in MeVSK IV Spectrum

0.54
0.52
0.5
0.48
0.46
0.44
0.42
0.4
0.38
0.36

0.7
0.6
0.5
0.4
0.3
0.2

8

Data/MC (unoscillated)

Data/MC (unoscillated)

0.4

SK II Spectrum

0.7

4

6

8

10

12

14

16
18
Ekin in MeV

4

6

8

10

12

14

16
18
Ekin in MeV

Figure 8.8: SK-I, II, III and IV recoil electron spectra. The green (blue) shape is the best
fit to SK data using the global solar (solar+KamLAND) best-fit oscillation parameters. Theorange (black) line is the best fit to SK data with a general parabolic (cubic) Pee survival
probability.
other words the νe survival probability Pee . Also shown in Figure 8.8 are predictions assuming best fit oscillation parameters coming from the global solar neutrino (solar+KamLAND)
analysis in green (blue). Also shown are fits to the spectra assuming a general parabolic(cubic) survival probability in orange (black).
While it is convenient to test the SK spectra (survival probability) against predicted
spectra resulting from the assumption of neutrino flavour oscillation, it is also possible to fit
the spectra to more general functions. In order to discuss the energy dependence of the solar
neutrino flavour composition in a general way, SNO [20] has parameterized the νe survival

160

probability Pee by a quadratic function centered around 10 MeV. This is described byPee (Eν ) = c0 + c1

Eν
MeV



2
Eν
− 10 + c2− 10 .
MeV(8.1)

This parameterization does not very closely describe the expected MSW resonance oralternative neutrino oscillation scenarios, such as non-standard interactions [48] or massvarying neutrinos [49]. However, it is both simple, and SNO found it unbiased. As alternatives to the quadratic approximation, the SK data is also fit assuming an exponential
and a cubic extension of the quadratic fit to the survival probability. The exponential fit isparameterized bye1
Pee (Eν ) = e0 +
e2


e2

e

Eν

MeV


−10

!
−1 .

(8.2)

The parameterization is chosen in this way to allow direct comparison of e0 and e1 to thequadratic coefficients c0 and c1 . The parameter e2 controls the “steepness” of the exponential
fall (or rise). We allow either an “upturn” or a “downturn” in the fit. Both the exponential
and the cubic parameterizations can accurately describe the MSW resonance curve (for bestfit oscillation parameters chosen by the global solar and solar+KamLAND analyses), as seenin Figure 8.9. Table 8.3 lists the quadratic, exponential and cubic coefficients which best
resemble either of the two MSW resonance curves. Although the quadratic approximation
does not describe well the MSW resonance curves, SK data is fit to this assumption to easethe comparison to SNO’s results.
Table ?? lists the spectral data of each SK phase (shown in Figure 8.8), used in the
combined spectrum analysis. The given uncertainties include the statistical uncertainties,
as well as the energy uncorrelated systematic uncertainties. Figure 8.10 shows the resulting
allowed areas of the exponential coefficients e1 and e2 based on the SK spectral data fit. The“base-line” (average Pee ) e0 is marginalized, as e0 is determined by the comparison of theelastic scattering rate in SK and the SNO NC interaction rate on deuterium. Figure 8.11
161

Pee

0.44
0.42
0.4
0.38
0.36
0.34
0.32
0.3

4

6

8

10

12

14
Eν in MeV

Figure 8.9: νe survival probability Pee based on the solar global best-fit parameters (thicksolid black) and the solar+KamLAND best-fit parameters (thick solid blue). The solid red(cyan) line is the best exponential approximation to the thick solid black (blue) line. The
dashed (dotted) orange line is the best quadratic (cubic) approximation to the thick solid
black. The dashed (dotted) green line is the best quadratic (cubic) approximation to thethick solid blue line.

Table 8.3: Best approximations of the MSW resonance curves when using exponential andpolynomial parameterizations of Pee .

expon. e0
expon. e1
expon. e2
expon. χ2 , ∆χ2polyn. c0
polyn. c1
polyn. c2
polyn. c3
polyn. χ2 , ∆χ2

sin2 θ12 = 0.304∆m221 = 7.41 · 10−5 eV20.3205−0.0062−0.270771.89, 1.28
0.31940.3204−0.0071−0.0059+0.0012+0.00090−0.0001
71.92, 1.31 71.94, 5.37

162

sin2 θ12 = 0.314∆m221 = 4.90 · 10−5 eV20.3106−0.0026−0.3549
70.96, 0.35
0.30950.3105−0.0033−0.0021+0.0008+0.0005
0−0.000170.82, 0.21 71.06, 4.49

Table 8.4: Elastic scattering rate ratios and energy uncorrelated uncertainties for each SK
phase.
Energy (MeV)
3.49-3.99
3.99-4.49
4.49-4.99
4.99-5.49
5.49-5.99
5.99-6.49
6.49-6.99
6.99-7.49
7.49-7.99
7.99-8.49
8.49-8.99
8.99-9.49
9.49-9.99
9.99-10.49
10.49-10.99
10.99-11.49
11.49-11.99
11.99-12.49
12.49-12.99
12.99-13.49
13.49-14.49
14.49-15.49
15.49-19.49

SK-I
−
−
+0.0430.453−0.042+0.0230.430−0.022
0.449±0.018
0.444±0.015+0.016
0.461−0.015
0.476±0.016+0.0170.457−0.016+0.0170.431−0.016+0.0180.454−0.017
0.464±0.019
0.456+0.021−0.0200.409±0.021
0.472+0.025−0.024+0.0280.439−0.026+0.0330.460−0.031+0.0390.465−0.036+0.0480.461−0.043+0.0640.582−0.057+0.0590.475−0.052+0.1200.724−0.102+0.1730.575−0.130

SK-II
−
−
−
−
−
−
0.439+0.050−0.0480.448+0.043−0.041
0.461+0.037−0.0360.473+0.036−0.0350.463+0.036−0.0340.499+0.038−0.0370.474+0.038−0.036
0.481+0.041−0.0390.452+0.043−0.0400.469+0.046−0.0430.482+0.052−0.0480.419+0.054−0.0490.462+0.063−0.0570.444+0.070−0.0620.430+0.066−0.0590.563+0.100−0.0870.648+0.123−0.103

SK-III−
0.448+0.100−0.0960.472+0.058−0.0560.420+0.039−0.0370.457+0.035−0.0340.433+0.023−0.0220.504+0.025−0.0240.424+0.024−0.0230.467+0.024−0.0230.469+0.026−0.0250.420+0.026−0.0250.444+0.029−0.0270.423+0.031−0.0290.529+0.037−0.0350.481+0.041−0.0370.391+0.044−0.0400.479+0.055−0.0490.425+0.061−0.0530.400+0.073−0.061
0.422+0.093−0.074
0.663+0.110−0.0930.713+0.201−0.1500.212+0.248−0.122SK-IV
0.511+0.069−0.0680.429+0.034−0.0330.498±0.026
0.464±0.016
0.437±0.014
0.451+0.014−0.0130.455±0.017
0.450±0.015
0.480±0.016
0.455+0.017−0.016
0.456+0.018−0.0170.433+0.019−0.0180.429+0.020−0.0190.411+0.022−0.0210.439+0.026−0.0240.456+0.029−0.0280.423+0.033−0.0300.397+0.038−0.0340.459+0.049−0.044
0.461+0.062−0.0540.514+0.063−0.0550.444+0.107−0.0850.432+0.181−0.127

Table 8.5: SK measured exponential and polynomial fit coefficients and correlations. Also
given are SNO’s quadratic fit coefficients (slightly different than the published values since
the day/night asymmetry is not fit), as well as SK and SNO combined measured quadratic
fit coefficients and correlations.
Data sete0e1e2
e0 -e1 corr.
SK (comb) 0.343 ± 0.022 +0.0038 ± 0.0056 no constraint+0.016
c0
c1
c2
c0
c1
c2
c3
SK (comb) 0.343 ± 0.024 +0.0038 ± 0.0067 −0.0000 ± 0.0031 0.323 ± 0.025 −0.034 ± 0.017 0.0084 ± 0.0054 0.0043 ± 0.0022
c01−0.164−0.3791+0.264−0.479−0.342c1−0.1641+0.635+0.2641−0.610−0.922c2−0.379+0.6351−0.479−0.6101+0.811c3−0.342−0.923+0.8111c0 -c1 corr.
c0 -c2 corr.
c1 -c2 corr.
SNO
0.315 ± 0.017 −0.0007 ± 0.0059 −0.0011 ± 0.0033−0.301−0.391−0.312
SK+SNO 0.311 ± 0.015 −0.0052 ± 0.0036 −0.0005 ± 0.0019−0.440−0.412+0.264

163

2

e2

∆χ

8 3σ
6
4 2σ
2
0.3 1σ
0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
1σ

-0.04

0.0

2σ

3σ

0.04 2 4 6 82e1∆χFigure 8.10: Allowed areas of the shape parameters (e1 and e2 ) of an exponential fit to thesurvival probability Pee of SK data. The parameter set corresponding to the solar neutrino(solar+KamLAND) data best fit is indicated by the dark green (blue) star.

compares the allowed one σ band of survival probabilities coming from the quadratic (blue),
cubic (green) and exponential (red) fits to the SK spectral data. Table 8.5 gives the resulting
best-fit coefficients and correlations coming from the quadratic, cubic and exponential fits
to the SK spectral data, as well as the best quadratic fits to SNO data and SK plus SNO
data.
The SK spectral data χ2 of the full MSW calculation is compared to that of the best
exponential, cubic and quadratic function fits. Table 8.6 compares the various χ2 ’s comingfrom the SK spectral data fits to predictions based on MSW oscillation parameters comingfrom the global solar (solar+KamLAND) best-fit, as well as those for the best-fits to the
exponential, cubic and quadratic functions. The final row of Table 8.3 gives the χ2 from thebest exponential, quadratic and cubic approximations of the MSW resonance curve, as wellas the difference in χ2 coming from the exponential, quadratic and cubic best fits to the SK
spectral data.

164

Pday
ee

0.6
0.5
0.4
0.3
0.2
0.1
0.0
3

4

5

6

7

8

9

10 11 12 13 14 15
Eν in MeV

Figure 8.11: Allowed survival probability 1 σ band from SK data. The red (blue) area isbased on an exponential (quadratic) fit, the green band is based on a cubic fit. The 8 B flux
is constrained by the NC measurement from SNO. The absolute value of the 8 B flux doesn’t
affect much to the shape constraint, just the average value.

Table 8.6: Spectrum fit χ2 comparison.
Fit Parameterssin2 θ12 = 0.304∆m2 = 7.41 · 10−5 eV2
sin2 θ12 = 0.314∆m2 = 4.90 · 10−5 eV2e0 = 0.340 e1 =+0.0040 e2 = 0
c0 = 0.340 c1 =−0.0040 c2 = 0
c3 = 0c0 = 0.322 c1 =−0.033 c2 = 0.008
c3 = 0.0042
Just D/N
Flat suppression

χ2

Φ8 B cm−2 sec−1

Φhep cm−2 sec−1

71.95

5.40 · 106

15.7 · 103

70.76

5.53 · 106

14.8 · 103

70.61

5.24 · 106

15.2 · 103

70.61

5.24 · 106

15.2 · 103

66.57

5.25 · 106

2.1 · 103

72.31
71.50

5.49 · 106
5.58 · 106

13.3 · 103
13.7 · 103

165

Data/MC (unoscillated)

SK I/II/III/IV LMA Spectrum

0.6

0.58
0.56
0.54
0.52
0.5
0.48
0.46
0.44
0.42
0.4

4

6

8

10

12

14

16

18
Ekin in MeV

Figure 8.12: SK-I+II+III+IV recoil electron spectrum. The green (blue) shape is the best
fit to SK data using the global solar (solar+KamLAND) best-fit oscillation parameters. Theorange (black) line is the best-fit to SK data with a general parabolic (cubic) Pee survival
probability.

The MSW resonance curve “upturn” below ∼6 MeV expected from the solar+KamLAND(solar global) best-fit parameters is slightly favored by either the exponential or the quadratic
fit. For the cubic fit, the MSW resonance shape is disfavored, which is in part driven byfluctuations above ∼13 MeV, where both statistical and systematic uncertainties are large.
These fluctuations cannot be accommodated by the other two parameterizations, so theirbest fit is about 3.65 units of χ2 larger than the cubic best fit. If this shift is subtracted,
the cubic fit significance becomes quite similar to that of the exponential and quadratic fits.
Figure 8.12 shows the statistically combined SK-I, II, III and IV spectrum (black points),
along with the predictions coming from best fit MSW oscillations parameters (when using
either the best-fit from the global solar (green) or solar+KamLAND (blue) analyses), and thequadratic (cubic) best fit to the SK spectral data in orange (black). This figure is only used
for demonstration purposes, since the actual fit to the SK spectra is not done by combing
the spectra and then making the fits, but by fitting each spectrum to its prediction and then
combing the χ2 ’s afterwards.

166

8.5 SK+SNO Spectrum Analysis
Finally, the combination of the SK spectral fit to a general parabolic survival probability
can be combined with that of SNO. The left panel of Figure 8.13 shows the allowed shapeparameters (c2 vs. c1 ) coming from the quadratic fit to the SK spectral data (SNO data) ingreen (blue). The right panel shows the allowed slope (c1 ) and baseline (c0 ). The results of
the SNO fit shown here slightly differ from the published results, as we assume standard MSW
day/night effects rather than making a linear fit to the day/night asymmetry of the survival
probability, as SNO did in [20]. The precision of the SK constraint is similar to that basedon the SNO data, while both are statistically consistent. Since SK’s correlation between c1
and c2 is opposite to that of SNO’s, a combined fit is rather powerful in constraining theshape. The c1 -c2 correlation becomes much smaller.
The combination of SK and SNO data not only significantly increases the precision of
the c0 determination, but the uncertainties of the shape are very much reduced. While SK(SNO) data by itself slightly prefers an “upturn” (“downturn”), the combined fit favors aslightly stronger “upturn”. SK data generally likes a higher baseline than SNO’s. SNO’s
sensitivity is dominated by the charge current interactions, which preserve the neutrinoenergy. However, the nuclear threshold energy takes away some of the advantage over SK’s
higher statistics in the elastic scattering data. As a consequence, SNO’s uncertainties are
smaller at higher neutrino energy, while SK’s uncertainties are smaller at lower neutrinoenergy. Therefore, the combined fit follows the higher (lower) baseline of SK (SNO) as low(high) neutrino energy, and the end result is the favoring of a stronger “upturn”. The left
panel of Figure 8.14 displays the one σ allowed bands coming from the quadratic fits to theSK (green) and SNO (blue) spectral data separately. The combination of SK and SNO canbe seen in the right panel (red). The final row of Table 8.5 gives the best-fit coefficients andcorrelations for the combined fit.

167

2

2

8 3σ
6
4 2σ
2

∆χ

∆χ

8 3σ
6
4 2σ
2

1σ

c2

c1

1σ

0.03
0.01

0.02
0.01

0

0
-0.01

-0.01

-0.02
-0.03
1σ

-0.04

0.00

2σ

3σ

1σ

0.04 2 4 6 8
2
c1
∆χ

0.3

0.4
c0

2σ

3σ

2468
2
∆χ

Pday
ee

Pday
ee

Figure 8.13: Allowed areas of the shape parameters c1 vs. c2 (c1 vs. c0 ) of a quadratic
fit to the survival probability Pee of SK (green) and SNO (blue) data on the left (right).
Also shown is a combined fit (red). The parameter set corresponding to the solar neutrino(solar+KamLAND) data best-fit is indicated by the dark green (blue) star.

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0.0
3

4

5

6

7

8

9

10 11 12 13 14 15
Eν in MeV

0.0
3

4

5

6

7

8

9

10 11 12 13 14 15
Eν in MeV

Figure 8.14: Allowed survival probability 1 σ band from SK (green) and SNO (blue) data.
The red band gives the combination of SK and SNO. Also shown are predictions based on
the best-fit parameters from the global solar (green) and solar+KamLAND (blue) analyses.

168Chapter 9
Solar Neutrino Day/Night Flux
Asymmetry Analysis
9.1 The Solar Neutrino Day/Night Flux Asymmetry
The matter density of the Earth affects solar neutrino oscillation while the Sun is below
the horizon. This so called “day/night effect” will lead to an enhancement of the νe flavour
content during the nighttime, for most MSW oscillation parameters. This is especially true
for those parameters within the favored LMA region. The solar zenith angle θz is definedas the angle between the vector from the Sun to the reconstructed vertex and the detector
z-axis (defined in Figure 7.1), at the time of each event. The most straight-forward test of
the day/night effect uses θz to separately measure the solar neutrino flux during the day ΦD(defined as cosθz ≤0) and the night ΦN (defined as cosθz >0). The day/night asymmetry(ΦD − ΦN )
ADN = 1
(ΦD + ΦN )
2

(9.1)

defines a convenient measure of the size of the effect. SK is able to simply search for thedifference in the day and night fluxes by dividing the data into separate bins for the day
and night times (zenith bins), and then running the extended maximum likelihood signal
extraction method (see Section 7.1) for each zenith bin. Because of various systematic effects
from making the tight fiducial volume cut, and the fact that the day/night effect is smallerat the lowest 8 B solar neutrino energies, SK searches for the day/night asymmetry for recoil
electron kinetic energies of 4.5-19.5 (6.5-19.5) MeV during SK-I, III and IV (II).

169

9.2 The Solar Neutrino Day/Night Amplitude Fit
To reduce systematic effects, increase statistical precision and to take into account thedifferences in the zenith angle distribution of the solar neutrino flux coming from different
values of the oscillation parameters, a more sophisticated method to search for the day/nightflux asymmetry can be used. This is termed the “day/night amplitude fit”. Details of thismethod can be found in [50]. The idea behind the method is to extend the standard SK
signal extraction likelihood (Equation 7.1) so that it can allow for a time variation of thesolar neutrino signal. This is done by modifying the likelihood to

−(

L=e

P

i Bi +S)

N
ni
bin Y
Y

(Bi · bij + S · Yi · sij · zi (α, t)),(9.2)

i=1 j=1

where z(α, t) is the new signal time variation factor.
The time variation factor can add any time variable component to the solar neutrino flux,
including things like Earth orbit eccentricity, differences in the day and night MC simulation
selection efficiencies (directional dependences of event reconstruction in the detector) andof course the matter enhanced day/night asymmetry (solar zenith angle variation). As thistime dependent signal factor will include effects coming from neutrino oscillation (as well as
those not coming from neutrino oscillation), it will depend on the exact values chosen for theoscillation parameters. It is therefore convenient to break it up into two pieces. The form ofz(α, t) is then given aszi (α, t) =

(t)
− 1)
1 + α((1 + ai ) rriavei

1 + αai× zexp (t).

(9.3)

The factor zexp (t) accounts for expected time variation components, such as the differencein the expected neutrino flux coming from the inverse square of the distance between the Sun

170

Days/bin

1.4

1.2

1

0.8

0.6

0.4

0.2

0

0

200

400

600

800

1000
cz binFigure 9.1: The livetime in each cosθz (Cz ) bin (in days/bin). Cz bin = 1 and 1,000 correspond to cosθz = 0.001 and 1, respectively.

and Earth dependence on the flux. There is also the differences in detector efficiencies during
the day and night (coming from directional dependences of the detector). This expected time
variation factor does not depend on the neutrino oscillation parameters, and amounts to a
constant factor for each event in SK.
The remaining piece of z(α, t) takes into account the solar zenith zenith variation of theinteraction rate, coming from matter enhanced neutrino oscillation. It will therefore dependon the selected values of the oscillation parameters and the solar zenith angle (t can be
replaced by cosθz ). riave is the livetime weighted total average rate, which takes the SKIV solar zenith angle binned (one day bin and 1,000 night bins) livetime into account (seeFigure 9.1). riave is described byriave =Zri (Cz )l(Cz )d(Cz ),
all

171

(9.4)

where Cz ≡ cos θz and l(Cz ) is the livetime for each solar zenith angle bin. ri (t) (or ri (Cz ))is the solar neutrino interaction rate as a function of time (solar zenith angle) for the i-thenergy bin. For the sake of measuring the day/night asymmetry, the interaction rate during
the day riday and the interaction rate during the night rinight are defined as
riday =

Z

rinight =ri (Cz )l(Cz )d(Cz ),Z

dayri (Cz )l(Cz )d(Cz ).

(9.5)

night

We can now define the expected day/night asymmetry (in energy bin i) in terms of the
day and night interaction rates
rday − rnight
Ai = 1 i day i night .
(r + ri )
2 i

(9.6)

However, this theoretical day/night asymmetry only takes into account the expected
effects from neutrino oscillation, and does not take into account any difference in the livetime
between the day and night. Therefore we use the ai factor in Equation 9.3, which gives theeffective day/night asymmetry in SK, and comes from the result of folding the theoreticalday/night asymmetry together with the day/night livetime asymmetryLD − LN.
(LD + LN )2

LDN ≡ 1

(9.7)

The day and night livetimes have now been corrected for difference in day and night MCsimulation efficiencies. The effective day/night asymmetry is then11
ai = Ai × LDN .
2
2

(9.8)

The final parameter in Equation 9.3, α, is a free parameter which is termed the “day/night
amplitude”. This works as a scaling factor to the amplitude of the solar zenith angle distri172

butions. This factor will not change the expected shape of the solar zenith angle interaction
rate distributions, but will allow for the size of the effect to be adjusted.
The form taken for the solar zenith angle dependent piece of the time variation signal
factor was chosen so that the total SK livetime remains unchanged, and so that ri0 (α, t) =zi (α, t)×riave has the same average total rate riave . In particular, ri0 (0, t) = riave is independentof t (no day/night asymmetry) and ri0 (1, t) = ri (t) (returns the expected day/night asymmetry). The likelihood function in Equation 9.2 is then maximized with respect to the signal
S, the backgrounds Bi and the day/night amplitude α. The measured day/night asymmetry
coming from the day/night amplitude fit then found as

ADN = Ai × α ± Ai × σα .

(9.9)

Here σα is the statistical uncertainty on α coming from the maximum likelihood method. Ai
is the theoretically predicted day/night asymmetry, which naturally depends on the valuesof the oscillation parameters.
For a given set of oscillation parameters, the interaction rate as a function of the solar
zenith angle (and energy) ri (Cz ) can be predicted. This is based on the solar zenith angle
dependence of the νe survival probability. There are three steps in making the survival
probability predictions. First νe created in the interior of the Sun must be propagated tothe surface of the Sun. Then they must be propagated from the Sun to the Earth, andfinally through the Earth (during the night time) into the SK detector. The first step isdone by producing νe ’s at various positions throughout the Sun, according to the production
distributions from the SSM. They are then propagated to the surface of the Sun using
the neutrino wave-function and including two-flavour oscillations effects. Then an angularaverage is taken for each production radius in the Sun. An adiabatic approximation of theneutrinos propagating through the Sun is used. This gives the probability P2 for an νe173

produced at a specific radius in the Sun, to end up in the second mass eigenstate at thesurface of the Sun.
The propagation of the neutrinos from the surface of the Sun to Earth is easily done.
The Sun-Earth distance is on the same scale as the solar neutrino oscillation length, so thevacuum oscillations will average out. For the propagation of the neutrinos through the Earth,the neutrinos start out purely in the second mass eignestate, after encountering the MSW
resonance in the Sun. They are then propagated through the Earth, with matter densities
and boundaries inside the Earth modeled according to the PREM model [51]. Two-flavour
oscillations effects are taken into account while propagating the neutrino wave-function.
This will give the probability of the second mass eignestate to end up an νe inside the SK
detector, Pe . Finally Pe and P2 are combined together, while averaging over the different
solar production radii. Different solar neutrino types are produced with different ratios at
different solar radii. We do this averaging for 8 B and hep solar neutrinos. This gives the νesurvival probability

Pee (cos θz ) = (1 − P2 )(1 − Pe (cos θz )) + P2 Pe (cos θz ).

(9.10)The predictions of the solar zenith angle dependence on the νe survival probability are
shown in Figure 9.2, for oscillations parameters chosen by the global solar best-fit (∆m221 =4.90×10−5 eV2 and sin2 θ12 =0.314). To take the survival probability predictions to predictionsof the interaction rate for the i-th energy bin, we use
Z Ehigh,iri =Elow,idσνµ,τdσνe+ (1 − pe )dEdEν φ(Eν )dEe R(Ee , E) pe
dEe
dEe
Eν
Ee
Z

Z

νe
R(Ee , E) is the energy resolution function, dσanddEedσνµ ,ντdEe


.

(9.11)

are the differential cross sections

for either (νe -e) or (νµ,τ -e) elastic scattering, with incoming neutrino energy Eν , recoil electron energy Ee and solar neutrino flux φ(Eν ). The survival probabilities are calculated for174

0.425

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

Oscillated MC/MC

Oscillated MC/MC

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

0.42

0.45
0.445
0.44
0.435

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

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

0.43
0.425

0.415

0.42
0.41

0.415
0.41

0.405

0.405
0.4
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.4
-1

0.8
1
Cosθz

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8
1
Cosθz

Figure 9.2: Predicted solar zenith angle time variation of the ratio of the oscillated rate to
the unoscillated rate (∆m221 =4.90×10−5 eV2 and sin2 θ12 =0.314).

1,001 solar zenith angle bins, giving very smooth distributions to use inside the unbinned
maximum likelihood function. One bin is used for the daytime and the remaining 1,000 areused for the nighttime.
These zenith angle predictions are then used in the time variation signal factor appearing
in the modified maximum likelihood formula (Equation 9.2) when extracting the day/night
amplitude parameter. It should be noted here that to go from the two-flavour neutrino
approximation to the full three-flavour case, we can simply replace the electron density
in the Sun and Earth with Ne → Ne cos2 θ13 and modify P2 → P2 cos4 θ13 + sin4 θ13 [52].
This can be done because we have made the assumption that the neutrinos will propagate
adiabatically through the Sun and Earth. However, since recent measurements of θ13 [31, 32]
give an average value of sin2 θ13 ≈ 0.025, the effect on the expected solar neutrino day/nightflux asymmetry coming from the inclusion of all three flavours of neutrinos is negligible.
It is also convenient at this point to calculate the total number of expected 8 B andhep solar neutrinos, when taking neutrino oscillation into account (see Chapter 10). With
the precise solar zenith angle binning of the livetime and predicted interaction rates, andsticking the predicted 8 B and hep solar neutrino fluxes into Equation 9.11, the total number

175

of oscillated 8 B and hep elastic scattering events in SK are calculated as

Biosc =

1001
X
τ (Cz )
Cz =1

Hiosc

=

τtot

1001
X
τ (Cz )
Cz =1

τtot

8

ri B (Cz )(9.12)

rihep (Cz ).

(9.13)

τ (Cz ) is the livetime in each of the cosθz bins, which are obtained from the livetime distribution of the data (Figure 9.1), and τtot is the total livetime. These will be used in Chapter 10
when combing all of the SK data together, and then with other solar neutrino experiments
and KamLAND, to perform neutrino oscillation analyses.

9.3 Solar Neutrino Day/Night Flux Asymmetry Systematic
Uncertainty
9.3.1 Energy Scale Systematic Uncertainty
True day (night) solar neutrino events will mostly be going in the downward (upward)
direction, and so the directional dependence of the SK light yield (energy scale) will change
the observed interaction rate as a function of solar zenith angle and energy. To quantify the
directional dependence of the energy scale, the energy of the DT-produced 16 N calibration
data and MC simulated events were compared as a function of the reconstructed detector
zenith angle (Figure 4.22). Using the fit from this comparison and the 8 B MC simulated
events, a profile of the energy scale dependence was taken as a function of the solar zenith
angle, and the same fit used in Fig. 4.22 was again applied. The fit function was then applied
to the MC simulated expectation in the unbinned amplitude fit (explained in section 9.2),
while taking energy correlations into account. The resulting 0.05% change in the equivalent176

day/night asymmetry is taken as the systematic uncertainty coming from the directional
dependence of the energy scale. The systematic uncertainty of the day/night asymmetry foreach SK phase can be seen in Table 9.1. The large reduction seen from SK-I to SK-III and
IV comes from the introduction of the z-dependent water transparency parameter β within
the MC simulation.

9.3.2 Energy Resolution Function Systematic Uncertainty
Throughout the different phases of SK, the energy resolution function, which relates thetrue and reconstructed recoil electron energies was found by two slightly different methods.
During the SK-I and SK-IV phases, the 8 B MC simulated events were used to set up a
“transfer matrix” relating reconstructed to true recoil electron energy (and reconstructed
recoil electron energy to neutrino energy). By construction, this method considers the effectof all analysis cuts on energy resolution. For the SK-II and III phases, dedicated monoenergetic MC simulated events were produced to parameterize the energy resolution with aGaussian function, modeling only some analysis cuts. The two methods produce slightly
different results. In particular, the predicted day/night asymmetries differ by 0.05%. To
estimate the systematic uncertainty on the day/night asymmetry, coming from the energy
resolution function, the amplitude fit was performed using both methods, and the 0.05%
difference is taken as the systematic uncertainty.

9.3.3 Background Shape Systematic Uncertainty
Although there is only one background component fit in each energy bin of the day/night
amplitude fit (any time dependence of the background should be much slower than theday/night variation), different cosθsun background shapes must be used for different solar
zenith angle bins. We use one for the day and six for the night (in accordance with Ta177

ble 9.2). The systematically different shapes come from the detector’s directional bias whenreconstructing background events (upward, downward and horizontal directions are favored).
The background is first fit as functions of the detector zenith and azimuthal angles. Thesefits also yield a covariance matrix V for the fit parameters. The parameters of each of thezenith and azimuthal fits are varied by the 1 sigma statistical deviation, one at a time, givinga new background shape for each solar zenith angle bin. Because the background distributions are calculated as projections of the detector zenith and azimuthal angles on the solar
directions, the shape deviations as a function of solar zenith angle are fully correlated andmust be varied simultaneously. The day/night amplitude fit is then re-run for each set of new
background shapes. The difference in the mean value is taken as the error of the day/night
asymmetry due to that particular zenith or azimuthal fit parameter. These errors are thenpropagated to a total systematic uncertainty using the covariance matrix V coming fromthe detector coordinates fits. As seen in Table 9.1, the total uncertainty on the day/night
asymmetry coming from the background shapes is 0.6% across all SK phases, and is thelargest contribution to the systematic uncertainty.

9.3.4 External Event Cut Systematic Uncertainty
The vertex shift and angular resolution difference between data and MC can cause abias to the external event cut efficiency. We artificially shift the reconstructed vertex anddirection and estimate the fraction of events which are rejected by the external event cut
during the daytime and nighttime. The estimated systematic uncertainty is ±0.1% for SK-IV
and ±0.2% for SK-III, as seen in Table 9.1.

178

9.3.5 Earth Model Systematic Uncertainty
When searching for the day/night effect, different models of the Earth’s density profile
can change the signal rate zenith profiles, thus leading to changes in the measured day/night
asymmetry value. For this reason it is essential to model the earth as precisely as possible,
most frequently done using the PREM model [51] and an Earth which is assumed to be
spherical. A spherical description of Earth using the equatorial radius can lead to a significantly different day/night effect, compared to a spherical description using an average radius.
To better represent the Earth we have modeled an ellipsoidal Earth, using the equatorial
and polar radii as the semi-major and semi-minor axes of an ellipse, and rotating the ellipse
around its minor axis. The spherical PREM model density boundaries are then mapped
accordingly.
Due to SK’s location on Earth, when using the above procedure of modeling an ellipsoidal Earth, the rotational symmetry about the detector azimuthal angle goes away andthe day/night zenith amplitude fit must take into account the change in the expected signal
rate as a function of both the zenith and azimuthal angles. This was done by varying theazimuthal angle and the zenith angle when tracing neutrinos through the Earth, and thenusing the detector livetime to average over the azimuthal angle. The resulting expected
zenith dependent signal rates were then used in the day/night amplitude fit, along withthose while assuming a spherical Earth. The 0.01% change in the day/night asymmetry istaken as the systematic uncertainty coming from the Earth shape.
As a final step in estimating the systematic uncertainty coming from the model of the
Earth, the PREM model was replaced with the more recent PREM500 model [53]. This
gives an updated and more detailed description of the density profile of Earth. This resulted
in a 0.01% shift in the measured day/night asymmetry, and when added in quadrature to
the uncertainty coming from the Earth shape, 0.014% gives the total estimated uncertainty

179

Table 9.1: Day/night asymmetry systematic uncertainties by SK phase. The total is foundby adding the contributions for each phase in quadrature.

Energy Scale
Energy Resolution
Background Shape
External Event Cut
Earth Model
Total

SK-I SK-II SK-III
0.8% 0.8%
0.2%
0.05% 0.05% 0.05%
0.6% 0.6%
0.6%
—
—
0.2%
0.01% 0.01% 0.01%
1.0% 1.0%
0.7%

SK-IV
0.05%
0.05%
0.6%
0.1%
0.01%
0.6%

coming from the Earth model.

9.3.6 Summary of the Systematic Uncertainty
The total estimated systematic uncertainty on the measured day/night asymmetry (foreach SK phase) is calculated by adding the components together in quadrature. The result
can be seen in Table 9.1. The major improvement seen in SK-III and IV comes from theintroduction of a z-dependent water transparency parameter β into the MC simulation,
giving a much better handle on the directional dependence of the energy scale. Because theenergy scale and the background shape are the dominating contributions to the systematic
uncertainty, and each is measured independently for each phase of SK, the error on theday/night asymmetry is uncorrelated between SK phases.

180

Table 9.2: The observed zenith angle dependence of event rates in each energy region at 1
AU. Errors are statistical uncertainties only. The reduction efficiencies are corrected and the
expected event rates are for a flux of 5.25×106 cm−2 sec−1 .
Observed Rate
Expected Rate
8B
EnergyDAY
MANTLE1 MANTLE2 MANTLE3 MANTLE4 MANTLE5 CORE
hep
(MeV)
cosθz :-1∼00∼0.160.16∼0.33 0.33∼0.50 0.50∼0.67 0.67∼0.84
0.84∼1
+17.1
+12.9
0.326
106.5+14.9
83.7+12.9
94.2−11.9
68.1+13.7
67.2+14.7
4.49 − 4.99
76.7+5.4−13.8 122.0−15.8 168.0
−12.0
−12.4
−13.2
−5.2
+10.7
+10.2
+9.2
+9.4
+10.5
+11.3
0.612
132.2
141.8
137.6
135.2
130.5
133.2
4.99 − 5.99 126.5+4.2−10.2 291.5
−9.7
−8.8
−9.0
−10.0
−10.7
−4.1
+10.2
+10.2
0.803
144.2+8.9
154.5+8.3
156.5+8.6
135.1+9.4
5.99 − 7.49 139.2+3.8−8.5 169.5−9.7 320.6
−7.9
−8.2
−8.9
−3.7 145.5−9.7
+7.1
+6.1
+5.8
+5.9
+6.7
+7.1
+2.7
99.7−6.7 197.2 0.648
86.1−5.7
93.2−5.5
89.1−5.6
90.7−6.3
89.2−6.6
7.49 − 8.99
93.7−2.7+4.8
0.617
63.9+5.0
55.1+4.3
54.0+4.0
48.0+4.0
57.9−4.4
55.7+5.1
8.99 − 10.99 52.1+1.8−4.7 122.4
−4.0
−3.7
−3.7
−4.7
−1.8
+2.1
+2.0
+2.1
+2.5
+2.6
+0.9
36.1
0.364
17.3+2.5
14.9−1.9
14.7−1.7
15.3−1.8
17.3−2.1
17.4−2.3
10.99 − 12.99 15.5−0.9
−2.2
+1.08
0.204
4.36+1.31
3.77+1.13
4.16+1.03
2.50+0.92
2.52−0.73
5.69+1.55
12.99 − 15.49 3.84+0.46−0.98 7.41
−0.84
−0.80
−0.66
−1.18
−0.40

9.4 Solar Neutrino Day/Night Flux Asymmetry
Results
9.4.1 Straight Day/Night Flux Asymmetry Results
The SK-IV livetime during the day (night) is determined as 626.4 (679.9) days. The
extracted solar neutrino flux between 4.5 and 19.5 MeV, and assuming no oscillation, ismeasured as ΦD = (2.29 ± 0.03(stat.) ± 0.05(syst.)) × 106 cm−2 sec−1 during the day, andΦN = (2.42 ± 0.03(stat.) ± 0.05(syst.)) × 106 cm−2 sec−1 during the night. Figure 9.3 shows
the flux (between 4.5 and 19.5 MeV assuming no oscillation) as a function of cosθz , divided
into five day and six night bins (corresponding to the MANTLE 1-5 and CORE definitions ofTable 9.2). Figure 9.4 shows the solar zenith angle variation of the ratio of the measured rate
to the unoscillated MC simulated rate (assuming Φ8B =5.25×106 cm−2 sec−1 ), for the sevenenergy regions shown in Table 9.2. Overlaid are the expected solar zenith angle variations forthe best fit oscillation parameters coming from the global solar (solar+KamLAND) analysis
in red (blue). Table 9.2 lists the data used to make Figure 9.4, where the errors given are
statistical uncertainties only. The measured day/night asymmetry for SK-IV is found to be(-5.3±2.0±1.4)%. The middle column of Table 9.3 gives the day/night asymmetry for all

181

6

2

Flux (x10 /cm /s)

2.8

2.6

2.4

2.2

Day
Night

2

All

Day

-1

Night

0

1
cosθz

Figure 9.3: The solar zenith angle dependence of the solar neutrino flux (4.49-19.49 MeV).
The day data is sub-divided into five bins, while the night data is divided into six bins.
Solar neutrinos in the last night bin pass through the Earth’s core. The error bars show are
statistical errors only.

SK phases, with the final row giving the combination of all four phases. This combination
takes statistical and systematic uncertainties into consideration. The SK combined measured
day/night asymmetry is then found to be

ADN = (−4.2 ± 1.2(stat.) ± 0.8(syst.))%.

(9.14)

This combined SK measurement of the day/night asymmetry gives evidence for a non-zerosolar neutrino day/night effect, at the 2.8 σ level. This significant evidence for the existenceof the solar neutrino day/night effect is further backed up by the results in the following
sections.

182

Data/MC (unoscillated)

0.7

0.7

0.6
0.5

0.6

0.4
0.5

0.6
0.5

0.4

0.4

0.3

0

1

0

1

0

1
cosθz

0

1

Figure 9.4: Data/MC simulation (unoscillated) dependence on the solar zenith angle for various energy regions (zenith angle and energy bins defined in Table 9.2). The MC simulation
assumes a 8 B flux of 5.25×106 cm−2 sec−1 . Overlaid in red (blue) are predictions when using
the global solar (solar+KamLAND) best-fit oscillation parameters and the assumed neutrino
fluxes fit to best describe the data. The error bars shown are statistical uncertainties only.

Table 9.3: Day/night asymmetry for each SK phase, coming from separate day and night
rate measurements (left column) and the amplitude fit (middle column), with the expected
day/night asymmetry in the right column. The uncertainties shown are statistical and systematic. The final row gives the combination of all four phases, for both methods. The entire
middle and right columns assume ∆m221 =5.90×10−5 eV2 , sin2 θ12 =0.314 and sin2 θ13 =0.025.
ADN ±(stat.)±(syst.) ADN ±(stat.)±(syst.)
SK-I-2.1±2.0±1.3%-2.0±1.7±1.0%
SK-II-5.5±4.2±3.7%-4.3±3.8±1.0%
SK-III-5.9±3.2±1.3%-4.3±2.7±0.7%
SK-IV-5.3±2.0±1.4%-3.4±1.8±0.6%
Combined-4.2±1.2±0.8%-3.2±1.1±0.5%183

Expected ADN
-3.3%
-3.6%
-3.2%
-3.3%
-3.3%

9.4.2 Day/Night Amplitude Fit Results
As described in Section 9.2, the day/night amplitude parameter extracted from the unbinned maximum likelihood fit to the solar zenith angle variation of the interaction rate scales
the expected day/night asymmetry (for a given set of oscillation parameters) to give themeasured day/night asymmetry ADN . The middle column of Table 9.3 gives the day/night
amplitude fit results for each SK phase, for oscillation parameters of ∆m221 =4.90×10−5 eV2 ,sin2 θ12 =0.314 and sin2 θ13 =0.025. It is shown in the form of the equivalent day/night flux
asymmetry ADN . The actual amplitude and the uncertainty associated with that amplitude, extracted from the day/night amplitude fit, can be calculated by dividing the middle
column of Table 9.3 by the right column. The resulting SK combined measured day/night
asymmetry coming from the day/night amplitude fit is

ADN (∆m221 = 4.90 × 10−5 eV2 ) = (−3.2 ± 1.1(stat.) ± 0.5(syst.))%.

(9.15)

This is assuming sin2 θ12 =0.314 and sin2 θ13 =0.025. The expected value is -3.3%. If the valueof ∆m221 is shifted to a value preferred by the fit to all solar neutrino data and KamLAND, theexpected day/night asymmetry as SK goes to -1.7%. The SK combined measured day/night
asymmetry coming from the day/night amplitude fit then goes to

ADN (∆m221 = 7.41 × 10−5 eV2 ) = (−3.0 ± 1.0(stat.) ± 0.5(syst.))%.

(9.16)

This second method for extracting the solar neutrino day/night asymmetry also gives
evidence for a non-zero day/night effect, this time at the 2.7 σ level (for oscillation parameters
preferred either by SK or the solar+KamLAND analyses). This is clearly demonstrated in
Figure 9.5, where the SK combined day/night asymmetry as measured by the day/night
amplitude fit, is shown as a function of ∆m221 . The light (dark) gray band gives the 1184Day/Night Asymmetry (%)

SK-I,II,III,IV Combine Day/Night Asymmetry
1

0

-1

-2

-3

-4

-5

2

4

6

8

10

12

14

16

18

20
22
∆ m221 (10-5eV2)

Figure 9.5: Measured day/night asymmetry (fitted day/night amplitude times the expected
day/night asymmetry (red)) vs. ∆m221 , for sin2 θ12 =0.314. The light (dark) gray band givethe 1 σ statistical (statistical plus systematic) uncertainty. Overlaid are the allowed 1 σ
ranges from solar neutrino data (green band) and KamLAND (blue band).

σ statistical (statistical plus systematic) uncertainty and the red line gives the predicted
day/night asymmetry. Where the SK best fit line crosses the expectation, the value of thefitted day/night amplitude is one. Also overlaid in this figure is the 1 σ allowed regionsin ∆m221 , coming from the solar global fit (green) and from KamLAND (blue). Figure 9.6
shows the SK combined results of the day/night amplitude fit as a function of sin2 θ12 , shownas the equivalent day/night asymmetry. As seen in the figure, there is no dependence of themeasured day/night asymmetry on θ12 . This is true for values of θ12 contained within the
LMA region.
The day/night amplitude fit method was also run for each energy bin individually, foreach phase, and then combined statistically to give the measured day/night asymmetry as
a function of recoil electron kinetic energy. The SK combined results of this energy binned
day/night amplitude fit can be seen in Figure 9.7, with statistical uncertainties shown only.
The red line gives the expectation assuming the same oscillation parameters as in Table 9.3.
A general tendency of the data to follow the expectation can be seen here.

185

Day/Night Asymmetry (%)

SK-I/II/III/IV Combine Day/Night Asymmetry
1

0

-1

-2

-3

-4

-5
0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5
sin2θ12

Day/Night Asymmetry (%)

Figure 9.6: Dependence of the day/night asymmetry on sin2 θ12 for ∆m221 =4.90×10−5 eV2 .
The light (dark) gray band shows the 1 σ statistical (statistical plus systematic) uncertainty.
The red line shows the expected day/night asymmetry.

20

0

-20

-40
5

10
15
Recoil Electron Kinetic Energy (MeV)

Figure 9.7: SK combined day/night asymmetry vs. energy (measured day/night amplitude
times the expected asymmetry (red)) for ∆m221 =4.90×10−5 eV2 and sin2 θ12 =0.314. The
error bars shown are statistical uncertainties only.

186

9.5 SK+SNO Day/Night Asymmetry Results
The SNO experiment performed a search for the day/night variation of the chargedcurrent interaction rate on deuterium [20]. It defines the day/night asymmetry of the νesurvival probability as
D
N
(Eν )
(Eν ) − Pee
Pee.
Aee (Eν ) = 2 N
D
Pee (Eν ) + Pee (Eν )(9.17)

It then models that asymmetry with the function a0 + a1 (Eν − 10M eV ). SNO measuresa0 = −0.046 ± 0.034 and a1 = 0.016 ± 0.027. However, this measurements cannot bedirectly compared to the SK measurement, as the expected day/night asymmetry depends
upon the detector location on Earth. It is then necessary to cast the SNO measurement
into a day/night amplitude type measurement, similar to SK. Using ∆m221 = 4.90 × 10−5eV2 and sin2 θ12 = 0.314, one can calculate a scaling factor between the measured andexpected day/night asymmetry (the SNO day/night amplitude), which can in turn then becompared to the SK measurement coming from the day/night amplitude fit. In this way the
SNO measurement can be expressed as (-3.5±3.1)% (day/night asymmetry of the survival
probability at 10 MeV neutrino energy). In a similar fashion, the SNO measurement canbe converted to an SK equivalent day/night asymmetry of (-2.0±1.7)%, by multiplying the
SNO day/night amplitude by the SK expected day/night asymmetry. In this way, SK and
SNO combine together to measure the SNO day/night survival probability asymmetry at
10 MeV (neutrino energy) as (-4.62±1.74)%. The SK equivalent day/night asymmetry ismeasured as

ADN (∆m221 = 4.90 × 10−5 eV2 , sin2 θ12 = 0.314) = (−2.9 ± 1.0)%.

(9.18)

This SK and SNO combined measurement of the solar neutrino day/night flux asymmetry
includes both statistical and systematic uncertainties. It gives strong evidence for a non-zero

187

Day/Night Asymmetry (%)

SK-I/II/III/IV + SNO Day/Night Asymmetry
1

0

-1

-2

-3

-4

-5

2

4

6

8

10

12

14

16

18

20
22
∆ m221 (10-5eV2)

Figure 9.8: SK+SNO measured day/night asymmetry (fitted day/night amplitude times the
expected day/night asymmetry (red)) vs. ∆m221 for sin2 θ12 =0.314. The light (dark) grayband gives the SK plus SNO (SK) 1 σ statistical plus systematic uncertainty. Overlaid are
the allowed ranges from solar neutrino data (green band) and KamLAND (blue band).

day/night effect, at the 2.9 σ level. The expected day/night asymmetry for the above oscillation parameters is -5.76% (10 MeV day/night survival probability asymmetry) or -3.3% (SKday/night asymmetry). The SK and SNO combined measured day/night asymmetry (SKequivalent) can be seen in Figure 9.8, as a function of ∆m221 (sin2 θ12 =0.314, sin2 θ13 =0.025).
The light (dark) shaded gray region shows the SK plus SNO (SK) 1 σ statistical plus systematic uncertainty of the measured day/night asymmetry. Overlaid in red is the expected
SK day/night asymmetry. Also given is the 1 σ allowed region in ∆m221 coming from theglobal solar (solar+KamLAND) best-fit in green (blue).

188

Chapter 10
Solar Neutrino Oscillation
Analysis10.1 Spectrum Shape Oscillation Analysis
The comparison between the observed and predicted spectra is done by fitting each of thefour SK energy spectra to their individual predictions. The predictions are made according
to the method outlined in Section 9.2. This is done for oscillation parameters within the
MSW oscillation plane (10−11 eV2 ≤ ∆m221 ≤ 10−3 and 10−4 ≤ sin2 θ12 ≤ 1), and for values ofsin2 θ13 between 0 and 0.035. Although the analysis is carried out assuming three-flavours ofneutrinos, and without prior constraint on θ13 , it can effectively be turned into a two-flavour
style analysis by setting θ13 = 0.
The measured elastic scattering rate dpi ± σip in energy bin i and SK phase p are fit to
the calculated rates rip = βbpi + ηhpi . The 8 B and hep rates are calculated according to themethod outline in Section 9.2. The flux parameter β (η) scales bpi (hpi ), the elastic scattering
rate of the 8 B (hep) solar neutrinos. The rate and spectrum χ2 of SK phase p is then

χ2p =

2
Np  pXd − fp (ν, p , ρp )rp (sin2 θ12 , sin2 θ13 , ∆m2 )
e

e

21

σep

e=1

.

(10.1)

fp (ν, p , ρp ) is the spectral distortion factor resulting from the neutrino spectrum uncertainty,
described by the nuisance parameter ν and the SK energy scale (resolution) uncertainty ofphase p, described by p (ρp ). This χ2 can be written in a quadratic form using the 2×2

189

p
p
curvature matrix Cp and the best-fit flux parameters βminand ηmin
as



p



β − βmin pp
χ2p = χ2p,min + αp (β − βmin, η − ηmin
) · Cp · ,pη − ηmin(10.2)

if αp =1.
However, when adding the χ2 of the SK phases together (χ2sum =P42
p=1 χp ) we scale Cp by

2
σp,stat. This is done to account for the additional systematic uncertainty
2p,stat +σp,systchanging αp to σ2of the total rate (integrated over all energies, see Table 7.3) not covered byP

σip . Scaling theχ2 of each phase in this way has the effect of preserving the minimum χ2 and the best-fit flux
parameters, but enlarges the uncertainty with which the χ2 constrains the flux parameters.
The SK combined χ2 is then obtained from the χ2 sum by minimizing over the nuisance andflux parameters in
"
χ2 =

Min

ν,p ,ρp ,β,η

ν2 +

4
X
p=1

(2p + ρ2p ) +β − β0σβ2+η − η0ση2#+ χ2sum .

(10.3)

When the fit is done initially, the constraining term for the neutrino flux parameter β(second term) is left out and we only slightly constrain η to (8±16)×103 cm−2 sec−1 . This
allows for the minimization of the χ2 , taking only the spectral shapes into account. A χ2
map can then be made for the entire MSW oscillation plane, and by using the standard∆χ2 method, the excluded/allowed regions can be drawn. The resulting excluded contours
coming from the spectrum shape fitting of all four phases of SK can be seen in Figure 10.1.
The green lines mark the excluded regions at the 1, 3 and 5 σ confidence levels (shaded green
 2
2
13 −0.0242
region is 5 σ). In this figure the additional θ13 constraining term sin θ0.0026
is added
to the χ2 (motivated by the recent θ13 measurements).

190

2

∆m in eV

2

-3

10

-4

10

-5

10

-6

10

-7

10

-8

10

-9

10

-4

-3

10

10

-2

-1

10

10

1

10

2

10
tan (θ )2

12

Figure 10.1: Excluded contours of ∆m221 vs. tan2 θ12 from the SK-I/II/III/IV spectra (green)and day/night data (blue) at the 1, 3 and 5 σ confidence levels (5 sigma is filled). The unfilled
region gives the1 and 2 σ allowed
regions. The red contours are the combined fit. θ13 isconstrained by2

sin2 θ13 −0.0242
0.0026.

10.2 Solar Neutrino Flux Time Variation Oscillation Analysis
It is also possible to do an oscillation analysis using the SK time variation data. Thistakes into account the effects on neutrino oscillations coming from Earth’s matter (day/nighteffect). To do this the likelihood function used to extract the solar neutrino signal (Equation 7.1) is modified. It now includes a signal scaling function which reflects the deviationfrom the average interaction rate, arising from the solar zenith angle dependence. This scaling function is the simple ratio of the calculated expected interaction rate and the averageinteraction rate ri (Cz )/riave . ri (Cz ) and riave are calculated according to Equations 9.11 and9.4. The modified likelihood formula is written as

Lt.v. = e−(Pi Bi +S)

N
ni
bin Y
Y

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

i=1 j=1

191(10.4)

In this fashion, if ri (Cz )/riave =1, the original likelihood formula is recovered. To turn this
into a χ2 map, the difference in the log of the likelihood, with and without the time variation
factor included, is taken and multiplied by -2, for each set of oscillation parameters. This isgiven as∆χ2t.v. = −2(logL − logLt.v. ).

(10.5)

It is then possible to produce SK time variation excluded/allowed contours in the MSW
oscillation plane. This is done by subtracting the minimum value found for ∆χ2t.v. , from eachvalue of ∆χ2t.v. , for the different sets of oscillation parameters. The resulting excluded/allowed
contours can be seen in Figure 10.1. The blue lines mark the filled regions excluded by thetime variation data at the 1, 3 and 5 σ confidence levels (shaded blue region is 5 σ). The
unfilled regions are the 1 and 2 σ allowed contours. As in the case of the spectral fit, these2 2
13 −0.0242
to the χ2 .
results include the addition of the θ13 constraining term sin θ0.0026

10.3 SK Oscillation Analysis Results
With the χ2 maps from the SK spectrum shape and time variation fitting made, it is thenpossible to combine them together to get one unified χ2 map. This is done by the simple
addition of χ2spec and χ2t.v. . The constraining term β of the 8 B flux is now left in the spectral
χ2 (constrained to the SNO NC measurement (5.25±0.20)×106 cm−2 sec−1 ). The resulting
total SK χ2 is thenχ2SK = χ2spec + χ2t.v. +(β − 1)2 (η − 1)2+,
σ φ8 B
σφhep(10.6)

where σφ8 B =0.04 and σφhep =2.0, corresponding the SNO NC 8 B flux uncertainty [20] and the2

13 −0.0242 2
hep flux limit (φhep <2.9×SSM) [20]. This time the θ13 constraining term ( sin θ0.0026) isadded to the total χ2SK , but only once after the spectral and time variation χ2 ’s have been
added (this way it is not double counted).
192

From the minimum χ2SK value obtained, and using the standard ∆χ2 method, it is thenpossible to plot the excluded/allowed regions in the MSW oscillation plane. The resulting
excluded/allowed contours are shown in Figure 10.2. The shaded blue area gives the exclude
regions at the 1, 3 and 5 σ confidence levels, coming from the combination of the SK spectral
and time variation data. The unfilled blues lines gives the 1 and 2 sigma allowed regions
attained from the same data. The green lines (area) give(s) the allowed regions coming fromthe addition of the flux constraining terms into χ2SK , at the 1-5 (3) σ confidence levels. SK
data strongly favors the LMA region best-fit parameterssin2 θ12 (SK) = 0.341+0.029−0.025(10.7)

2∆m221 (SK) = 4.8+1.8−0.9 eV(10.8)

with sin2 θ13 =0.025, by more than 3 σ. It also strongly favors a large mixing solution (LMAor LOW regions), by almost 5 σ, over small mixing (SMA region).

10.4 SK+SNO Oscillation Analysis
Since SK and SNO both measure 8 B neutrinos across a similar energy range, the combined
analysis will profit from correlations, and is thus better than the mere addition of their χ2 ’s.
The result of SNO’s fit to the energy dependence of the νe survival probability can be castinto a χ2 of the same form as Equation 10.2. To do this the day/night asymmetry of theνe survival probability also included by SNO must be left out by setting a0 and a1 equal tozero. To make sure this definition of the SNO χ2 really reproduces the SNO results, the χ2 isminimized in the same fashion as for each SK phase, resulting in the 1-5 σ allowed regions,outlined by the blue lines in Figure 10.3 (shaded blue region gives 3 σ level). These allowed
regions very closely match the SNO published results [20]. Also overlaid in this figure (green)is the allowed regions from the SK only analysis. Comparing SK and SNO, we see that SNO
193

2

∆χ

8 3σ
6
4 2σ
2

2

∆m in eV

2

1σ

10

-4

10

-5

10

-6

10

-7

10

-8
-9

10 -4
10

10

-3

10

-2

10

-1

1σ

1

2σ

3σ

10
2468
2
2
tan (θ12)∆χFigure 10.2: Excluded and allowed contours of ∆m221 vs. tan2 θ12 from the SK-I/II/III/IV
spectra and day/night data (blue). The shaded blue region gives the excluded regions (1-5 σ)and the unfilled regions are the 1 and 2 σ allowed regions. Green shows the same data, butnow with the 8 Bflux constraint included, at the 1-5 σ confidence levels. θ13 is constrainedbysin2 θ13 −0.0242
0.00262.

gives a more precise measurement of the mixing angle (sin2 θ12 = 0.299+0.023−0.020 ), while SK gives−5
the more precise measurement of the mass squared splitting (∆m221 =4.8+1.8
eV2 ).
−0.89 ×10

Also, while SK is able to exclude the SMA (LOW) region at more than 5 σ (3 σ), SNO isonly able to exclude the SMA region at the 3 σ level. SNO gives 1-5 σ allowed areas at the
LOW solution.
Finally, to combine the SK and SNO analyses together, the SK-like SNO χ2 is added to
the SK χ2 , treating the SNO experiment just like a fifth phase of SK. The θ13 constraining
term is again added to the total χ2 at the end, to avoid double counting. The resulting 1-5
σ allowed regions in the MSW oscillation plane are shown by the red lines in Figure 10.3
(shaded red region gives 3 σ). In this way SK and SNO uniquely select the LMA region at

194

2

∆χ

8 3σ
6
4 2σ
2

2

∆m in eV

2

1σ

10

-4

10

-5

10

-6

10

-7

10

-8
-9

10 -4
10

10

-3

10

-2

10

-1

1σ

1

2σ

3σ

10
2468
2
2
tan (θ12)∆χFigure 10.3: Allowed contours of ∆m221 vs. tan2 θ12 from the SK-I/II/III/IV spectral andday/night data using the 8 B flux constraint (green), and the SNO results (blue), at the 1-5
σ
The SK and SNO combined result is shown in red. θ13 is constrained by confidence levels.
sin2 θ13 −0.0242
0.00262.

the 5 σ confidence level, with best-fit oscillation parameters chosen as

sin2 θ12 (SK+SNO) = 0.308 ± 0.014
2−5∆m221 (SK+SNO) = 6.0+1.2−1.0 × 10 eV .

(10.9)(10.10)10.5 Solar Neutrino Global Oscillation Analysis
The results from the Homestake experiment, gallium experiments (SAGE, Gallex/GNO)
and Borexino are added to the χ2 of the SK and SNO combined oscillation analysis. Theχ2 values of these experiments are based on two equivalent ways of defining the χ2 function

195[54], described by
N (=3)

χ2rate =

X

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

(10.11)n,m=1

N is the number of experiments, in this case N=3 (Cl, Ga (all three Ga experiments are com2
bined into one) and the Borexino experiment). σnm
is a matrix of the squared uncertainties,

which are the quadrature sum of the correlated and uncorrelated uncertainties among theexperiments. Rexpt is the observed rate and and Rtheor is the predicted event rate according
to the SSM. The solid green lines in Figure 10.4 show the allowed regions of all solar neutrino data combined, at the 1-5 σ confidence levels (shaded green area shows the 3 σ allowed
region). The dashed green lines in this figure give the almost identical SK plus SNO only fit
for comparison. The best-fit values of the oscillation parameters coming from fit to all solar
neutrino data are found to besin2 θ12 (solar) = 0.307 ± 0.014(10.12)∆m221 (solar) = 5.90 ± 1.1 × 10−5 eV2 .

(10.13)

It is further possible to combine the results of the solar neutrino global analysis with theresults from the KamLAND experiment (1-3 σ KamLAND only allowed regions are shownin blue in Figure 10.4). This is also done by the combination of the χ2 maps, once again
producing allowed areas in the MSW oscillation plane. The red lines in Figure 10.4 give theallowed regions coming from the combination of all solar neutrino data and the KamLAND
experiment, at the 1-3 σ confidence levels (shaded red area gives the 3 σ allowed region).
The best-fit oscillation parameters chosen by the solar+KamLAND oscillation analysis are

sin2 θ12 (solar+KamLAND) = 0.304 ± 0.013
2−5∆m221 (solar+KamLAND) = 7.45+0.20−0.19 × 10 eV .

196

(10.14)(10.15)

There is a slight tension between the solar neutrino and KamLAND reactor-antineutrino
measurements of ∆m221 , mostly coming from the SK day/night asymmetry measurement.
Even though the expected day/night asymmetry agrees well within the one σ band of themeasured day/night asymetry for the values of ∆m221 preferred by either the solar neutrino
or KamLAND data, SK and other solar neutrino data prefer values of ∆m221 that are smaller
than those which KamLAND’s data prefers.
It is also possible to remove the θ13 constraining term from the SK plus SNO χ2 to makean unconstrained fit on the θ13 . The resulting allowed contours in the sin2 θ13 vs. sin2 θ12plane are shown in Figure 10.5. The green (blues) lines give the 1-5 (1-3) σ confidence levels
coming from the global solar (solar+KamLAND) fit. Also shown by the red lines are theallowed areas coming from the combination of the solar neutrino data with the KamLAND
result, at the 1-3 σ confidence levels. The shaded regions in this figure give the 3 σ confidence
level. The combined solar neutrino data and KamLAND results give an ∼2 σ significancefor a non-zero θ13 .

197

2

∆χ

8 3σ
6
4 2σ
2

∆m in 10 eV

2

1σ

-5

18
16

2

14
12
10
8
6
4
2
1σ

0.1

0.2

0.3

0.4

0.5
2
sin (θ12)

2σ

3σ

2468
2
∆χ

Figure 10.4: Allowed contours of ∆m221 vs. sin2 θ12 from solar neutrino data (green) at 1-5
σ and KamLAND data (blue) at the 1-3 σ confidence levels. Also shown are the combined
results in red. For comparison, the almost identical results
combined fit are of the SK+SNO

sin2 θ13 −0.0242
0.0026

2

.

8 3σ
6
4 2σ
2
0.2 1σ2sin (θ13)∆χ

2

shown by the dashed dotted lines. θ13 is constrained by

0.1

0.0
0.0

1σ

0.1

0.2

0.3

2σ

3σ

0.4
0.5 2 4 6 822sin (θ12)∆χFigure 10.5: Allowed contours of sin2 θ13 vs. sin2 θ12 from solar neutrino data (green) at 1-5
σ, and KamLAND data (blue) at the 1-3 σ confidence levels. Also shown are the combined
results in red.

198

Chapter 11
Discussion
11.1 Conclusion
The beginning of the fourth phase of Super-Kamiokande was marked by the complete
replacement of the front-end electronics. This allowed for an improvement not only in thedescription of the peak of the timing residuals, but also an improvement in the comparison of the data and MC simulation timing residuals. This lead to an improvement in thevertex resolution, across all energies. Along with the improvement of the water circulation
system, leading to a lower background level in the central region of the detector, and a reoptimization of the cut criteria for SK-IV, the low energy threshold was reduced down to 3.5
MeV recoil electron kinetic energy. This is an improvement from the 4.0 MeV threshold set
in SK-III and the 4.5 MeV threshold of SK-I. The above improvements, in conjunction withimprovements to the description of the water transparency (position and time dependence),
the introduction of a new parameter to evaluate the amount of multiple Coulomb scattering(multiple scattering goodness (MSG)) and using the entire livetime for all energy bins, haveallowed for the reduction of both the statistical and systematic errors.
The total systematic uncertainty on the measured solar neutrino flux was estimated as±1.7% for the SK-IV analysis. This is much improved from the (+3.5%−3.2% ) in SK-I and ±2.2%
in SK-III. The extracted solar neutrino flux during the SK-IV phase is in good agreementwith the flux extracted from the previous three phases. When the data from all four phases
is combined together, SK has observed more than 60,000 solar neutrino events between 4.0
and 19.5 MeV. This is by far the largest sample of solar neutrinos to date. This observed
199

number of events corresponds to the total SK combined measured flux ofΦ8 B = (2.37 ± 0.014(stat.) ± 0.04(syst.)) × 106 cm−2 sec−1 .

(11.1)A χ2 fit of the SK spectral data for each of the four phases was performed. The spectrumof each phase was fit to various predictions resulting from MSW oscillations, as well asassuming more general functions of the quadratic, cubic and exponential type. From thesefits, it was shown that the SK spectral data slightly favors an “upturn”, when using the best
exponential or quadratic fit.
The SK combined spectral fit to a general parabolic survival probability was also combinedwith the similar analysis of SNO. This can be very beneficial to do, as SK measures lower
energy neutrinos with greater precision, while SNO measures higher energy neutrinos withgreater precision. The two experiments separately, slightly prefer either an “upturn” or a“downturn” in the νe survival probability, when going from higher energies to lower energies.
But because SK prefers a larger baseline coefficient (c0 ), and measures the low energy endof the spectrum more precisely (SNO measures the high energy end more precisely), the net
effect of combing the two experiments is the increased favoring of an “upturn”.
The solar neutirno flux day/night asymmetry was measured in SK by means of two
methods. The first looks at the event rates during the day and night times separately. The
second method makes an unbinned maximum likelihood fit to the solar zenith angle variationof the event rate. The SK-IV solar neutrino flux day/night asymmetry (for energies between
4.5 and 19.5 MeV) was measured with the greatest precision among all phases. This comesfrom the reduction in the energy scale systematic uncertainty on the day/night asymmetry,
due to the inclusion of the z-dependent water transparency parameter beginning with SK-III.
Another factor is the large livetime of SK-IV (comparable to SK-I) spanning across all energybins, with the lowest and most stable background rates to date. By using the first method200

of measuring the day/night asymmetry, and combing the results of SK-IV with the previous
three phases, the day/night asymmetry (oscillation parameter independent) is measured as

ADN = (−4.2 ± 1.1(stat.) ± 0.8(syst.))%.

(11.2)

This gives a non-zero measurement of the solar neutrino day/night effect, at the 2.8 σ level.
The solar neutrino flux day/night asymmetry results of the second method (day/night
amplitude fit) are dependent on the choice of oscillation parameters, although there is very
little dependence of the day/night asymmetry on sin2 θ12 and sin2 θ13 , within the LMA region. Taking the values of sin2 θ12 =0.314 and sin2 θ13 =0.025, the combined resulting effective
day/night asymmetry from all four phases of SK is found to be either

ADN (∆m221 = 4.90 × 10−5 eV2 ) = (−3.2 ± 1.1(stat.) ± 0.5(syst.))%,(11.3)ADN (∆m221 = 7.41 × 10−5 eV2 ) = (−3.0 ± 1.0(stat.) ± 0.5(syst.))%.

(11.4)orThis gives a 2.7 σ significance for a non-zero day/night effect. The first value of ∆m221
is preferred by SK data (expected day/night asymmetry=-3.3%), while the second value
is preferred by the solar+KamLAND oscillation analysis (expected day/night asymmetry=1.7%). While both of these results are consistent with expectations within ∼1 σ, the value of∆m221 chosen by SK slightly favored. When this measurement of the day/night asymmetry is
combined with SNO’s, the resulting SK equivalent solar neutrino day/night flux asymmetry
is measured as

ADN (∆m221 = 4.84 × 10−5 eV2 , sin2 θ12 = 0.314) = (−2.9 ± 1.0)%.

201(11.5)

This combined measurement gives 2.9 σ significance for a non-zero day/night effect. This
result, along with the SK combined day/night asymmetry measured by the difference inthe day and night time interaction rates, gives the first significant evidence for the solar
neutrino day/night effect. This is also the first significant evidence for matter enhanced
neutrino oscillation. While previous experiments have been able to confirm the existenceof neutrino oscillation, no experiment has seen significant deviation from purely vacuum
oscillation expectations.
The SK spectral, time variation and rate data were then used to execute an oscillation
analysis for parameters within the MSW oscillation plane. Using only SK data and placing
a constraint on the observed solar neutrino flux, a large mixing solution (LMA or LOW) is
favored over a small mixing solution (SMA), at better than the 5 σ confidence level. SK
data selects the LMA solution over the LOW solution at more than 3 σ, with the best-fit
oscillation parameters chosen as

sin2 θ12 (SK) = 0.341+0.029−0.025−5∆m221 (SK) = 4.8+1.8
eV2 .
−0.9 × 10(11.6)(11.7)The SK data was combine with the SNO data, taking advantage of correlations between
the two experiments. In this manner SK and SNO uniquely select the LMA solution

sin2 θ12 (SK+SNO) = 0.308 ± 0.014−5∆m221 (SK+SNO) = 6.0+1.2
eV2 ,−1.0 × 10(11.8)(11.9)

at more than 5 σ. The inclusion of other solar neutrino data was also carried out, adding inresults coming from the Homestake, gallium and Borexino experiments. The impact of theadditional experiments on the allowed LMA region is very minor, only slightly shrinking the
top portion of the contour. The best-fit oscillation parameters coming from the global solar
202neutrino data fit aresin2 θ12 (solar) = 0.307 ± 0.014(11.10)∆m221 (solar) = 5.90 ± 1.1 × 10−5 eV2 .

(11.11)

Finally the oscillation analysis of KamLAND was combined with the global solar neutrino
oscillation analysis. There is a slight tension between the two results, mostly coming from theresult of the SK day/night asymmetry measurement. This causes the solar global analysis
to prefer a slightly smaller value of ∆m221 than KamLAND. However, KamLAND gives amore precise measurement of ∆m221 , and therefore the global solar neutrino plus KamLAND
best-fit oscillation parameters are found to besin2 θ12 (solar+KamLAND) = 0.304 ± 0.013−5∆m221 (solar+KamLAND) = 7.45+0.20
eV2 .
−0.19 × 10

(11.12)(11.13)11.2 Future Outlook for Solar Neutrino Measurements
11.2.1 Future Solar Neutrino Measurements at Super-Kamiokande
The Super-Kamiokande solar neutrino data set has contributed many pieces to the solution of the solar neutrino problem. But this is not the end of SK. The experiment continues
to run with plans to collect data for a minimum of the next ten years. The solar neutrino
analysis of SK will be pushed on many different fronts in the years to come. One of two
main goals will be to push the energy threshold lower, probing deeper into the expected
MSW resonance curve, in hopes of observing the infamous “upturn”. The SK and SNO
combined analysis is giving a first sign that there is an upturn, and a larger, more precise203

SK data set, with a lower energy threshold will prove to be very powerful. It is believed thata better than 3 σ significance for the presence (or non-presence) of the MSW “upturn” canbe achieved. There is also the expected distortion of the observed neutrino spectrum comingfrom the contribution of hep neutrinos. With more statistics, and even better control ofdetector systematics, SK may eventually be able to make a measurement of the hep solar
neutrino flux, by measurement of the high energy spectrum distortions.
The second focus of SK in the next ten years will of course be to increase statistics,
while lowering systematic uncertainties, leading to the eventual 4 or 5 σ measurement of
the day/night effect. With the significance for a non-zero day/night effect already justbelow 3 σ, from SK data alone, another ten years of SK data with the current detector
configuration, and assuming the current value of the measured day/night flux asymmetry is
the true value, can lead to a better than 4 σ measurement. But SK refuses to remain in thecurrent configuration, as upgrades are already planned.
The first effort to upgrade SK is the addition of a new data acquisition system, developed
especially for low energy reconstruction. It is known as the wide band intelligent trigger(WIT). WIT is a fast online vertex reconstruction trigger system, used to make a pre-cut on
the massive number of very low energy events (<3.5 MeV), which are almost all coming fromradioactive backgrounds. Details of the WIT system are given in [55]. After the WIT systemdoes the fast online vertex reconstruction, based on the maximum likelihood fit to the timing
residuals, it makes various cuts on the fiducial volume, number of hits within a given time
window, and potentially the clustering of hits. This will allow the low energy threshold to belowered, without a large introduction of background into the final data sample. It is believedthat after the WIT system is installed, the low energy threshold will go to 3 MeV, a limit
met due to the inability to reconstruct events below 3 MeV. However, there is a proposal
being thrown around the table to increase the photo-cathode coverage of SK, giving more
hit PMTs per MeV of recoil electron energy and also increasing the energy resolution. This204

has the possibility of pushing the reconstruction limit down.
A second effort currently being pursued is the further reduction of the low energy background in the fiducial volume. It is believed that the majority of this very low energy
background comes from the decay chain of radon. The radon is believed to either be comingfrom the surfaces of the PMT glass or the FRP cases which house the PMTs, or that is being
injected into the detector with the water coming from the water purification system. Thereis a current study underway to discover which of the two cases is really taking place within
SK, and a plan of attract is being established in either case. For the first case, in which
the radioactive background emanates from the detector component surfaces, SK will plan toplace acrylic plates over the entire PMT ID structure. This will completely seal off the ID
from all detector components. The hope is that this “acrylic vessel” will block the radonfrom entering the fiducial volume. If the radon is being injected with the purified water, thenit may be possible to redesign the water system with a special stage to specifically remove
radon. Either way, it is believed that the reduction of half of the low energy backgroundcontaminating the fiducial volume may be possible.
Finally, there is a continuous research and development effort to dope SK with gadolinium(Gd) salt. This effort, currently called EGADS (Evaluating Gadolinium’s Action on Detector
Systems), consists of a 200 ton stainless steel test tank. This test tank is going to be
instrumented with 240 50 cm PMTs (SK type PMTs), and then doped with Gd sulfate.
This will allow for the efficient detection of neutrons within a water Cherenkov detector. So
far the tests at EGADS have been succesful, and within the next year the EGADS team is
expected to give results which show that doping SK with Gd sulfate will give the desired
physics results.
One of the main advantages of being able to efficiently detect neutrons in SK is that
antineutrinos, which interact mainly due to inverse β-decay and generate a positron andneutron, can be separated from neutrinos. This could lead to an improvement in the purity of205the solar neutrino data sample, by separating not only the antineutrino events, but also otherevents which may have an accompanying neutron. This, and the other efforts being pursued
by the SK collaboration, will make the next ten years of SK solar neutrino measurements
quite exciting.

11.2.2 Other Current and Future Solar Neutrino Experiments
Of course measuring the 8 B solar neutrino spectrum distortions and day/night flux asymmetry are not the only avenues in which to probe the solar interior. As neutrinos areproduced at other points in the pp chain, as well as at multiple points within the CNO cycle,
it is advantageous for other experiments to make measurements of these neutrinos. The
most sought after solar neutrino measurement is now the measurement of the CNO cycle
neutrino flux. Because the three neutrino species coming from the CNO cycle have very
similar endpoints energies, it may be very difficult to disentangle the three fluxes from one
another, and so a first goal is to make the combine flux measurement. A measurement of
the CNO solar neutrino flux can be used to calculate the amount of heavy elements in thesun, an important input in standard solar models.
Currently this measurement has been out of reach in the Borexino detector, due to highbackground levels, coming mostly from the different isotopes of carbon. But Borexino has
been working to further purify the scintillator, reducing these backgrounds. Borexino plans
to continue data taking in hopes of reducing their upper limit on the CNO solar neutrino
flux, and possibly even make a measurement [56]. The future of Borexino will also include
increasing the precision of their 7 Be, pep and low energy 8 B solar neutrino measurements.
It will also attempt to make a measurement of the pp solar neutrino flux, and possibly
spectrum, although the large amount of 14 C background may make this impossible.
A second experiment, currently in the works, is an extension of the SNO experiment [57],206

in which the SNO detector will be filled with liquid scintillator and coined SNO+. The main
focus of the detector will be to observe neutrino-less double β decay, but it will still havethe ability to observe solar neutrinos. Because of the extreme depth of the SNO+ detector,
the long-lived spallation product 11 C, which is the background contaminating the Borexino
detector in the search for CNO neutrinos, will be greatly reduced, along with all otherspallation products. This could make it possible for SNO+ to make the first measurementof the CNO solar neutrino flux. The lower background rate coming from spallation products
could also allow SNO+ to make the most precise measurement of the 1.4 MeV pep solar
neutrinos. SNO+ may also be able to measure 7 Be and 8 B neutrinos, and while time variation
measurements may be possible, the experiment may not have much to say about the spectral
shape of the 8 B neutrinos.
Finally there are a few very large scale detectors being planned or envisioned. The mostprobable of these experiments, is an extension of the SK detector to a 20 times larger
detector, call Hyper-Kamiokande (HK) [58]. This project is making rapid progress, and is
expected to make a formal proposal to the Japanese funding agencies soon. There are still
many unknowns about the exact location and performance of the detector, but it is for surethat the gigantic size of the detector will allow for a huge increase in the observed event
rate. It is predicted that HK could make a measurement of the solar neutrino day/night flux
asymmetry, at better than 5 σ within 10 years. This is assuming that the great experience ofworking with the SK detector translates over to the HK detector, and detector systematics
are understood at the same level as SK. It also assumes that the current measured day/night
asymmetry is reasonably close to the true value. Also the large number of statistics could
make it possible to investigate other types of time variation.
There is also hope that some of the unknown things about HK, like PMT coverage anddetector depth, will be decided so that they help the solar neutrino analysis. The currently
proposed depth will give HK ∼10 times the rate of muons in the detector, which may not207kill the solar neutrino analysis, but can definitely make it more difficult. The number ofspallation products is expected to increase by a factor of 20, at the most, but this may be
reduced with further improvements to the spallation cut. The exact number of PMTs is stillundecided, although current studies being done tend to imply a photo-cathode coverage on
the order of 20% is likely. This would give the solar neutrino analysis a low energy thresholdof ∼7 MeV (similar to SK-II), killing any hope of measuring the MSW resonance curve.
However, if the coverage were to end up similar to the 40% currently in SK, this would makeHK a nice detector for probing the MSW resonance, and to possibly make a measurementof the hep solar neutrino flux.

11.3 Closing Words
The study of solar neutrinos has provided many important insights to unlocking the secretsof the energy production method within the core of the Sun, and in helping to establish andsolve the mystery which is neutrino oscillation. Discovery of the solar neutrino problem
led to an intense study of both the SSM and the SM of particle physics, giving rise to thequestion of whether or not neutrinos can change flavours, as a consequence of having a finite
mass. It was in fact SK which gave the first evidence for neutrino oscillation (coming fromthe atmospheric neutrino analysis), and in combination with SNO, finally concluded thatthe solution to the solar neutrino problem was indeed neutrino flavour conversion. The SNO
NC measurement and the oscillation solution to the solar neutrino problem has also brought
the solar neutrino flux measurements into agreement with SSM predictions. This has proventhe validity of the SSM description of the energy production method within the Sun. The
continued realtime solar neutrino measurements are the only way to probe the core of theSun, and to truly know that it is still producing energy.
With the results of SK and other solar neutrino experiments, the theory of MSW oscil208

lation has become the favorite solution to oscillation analyses. SK-I-IV has finally provided
evidence for the a signature of solar neutrino oscillation (and matter enhanced neutrino oscillation in general), with the measurement of the solar neutrino day/night flux asymmetry.
But this is not the end of the solar neutrino story, as there remains unmeasured solar neutrino species, which have the potential to unlock further secrets of the solar interior, and tofurther our understanding of neutrino oscillation. The coming years will be met with more
exciting solar neutrino measurement results, not only from SK, but from other current andenvisioned experiments as well.

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