U C,I

Solar Neutrino Measurement from the Second Phase of the Super-KamiokandeExperiment

D

submitted in partial satisfaction of the requirementsfor the degree of

D P

in Physics

by

John Parker Cravens

Dissertation Committee:Professor Henry W. Sobel, Chair

Professor Steven BarwickProfessor Arvind Rajaraman

2008

c© 2008 John Parker Cravens

The dissertation of John Parker Cravensis approved and is acceptable in quality and form for

publication on microfilm and in digital formats:

Committee Chair

University of California, Irvine2008

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For Mina.

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A

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1 N 11.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Formalism of Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Neutrinos in the Standard Model . . . . . . . . . . . . . . . . 31.2.2 Neutrino Mass . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.3 Majorana Mass . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.4 See-saw Mechanism . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Neutrino Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3.1 Weak Interaction . . . . . . . . . . . . . . . . . . . . . . . . . 91.3.2 Neutrino Flavor Change . . . . . . . . . . . . . . . . . . . . . 111.3.3 Neutrino Flavor Change in Matter . . . . . . . . . . . . . . . 13

2 S N 172.1 The Standard Solar Model . . . . . . . . . . . . . . . . . . . . . . . . 172.2 The Solar Neutrino Problem . . . . . . . . . . . . . . . . . . . . . . . 22

2.2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.2 The Oscillation Hypothesis . . . . . . . . . . . . . . . . . . . 23

2.3 Solar Neutrino Experiments . . . . . . . . . . . . . . . . . . . . . . . 272.3.1 Homestake . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.3.2 Kamiokande . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.3.3 GALLEX/GNO & SAGE . . . . . . . . . . . . . . . . . . . . . 282.3.4 Super-Kamiokande . . . . . . . . . . . . . . . . . . . . . . . . 292.3.5 Sudbury Neutrino Observatory . . . . . . . . . . . . . . . . . 292.3.6 Summary of Current Solar Neutrino Data . . . . . . . . . . . 30

3 T S-K-II D 343.1 Experiment Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 343.2 Detection Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2.1 Neutrino Interaction . . . . . . . . . . . . . . . . . . . . . . . 373.2.2 Cherenkov Radiation . . . . . . . . . . . . . . . . . . . . . . . 37

3.3 Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.3.1 Tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

iv

3.3.2 Inner Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.3.3 Outer Detector . . . . . . . . . . . . . . . . . . . . . . . . . . 393.3.4 Photomultiplier Tubes . . . . . . . . . . . . . . . . . . . . . . 41

3.4 Purification Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.4.1 Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.4.2 Air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.5 Data Acquisition System . . . . . . . . . . . . . . . . . . . . . . . . . 473.5.1 ID DAQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.5.2 OD DAQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.5.3 Trigger System . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.6 Water Transparency Measurement . . . . . . . . . . . . . . . . . . . 513.6.1 Laser Light Injection . . . . . . . . . . . . . . . . . . . . . . . 513.6.2 Time Variation . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.7 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.7.1 PMT Calibration . . . . . . . . . . . . . . . . . . . . . . . . . 553.7.2 Energy Calibration . . . . . . . . . . . . . . . . . . . . . . . . 61

4 SK-II S E 684.1 Event Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.1.1 Vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.1.2 Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.1.3 Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.2 Data Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.2.1 Noise Reduction . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.3 Spallation Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.3.1 Muon Track Reconstruction . . . . . . . . . . . . . . . . . . . 834.3.2 Selection of Spallation Events . . . . . . . . . . . . . . . . . . 84

4.4 Timing & Hit Pattern Cut . . . . . . . . . . . . . . . . . . . . . . . . 864.5 Gamma-ray Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.6 Final Data Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.7 Monte Carlo Event Simulation . . . . . . . . . . . . . . . . . . . . . 89

4.7.1 Neutrino Rate and Interaction . . . . . . . . . . . . . . . . . 914.7.2 Detector Response . . . . . . . . . . . . . . . . . . . . . . . . 934.7.3 Energy Resolution Function . . . . . . . . . . . . . . . . . . . 95

4.8 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.8.1 Signal Extraction . . . . . . . . . . . . . . . . . . . . . . . . . 984.8.2 Day-Night and Seasonal Variation . . . . . . . . . . . . . . . 1014.8.3 Energy Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.9 Summary of Systematic Errors . . . . . . . . . . . . . . . . . . . . . 106

5 O A 1095.1 Analysis of the SK-II χ2 . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.1.1 Oscillated Rate Predictions . . . . . . . . . . . . . . . . . . . 1115.1.2 SK-II Systematic Error Treatment . . . . . . . . . . . . . . . . 1145.1.3 Time-Variation Analysis . . . . . . . . . . . . . . . . . . . . . 118

v

5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1195.2.1 SK-II Spectrum & Time-Variation Fits . . . . . . . . . . . . . 1205.2.2 SK-II Rate Constrained Fit . . . . . . . . . . . . . . . . . . . . 122

5.3 Combining with Other Solar Experiments . . . . . . . . . . . . . . . 1235.3.1 Sudbury Neutrino Observatory . . . . . . . . . . . . . . . . . 1235.3.2 Radiochemical Experiments & KamLAND . . . . . . . . . . 126

5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

B 130

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2.1 The CNO cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 The pp chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3 The solar electron density profile . . . . . . . . . . . . . . . . . . . . 242.4 The MSW effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.5 Oscillation parameter solutions allowed by solar neutrino data . . . 262.6 BP2004 SSM solar fluxes . . . . . . . . . . . . . . . . . . . . . . . . . 312.7 BP2004 SSM compared with experimental data . . . . . . . . . . . . 33

3.1 SK in Japan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2 The SK detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.3 Photograph of the ID . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.4 Diagram of an ID PMT. . . . . . . . . . . . . . . . . . . . . . . . . . . 423.5 The PMT quantum efficiency and Cherenkov spectrum . . . . . . . 433.6 Water purification system . . . . . . . . . . . . . . . . . . . . . . . . 453.7 Radon levels at SK . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.8 Air purification system . . . . . . . . . . . . . . . . . . . . . . . . . . 463.9 ATM schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.10 ID DAQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.11 SK-II laser system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.12 Laser light timing distributions . . . . . . . . . . . . . . . . . . . . . 543.13 SK-II light scattering coefficients . . . . . . . . . . . . . . . . . . . . 553.14 PMT acceptance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.15 Water transparency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.16 Xenon lamp calibration system . . . . . . . . . . . . . . . . . . . . . 583.17 Relative gain by xenon lamp calibration . . . . . . . . . . . . . . . . 593.18 The Ni-Cf source apparatus . . . . . . . . . . . . . . . . . . . . . . . 603.19 The PMT charge distribution . . . . . . . . . . . . . . . . . . . . . . 603.20 The N2 laser calibration system . . . . . . . . . . . . . . . . . . . . . 613.21 Timing-charge map . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.22 The LINAC calibration system . . . . . . . . . . . . . . . . . . . . . 633.23 LINAC energy scale and resolution differences in LINAC data and

MC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.24 DTG setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.25 DTG energy scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.1 Vertex resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.2 Directional likelihood function . . . . . . . . . . . . . . . . . . . . . 724.3 Directional resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.4 SK-II energy scale as a function of time . . . . . . . . . . . . . . . . . 764.5 Time difference cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.6 OD trigger cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.7 Ambient electronic noise cut . . . . . . . . . . . . . . . . . . . . . . . 79

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4.8 Flasher cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.9 Fiducial volume cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.10 Muon track reconstruction . . . . . . . . . . . . . . . . . . . . . . . . 854.11 PMT timing and hit pattern cut . . . . . . . . . . . . . . . . . . . . . 874.12 Gamma-ray cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.13 Data reduction summary . . . . . . . . . . . . . . . . . . . . . . . . . 904.14 Data reduction efficiency summary . . . . . . . . . . . . . . . . . . . 904.15 Neutrino distributions, ν − e− elastic scattering cross sections, and

expected recoil electron spectra . . . . . . . . . . . . . . . . . . . . . 924.16 The SK-II light absorption coefficient . . . . . . . . . . . . . . . . . . 944.17 Energy resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.18 Predicted recoil electron spectrum . . . . . . . . . . . . . . . . . . . 974.19 Solar angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.20 The expected signal shape . . . . . . . . . . . . . . . . . . . . . . . . 1004.21 The extracted signal in the solar direction . . . . . . . . . . . . . . . 1014.22 The extracted signal by energy bin . . . . . . . . . . . . . . . . . . . 1024.23 Flux time variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.24 Flux time variation compared with solar activity . . . . . . . . . . . 1064.25 SK-II energy spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.1 Live time and solar zenith rates . . . . . . . . . . . . . . . . . . . . . 1135.2 Electron densities of the Sun and Earth . . . . . . . . . . . . . . . . . 1135.3 Energy correlated systematic errors on predicted spectrum . . . . . 1155.4 Spectrum χ2 map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.5 Spectrum and time variation χ2 map . . . . . . . . . . . . . . . . . . 1225.6 8B flux constrained χ2 map . . . . . . . . . . . . . . . . . . . . . . . . 1245.7 SK-SNO combined χ2 map . . . . . . . . . . . . . . . . . . . . . . . . 1275.8 Global χ2 map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

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2.1 BP2004 SSM fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2 Summary of experimental flux values . . . . . . . . . . . . . . . . . 32

4.1 SK-II observed and expected rates . . . . . . . . . . . . . . . . . . . 1054.2 SK-II systematic error . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.3 Energy correlated systematic errors for each energy bin . . . . . . . 108

ix

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I wish to thank Dr. Michael Smy for his tremendous help on the analysis and thedrafting of this thesis. He was a great model to follow and I continually admiredhis abilities as an experimentalist.

I wish to thank Prof. Hank Sobel for his support. I had many chances to at-tend and speak at conferences as well as traveling to Irvine to work with Dr. Smyon the analysis. He never restricted my path to this degree.

I wish to thank Prof. Masayuki Nakahata for letting me work with the solarneutrino group in Japan and also letting me participate as a presenter in local con-ferences. My time in Japan was unique and I hope to spend more time there as ascientist.

These three men are only the principle people during my graduate career. Thereare so many more that I want to thank but are too numerous to list here. To allthose I had the honor of knowing and working with, I will always remember youfondly.

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J P C

1998 B.S. in Physics, Southern Methodist University, Dallas

2003 M.S. in Physics, University of California, Irvine

2008 Ph.D. in Physics, University of California, Irvine

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A D

Solar Neutrino Measurement from the Second Phase of the Super-KamiokandeExperiment

By

John Parker Cravens

Doctor of Philosophy in Physics

University of California, Irvine, 2008

Professor Henry W. Sobel, Chair

The second phase of the Super-Kamiokande experiment aimed at the continu-

ation of the solar neutrino measurement after the 1496-day first phase. However,

the second phase operated with a photocathode coverage 47% of the first phase’s.

This reduction in sensitivity prompted the development of new analysis tools and

created larger estimations of systematic errors. Despite these changes, the second

phase solar neutrino data showed consistency with the first phase and no indica-

tion of systematic tendencies between the two phases was present. An oscillation

analysis of the second phase resulted in reduced exclusion power of the neutrino

mixing angle and mass difference parameter set. However, a rate constrained com-

bined oscillation analysis of both phases continues to favor the Large Mixing Angle

solution at 95% confidence level.

xii

C 1

N

The study of neutrinos is currently an active topic. Both theoretical and experi-

mental research is being conducted to unlock the secrets of neutrinos and neutrino

mass. The phenomenon of neutrino oscillation is currently the only way experi-

mentalists can conclude that neutrinos have mass but are still unable to reveal their

definite quantities. Theoreticians, however, have laid the groundwork that could

possible describe why neutrinos have the masses they do and the consequence that

is oscillation.

1.1 H

The neutrino was the first particle to be proposed that was not constituent of atomic

matter. This proposal came in the last month of 1930 from Wolfgang Pauli in at-

tempt to reconcile the canon of conservation of energy with the results of James

Chadwick’s 1914 214Pb and 214Bi decay experiments. In these experiments, Chad-

wick demonstrated the decaying β particles were continuous in energy, not discreet

as expected in the burgeoning quantum theory at the time. The introduction of a

weakly interacting neutral fermion seemingly solved the discrepancies although

presented a greater challange: how to detect it.

The newly proposed neutral fermion was dubbed ”neutrino” by Enrico Fermi

and almost all serious scientific efforts aimed at the neutrino were through theoret-

ical research, mainly by Fermi and his theory of β-decay. It wasn’t until 1956 that

Frederick Reines and Clyde Cowan announced their detection of reactor antineu-

trinos from the Savannah River reactor in South Carolina. Only one neutrino type

1

(or flavor) was believed to exist.

Earlier in 1947, with the observation of the decay π+ → µ+, energy conservation

was again seemingly violated but this could now be easily remedied by the as-

sumption that an invisible neutrino carried off the remaining energy. It would not

be shown until 1962 (after the detection by Reines and Cowan) in an experiment

by Leon Lederman, Melvin Schwartz and Jack Steinberger, working at the then-

powerful AGS, that the neutrinos in pion decay were in fact distinguishable from

those in β-decay. The new neutrino was named the muon neutrino and the origi-

nal one the electron neutrino due to the interactions with their respective charged

leptons.

This brought rise to the idea of lepton family in which each charged lepton has

a chargeless neutrino as its partner. Also with this grouping came conservation of

lepton family number. Lepton number conservation (where leptons would have

L = 1 and antileptons L = −1 and their total value was conserved during a reaction)

was already proposed in 1953 to account for missing decay modes. But other

modes, such as µ → e− + γ, while allowed by lepton number conservation, had

experimental cross section limits that were exceedingly small. With two families, it

was possible to assign Le and Lµ for each the electron and muon flavors respectively.

The lepton family numbers were postulated to be conserved quantities in addition

to global lepton number. This eliminated the possibilities of numerous processes.

A third family would soon appear with the discovery of the tau lepton in 1975

by Martin Perl. It was again assumed that energy conservation in tau decays was

mediated by a neutrino. On July 21st, 2000, the neutrino beam experiment DONUT

at Fermilab announced that the tau neutrino had successfully been detected.

Thirty-two years earlier, neutrinos from the Sun were detected by Ray Davis and

his Homestake experiment making it the first detection of non-terrestial neutrinos.

In the abandoned Homestake gold mine in South Dakota, USA, electron neutrinos

2

were detected through the reaction 37Cl + νe →37Ar + e−. This was ironic since

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

νe →37Ar + e− [1], a process which violates global lepton number. Homestake

operated continuously from 1967 until 1994.

What Homestake’s results concluded were that the observed solar neutrino

flux was three times lower than what is predicted by theoretical solar models. A

possible explanation was the solar models were flawed. However, after constant

revision of solar theory and with the measurements of proceeding solar neutrino

experiments, this discrepancy only got worse [2]. Another possibility was that the

electron neutrinos from the Sun may transition to another flavor where the detectors

are not sensitive or have limited sensitivity. This is possible if neutrinos were

ascribed some mass. The theory of massive neutrinos ”oscillating” was already

developed by Z. Maki, M. Nakagawa, and S. Sakata in 1962 and independently by

B. Pontecorvo in 1967. But previous neutrino measurements gave no indication of

mass, nor is it required by the Standard Model. What Homestake and eventually

other solar, atmospheric, and accelerator neutrino experiments did was present the

possibility of neutrino mass.

1.2 F N

1.2.1 N SM

In the fermion group of the Standard Model (SM) of particles and interactions,

fundamental particles are divided into three generations labeled simply by order.

Each generation contains 2 quarks and 2 leptons. The particles in the lepton sector

of the 3 generations can be grouped into electron, muon, and tau flavors. They

consist of the electron and electron neutrino in the first generation (electron flavor),

muon and muon neutrino in the second generation (muon flavor), and tau and tau

3

neutrino in the third generation (tau flavor).

With the absence of charge, neutrinos interact with themselves and other matter

only through the weak and gravitational interactions. There are two classes of

weak interactions characterized by their exchange bosons. The charged current

(CC) interaction is mediated by the massive W± boson and the neutral current

(NC) interaction is mediated by the massive Z boson.

Neutrinos are assumed to be massless in the SM. Along with their ~/2 spin

angular momentum, this constrains neutrinos to be in a state of definite helicity.

Furthermore, in agreement with experimental results and the accepted form of the

weak interaction theory, neutrinos have negative helicity while their antineutrino

partners have positive helicity. No positive (negative) neutrino (antineutrino) states

exist in the SM.

1.2.2 NM

Although the SM does not account for massive neutrinos, mass can be introduced

into its current framework. This requires the inclusion of a chirally right-handed

neutrino field which could give rise to a positive helicity neutrino state depending

on an observer’s inertial reference frame. The procedure is analogous to the way

the charged leptons acquire their masses: through coupling with the Higgs field.

The relevent term in a massive neutrino-Higgs Lagrangian density is

Lmass = −yφ0ν′Rν′

L + h.c., (1.1)

where y is the Yukawa coupling constant, φ0 is the vacuum expectation value, and

h.c. denotes the hermitian conjugate. The neutrino mass is recognized as mν = yφ0.

The vacuum expectation value φ0 is 174 GeV and, as the same for the charged

leptons, y is determined experimentally.

4

Unfortunately, there is no direct experimental measurement of definite neutrino

mass. There only exists mass squared differences (∆m2) between neutrinos of

different generations. But these limited results do suggest a definite mass around

the value of 0.05 eV. This makes the order of magnitude of y 10−13.

1.2.3 MM

The Yukawa coupling constant y is inexplicably small, especially given the sizes of

the charged lepton couplings. In an effort to give better reason for small neutrino

masses, a second mass term can be introduced into the mass Lagrangian. The

nature of this term assumes that νL and νR are not independent fields but are

somehow related. This comes in the form of the Majorana Condition and is given

explicitly as

ν′R(L) = Cν′TL(R), (1.2)

where C is the charge-conjugate matrix and T represents transpose. Equation 1.2

can be made slightly clearer by recognizing the right side is the definition of the

charge conjugate field, νc. This is the first major departure from the SM. For a

neutrino field with both chirally left- and right-handed fields ν = νL + νR, charge

conjugation returns the same field (νc = ν). This is identical to saying a particle

is its own antiparticle. If neutrinos have this property, they are called Majorana

neutrinos. When a neutrino and its antineutrino have distinction, they are known

as Dirac neutrinos.

Using the Majorana Condition, the mass term of 1.1 can be rewritten as

Lmass = −12

mR(ν′R)cν′R + h.c., (1.3)

where the factor 1/2 is introduced to avoid double contributions of the same field.

The mass mR is known as a right-handed Majorana mass. (The same can be

5

accomplished with νL and mL but is omitted for brevity in the following discussion.)

The mass of Equation 1.1 is known as a Dirac mass (mD = yφ0).

1.2.4 S-M

For simplicity, only a single flavor will be considered and will regard the two other

flavors as nonexistent. The new Majorana mass term can be added to the Dirac

mass Lagrangian to obtain

Lmass = −mDν′Rν′

L +mR

2(ν′R)cν′R + h.c. (1.4)

Using the Majorana Condition for the fields associated with the Dirac mass mD, the

Lagrangian can be written in matrix form:

Lmass = −12

((ν′L)c ν′R

) 0 mD

mD mR

ν′L

(ν′R)c

+ h.c. (1.5)

From Equation 1.2, it can be seen that the left and right field vectors in the above

expression each have two identical components making the leading factor of 1/2

necessary. For the mass matrix M (the matrix in the above equation with elements

0, mD, and mR), it is no longer diagonal due to the presence of the Dirac mass mD.

To find the mass of the neutrino, M must be diagonalized and can be done so by

the common method of eigen decomposition. The resulting diagonal elements are

m1,2 =12

(mR ±

√(mR)2 + 4m2

D

). (1.6)

6

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

eigenvalues

m1 =m2

D

mR, (1.7)

m2 = mR. (1.8)

By Equation 1.6, m2 is negative, but since the eigenvalues represent real, physical

masses, m2 is made positive.

To avoid forcing m2 to be positive, eigenvalues greater than zero can be obtained

by a suitable choice of rotation matrix V to be operated on the mass matrix M [3]:

VTMV =

1 −iρ

ρ 1

0 mD

mD mR

1 ρ

−iρ 1

= m2

D/mR 0

0 mR

, (1.9)

where ρ = mD/mRM and again assuming mR � mD.

With the matrix notation of Equation 1.9, it is possible to write the mass La-

grangian in term of mass eigenvalues and their corresponding eigenstates:

Lmass = −12

((ν′L)c ν′R

)VV−1

0 mD

mD mR

VV−1

ν′L

(ν′R)c

+ h.c.|V

= −12

(ν1 ν2

) m2

DmR

0

0 mR

ν1

ν2

,where h.c.|V signifies that the V rotation is applied to the hermitian conjugate. The

components ν1 and ν2 are recognized as neutrino mass eigenstates with masses

m2D/mR and mR, respectively. They are defined in terms of ν′L and ν′R as

ν1

ν2

= V−1

ν′L

(ν′R)c

+V−1

(ν′L)c

ν′R

. (1.10)

7

What has happened is by assuming one neutrino field ν′ = ν′L + ν′

R and two

mass terms in the Lagrangian mD and mR, two mass eigenstates appear. With the

condition that mR be considerably larger than mD, the eigenstate masses are likewise

very small and very large. Furthermore, it is exactly due to mR’s relatively large

value that m1 is small. This is known as the see-saw mechanism and is an attractive

explanation of the observed smallness of neutrino mass. A small Yukawa coupling

is not necessary and mD can be assumed to be of the same scale as the charged

leptons and quarks. There exists no constraints on the size of mR and it can be

chosen accordingly. As an example [4], suppose mD is 175 GeV, the same mass as

the top quark. If mR is given an order of magnitude of around 1015, m1 becomes

∼ 0.03 eV, a very likely mass for a light neutrino.

Several questions arise from this formulation. One is the prediction of a heavy

neutrino which is as-of-yet unobserved. By dividing ν1 and ν2 into left and right-

handed components, Equation 1.10 can be rearranged to

ν′L = V11ν1,L + V12ν2,L, (1.11)

ν′R = V21ν1,R + V22ν2,R. (1.12)

The left-handed component which participates in weak interactions is mostly ν1,L

with mass m1 since V12 = mD/mR is a small parameter. On the other hand, ν′R is

mostly ν2,R with mass mR. But ν′R does not take place in weak interactions and is

regarded as sterile. Therefore, if a heavy neutrino does exist, it might be impossible

to detect.

A second concern is that of a Majorana-type neutrino. By charge-conjugating

Equation 1.10, it can be seen that ν1 and ν2 are indeed equal to their antiparticles.

But it has repeatedly been observed that weak interactions favor different outcomes

depending on whether a neutrino or antineutrino contributed to the process. This

8

can be rectified by realizing that a massive neutrino can be in two different helicity

states and that the differences in processes involving neutrinos or antineutrinos

depend only upon the Majorana neutrino’s helicity.

Thirdly, since a Majorana neutrino is its own antiparticle, lepton number con-

servation is clearly violated. There may not be a solution for this but given the

new physics possible with massive neutrinos, lepton number violation could just

as well be tolerated, explained, or discarded in a broader theory. The exceptions to

the rules of the SM that massive neutrinos bring can be far-reaching and worthy

of new experiments to test such theories as Majorana neutrinos and the possible

existence of a heavy sterile neutrino. But in this section, only one neutrino field was

considered. When extending the argument for three neutrinos flavors of the three

fermion generations, striking new behavior not present in the SM is uncovered.

1.3 N O

1.3.1 W I

The one flavor model in the previous section was seen to be a combination of two

massive neutrinos, an active one and a sterile one. Given the small contribution

of the sterile neutrino in the weakly interacting field component ν′L, ν2 will be

discarded. Equation 1.11 now reads ν′L = V11ν1,L = ν1,L and is purely light in mass.

It also states the mass eigenstate ν1 is also an eigenstate of the charged current weak

interaction

LW = −g√

2W−

ρ¯′Lγρν′L + h.c., (1.13)

where W−

ρ is the massive exchange boson and `′L is the left-handed charged lepton

partner of ν′L. This can be extended to the three known flavors by rewriting the

9

flelds `′L and ν′L as vectors composed of their mass eigenstates:

`′L = AL

`L,e

`L,µ

`L,τ

, ν′L = Z

νL,1

νL,2

νL,3

. (1.14)

The left side represents the left-handed field in terms of the charged leptonic mass

eigenstates `e, `µ, and `τ. The 3 × 3 hermitian matrix AL [5] diagonalizes the mass

matrix for the charged leptons in much the same way as V did for the one flavor

neutrino case in Equation 1.9. The resulting masses are me, mµ, and mτ.

The right side of Equation 1.14 is the left-handed neutrino field in terms of three

light neutrino mass eigenstates with masses m1, m2, and m3. The 3 × 3 matrix Z is

analogous to the matrix element V11 in Equation 1.11. By writing ν′L as the combina-

tion of three light neutrino mass eigenstates, it is implied that the diagonalization

of the mass matrix resulting from the see-saw mechanism for three active neutrinos

produces three additional heavy neutrinos ν4, ν5, and ν6. They are discarded for

the same argument given at the beginning of this section.

The charged current interaction Lagrangian can now be written for three flavors

as

LW = −g√

2W−

ρ

¯L,e

¯L,µ

¯L,τ

T

γρA†LZ

νL,1

νL,2

νL,3

+ h.c.,

or more compactly as

LW = −g√

2W−

ρ

∑α=e,µ,τ

3∑i=1

¯`L,αγρ(A†LV

)αiνL,i + h.c. (1.15)

Here, the charged lepton `L of flavor α interacts via the charged boson W− with

not one neutrino of mass-type i but a mixture of all three neutrinos. This mixture

10

is defined by the leptonic mixing matrix U = A†Z. The mixed neutrinos can then

be labeled as new states to correspond with the charged lepton they interact with:

να =∑3

i=1 UαiνL,i where α = e, µ, τ for electron, muon, and tau respectively.

1.3.2 N F C

With three mass-type neutrinos now written in the flavor basis να, it becomes clear

that the creation and annihilation of a neutrino by weak interactions does not

take place when the neutrino is in a definite mass state. As a consequence, since

a neutrino of flavor α is a superposition of mass eigenstates, the time evolution

of a flavor state can induce a transition to another flavor state. This can best be

demonstrated by starting with the expressions of flavor states and mass states in

braket notation:

|να〉 =3∑

i=1

Uαi|νi〉, |νi〉 =

3∑β=1

U∗βi|νβ〉, (1.16)

where the handedness subscript L has been dropped for brevity.

A neutrino is produced weakly and is initially (t = 0) in flavor state α. By

applying the time evolution operator, it is possible to render the time-dependent

expression

|να(t)〉 =3∑

i=1

Uαie−iEit|νi〉, (1.17)

from which the probability that the neutrino at some later time t will be in another

flavor state β:

Pνα→νβ = |〈νβ|να(t)〉|2 =

∣∣∣∣∣ 3∑i=1

U∗βie−iEitUαi

∣∣∣∣∣2. (1.18)

For relativistic neutrinos, the energy component Ei can be approximated by

Ei =√|p|2 +m2 ' |p| +

m2i

2|p|' E +

m2i

2E. (1.19)

11

Along with t = v/L ' c/L where L is the distance from the neutrino source to the

point of observation and c is the speed of light (c = 1 in current units), this makes

the transition probability

Pνα→νβ = |〈νβ|να(t)〉|2 =

∣∣∣∣∣ 3∑i=1

U∗βie−iEte−i

m2i

2E LUαi

∣∣∣∣∣2. (1.20)

Before expanding the absolute square, it will benefit the discussion by simpli-

fying to the case of two mass-type neutrinos (i = 1, 2). The leptonic mixing matrix

can then be written as

U =

cosθ sinθ

− sinθ cosθ

. (1.21)

Equation 1.20 becomes

Pνα→νβ = sin2 2θ sin2(∆m2 L

4E

)(1.22)

for two neutrinos. The survival probability (the chance a neutrino of flavor α will

remain flavor α after some time t) is likewise given by

Pνα→να = 1 − Pνα→νβ = 1 − sin2 2θ sin2(∆m2 L

4E

), (1.23)

where ∆m2 = m21 −m2

2. The sin2 2θ term describes the amount of ”mixing” between

the two mass eigenstates and θ is hence known as the mixing angle. The mass

squared difference ∆m2 in the second sine term, along with E, determines the

frequency of flavor change. It can therefore be seen why flavor change is also

referred to as neutrino oscillations.

It is interesting to note that neutrino oscillation is purely a consequence of

massive neutrinos. By setting m1,2 = 0, the chances a neutrino will not oscillate are

100%, making any in-flight transition impossible.

12

The mixing angle and mass-squared difference are quantities that are currently

experimentally determined. By measuring only neutrino rates in disappearance

experiments (where a lower-than-expected rate is obtained for a given neutrino

flavor) or appearance experiments (where a non-negligible rate is obtained for a

flavor not expected when considering no oscillations), individual masses cannot

be determined. In addition, an experiment’s L/E (L-over-E) value specifies to what

sensitivity ∆m2 can be measured [6]. This sensitivity is the value ∆m2 for which

∆m2L2E

∼ 1. (1.24)

Also dependent on L/E and ∆m2 is the ability to detect neutrino oscillations at

all. Given a length L0 = 4πE/∆m2, the survival probability becomes

Pνα→να = 1 − sin2 2θ sin2(π

LL0

). (1.25)

If L � L0, the period of oscillation becomes large, driving sin2(πL/L0) to zero and

thereby making detection extremely difficult. On the other hand, if L � L0, the

frequency becomes high and only the mixing angle values can be probed. Since

∆m2 is determined by nature, care must be taken when choosing detector location

for a given source energy in baseline oscillation experiments. For natural sources

such as cosmic ray-induced atmospheric neutrinos and solar neutrinos, a little bit

of luck is also helpful.

1.3.3 N F C M

The formulism above only concerns neutrinos traveling in vacuum. When matter is

traversed, neutrinos interact with the ambient particles via charged W and neutral

Z boson exchange. This can lead to forward elastic scattering where an interaction

13

potential Vint gives an added contribution. Ordinary matter (matter composed of

electrons, neutrons, and protons) will be considered making the W boson-mediated

processes only affect the electron neutrino. The Z boson-mediated processes affect

all flavors equally. To write an explicit form of Vint, the leptonic mixing matrix will

be assumed to describe mixing between mass eigenstates |ν1〉 and |ν2〉 and flavor

states |νe〉 and |νµ,τ〉. The actual form of U in Equation 1.21 does not change.

Before constructing Vint, the Hamiltonian in the flavor basis will be obtained. The

energies Ei of the previous section are energy eigenvalues of the time-dependent

neutrino mass eigenstates |νi(t)〉. Therefore, it is possible to write the Schrodinger

equation

iddt|νi(t)〉 = H|νi(t)〉 = Ei|νi(t)〉, (1.26)

where H is diagonal with elements Ei. To express this in the flavor basis, one needs

only to apply the leptonic mixing matrix to the above expression to yield

iddt

U|νi(t)〉 = UHU†U|νi(t)〉 = H(α)|να(t)〉, (1.27)

where H is now rotated to

H(α) = EI +∑

m4E

I +∆m2

4E

− cos 2θ sin 2θ

sin 2θ cos 2θ

. (1.28)

Now it is possible to add the matter dependent potential Vint to H(α). As stated

earlier, the electron neutrino is exclusive in its interactions with ambient electrons

via W exchange. And, obviously, the traversed matter’s electron density is a factor

in the potential. Therefore, Vint can be written as

Vint =√

2GFNnI +√

2GFNpI +√

2GFNeI +√

2GFNe

1 0

0 0

, (1.29)

14

where GF is the Fermi constant and Nn, Np, and Ne are the neutron, proton, and

electron number densities respectively. Dropping identity matrix I terms in H(α)

and Vint (they do not contribute effects to neutrino flavor change), the Hamiltonian

in matter becomes

H′ = H(α) +Vint =∆m2

4E

− cos 2θ sin 2θ

sin 2θ cos 2θ

+ √2GFNe

1 0

0 0

. (1.30)

This Hamiltonian can then be diagonalized back into the mass basis but with

eigenvalues

EM,1,2 =∆m2A

2E±∆m2

4E

√(cos 2θ − A)2 + sin2 2θ. (1.31)

An effective mass squared difference in matter can be identified as

∆m2M = ∆m2

√(cos 2θ − A)2 + sin2 2θ. (1.32)

The term A = 2√

2EGFNe/∆m2 is a unit-less parameter that is regarded as a gauge

of the influence matter has on oscillation. By taking the limit A → 0, vacuum

oscillation is restored. An effective mixing angle in matter is likewise written as

sin2 2θM =sin2 2θ

sin2 2θ + (cos 2θ − A)2. (1.33)

The effective mixing angle and mass squared difference can replace their vacuum

counterparts sin2 2θ and ∆m2 in Equation 1.23 to describe the survival probability

of νe propagating through matter of constant electron density Ne. Matter effects

play an important role in the way neutrinos behave in the dense environments of

the solar and terrestrial interiors.

The possibility exists when A = cos 2θwhich can arise when the electron number

density reaches a critical value Ne,crit. In this case, the mixing angle in vacuum

15

could be small but propagation through matter promotes the mass eigenstates to

maximal mixing (θM = π/4). This is known as the MSW effect and was first realized

by Mikheev, Smirnov, and Wolfenstein [7]. It will soon be explored in the context

of propagation in the Sun.

16

C 2

S N

The detection of solar neutrinos first occurred in 1967 with the Homestake exper-

iment of the late Raymond Davis. Since then, solar neutrinos have been detected

almost continuously by various other experiments. Neutrinos are the only particles

which allow researchers to peer inside the Sun to decipher its mechanisms. This

has been the historical basis of early solar neutrino detection. Nowadays, solar

neutrinos are being studied themselves to determine their properties and, on a

larger scale, the possibility of new fundamental physics. This is precisely the aim

of the latest research in solar neutrinos including Super-Kamiokande. Included

in this chapter is a general description of the production of neutrinos inside the

Sun as theorized by the Standard Solar Model, a brief history of the solar neutrino

problem, how neutrino oscillation from the previous chapter can be applied to

solar neutrinos, and a synopsis of the solar neutrino experiments included in this

analysis.

2.1 T S SM

The late John Bahcall, the eminent solar neutrino physicist and author of several

well-regarded solar models, gives his definition of the Standard Solar Model (SSM):

By the Standard solar model, we mean the solar model which is con-

structed with the best-available physics and input data. All of the solar

models we consider ... are required to fit the observed luminosity and

radius of the sun at the present epoch, as well as the observed heavy

element to hydrogen ratio at the surface of the sun [8].

17

One of the principle goals of the SSM is to accurately model the energy production

that occurs in the core of the Sun. Since this energy is released in the form of photons

and neutrinos, understanding the production mechanisms can yield predictions of

the solar neutrino flux. Comparisons can then be made with experimental data to

test the accuracy of the model or, conversely, the completeness of neutrino theory.

The SSM considered in this work is BP2004 [9]. To begin constructing a solar

model, three assumptions are made: 1.) the Sun evolves in hydrostatic equilibrium,

2.) solar energy is transferred via radiation, conduction, convection, or neutrino

loss, and 3.) thermonuclear reactions are the only form of energy production inside

the Sun [10] [11]. The last assumption is the one of interest due to its correlation to

neutrino production.

The net thermonuclear reaction that takes place in the Sun’s core is the fusion of

four protons and two electrons which forms a 4He nucleus along with two electron

neutrinos and Q = 26.73 MeV of energy:

4p + 2e− → 4He + 2νe +Q.

This happens via two groups of reactions: the proton-proton (pp) chain and the

Carbon-Nitrogen-Oxygen (CNO) cycle.

The CNO cycle is the lesser contributor of the two, producing only 1.6% of the

Sun’s total output energy. This is due to the Sun’s core temperature being lower

than is required for the CNO chain to acquire dominance (this is around 1.8×107 K

whereas the core temperature is 1.5×107 K) [12]. By way of carbon, nitrogen and

18

oxygen catalysts, there are three reactions that produce neutrinos:

13N→13 C + e+ + νe

15O→15 N + e+ + νe

17F→17 O + e+ + νe.

The upper boundaries of the emitted neutrino energies are 1.20 MeV, 1.73 MeV, and

1.74 MeV respectively. With these low energies coupled with the low percentage of

total solar luminosity, the neutrino flux of the CNO cycle does not have a significant

impact on the study of solar neutrinos. Figure 2.1 shows the CNO cycle.

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

The pp chain, making up for the remaining 98.4% of solar energy, can further be

broken down into four branches: pp-I, pp-II, pp-III, and hep. Of the processes leading

19

up to the formation of 4He in these branches, five produce neutrinos. In contrast

to the CNO cycle neutrinos, those from the pp chain are the major contributors to

solar neutrino detection on Earth. The pp chain’s neutrino producing reactions are

named pp, 7Be, pep, 8B, and hep and are listed below.

p + p →2H + e+ + νe,

7Be + e− → 7Li + νe,

p + e− + p →2H + νe,

8B →8Be + e+ + νe,

3He + p →4He + e+ + νe.

with neutrino energies <0.42, 0.86, 1.44, ≤14.06, and ≤18.77 MeV respectively.

Figure 2.2 shows the pp chain.

In total, eight reactions (including the CNO cycle) give rise to the predicted

neutrino fluxes of the BP2004 SSM as summarized in Table 2.1. The large errors in

the CNO and 8B fluxes come from the uncertainties in chemical composition and is

20% for 8B and around 35% for CNO. Another large uncertainty is in the 3He→4He

reaction rate and is 7.5% for 8B and 8.0% for 7Be [9].

Table 2.1: Table of the BP2004 SSM predicted flux values for each solar neutrinosource. All values are in units of cm−2 s−1.

source BP2004pp 5.94(1 ± 0.01) × 1010

pep 1.40(1 ± 0.02) × 108

hep 5.94(1 ± 0.16) × 103

7Be 5.94(1 ± 0.12) × 109

8B 5.94(1 ± 0.23) × 106

13N 5.94(1+0.37−0.35) × 108

15O 5.94(1+0.43−0.39) × 108

17F 5.94(1 ± 0.44) × 106

20

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

21

2.2 T S N P

2.2.1 B

A problem arose in 1968 when the Homestake experiment detected a neutrino flux

of about one third of that of the SSM. Of course, investigations began into the

detection efficiencies and SSM assumptions (or lack of) but ultimately yielded no

adequate explanation for the deficit. The newer Kamiokande experiment began

collecting data and in 1989 reported a flux slightly over half of the expected SSM

value. In 1992, the GALLEX collaboration said they also observed a flux 65% of the

SSM (closely related GNO and SAGE also reported similar deficits). After much

struggle to bring the SSM into agreement with experiment, an inconsistency uncov-

ered in the temperature dependence of the solar neutrino flux and the current three

observed values pointed to the very likely possibility that fundamental particle

physics needed to be revised and not the SSM.

The Homestake experiment detected neutrinos from a 0.814 MeV threshold. This

includes 7Be, pep and large portions of the 8B and hep fluxes. Kamiokande detected

only the upper tails of 8B and hep from a 7.5 MeV threshold. Given that the 8B

neutrino flux dependence is heavily dependent on the Sun’s core temperature (as

T25core [13]), and less dependent for 7Be (T11

core), a slight temperature adjustment in

the SSM could result in agreement with Kamiokande’s 8B measurement. However,

the 7Be flux becomes not as strongly suppressed as Homestake would suggest with

their greater flux deficit. On the other hand, the pep and pp flux are inversely

proportional to core temperature (∼ T−1core) but pp-sensitive GALLEX indicates flux

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

property.

22

2.2.2 T O H

The solution to the solar neutrino problem came after considering the possibility

of neutrino flavor change. As stated earlier, the theory of neutrinos oscillating

between flavor states was proposed by Maki, Nakagawa, and Sakata in 1952 and

later Pontecorvo in 1967. If the solar neutrino data fit the oscillation hypothesis, it

would mean solar neutrinos were endowed with finite mass.

A simple example of solar neutrino oscillations is the transition of νe produced

in the core of the Sun to νµ or ντ during its flight to the Earth. Since this takes

place in the vacuum of space, the survival probability of Equation 1.23 with L =

149.6 × 106 km (1 au, the Earth-Sun distance) can be used. Additionally, if L is

substituted with

L = (1 au) ×[1 − ε cos

(2π

tT

)], (2.1)

where ε = 0.0167 is the eccentricity of the Earth’s orbit and T = 1 year, a seasonal

pattern could be observed in the rates throughout the year. This possible scenario

is known as the vacuum (VAC) solution.

Another possibility also exists when considering matter effects. The electron

density at the Sun’s core is Ne,core ' 150 g/cm−3 and Equations 1.32 and 1.33 are

used in 1.23 to calculate the survival probability. But the electron density outside of

the core is not constant and instead follows a roughly inverse exponential trend (see

Figure 2.3). It is then possible for Ne,Sun(r) (where r is the ratio of the distance from

the core to the solar radius and making Ne,Sun(r = 0) = Ne,core and N(r = 1)e,Sun = 0)

to equal Ne,crit, the resonance condition (MSW effect) mentioned in Section 1.3.3.

The neutrino mass eigenstates in matter are νM,1 and νM,2 and correspond to

energy eigenvalues given in 1.31. By applying the leptonic mixing matrix of 1.21

(with the substituted effective mass splitting and mixing angle) to the matter mass

23

distance/sun radius

e- den

sity

in m

ol/c

m3

10-2

10-1

1

10

10 2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 2.3: The Sun’s electron density profile as a function of distance from the coreto the surface (defined as r in the text).

eigenstates, it can be seen that

νe = νM,1 cosθM + νM,2 sinθM, (2.2)

νµ,τ = −νM,1 sinθM + νM,2 cosθM. (2.3)

At the large density Ne,core in the core, the effective mixing angle is pushed to π/2,

making core-born νe mostly νM,2. If the neutrino propagates adiabatically to the

surface of the Sun, it will stay in state νM,2. But as Ne,Sun(r) decreases, so will the

effective mixing angle and thus increase the contribution of νµ,τ in νM,2. At the

resonance Ne,Sun(r) = Ne,crit, the two flavor states are maximally mixed, and at the

surface where N(r = 1)e,Sun = 0, νM,2 has become the vacuum state ν2. The neutrino

does not oscillate after emergence from the Sun. If the neutrino propagates through

the Sun non-adiabatically, it can possibly jump into the νM,1 state at the resonance

and emerge as ν1. Figure 2.4 shows the vacuum eigenstates (expressed as m2) as a

function of solar electron density.

There exists the increased possibility of the ν2 mixture converting back into νe

and is a consequence of interactions with Earth’s matter. In this case, an experiment

sensitive to solar neutrinos with the capabilities of realtime detection might see a

24

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

slightly higher nighttime rate when compared to the daytime rate. As with the

Sun, the Earth’s density as a function of radius is not constant but is at values less

than an order of magnitude of the Sun’s. There is no resonance inside the Earth.

The combinations of (vacuum) mixing and mass-squared difference in the regime

where matter effects are important is the plane typically above ∆m2 > 10−9 eV2. Re-

gions on the plane allowed by solar neutrino data are known as the small mixing

angle (SMA), the large mixing angle (LMA) and the low solutions1. All four solu-

tions are shown in Figure 2.5.

1The low (LOW) solution refers to the small ∆m2 values for matter oscillations and the relativelylow possibility of it being the solution when considering SMA and LMA.

25

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

26

2.3 S N E

2.3.1 H

The radiochemical experiment Homestake (1967-1994), located in the South Dakota

Homestake gold mine at 1478 m (4200 meters water equivalent2) depth, employed

the reaction 37Cl + νe →37Ar + e− by filling a tank with 615 tons of C2Cl4 (tetra-

chloroethylene) with a 24.23% concentration of the 37Cl isotope and letting it sit

for around 30 days to capture solar neutrinos. Argon (which contained an ad-

mixture of the 37Ar isotope after exposure) was then removed from the tank by

purging it with helium gas for 20 hours. Residual tetrachloroethylene and other

active gases were removed and the argon was extracted (95% efficient). The 37Ar

isotopes were counted by the emission of Auger electrons (81.5% at a total energy

of 2.823 keV) and X-rays (8.7%) when 37Ar decayed back into 37Cl. It was a total rate

measurement unable to distinguish direction or energy of the captured neutrinos.

2.3.2 K

Kamiokande (1987-1995, solar measurement) was the first solar neutrino exper-

iment after Homestake and was started 20 years later. The original purpose of

Kamiokande was the detection of proton decay to test certain Grand Unified The-

ories. It was upgraded in 1987 to facilitate the collection of solar neutrinos. The

detector was a cylindrical tank located 1000 m (2700 mwe) below ground in the

Kamioka mine in Japan and was filled with 3000 tons of pure water surrounded by

948 photomultiplier tubes (PMT) of 50 cm diameter. The PMTs collected Cherenkov

light from electrons recoiling from elastic scattering with solar neutrinos. The reac-

tion νe + e− → νe + e− was the dominate process but Kamiokande was also sensitive

2Meters water equivalent (mwe) is the corresponding distance in water that would provide equalshielding.

27

to νµ,τ + e− → νµ,τ + e− but at a much lower rate and it could not be discriminated.

Kamiokande was a realtime experiment which could discern the direction and

energy of recoil electrons from neutrino scattering, but limited statistics hid the ap-

pearance of possible spectral distortions which are a telltale sign of the parameters

that govern neutrino oscillation.

2.3.3 GALLEX/GNO & SAGE

Like Homestake, GALLEX (May 1991-June 1997) was a radiochemical experiment.

It detected solar neutrinos through the reaction 71Ga + νe →71Ge + e− which

has a threshold of Eν=0.233 MeV. This made the detection of all solar neutrino

sources possible. Located in the underground national laboratory of Gran Sasso

in Italy, GALLEX had 101 tons of liquid gallium chloride solution (GaCl3-HCl)

under an overhead of 3300 mwe. After exposures lasting about 30 days, 71Ge

was chemically extracted and proportional counters waited for its decay back into

71Ga. Improvements to the extraction process gave birth to GNO (May 1998 - April

2003), a very similar experiment using the same detector as GALLEX. In total,

GALLEX/GNO performed 123 extraction runs.

SAGE (January 1990-December 2001) was a joint Soviet-American Gallium Ex-

periment (later Russian-American Gallium Experiment) which employed the same

reaction as GALLEX/GNO. It was located at the Baksan Neutrino Observatory 2000

m (4700 mwe) under the Caucasus Mountains dividing Asia and Europe. Its vol-

ume was a total 50 tons shielded by 60 cm low-radiation concrete and a 6 mm steel

shell. By the end of the experiment, SAGE had conducted nearly 100 successful

solar neutrino measurements.

28

2.3.4 S-K

Super-Kamiokande (SK, April 1996-present) was initiated in June, 1996 and is the

successor of the Kamiokande experiment employing the same Cherenkov radiation

detection technique. In the solar neutrino analysis, the larger volume of SK gave

a statistical advantage over Kamiokande allowing discrimination of the neutrino

energy spectrum shape. With a shape identified, it would then be possible to distin-

guish a favorable solution out of the possible four set forth by oscillation theory and

earlier solar experimental data. This came to fruition with the publication stating

that the SMA and vacuum solutions were disfavored by not observing distortions

in the spectrum shape [14]. Later, after further exploiting SK’s strength of improved

statistics, it was announced that the LMA solution was favored, and with the com-

bination of all solar neutrino experimental results, LMA was the unique solution

to the solar neutrino problem at very high confidence [15]. The SK experiment is

currently divided into three stages: SK-I, SK-II, and SK-III.

2.3.5 S N O

The Sudbury Neutrino Observatory (SNO, 1999-2006) was the first experiment

able to measure the total solar neutrino rate and compare it to the rate of νe that

did not undergo oscillation. Located 2092 m (6010 mwe) below the surface in the

Creighton mine in Sudbury, Canada, SNO was a water Cherenkov detector that

used heavy water (D2O) as its target. The neutrino rates from the 8B neutrino flux

were measured using the following charged current (CC), neutral current (NC),

29

and elastic scattering (ES) processes:

CC : νe +D → p + p + e−, (2.4)

NC : νx +D → n + p + νx, (2.5)

ES : νx + e− → νx + e−, (2.6)

where x = e, µ, τ.

By subtracting the νe CC rate from the total NC rate, SNO could show that

two-thirds of the neutrinos from the Sun were changing into flavors not sensitive

to the CC reaction in the detector. Likewise, the ES-minus-CC weighted difference

also pointed to neutrino flavor change and was in good agreement with the NC-

minus-CC rate. The NC rate confirmed experimentally the theoretical SSM model

on neutrino emission for the 8B flux.

SNO was in three phases: Phase I using pure D2O [16], Phase II using NaCl dis-

solved in the heavy water (the salt phase) [17], and Phase III with the salt removed

and an array of 300 3He proportional counter tubes added. Each successive phase

was an effort to improve upon the measurement of the NC total rate by increasing

both the neutrino capture cross section and collection efficiency. At the time of this

writing, Phase III results have not been published.

2.3.6 S C S N D

This section presents a comparison in graphical and tabular form of the SSM pre-

dicted flux values and those measured by experiment.

30

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

31

Tabl

e2.

2:Fo

reac

hex

peri

men

tdes

crib

edin

Sect

ion

2.3,

the

neut

rino

dete

ctio

nre

acti

on,n

eutr

ino

ener

gyth

resh

old,

mea

sure

dflu

x,an

dpe

rcen

tage

ofth

eBP

2004

SSM

are

liste

d.In

the

flux

colu

mn,

the

first

erro

rgi

ven

isst

atis

tica

l,th

ese

cond

issy

stem

atic

.In

the

BP20

04co

lum

n,th

est

atis

tica

lan

dsy

stem

atic

erro

rsha

vebe

enad

ded

inqu

adra

ture

.D

ata

are

take

nfr

om[6

][16

][17

][18

].ex

peri

men

tre

acti

onth

resh

old

flux

(∗SN

U,∗∗×

106

cm−

2s−

1 )%

ofBP

04H

omes

take

ν e+

37C

l→

e−+

37A

r0.

814

2.56±

0.16±

0.16∗

30.1±

2.7

GA

LLEX/G

NO

69.3±

4.1±

3.6∗

52.9±

4.2

SAG

Eν e+

71G

a→

e−+

71G

e0.

233

70.8+

5.3+

3.7∗

−5.

2−

3.2

54.0±

4.0

Kam

ioka

nde

6.7

2.80±

0.19±

0.33∗∗

48.4±

6.6

SK-I

ν x+

e−→ν x+

e−4.

72.

35±

0.02±

0.08∗∗

40.6±

1.4

ν e+

D→

p+

p+

e−6.

91.

76+

0.06

−0.

05±

0.09∗∗

30.4±

1.9

SNO

-D2O

ν x+

D→

n+

p+ν x

2.22

45.

09+

0.44+

0.46∗∗

−0.

43−

0.43

87.9±

11.1

ν x+

e−→ν x+

e−5.

72.

390.

240.

23±

0.12∗∗

41.3±

4.7

ν e+

D→

p+

p+

e−6.

91.

68±

0.06+

0.08∗∗

−0.

0929.0±

1.7

SNO

-NaC

lν x+

D→

n+

p+ν x

2.22

44.

94±

0.21

0.38∗∗

−0.

3485.3±

7.5

ν x+

e−→ν x+

e−5.

72.

35±

0.22±

0.15∗∗

40.6±

4.6

32

pp, pep 7Be 8B CNO experiment

Cl H2O D2O D2OGa

0.301±0.027

0.406±0.014

0.484±0.066

0.529±0.042

0.540±0.040

0.290±0.017

0.853±0.075

SK Kam

ioka

nde

SAG

E

GA

LLE

X/G

NO

SNO

SNO

Hom

esta

ke

1.0± 0.23 1.0± 0.23 1.0± 0.231.0+0.09

!0.08

1.0± 0.21

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

33

C 3

T S-K-II D

Particle physics today is at the stage where theory outpaces experiment. To con-

firm or exclude the myriad of hypotheses by experimental means has often re-

quired increasingly higher energies and larger detectors. As a result, collabora-

tions of physicists form and grow, each contributing a share of responsibility to

develop and maintain what possibly could be a mammoth scientific instrument.

Super-Kamiokande was borne out of the desire to improve the already successful

Kamiokande measurement. Sixteen times larger than its predecessor, the SK exper-

iment has operated almost continually since June, 1996 and is presently in its third

stage. The current chapter aims to present the construction and detection methods

of the second stage detector, Super-Kamiokande-II1.

3.1 E O

The Super-Kamiokande experiment utilizes a large water Cherenkov imaging de-

tector located in a zinc mine in Hida City (formally Kamioka Town), Gifu Prefecture,

Japan. The mine is owned and operated by Kamioka Mining and Smelting Com-

1The Super-Kamiokande-II detector was the result of a partial recovery from a November 12th,2001 accident which destroyed over half of the PMTs in SK-I (details on the SK-I detector canbe found in [21]). Reconstruction was finished in September 2002 but with 19% photo-cathodecoverage compared with SK-I’s 40%.

The cause of the accident was determined to be an imploding PMT located at the bottom of thedetector. This created a propagating shock-wave in water. The SK-I tank at that time was fillingwith water after a recent electronics upgrade and was approximately two-thirds full. The resultingshock-wave destroyed 6,777 out of the 11,146 PMTs, virtually all PMTs below the water line.

To help ensure such an accident would not occur again, the PMTs were encased in blast shieldsconsisting of a fiber-reinforced plastic shell with a clear acrylic cover over the PMT photo-sensitivearea. After SK-II completed in October 2005, full restoration of the PMTs took place and SK-III turnedon in July, 2006. SK-III currently has 11,129 PMTs and are all encase in blast shields. Information istaken from [22][23]

34

pany. The detector is 1,000 meters (2,700 mwe) under the summit of Mt. Ikeno.

The coordinates are 36◦25’33” N, 137◦18’37” E and is 371.8 m above sea level. It

is approximately 250 km from Japan’s capital of Tokyo. Figure 3.1 shows SK’s

location in Japan.

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

The detector is the largest experiment currently under task at the Kamioka

Observatory, part of the University of Tokyo’s Institute for Cosmic Ray Research

(ICRR). SK and other experiments at the underground observatory take advantage

of the reduced cosmic ray muon rate (2.2 Hz) which is a background in many

measurements. The types of analyses performed at SK are atmospheric neutrino,

solar neutrino, supernova neutrinos (realtime and relic), proton decay, and other

topics.

The experimental site is accessed by a 1.5 km horizontal tunnel that is traveled

by reduced emission, non-spark ignition diesel vehicles. The SK site is an enclosed

area consisting of a control room with data collecting and monitoring computers

and a dome area immediately above the SK tank. Electronics huts which house

35

the equipment to power and read the information of the light collecting PMTs

are located in this area as well as various calibration tools and equipment. An

electron linear accelerator (LINAC) is housed in a room to the side of the dome

area. Figure 3.2 shows the SK site and tank.

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

The SK collaboration of researchers and engineers numbers about 140 and come

from five countries: Japan, US, Korea, China, and Poland. The SK experiment is

largely funded by Japan’s Ministry of Education, Culture, Sports, Science and Tech-

nology and in part by the U.S. Department of Energy and the U.S. National Science

Foundation. Additional funds come from the Korean Research Foundation, the

Korean Ministry of Science and Technology, and the National Science Foundation

of China.

36

3.2 DM

3.2.1 N I

Solar neutrino detection at SK is achieved by the collection of Cherenkov light

generated by recoil electrons in water scattered by incoming solar neutrinos. The

scattering is elastic, meaning the neutrino and electron are retained throughout the

interaction. SK is sensitive to all three types of neutrinos:

νe + e− → νe + e−,

νµ + e− → νµ + e−,

ντ + e− → ντ + e−.

All processes are neutral current processes but only the νe can interact with the

electron via charged current as well. This implies a larger cross section for νe and

makes it possible to measure solar νe disappearance since contamination from νµ,τ

interactions is small. The total cross sections for 10 MeV incident neutrinos are

σtot =

8.96 × 10−44 cm2 for νe

1.57 × 10−44 cm2 for νµ,τ. (3.1)

The scattering is highly forward making the recoil electron’s direction is more

uncertain by multiple scattering rather than the variance of kinematic angles.

3.2.2 C R

Cherenkov radiation is light generated when a charged particle travels in a medium

of refractive index n faster than the speed of light c in that medium (v > c/n). For a

recoil electron to emit Cherenkov light in water, its energy must be greater or equal

to the Cherenkov threshold energy of 0.767 MeV.

37

The differential number of photons generated per unit wavelength dλ per unit

distance dx the electron2 travels is

d2Ndxdλ

=2παλ2

(1 −

1(n(λ)β)2

)=

2παλ2 sin2 θC, (3.2)

where β = v/c and α is the fine structure constant. Here, the refractive index n is a

function of wavelength. The total number of photons is found by integrating over

travel distance and all wavelengths. The right side of 3.2 has an angle known as the

Cherenkov opening angle and describes the direction the photons travel relative

to the path of the electron. The resulting radiation forms a cone with opening 2θC

around the electron trajectory. The opening angle can alternatively be written

cosθC =c

n(λ)v, (3.3)

and can be easily seen to vary as a function of electron velocity. For a recoil electron

in water, θC ∼ 42◦.

3.3 D

3.3.1 T

The SK detector is a stainless steel cylindrical tank 41.4 m tall and 39.3 m in diameter.

It is in a rock-hewn cavity with concrete filling the narrow area between the tank

and rock. Two meters from the inside wall is a 1 m wide trussed support structure

that climbs the entire height of the detector. There is also a trussed ”floor” and

”ceiling” 2 meters from the bottom and top of the tank. This support structure

essentially divides the detector into two sections: inner detector (ID) and outer

2This discussion is valid for all charged particles. Electrons are used as an example to keep incontext with SK’s solar neutrino detection method.

38

detector (OD). Its purpose is to have mounted on it the photomultiplier tubes

(PMT) that are responsible for light collection. Also, opaque sheets are attached to

the structure to function as an optical division between the ID and the OD.

3.3.2 I D

The ID consists of the space inside the support structure. On the structure are

mounted 5182 50 cm PMTs in a checker board pattern (see Figure 3.3). Each PMT

is encased in a fiber-reinforced plastic shell capped with a clear acrylic dome over

the PMT’s photo-sensitive surface. The purpose of the shield is to protect against

propagating shock-waves that might occur if a PMT should succumb to fatigue

and implode under the pressure of water and vacuum. At normal incidence, the

acrylic domes have a transparency better than 98% at 400 nm in light wavelength.

At 300 nm, it is about 86%. The PMTs are sensitive in the range of 300-600 nm. To

reduce reflected light in the ID, black polyethylene sheets are placed on all areas of

the support structure where no PMT is mounted. The total PMT coverage of the

SK-II detector is 19%.

3.3.3 O D

The OD consists of the space between the support structure and the stainless steel

wall. It acts as a veto detector for incoming muons and a water shield from neutrons

and gamma rays emitted from the surrounding rock. On the support structure are

mounted 1885 20 cm PMTs in a grid pattern. The OD PMTs are not encased in blast

shields. White Tyvec R© sheets are placed on all areas where no PMT is mounted to

increase light collection from light reflection.

39

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

40

3.3.4 P T

The ID is equipped with Hamamatsu R3600 PMTs with a photo-sensitive surface

(the photocathode) 50 cm in diameter (Figure 3.4. They were originally developed

for the Kamiokande experiment. The bulbs of the PMTs are hand blown from

borosilicate glass to a thickness of about 5 mm. The inner surface of the glass is

coated with Bialkali (Sb-K-Cs) photocathode and is matched so its sensitivity to

light (quantum efficiency, or QE) coincides with the peak of the Cherenkov light

spectrum. The QE is about 21% at the peak between wavelengths 360 nm and 400

nm (see Figure 3.5). The 11 stage ”Venetian blind” dynode structure multiplies the

ejected electron (photoelectron, or p.e.) from the photo-cathode to create a gain of

107 electrons through an applied voltage of 1700 V to 2000 V. This signal can now

be read out via a 70 m coaxial cable that winds up the support structure into the

dome area and into one of four electronic huts.

The OD PMTs are Hamamatsu R1408 PMTs with a photo-cathode of 20 cm in

diameter. An acrylic wavelength shifting plate of dimensions 60×60×60 cm3 and

doped with 50 mg/l of bis-MSB is attached to the bulbs of the OD PMTs. The plates

serve the function of absorbing light in the ultraviolet and shifting it to blue-green

to match the peak sensitivity of the PMTs. This improves the collection efficiency

by a factor of 1.5.

3.4 P S

3.4.1 W

Removing impurities from the tank water is important to increase the water trans-

parency which directly affects the collection of Cherenkov light from events. Some

of the impurities may also be radioactive (222Rn) and contribute to unwanted back-

41

phot

osen

sitiv

e ar

ea >

460

φ

<φ

520

7000

0~

720~

)20610(

φ 82

2

φ 25

410

φ 11

6ca

ble

leng

th

water proof structure

glass multi-seal

cable

(mm)

Figure 3.4: Diagram of an ID PMT.

42

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

ground events. The SK tank water is originally drawn from two streams in the

Kamioka mine that are sourced by the natural seepage of rain and snow melt

through the rock. It is then pumped into a water purification system that removes

contaminants before placement in the tank. This water is then recirculated at a rate

of about 35 tons per hour. The purification process is listed below and the system

setup is shown in Figure 3.6.

1 µ

A series of filters removes small particle in the water.

1

The heat exchanger functions to cool the water to reduce the growth of bacteria

and PMT dark noise rate.

43

C

The polisher removes Na+, Cl−, Ca+2, and other heavy ions.

R-

This step dissolves reduced-radon air into the water to increase the efficiency of

the vacuum degasifier.

R

An additional filter to further remove contaminants.

V

The degasifier removes radon gas and oxygen dissolved in the water. The efficiency

of extraction of radon is about 96%.

2

The second heat exchanger removes heat in the water mainly coming from the

prior filtering processes.

U

The ultra filter further removes contaminants down to sizes of 10 nm in diameter.

M

A second degasifier, it removes radon gas with an 83% efficiency.

After filtering, the water’s radon concentration is reduced from about 2 mBq/m3 to

0.4±0.2 mBq/m3. The water is then introduced into the SK tank from the bottom.

44

SK TANK

REVERSE

OSMOSIS

BUFFER

TANK

PUMP

PUMP

PUMP

FILTER

(1µm Nom.)

UV

STERILIZER

HEAT

EXCHANGER

VACUUM

DEGASIFIER

CARTRIDGE

POLISHERULTRA

FILTER

REVERSE

OSMOSIS

PUMP

RN-LESS-AIR

DISSOLVE TANK

RN-LESS-AIR

SUPER-KAMIOKANDE WATER PURIFICATION SYSTEM

MEMBRANE

DEGASIFIER

HEAT

EXCHANGER

Figure 3.6: Schematic of the SK water purification system.

3.4.2 A

The ambient air in the mine is heavily concentrated with radon emanated from

the surrounding rock. In the summer months, radon levels are around 2000-3000

Bq/cm3 while in the winter months they are around 100-300 Bq/m3. This seasonal

dependence is shown in Figure 3.7. Due to the air outside the mine being at a near

constant of 10-30 Bq/m3 during the year, the outside air is pumped into the SK

dome area. This is accomplished by a housing of air pumps, filters, and a cooler

located outside the mine that draws fresh air and channels it into the SK dome area

via an air duct at 50 m3 per minute. As a result, radon concentrated mine air is

displaced and the radon levels are typically 20-30 mBq/m3.

Reduced-radon air is also pumped into the ∼ 60 cm air gap between the water’s

surface and the top of the stainless-steel tank roof. Radon is further reduced by

an air purification system (shown in Figure 3.8) that reduces the air to less than 3

mBq/m3. This system consists of a compressor (to 7.0 8.5 atm), 0.3 µm filter, buffer

tank, air drier (to improve efficiency of radon removal and to remove CO2), carbon

45

0

500

1000

1500

2000

2500

3000

3500

01/01/00 03/01/00 05/01/00 07/01/00 09/01/00 11/01/00 01/01/01

rado

n le

vel [

Bq/

m^3

]

2000 Radon monitor readings at SuperK

Mine Air (at sink)Control Room Air

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

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

SUPER-KAMIOKANDE AIR PURIFICATION SYSTEM

COMPRESSOR

AIR FILTER

(0.3mm)

BUFFER

TANK

AIR DRIER

CARBON

COLUMN

HEAT

EXCHANGER

COOLED

CHARCOAL

(-40 oC)

AIR FILTER

(0.1mm)

CARBON

COLUMNAIR FILTER

(0.01mm)

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

46

3.5 D A S

3.5.1 ID DAQ

The first task of ID data acquisition (DAQ) system is the collection of PMT signals3.

This is done by Analog Timing Modules (ATM). They are electronic boards that

receive signals from PMTs via 12 channels (one PMT per channel). Each channel

has a charge dynamic range of 550 pC at 0.2 pC (∼0.1 p.e.) resolution and a time

dynamic range of 1.2 µs at 0.3 ns resolution. Once a PMT receives a hit, it sends its

charge information to the ATM channel where it is amplified 100 times and then

split into four signals. One signal is sent to PMTSUM which is an analog sum of

the 12 channels used for the Flash ADC DAQ. More information on Flash ADC can

be found elsewhere [25]. Another signal is sent to a discriminator to compare the

voltage with a preset threshold. The two other signals are sent to two Charge to

Analog Converters (QAC). The presence of two QACs (and two TACs, explained

in the next paragraph) is to reduce electronics dead time for quick succession of

events such as a muon followed by its decay electron or supernova neutrino bursts.

The signal sent to the discriminator is measured against a threshold of 0.25 p.e. If

it exceeds this threshold, a HITSUM signal is generated with a 15 mV pulse height

and 200 ns width. This represents one hit PMT and is sent to the global triggering

system for collection with other ID PMT hits. At the same time of the generation of

the HITSUM signal, the QACs are instructed to open a 400 ns wide gate to store the

charge information. Likewise, Time to Analog Converters (TAC) are instructed to

open a gate to later be closed by the global trigger. There are two TACs per channel

to reduce electronics dead time.

Once all the PMT hits are assembled and pass a predefined threshold, the global

trigger will send out signals to all the ATMs to save the charge and timing in-

3Information on the DAQ was retrieved primarily from [24].

47

formation they have collected. At this time, the TAC closes its gate marking the

end of all hits. The QAC and TAC information is then digitized by the Analog to

Digital Converter (ADC) and sent to the First In First Out (FIFO) internal memory

of the ATM for storage. If the total of PMT hits do not exceed the threshold, the

global trigger does not send out signals to the ATMs and the TAC gate closes at its

maximum width of 1.2 µs. All information in the TACs and QACs are discarded.

Figure 3.9 shows a schematic of the analog input of one ATM channel.

DISCRI.

THRESHOLD

PMT INPUT

DELAY

CU

RR

EN

TS

PL

ITT

ER

&

AM

PL

IFIE

R

PED_START

TRIGGER

to PMTSUM

to HITSUMW:200nsHITOUT

SELF GATE

START/STOP

TAC-A

QAC-A

TAC-B

QAC-B

CLEAR

CLEAR

GATE

GATE

QAC-B

TAC-B

QAC-A

TAC-A

to ADC

HIT

START/STOP

W:300ns

ONE-SHOT

GATE

Figure 3.9: The analog input of one ATM channel.

Each ATM board is housed in a TRISTAN KEK Online (TKO) crate (20 boards

maximum) along with GO/NoGo (GONG) and Super Control Header (SCH) mod-

ules. The GONG module serves to distribute the global trigger signal to all 20 ATM

boards while the SCH module collects the digitized ATM FIFO data to send to the

Super Memory Partner (SMP) module. There is one SMP module for every TKO

crate and are housed in Versa Module Europe (VME) crates (six per VME crate).

The SMP modules then relay the data to computer servers finally to be collected

by an online host computer. The data from all hit PMTs is assembled into events

with various information relating to charge, time, number of hit PMTs, location of

48

hit PMTs, etc. The events are then sent to the offline system. Figure 3.10 shows the

organization of the SK DAQ.

In total, there are 12 channels per ATM board and 20 ATM boards per TKO

crate. 12 TKO crates, 2 VME crates with 6 SMP modules, and 2 computer servers

are housed in one electronics hut. There are 4 electronics hut and one central hut

that houses the online host computer and global trigger hardware. These 5 huts

are in the dome area immediately above the SK tank. Since there are 5182 PMTs in

SK-II, each ATM has an average of 5.4 live channels out of a possible 12.

3.5.2 OD DAQ

The OD DAQ serves to record hits in the OD that come before and after ID global

trigger events. The OD PMT signals are sent to Charge to Time Converters (QTC)

where an OD HITSUM is generated (0.25 p.e.) threshold. If the OD trigger, which

serves the same purpose as the ID global trigger, issues a signal, the OD HITSUM

data is digitized and saved to a TDC module. The TDC can store up to 8 QTC

pulses at a 0.5 ns resolution. Its timing dynamic range is 16 µs with 10 µs before

and 6 µs after the ID global trigger. The data is saved only if an ID global trigger

was issued. The OD information is then sent to the online computer and merged

with the event information of the ID.

3.5.3 T S

The HITSUM signal generated by the ATM boards are collected at the global

triggering system in the central hut. This triggering system adds all HITSUMs to

create a total HITSUM. If the total HITSUM exceeds 110 mV (roughly 7 hits), then

a global trigger is issued.

There are 3 main levels to the triggering system. They are the Super Low Energy

49

20-inch PMT

ATM

x 240

x 20

ATM

GONG

SCH

interface

SMP

SMP

SMP

SMP

SMP

SMP

TRIGGER

HIT INFORMATION

x 6

Analog Timing Module

Ultra Sparc

VME

online CPU(slave)

online CPU(slave)

online CPU(slave)

online CPU(slave)

online CPU(slave)

online CPU(slave)

online CPU(slave)

Ultra Sparc

VME

online CPU(slave)

Analog Timing Module

TKO

TKO

Ultra Sparc

Ultra Sparc

Ultra Sparc

Ultra Sparc

Ultra Sparc

Ultra Sparc

Ultra sparc

Ultra Sparc

interface

online CPU(host)

online CPU(slave)

VME

FDDI

FDD

I

TRIGGER

Super Memory Partner

Super Memory Partner

SMP x 48 online CPU(slave) x 9

PROCESSOR

TRG

interrupt reg.

20-inch PMT

ATM

x 240

x 20

ATM

GONG

SCH

20-inch PMT

ATM

x 240

x 20

ATM

GONG

SCH

20-inch PMT

ATM

x 240

x 20

ATM

GONG

SCH

interface

SMP

SMP

SMP

SMP

SMP

SMP

PMT x 11200 ATM x ~1000

Figure 3.10: The layout of the ID DAQ.

50

(SLE) trigger, the Low Energy (LE) trigger, and the High Energy (HE) trigger. An

event is said to be an SLE, LE, or HE event if the total H ITSUM exceeds 110 mV, 152

mV, or 180 mV respectively. A fourth, independent trigger monitoring OD PMT

hits registers an event if there are more than 20 OD hits.

The SLE triggered events are mostly radioactive background from the ID walls

and PMTs and gamma rays from the outer rock wall. To save limited storage

space, these events need to be filtered. This is accomplish by a fast online vertex

reconstruction algorithm. Originally SK-I’s standard vertex reconstruction tool, it

was modified and optimized for SK-II to use online (see Section 4.1.1 for details

on the fitter). Those events that reconstruct outside the fiducial volume (defined

as 2 meters from the ID wall) are rejected. All others are recorded for later offline

analysis. The SLE trigger and the vertex reconstruction algorithm are collectively

known as the Intelligent Trigger (IT). The reduction in data is around a factor of 6.

3.6 W TM

3.6.1 L L I

To measure the effects of light scattering and absorption in water, SK-II has an

automated system of 8 laser light injectors tunable to different wavelengths: 337

nm, 371 nm, 400 nm, and 420 nm. The injection points are diagramed in Figure 3.11.

Light is injected at regular intervals and the total charge Q is collected on the

opposing wall. This is direct, unimpeded light. Reflected light is collected by

counting the number of hit PMTs on all other surrounding walls. The ratio of

hit-PMT/Qbottom is then plotted as a function of time and is used to adjust (tune) a

quantified model of intensity given by

I(x) = I0 ·1x2 · exp(−x/Latten), (3.4)

51

where I0 is the initial intensity and x is the distance from the light source. Latten is

the attenuation length and given as

Latten =λ4

αabs + αRayleigh + αMie. (3.5)

The α terms are the tunable coefficients and are adjusted to match the ratio-timing

distributions of laser data and MC. The αRayleigh and αMie coefficients correspond

to Rayleigh and Mie scattering respectively. The αabs coefficient corresponds to

absorption and is tuned with LINAC data instead (see Section 4.7.2 for a description

of absorption). The SK-II values are αRayleigh = 1.7 × 108 and αMie = 5.0 × 106. An

example of the laser timing distribution is shown in Figure 3.12. The first broad

peak in the timing distribution is due to light scattered from water. The second,

smaller peak is due to reflection off the acrylic blast shield, PMT, and black sheet.

The attenuation coefficients are plotted in Figure 3.13 with their λ−4 dependence.

3.6.2 T V

The water quality changes with time and needs to be accounted for during event en-

ergy reconstruction. Decay electrons from cosmic ray muons provide an abundant

and constant source to measure the SK water’s transparency length periodically.

The charge collected from a stopping muon’s decay electron is fitted to the function

Qi = Q0 ·f (θi)x2 · exp(−x/Latten), (3.6)

where, xi is the distance from the light source to the i-th PMT, Q0 is a constant,

and fi(θ) is the i-th PMT acceptance at angle θ (Figure 3.14). Since the Cherenkov

light cannot be isolated in frequency, Latten is measured for the entire (averaged)

spectrum. Latten as a function of time during the SK-II data taking period is shown

52

Top oldTop new

Barrel 1(1232.25 cm)

Barrel 2 (595.95 cm)

Barrel 3(-40.35 cm)

Barrel 4(-605.95 cm)

Barrel 5(-1242.25 cm)

Bottom

Figure 3.11: The positions of the 8 laser light injectors in the SK-II tank. ”Top old”refers to the original SK-I light injector which is still used in SK-II.

53

Radius= 10 m

Radius = 10 m

Number ofhit PMTs

Total Charge

Number ofhit PMTs

Total Charge

REAL DATAMC

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

54

10-4

10-3

10-2

10-1

1

200 250 300 350 400 450 500 550 600 650 700

wavelength (nm)

scat

teri

ng

-ab

sorp

tio

n c

oef

fici

ent

(1/m

)

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

in Figure 3.15

3.7 C

3.7.1 PMT C

Three types of calibration are performed on the PMTs: relative gain, absolute gain,

and timing. They are to insure that all PMTs give a uniform response to reduce

systematic bias. The information obtained by calibration data are used to adjust

each individual PMT.

55

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

56

80

85

90

95

100

105

110

115

120

2500 2600 2700 2800 2900 3000 3100 3200 3300 3400

elapsed day

wat

er tr

ansp

aren

cy (

m)

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

R G

The relative gain of a PMT is defined as

Gi =Qi

Q0 × f (θ)· l2

i · expli

L(3.7)

where Qi is the collected charge, Q0 is a constant normalization factor, li is the

distance from a light source to the PMT, L is the water attenuation length, and f (θ)

is the PMT acceptance function dependent on incident angle θ (Figure 3.14). The

light source used in calibration is a scintillating ball exposed to a xenon lamp and is

designed to emit light uniformly in the tank at a peak wavelength of 440 nm. Since

the gain is related to its applied high voltage as G α Va (a is a constant), the voltage

is tuned until all PMTs obtain a uniform collected gain. Xenon lamp calibration

data is taken at several positions in the tank and for several high voltage values.

The relative gain spread is realized at 7%. The remaining difference is compensated

57

for in the event reconstruction software. Figure 3.16 shows the setup of the xenon

lamp calibration system. Figure 3.17 shows the relative gain for all PMTs.

Xe Flash LampUV filter ND filter

Optical fiber

Pho

toD

iode

Pho

toD

iode

ADC

Monitor

PMT

Scintilator

Trigger20inchPMT

SK TANK

Scintilator Ball

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

A G

The intensity of a PMT signal is quoted in photoelectrons (p.e.) during analysis but

it is charge in pico-Coulombs that is read from the PMTs. Therefore, a conversion

factor needs to be determined at the single p.e. level to render meaningful physical

quantities. A Ni-Cf source is used for the absolute gain calibration. The source

apparatus is shown in Figure 3.18. 252Cf undergoes spontaneous fission and its

emitted fast neutrons are thermalized in water. The neutrons are then captured

on Ni wire which then emits gamma rays around 9 MeV in energy. The gamma

rays, now outside the source apparatus, Compton scatter with electrons, producing

Cherenkov light. The energy is low enough to assume each PMT is hit with a single

photon. A charge distribution is made (Figure 3.19) to find what mean charge

58

0

200

400

600

800

1000

1200

1400

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Relative PMT Gain

Nu

mb

er o

f P

MT

s in

Eac

h B

in

σgain = 7.0%

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

corresponds to these one p.e. hits. It is determined that 2.055 pC is equivalent to

one p.e. and is used as a conversion factor.

T C

The relative timing of all PMTs must be adjusted so that it is possible to reconstruct

event vertices and directions accurately. PMTs can display relative timing offsets

depending on the amount of charge a PMT collects. The charge dependence is

mainly due to ”slewing,” an effect in which greater charge hits are triggered earlier

than those with less charge. A fast pulsing light source is used to correct for this.

A N2 laser with pulse width ∼ 3 ns and attched to a dye laser module fires intense

light of 384 nm wavelength into two channels: one going to a monitoring system

and the other to a light-diffuser ball lowered into the SK tank (Figure 3.20). The

light is diffused without significantly affecting the time spread. The light intensity

is varied to correspond with a PMT signal of one p.e. to several hundred p.e. and

the PMTs are read. The distribution of timing and charge is plotted in Figure 3.21.

59

20m

m

200m

m

130m

m

125m

m

50mm

10m

m 30m

m

Nickel wire + Water

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

10-2

10-1

1

-2 0 2 4 6 8 10

1 p.e. distribution Charge (pC)

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

60

The errors on the measurements are the timing resolutions of the PMTs and are

typically ∼ 3 ns. The data are then used to counter-offset the PMTs to match one

time regardless of the differing amounts of charge they receive.

Diffuser Ball

Super-Kamiokande Detector

N2 LaserTrigger PMTND filter

LUDOXDiffuser Tip (TiO2)

Dye

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

3.7.2 E C

To better determine the oscillation parameters tan2 θ and∆m2, the SK solar analysis

looks at the shape of the recoil electron energy spectrum. The presence and nature

of distortions in the spectrum points to these parameters. It is therefore vital to cal-

ibrate the energy scale and energy resolution to a high degree or accuracy. SK uses

two calibration sources exclusively employed for the solar neutrino measurement:

monochromatic electrons and 16N decay products.

LINAC C

To better understand the recoil electron energies from scattered solar neutrinos, SK

first measures the energies of electrons from a controlled source: an electron linear

accelerator (LINAC) [26]. The LINAC used is a Mitsubishi ML-15MIII LINAC

which was originally manufactured for medical purposes. After purchasing, in-

61

880

890

900

910

920

930

940

950

960

970

980

Q (p.e.)

T (

nsec

)

1 10 100

Linear Scale Log Scale

5

Figure 3.21: The timing-charge map (TQ-map) of all ID PMTs. The circles representthe timing resolutions (∼3 ns).

stalling at the detector site, and performing modifications (which included reduc-

tion of the electron gun current to decrease the number of accelerated electrons),

the first successful regular operation of the SK LINAC was in September 1997. The

SK-II LINAC data taking was in March, 2004.

The LINAC accelerates electron ”bunches” using 2 µs-wide microwave pulses.

The number of electrons in a bunch (in the order of 106) is too high for calibration

purposes and is collimated down to a few electrons per bunch. The beam leaves

the radiation-shielded room where the LINAC is housed through an evacuated

beam pipe. To reach the SK tank, the beam must be steered and is done so by

three electromagnets (D1, D2, and D3, see Figure 3.22). The D1 magnet not only

bends the beam downward at a 15◦ angle but also acts as a momentum selector by

deflecting electrons with momenta other than the momentum desired. The beam

passes additional collimators before being bent 90◦ into the SK tank. The end of the

62

4000 cm

42

00

cm

D1 MAGNET

D2 MAGNETD3 MAGNET

BEAM PIPE

LINAC

TOWER FOR INSERTING BEAM PIPE

1300 cm

4

5

1

2

3 6

X

Z

Y

-12m -4m

+12m

-12m

0m

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

63

beam pipe is closed with a 50 µm-thick titanium cap which allows electrons to pass

without much attenuation. Also at the beam pipe’s end is a LINAC trigger counter

which can be connected to the main SK trigger system to discriminate calibration

events. The electron in-tank energies taken for SK-II are 5.8, 6.8, 8.8, and 13.4 MeV.

The energy calibration proceeds by reproducing downward-traveling electrons

of the LINAC energies taken in the MC detector simulation program4. These and

the LINAC calibration data events are then reconstructed with the same methods

of the solar neutrino analysis. The MC and data energy distributions are fitted

with a Gaussian function and are compared. Any position dependent deviation

is minimized by adjusting parameters in the MC simulation. These include water

scattering, water absorption, PMT and black sheet reflection, and others. Once

this minimum has been achieved, the absolute gain of the MC PMTs is adjusted

to eliminate any difference in the averaged MC energy and the averaged LINAC

data energy. The top plot of Figure 3.23 shows the MC energy scale deviation from

LINAC data. Using the equation

σ2 =118

18∑i=1

σ2i , (3.8)

where σi are the deviations for the 6 positions over 6.8, 8.8, and 13.4 MeV, an energy

scale systematic uncertainty is assigned to 1.4%.

Uncertainty in energy resolution can also be estimated. Energy resolution is the

one sigma value from the Gaussian fit and is compared with MC. An average value

of 2.5% deviation is obtained.

LINAC calibration data can also be used to estimate vertex and direction recon-

struction uncertainties. They are discussed in their respective sections.

4The LINAC energies are determined by aiming the beam into a Ge detector which has an energyresolution of 1.92 keV. This is done after each successful beam injection into the tank.

64

-0.04-0.03-0.02-0.01

00.010.020.030.04

4 6 8 10 12 14Energy (MeV)

(MC

-LIN

AC

)/L

INA

C

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

4 6 8 10 12 14Energy (MeV)

(MC

-LIN

AC

)/L

INA

C

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

65

DTG

The Deuterium-Tritium Generator (DTG) [27] is a neutron generator used to induce

the creation of 16N in the SK tank. 16N’s subsequent decay is then used as an energy

calibration source. See Figure 3.24 for a schematic of the DTG setup.

Y.Takeuchi@WIN’05 14June 8, 2005

16

n

nnn

nn N16

2 m

(a) (b) (c)

E =14.2 MeVn

O(n,p) N16

1616N calibrationN calibration

D + T ! 4He + n

n + 16O ! p + 16N

DT Generator

~106 neutrons / pulse

~1% of neutrons create 16N16N decay is precisely known.

66.2% 6.129MeV ! + 4.29MeV ",

28.0% 10.419 MeV ", etc.

Data taken at various positions.

Uniform direction complementary to LINAC calibration.

Figure 3.24: The setup of the DTG calibration apparatus. The generator is loweredinto the tank (a). Neutrons are created which interact with 16O in the tank andmake 16N (b). The generator is then lifted 2 meters to allow the decay of 16N toproceed with minimal photocathode shielding by the generator (c).

The production of this source starts with collision of deuterium and tritium

inside the source generator to create 4He and a neutron. This is done in pulses with

about 106 neutrons in each pulse. The neutrons then exit the generator into the

water and react with 16O and release 16N and protons. The 16N decays with a half-

life of 7.13 seconds with decay products of 6.13 MeV gamma rays and 4.29 MeV

beta particles (66.2% contribution) along with 10.42 MeV betas (28.0% contribution).

These events are reconstructed for energy and an absolute energy scale uncertainty

66

is found to be 1.2% and in good agreement with LINAC data and MC. Since the

DTG is relatively small and portable, data can be taken more often and in many

tank positions to better track the time and position dependence of the energy scale.

Also, since 16N decays isotropically, zenith angle dependence of the energy scale

can be measured and is found to be 0.5%. This is used as an uncertainty in the

day-night asymmetry value. Figure 3.25 shows the absolute and zenith energy

scales using DTG data and MC.

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

-12 0 12 -15

-12 0 12 15 -15

-12 0 12 15 -15

-12 0 12 15 -15

-12 0 12 15 -15

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

(a) (b) (c)

(1) (2) (3) (4) (5)

SOURCE z-position (m)

(MC

-SO

UR

CE

)/S

OU

RC

E

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1cos(zenith angle)

(MC

-SO

UR

CE

)/S

OU

RC

E

Figure 3.25: 16N energy scale deviation from MC. Representing the varyingtimes calibration data were taken, (1) Nov. 2003, (2) March 2004, (3) July2004, (4) Nov. 2004, (5) Sep. 2005. (a), (b), and (c) represent the x positions15.20 m, 10.96 m,−14.49 m respectively. All other calibration data were taken atx = 0.35 m. The bottom plot shows the 16N energy scale deviation from MC for 6selected zenith angles of the detector (-1 is down).

67

C 4

SK-II S E

SK-II’s high threshold of 7.0 MeV which corresponds to 8B and hep neutrino de-

tection is in an area of large, unwatned background. These include gamma rays,

nuclear spallation products, muon decays, radon decays, and electronic noise. To

successfully separate neutrino events from this background, highly efficient recon-

struction techniques need to be employed. It is then necessary to remove those

events which are most unlike those expected from neutrino-electon scattering. Fi-

nally, the identification of the recoil events from solar neutrinos is carried out by

a comparison of simulated events based on theoretical models and Monte Carlo

random generation.

4.1 E R

4.1.1 V

The determination of event vertex in SK-II is done by an algorithm specifically

developed to address the reduced light collection capability of SK-II relative to

SK-I. Since this algorithm marks a significant departure in the event reconstruction

methods of SK-I, it warrants a comparison.

The motivation for a new way to reconstruct event vertices comes from the

reduced photo-cathode coverage in SK-II and the SK-I standard vertex fitter’s

inability to perform well at lower energies. As an example, the efficiency of the

SK-I standard fit significantly drops at energies below the SK-I analysis threshold

of 5.0 MeV. With 40% photocathode coverage, this corresponds roughly to 25 PMT

68

hits. At 19% coverage in SK-II, 25 hits translates to 8 MeV. The SK-I and the new

SK-II standard vertex fitters function as follows.

The timing residual in an event is defined as the time difference between a PMT’s

hit time ti and the emission time t0 (fitted to minimize all timing residuals) minus

the time it would take Cherenkov light to reach that PMT given the event’s vertex

~v in the tank:

tresidual = (ti − t0) − |~v − ~hi|/c, (4.1)

where ~hi is the vector location of the hit PMT and c is the group velocity of light in

water.

In SK-I, vertex reconstruction is accomplished by selecting a limited number of

hit PMTs from an event (to reduce bias from PMT dark noise and scattered light hits)

and calculating a goodness relation based on the timing residuals of those selected

hits and a candidate vertex ~v. A systematic grid search of candidate vertices is

performed until the goodness reaches a maximum value. After that, the vertex

position is fine-tuned to further maximize the goodness. The SK-I reconstruction

will not attempt a vertex fit for less than 10 hits.

In contrast, SK-II uses all hits from an event to form the timing residuals for

determination of the vertex position. PMT dark noise is taken into account by

constructing a likelihood describing the shape of the timing residual distribution

from LINAC calibration data. This likelihood is then maximized from a vertex

search based not on a grid pattern but from a list of vertex candidates calculated

from PMT hit combinations of 4 hits each. The four-hit combinations each define

a unique vertex given their timing constraints. Any event with four hits or more is

reconstructed.

SK-II also makes use of the SK-I goodness-grid search method in its online and

initial offline analysis for filtering background events. The final reconstruction, or

the standard fit based on the residual likelihood method, is the final determination

69

of the vertex and can also be seen as a correction for any misreconstructed events

which survived the filtering process. Figure 4.1 shows the various vertex resolu-

tions for the SK-II vertex reconstruction. The uncertainty of the measured solar

neutrino rate due to systematic shifts in vertex position is estimated to be 1.1%.

SK-II also utilizes a fast fit online reconstruction method for pre-filtering low

energy events. Details can be found elsewhere [18].

0

50

100

150

200

250

300

5 6 7 8 9 10 11 12 13 14electron total energy (MeV)

vert

ex r

eso

luti

on

(cm

)ve

rtex

res

olu

tio

n (

cm)

standard fitgoodness fitfast fit

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

4.1.2 D

To recognize the Sun as the source of a neutrino scattering rate rising over a rel-

atively flat background rate, event direction reconstruction is important. To do

this, SK-II uses a maximum likelihood method that scans directions which corre-

spond with Cherenkov ring patterns. A function f (cosθdir) describes distributions

of opening angles between a candidate particle direction ~d and the reconstructed

vertex-to-hit PMT position (~v − ~hi). It is made by MC and an example for 10 MeV

70

electrons is shown in Figure 4.2. The effects of electron multiple scattering and light

scattering in water can be seen by the broad peak around the 42◦ Cherenkov angle.

Opening angles θi between the reconstructed vertex ~v and the vector normal to the

PMTs inward-facing surface and weighted by the PMT acceptance function a(θi)

are then summed as a product with f (cosθdir) in a likelihood function

L(~v) ≡N30∑

i

log( f (cosθdir))i ×cosθi

a(θi), (4.2)

where N30 is the number of hit PMTs with timing residuals within a 30 ns window

around tresidual = 0. The direction ~d is scanned at various levels of precision until

Equation 4.2 is maximized. An absolute angular resolution is then defined as the

maximum angular difference between 68% of the reconstructed and true event di-

rections as determined by MC. It is plotted as a function of energy in Figure 4.3.

The systematic uncertainty in the angular resolution is 6.0% when comparing reso-

lutions of data and MC. This results in a±3.0% error on the total flux measurement.

4.1.3 E

To approximate the energies of solar neutrinos, the recoil electrons off which they

scatter is reconstructed and acts as a good equivalence to neutrino energy since the

recoil electron mostly preserves the incident neutrino’s direction. Whereas vertex

and direction reconstruction rely on the location and pattern of Cherenkov light

received by PMTs, energy reconstruction is dependent on the intensity of that light.

Cherenkov photons from recoil electrons travel in the tank and are detected

by the 50 cm PMTs through generation of photoelectrons in the photocathode.

The number of Cherenkov photons are approximately proportional to the electron

energy and in turn proportional to the number of generated photoelectrons. But

given the poor charge resolution (∼ 50%) of the PMTs and the fact that electrons

71

0

0.2

0.4

0.6

0.8

1

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

rela

tive

pro

bab

ility

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

15

20

25

30

35

40

45

5 6 7 8 9 10 11 12 13electron total energy (MeV)

dir

ecti

on

al r

eso

luti

on

(d

egre

e)

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

72

below 20 MeV produce roughly one photon per hit PMT, the number of hit PMTs,

or Nhit, is counted instead.

To reduce the contamination of noise hits, Nhit is defined as the number of hits

whose timing residuals (Equation 4.1) fall within a 50 ns window1. But this alone

is not adequate for energy calculation. Corrections must be made for detector-

specific dependencies including PMT occupancy, late hits, dark noise, dead PMTs,

photo-cathode coverage, water transparency, and PMT gain. These are manifested

as an ”effective” hit sum

Neff =

Nhit∑i=1

((Xi + εtail − εdark) ×

Nall

Nlive×

1S(θi, φi)

× exp( ri

λeff

)× Gi(t)

). (4.3)

O

To estimate the effect of multiple photoelectrons in the i-th hit PMT, the occupancy

Xi is defined as

Xi =

− log (1−xi)

xixi < 1

2.5 xi = 1, (4.4)

where xi is the ratio of hit PMTs in a 3×3 area surrounding the i-th PMT over the

total number of live PMTs in the same area. The − log (1 − xi) term is then the

estimated number of photons per one PMT in that area and is determined from

Poisson statistics. The value when xi = 1 is assigned to 2.5. It should be mentioned

that in SK-II the usual number of PMTs in a 3×3 area is 5 (9 in SK-I) and is arranged

in a checkered pattern.

1Introduced in the last section was N30, the number of hits within a 30 ns window. The generalnotation in this work is the number of hits N subscripted by the width of the timing window innanoseconds. If Nhit is given without numerical scripting, 50 ns is assumed.

73

L H

To account for hits falling outside the 50 ns timing window, the term εtail = (N100 −

N50)/N50 is added to the occupancy. Such hits could arise from Cherenkov photons

being scattered and reflected and thus arriving late to the PMT surface.

D N

Dark noise is subtracted from the occupancy with the term εdark = (100 ns ×Nlive ×

rdark)/N50. Nlive is the number of functioning PMTs for a given run and rdark is the

measured dark rate for that run.

D PMT

To correct for non-functioning PMTs, the factor Nall/Nlive is applied. Nall is the total

number of PMTs and is 5182 in SK-II.

P-C C

The term 1/S(θi, φi) is to account for the direction-dependent photocathode cover-

age. It acts as a scaling function on the average coverage of 19%.

W T

The water transparency is accounted for by ri/λeff where ri is the distance from the

reconstructed vertex to the i-th hit PMT. λeff is the measured water transparency.

PMT G

The last factor, Gi(t), is the PMT gain correction. This is to adjust the relative gain

of the PMTs at the single photoelectron level. The differences in gain arise from the

74

various stock and fabrication dates of the PMTs.

After determining Neff, a corresponding value for an event’s energy must be

obtained. This is done by generating MC events at discrete input energies between

5 and 80 MeV, calculating their Neff values, and then interpolating the energy

function. This function is then used to calculate an event’s total energy in MeV.

The energy values are checked with LINAC and 16N calibration data and MC (see

LINAC and 16N sections).

Since the corrections in Neff depend on the water transparency, the reconstructed

energy also varies slightly with changing water quality. See Figure 4.4 for Neff as a

function of time for a given water transparency. When calculating energy for data

events, the water transparency value as determined by decay electrons from cosmic

ray muons is used as an input parameter. However, for MC events, the change

in water transparency is not simulated due to its relative stability. A calculated,

constant value of 101 m is used for all MC events in the SK-II solar neutrino analysis.

4.2 D R

The average LE trigger rate in SK-II is about 63 Hz. Therefore, each day 5.4×106

events above 8.0 MeV are expected in the SK-II tank. The vast majority of these

events are not recoil electron events from solar neutrinos (even considering all

energies, the BP2004 expected rate is 325.4 events per day). To separate the solar

neutrino events from unwanted events, or background, a series of reduction cuts

must be applied to the initial data set. These cuts are

1. Noise reduction, to reduce the contribution of triggered events originating

from PMTs and electronic components and to eliminate charged particles

coming from outside the detector. It also discards entire runs or subruns that

75

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

are deemed inadequate for the solar neutrino analysis.

2. Spallation cut, to reduce beta-decay and gamma ray events arising from

spallation products of incoming muons.

3. Timing and hit pattern cut, to verify the trustworthiness of reconstructed

events in effort to further eliminate noise events that might have passed

previous cuts.

4. Gamma cut, to cut gamma-ray events originating from the ID wall.

The reduction cuts utilize the information obtained from the PMT timing, charge,

and event reconstruction and are applied to all LE-triggered events as well as those

events that survived the SLE filtering process (see Section 3.5.3).

76

4.2.1 N R

Noise reduction is a multistep process that relies on a variety of information. It is

also known as first reduction since it is the first effort to reduce background. The

steps and their descriptions follow.

P C

To remove high energy events such as cosmic ray muons and atmospheric neutrino

events, a total p.e. cut is applied that keeps events registering p.e.≤500.

R A T

This is simply a removal of those events that might have an additional trigger in

coincidence with an LE or SLE trigger. These are mainly calibration events and

atmospheric analysis-related pedastal events.

C T D E

When two events follow closely in time, they may be a stopping muon and its

decay electron or other unwanted electronic noise such as signal reflection in the

PMT-ATM cable. To eliminate this background, when the event time difference

is less than 50 µs apart, they are removed. Figure 4.5 shows the time difference

between events in a representative run.

R OD- E

A charged particle such as a cosmic ray muon may enter the ID and leave its

signature in the OD. Therefore, when the OD registers an event (p.e.≥20), it is

removed. Figure 4.6 shows the reduction of events after the OD trigger cut.

77

1

10

10 2

10 3

10 4

10 5

2 3 4 5 6 7 8 9 10log(time difference (ns))

even

ts

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

10 2

10 3

10 4

0 10 20 30 40 50 60 70 80 90 100number of OD hit PMTs

even

ts

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

78

R A E N

Ambient noise from electronic components come from noisy PMTs or from various

effects originating in the dome area above the tank. This noise typically produces

an absolute PMT charge no greater than 0.5 p.e. A ratio can be formed by defining

N as the total number of hit PMTs with |QPMT| < 0.5 and S as the total number

of PMTs regardless of charge. A cut is made for those events with N/S < 0.55.

Figure 4.7 shows the ratio before and after reduction.

The same noise events are often clustered on one ATM. If the number of hit

PMTs on one ATM is greater than 95% of the total PMT channels on that ATM, then

the corresponding events are rejected. Figure 4.7 shows a typical distribution of

this percentage.

11010 210 310 410 510 6

0 0.2 0.4 0.6 0.8 1N/S ratio

even

ts

10 3

10 4

10 5

0 0.2 0.4 0.6 0.8 1ATM %

even

ts

Figure 4.7: Top: Events with a ratio of low charge PMTs to total PMTs less than0.55 are removed. The shaded area show the events that passed the cut. Bottom:Events with a large number of hit PMTs on one ATM are removed.

79

E F PMT

A flashing PMT is caused by an electrical discharge between dinodes and can be

caused by water seeping into the PMT housing. This can severely affect a normally

uniform trigger rate in the detector and hinder event reconstruction. Fortunately,

flashing PMTs with a high charge can easily be identified and immediately turned

off. Unfortunately, those that are not readily identifiable can remain in the data set.

To identify a flashing PMT for removal, its neighboring PMTs are scrutinized for

high charge. If the number of PMTs NPMT surrounding the PMT with the maximum

charge Qmax in an event exceeds the function

NPMT >

10 Qmax < 17.02

−0.51 ×Qmax + 18.75 17.02 ≥ Qmax < 32.59

2 32.59 ≥ Qmax

, (4.5)

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

02.5

57.5

1012.5

1517.5

2022.5

0 20 40 60 80 100120140160180200

10-1

1

10

10 2

10 3

10 4

10 5

02.5

57.5

1012.5

1517.5

2022.5

0 20 40 60 80 100120140160180200

10-1

1

10

10 2

10 3

10 4

10 5

NPMT

even

ts

QMAX -2000

-1500

-1000

-500

0

500

1000

1500

2000

0 500 1000 1500 2000 2500 3000x 10

3

r2 (cm2)

z (c

m)

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

80

R F F

Failed fits refer to the inability of the vertex reconstruction algorithm to locate a

suitable vertex for an event. At this stage and until the end of noise reduction,

the goodness-grid search algorithm of SK-I and optimized for SK-II, mentioned in

Section 4.1.1, is used to reconstruct events. Failure can arise from events having

less than the minimum 5 hits.

C B G C

The vertex reconstruction assigns a value which gives a numerical gauge of the

goodness of the determined vertex. Those events reconstructed with a goodness

less than 0.4 are rejected.

F V C

A fiducial volume cut is applied using the reconstructed vertex of events. Those

events falling within 2 m of the ID wall are rejected. This is to eliminate noise coming

from the wall (PMT support structure, PMT glass, black sheet, etc.). Figure 4.9

shows the z vs. r2 distribution of reconstructed events in the tank.

L E C

Finally, an energy cut removes all events (SLE and LE) below 6.5 MeV.

After the completion of all noise reduction cuts, various distributions of the

data (such as those presented in the above figures) are manually scanned for each

run. The purpose is to identify anomalies that might have passed the automated

reduction process. These include low rate flashing PMTs, higher than normal or

erratic time dependent event rates, high occurrences of noise in a single run or

subrun, and others. When such data are observed, the run (or subrun, depending

81

-2000

-1500

-1000

-500

0

500

1000

1500

2000

0 500 1000 1500 2000 2500 3000x 10

3

r2 (cm2)

z (c

m)

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

on the magnitude of contamination) they occupy is labeled as a bad run and is

discarded from the analysis. Short runs, typically under an hour, are discarded as

well.

The total systematic error due to differences in reduction efficiencies between

data MC is estimated to be ±1.0% on the total flux measurement.

4.3 S C

When cosmic ray muons travel in the SK tank, they break apart 16O nuclei and

create unstable nuclei known as spallation products. These products soon decay

releasing electrons, positrons, or gamma-rays with energies largely between 7 and

20 MeV. This obvious background to the solar neutrino signal must be reduced.

This is accomplished by taking advantage of the correlation spallation products

have with the initial muon.

82

4.3.1 M T R

To identify muon events, PMT charge is scrutinized: if the total charge in the tank

exceeds 3000 p.e. and at least one PMT has more than 100 p.e., then the event is

marked as an incoming muon. To reconstruct its track, the position of the earliest

hit PMT with at least 2 neighboring hit PMTs (hit within 5 ns of the first hit) is

identified as the entry point. The central point of all PMTs with more than 190 p.e.2

is identified as the exit point. The line from entry point to exit point is defined

as the muon track. The quality of the track is measured by defining Lent as the

minimum distance between each charge saturated PMT at the entry point and Lexit

as the maximum distance between each charge saturated PMT at the exit point.

The track is considered a good reconstruction if Lent > 300 cm and Lexit < 300 cm.

After the initial track reconstruction, the charge residual Qresidual of the muon

event is determined by

Qresidual = Qtotal − pL, (4.6)

where Qtotal is the total charge in the event, L is the reconstructed track length, and

p = 11.4 p.e./cm and is the average number of photoelectrons ejected per unit track

length of the traveling muon. If a muon event has a large charge residual, the first

reconstruction may not be trusted. If Qresidual > 12, 000 p.e., it is refitted with a

second algorithm regardless of the quality of the first reconstruction.

The second reconstruction utilizes the same entry point as explain above but

searches for an exit point by following the Cherenkov wake left by the muon. A

goodness relation is defined as

gµ =

∑i

1σ2

iexp

[−

12

(ti−T1.5σi

)2]∑

i1σ2

i

, (4.7)

2A PMT with more than 190 p.e. is said to be charge saturated.

83

where σi is the time resolution of the i-th hit PMT and T is the time the muon

entered the detector. The fitted entrance time ti is defined as

ti = Ti(~xi) −lµ(~xexit)

c−

lγ(~xexit)c/n

(4.8)

where ~xi is the location of the i-th hit PMT, Ti is the time of the i-th hit PMT,

~xexit is the exit point, lµ and lγ are the flight distance of the muon and Cherenkov

photons respectively, and n is the refractive index of water. Figure 4.10 shows the

components of the fitted entrance time equation. After searching candidate exit

points, the one that gives gµ > 0.85 is chosen.

4.3.2 S S E

After the passing of a muon in the tank, the resulting nearby spallation products

decay with half lives up to 14 seconds. Therefore, the decay products’ spatial and

temporal proximity to the muon track and event time can be used as variables in

a likelihood test. They are expressed as a difference of the event vertex and time

to that of a preceding muon (∆L and ∆T respectively). A third variable, the muon

event’s residual charge Oresidual (Equation 4.6), is also used in the likelihood and

serves as an indicator of the muon’s lost energy and hence possibility of creating

spallation products. When compared to solar neutrino events, spallation events

are expected to be spatially closer to a preceding muon track (small∆L), temporally

closer to a muon’s event time (small ∆T), and greater residual in the tank (large

Qresidual). Each candidate solar neutrino event (those that survived the various

noise cuts) are tested with the 200 preceding muon events. Those events with a

likelihood greater than 0.98 are rejected. It is possible to use the likelihood even

if the muon track failed to be reconstructed. In this case, ∆T and Oresidual are used

and events with a likelihood greater than 0.92 are rejected. Since this cut inevitably

84

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

85

results in signal loss, a dead time is calculated with a randomly collected sample of

events not correlated with muons. The effect this has on the final flux measurement

is estimated to be 0.4%.

4.4 T & H P C

After noise reduction and the spallation cut, many background events remain due

to mis-reconstruction inside the fiducial volume. The 2-dimensional timing-hit

pattern cut is designed to remove those events whose reconstruction should not

be trusted. An optimized hit PMT timing goodness is defined (Equation 4.9) by

comparing two timing residual Gaussian distributions, one with a width of σ = 5

ns to encompass selected hits and the other with a ω = 60 ns width characteristic

of the PMT timing resolution for a single photo-electron:

gt(~v) =Σe−

12

((τi(~v)−t0ω )+(

τi(~v)−t0σ )

)2

Σe− 12 (τi(~v)−t0ω )2

. (4.9)

The effective hit time is defined as τi(~v) = ti − |~v − ~hi|/c which is just the timing

residual tresidual of Equation 4.1 with added t0. The sums are over all hits.

The hit pattern goodness allows us to identify Cherenkov events by their

azimuthal-symmetric ring pattern from a reconstructed vertex and direction. All

others are labeled mis-reconstructed or non-Cherenkov events. A goodness func-

tion gp(~v) is defined for all directional events.

A cut on the goodness values is made in tandem using a hyperbolic radius of

g2t − g2

p > 0.25 and rejecting all other events. Figure 4.11 shows this background

reduction cut on data and 8B Monte Carlo in the 7.0-7.5 MeV bin. When the cut is

applied to LINAC data and MC, a total flux systematic error of ±3.0% is obtained.

86

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.810

-5

10-4

10-3

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.5

0.4

0.3

0.2

0.1

00.5 0.50 0

10-3

10-4

10-5

vertex (timing) goodness

dir

ecti

on

(p

att

ern

) g

oo

dn

ess data MC

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

87

4.5 G- C

Although a 2 m fiducial volume cut is applied to reject events that are coming from

the ID wall, similar events might remain inside the fiducial volume. To eliminate

them, an additional gamma-ray cut is applied using the energy, direction, and

vertex based on the standard fitter. With this information, events with direction

inward from the wall are rejected with the following criteria:

7.0 MeV < E ≤ 7.5 MeV and dwall < 11 m

7.5 MeV < E ≤ 8.0 MeV and dwall < 10 m

8.0 MeV < E ≤ 8.5 MeV and dwall < 8 m

8.5 MeV < E ≤ 9.0 MeV and dwall < 7 m

9.0 MeV < E ≤ 10.0 MeV and dwall < 5 m

10 MeV < E ≤ 20 MeV and dwall < 4 m,

where E is the reconstructed event energy and dwall is the distance from the wall

in the direction of the reconstructed event (Figure 4.12). This further reduces

the number of events around the areas of large z or r2 as seen in Figure 4.9. The

uncertainty on the total flux measurement is estimated as±1.0% for the gamma-ray

cut.

4.6 F D S

The SK-II data period lasted from December 10th, 2002 to October 6th, 2005. The

raw data sample before reduction contains 1.5×1010 events. This includes all events

before trigger cuts and low energy cuts. The first noise reduction (with bad run

cut) greatly reduces this total to 6.1×106 events. Figure 4.13 shows graphically the

88

Figure 4.12: dwall is defined as the distance from the wall in the direction of thereconstructed event.

contributions of succeeding cuts and Figure 4.14 shows the efficiencies on MC of

those cuts. The final solar sample event count is 117,696. Still, the majority of

events come from background that were not removed. The next step is to employ

Monte Carlo event simulations to separate the remaining background and solar

signal.

4.7 M C E S

To identify which recoil electron events are coming from scattered solar neutrinos, a

computer generated model is used for comparison with the reduced SK-II data set.

This model, based on theoretical rates, kinematical distributions and cross sections,

and a detector response simulation, is also valuable for determining reduction

efficiencies and, later, oscillation spectra.

89

10-2

10-1

1

10

10 2

10 3

10 4

10 15 20Energy (MeV)

Eve

nts/

day/

0.5M

eV

After noise reductionAfter spallation cutAfter timing-hit pattern cutAfter gamma cut

Figure 4.13: Summary of the data reduction steps.

0

0.2

0.4

0.6

0.8

1

4 6 8 10 12 14

After noise reductionAfter spallation cutAfter timing-hit pattern cutAfter gamma cut

Energy (MeV)

Effi

cien

cy

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

90

4.7.1 N R I

The first half of the SK-II Monte Carlo event simulation involves stating, statistically,

what recoil electron signal is expected due to scattering with neutrinos coming from

the Sun. This can be divided into three steps3, each determined separately:

1. The rate of neutrinos emitting from the Sun

2. The probability a neutrino of energy Eν will scatter with an electron

3. The resulting energy and direction of the recoil electron

The first step comes from the measured 8B neutrino spectrum of Ortiz [28] and

the total flux prediction of the BP2004 Standard Solar Model (SSM) [9]. Using

these, it is possible to state the total rate of solar neutrinos arriving at earth. But

since at SK-II solar neutrinos are only detectable indirectly by scattering with

electrons, a scattering rate must be included in the equation. Taken from [29], the

differential cross section dσ(Eν)/dE describes the probability a neutrino of energy Eν

will scatter off an electron with total energy E =√

p +m2. The neutrino spectrum

and differential cross section is shown in Figure 4.15. Now it is possible to define

a total rate of electrons scattered off solar neutrinos:

R8B = ΦSSM Ne

∫∞

0

∫∞

0dEνdEφ(Eν)

dσ(Eν)dE

, (4.10)

where φ(Eν) is the neutrino spectrum. The ΦSSM factor is the BP2004 8B flux pre-

diction and is 5.79 × 106 cm−2 s−1 . Ne is equal to 1.086 × 1034 and is the number

of electrons in the 32.4 kton volume of the SK-II inner detector. By summing

over all electron and neutrino energies, the total expected rate in the SK-II tank is

325.4 events per day. The errors on the neutrino spectrum and cross section come

3When considering neutrino oscillations, a fourth step will be interjected.

91

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

respectively.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 2.5 5 7.5 10 12.5 15 17.5 20neutrino energy (MeV)

emis

sion

pro

babi

lity

0

0.02

0.04

0.06

0.08

0.1

0 5 10 15 20electron total energy

diffe

rent

ial.

c.s.

(10

E-1

9)

10-5

10-4

10-3

10-2

10-1

1

10

10 2

0 2.5 5 7.5 10 12.5 15 17.5 20electron total energy

even

ts p

er d

ay (

32.4

kto

ns)

Figure 4.15: Left: The black solid line is the 8B neutrino spectrum from [28] and thered dotted line is the hep neutrino spectrum. Both are normalized to unity. Center:The black solid line is the νe − e elastic scattering differential cross section in unitsof cm2 MeV−1

× 10−42. The red dotted line is the elastic scattering differential crosssection for νµ,τ − e. Right: The black solid line is the expect number of events perday as a function of electron energy coming from scattering with 8B neutrinos. Thered dotted line is for scattering with hep neutrinos.

Since neutrinos from the hep chain are also in the energy range which SK-

II is sensitive, hep neutrinos are included in the simulation. The rate for hep is

constructed the same way as the 8B rate in Equation 4.10 but substituting φ(Eν) for

the hep neutrino spectrum φhep(Eν). To calculate the total rate of hep neutrinos in

the SK-II tank, the BP2004 SSM value of ΦSSM,hep = 7.88 × 103 cm−2s−1 is used and

comes to 0.64 events per day.

The third step in determining an expected recoil electron signal in SK-II is the

electron’s energy and direction. By the construction of Equation 4.10, the energy

information is already included. The integral over all electron energies is dropped

and R8B becomes r8B(E), a function of the recoil electron energy. The rate spectrum is

shown in Figure 4.15. The direction of the scattered electron begins by recognizing

the incident neutrino originates from the Sun for whose direction is constantly

known. Therefore, the incident neutrino’s path is along the same vector from the

92

Sun to the detector. The scattered electron’s direction is finally determined by the

kinematics of the elastic collision and its scattering angle with respect to the initial

neutrino direction. It is given by

cosθ =1 + m

Eν√1 + 2m

ET

, (4.11)

where m is the mass of the electron and ET is the kinetic energy of the recoil electron

(disregarding mass). In the azimuth, scattering is symmetric.

4.7.2 D R

Having obtained a model describing the rate of recoil electrons from 8B and hep

neutrinos, it can now be used as input information for the SK-II detector simulation

computer routines collectively called skdetsim for SK detector simulation. These

routines simulate the physical processes of energetic electrons in water and, in

general, the scattering and absorption of photons produced by these processes in

the SK-II tank.

The skdetsim program is based on CERN’s GEANT 3.23 package which allows

simulation of electromagnetic processes in the range of 10 keV to 10 TeV. After

instructing the relevant GEANT routines to simulate electrons in water with the

spectra plotted in Figure 4.15 with recoil direction 4.11 from the Sun-detector vector,

the simulation package tracks the electrons and models the physical reactions that

could appear during its journey. Such reactions include Cherenkov radiation, mul-

tiple scattering, ionization loss, gamma-ray production, bremsstrahlung radiation,

pair creation, and Compton scattering.

The skdetsim program’s next role is to simulate the scattering and absorption

of generated photons in the SK-II tank. To describe light scattering, the laser

data obtain in the water transparency measurement (Section 3.6.1) is used. For

93

absorption in the longer wavelengths above 350 nm, a third party independent

measurement [32] of pure water using an Integrating Chamber Absorption Meter

(ICAM) is used. For shorter wavelengths (λ < 350 nm), the MC energy output in the

simulated tank is highly dependent on the absorption coefficient and is carefully

tuned with LINAC calibration data. The most suitable coefficient is chosen by

minimizing the differences in reconstructed LINAC data and MC energies. This

minimum was found to be a 1.4% difference for 13.4 MeV and 8.8 MeV events

using no absorption below 350 nm. See Section 3.6.1 for a discussion of the water

transparency model with its tuned coefficient values. The absorption coefficient is

shown in Figure 4.16

10-4

10-3

10-2

10-1

1

200 250 300 350 400 450 500 550 600 650 700

wavelength (nm)

scat

teri

ng

-ab

sorp

tio

n c

oef

fici

ent

(1/m

)

Figure 4.16: The solid line is the total scattering and absorption coefficient de-scribing light-loss and scattering in water for SK-II MC. The dotted line is thecontribution of absorption only.

Having traversed the simulated water, the GEANT photons have the possibility

to reach the blast shield-encased PMTs4. The acrylic front shield introduces another

4The other possibility is light may travel to the inner detector black sheet. Although the blacksheet is meant to absorb photons, scattering can occur and is modeled after test data taken in theSK-II tank.

94

medium for which scattering and absorption could occur. Extensive studies were

conducted to determine the overall transparency of the acrylic shields; at normal

incidence transparency is better than 98% above 400 nm in wavelength and is about

86% at 300 nm.

Other light-interacting properties of the PMTs are also simulated using observed

or calculated values such as reflection off the PMT glass and absorption in the

photocathode. Once the photon is ready for conversion into a photoelectron,

PMT timing resolution, PMT collection efficiency, and PMT quantum efficiency are

adjusted appropriately in skdetsim.

Finally, a trigger simulator is applied to the MC-simulated data. The output

is identical in form to that of real data but consists solely of signals generated

from simulated recoil electrons off solar neutrinos. In addition, the original mo-

menta of these events are preserved and are used to gauge the performance of the

reconstruction tools and reduction cuts.

4.7.3 E R F

Although not utilized for the extraction of SK-II solar neutrino data, an analytical

function of the detector’s energy resolution can be constructed with previously

determined information. This function can be used to apply the SK-II energy

response to theoretical spectra for comparison with SK-II data.

The MC events used for the Neff-to-energy conversion function are this time

reconstructed for energy E′. The distribution of E′ follows a Gaussian shape. They

are then fitted to determine their one sigma values and the fitting error on those

values (δσ). Energy resolution over energy ((σ ± δσ)/E′) is plotted as a function of

E′ and is again fitted to obtain the function (in MeV)

σE(E) = 0.0536 + 0.5200√

E + 0.0458E, (4.12)

95

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

Equation 4.12 are shown in Figure 4.17.

0

1000

2000

0 10 20 30Reconstructed energy (MeV)

Nu

mb

er o

f ev

ents Mom=5MeV

7MeV

9MeV11MeV

13MeV15MeV

18MeV

22MeV26MeV

30MeV

0.1

0.2

0.3

0.4

0 10 20 30electron total energy (MeV)

E r

eso

luti

on

/E

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

This is used in a Gaussian probability density function

R(E,E′) =1

√2πσE

exp[−

(E′ − E)2

2σ2E

]. (4.13)

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

96

in SK-II as a function of measured energy E′:

r8B(E′) = ΦSSM Ne

∫∞

0

∫∞

0dEνdEφ(Eν)

dσ(Eν)dE

R(E,E′). (4.14)

The hep rate is determined in similar fashion by substituting ΦSSM and φ(Eν) with

their hep counterparts. Figure 4.18 shows the calculated rates of 4.14 in the standard

SK-II binning and compared to the rates as determined by skdetsim and event

reconstruction. The maximum difference in binned rates between both methods is

no larger than 7.5%.

1

10

10 2

10 15 20Energy(MeV)

Eve

nts/

kton

/yea

r

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

97

4.8 R

4.8.1 S E

Since the recoil electrons from solar neutrino scattering are strongly correlated with

the vector from the detector to the Sun, information on the Sun’s position and the

expected dependence the number of recoil events has on this direction can be used

to extract the data signal.

This is done by an extended maximum likelihood fit with data and MC distri-

butions. The likelihood function is

L = exp( Nbin∑

i=1

Bi + S) Nbin∏

i=1

ni∏j=1

(Bi · bi(cosθi j) + S · Yi · si(cosθi j

), (4.15)

where Nbin = 17 and is the number of bins between 7.0 MeV and 20.0 MeV and

ni is the number of observed events in the i-th energy bin. Bi is a free parameter

that represent the number of background events and S · Yi is the number of signal

events again for the i-th energy bin. The free parameter S is the total number of

solar neutrino events and Yi is the expected fraction5 of that total for each bin. The

solar angle dependent distributions bi(cosθSun) and si(cosθSun) (θSun ≡ θi j, shown

in Figure 4.19) describe the expected shapes of the background and recoil electron

signal respectively.

The background shape bi(cosθSun) has a non-flat distribution when viewed as

a function of solar angle θSun. This is due to the effective coverage of background

sources seen from various vantage points in a cylindrical detector. To estimate

this background, bi(cosθSun) is extracted from the data by fitting the zenith angle

and azimuthal event distributions with an eighth degree polynomial. The solar

neutrino events are those defined as cosθSun > 0.7 and are weighted to reduce

5As expected from the Ortiz [28] spectrum.

98

!Sun

SK

Sun

Figure 4.19: The solar angle θSun.

99

their contribution so the fitted function sees no solar peak. The background is then

re-expressed as a function of θSun.

The signal shape si(cosθi j) is obtained from MC and incorporates the effects of

electron multiple scattering and the energy resolution of the detector. This causes

a smearing of the peak in the direction of the Sun (cosθSun = 1.0) and is dependent

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

0

0.05

0.1

0.15

0.2

0.25

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

20 MeV

12 MeV

10 MeV

8 MeV

cosθsun

pro

bab

ility

den

sity

Figure 4.20: The expected cosθSun distributions for 4 energies and obtained fromMC.

The values Bi and S are then chosen so the likelihood has a maximum value. For

a live time of 791 days of SK-II data from 7.0 to 20.0 MeV, the extracted number of

signal events is 7212.8+152.9−150.9(stat.) +483.3

−461.6(sys.). This corresponding 8B flux is

(2.38 ± 0.05(stat.)+0.16−0.15(sys.)) × 106 cm−2sec−1.

The cosθSun distribution of data and the extracted signal and background can be

100

seen in Figure 4.21. Figure 4.22 shows the distribution divided into energy bins of

the recoil electron spectrum.

0

0.05

0.1

0.15

0.2

-1 -0.5 0 0.5 1cosθsun

Eve

nt/d

ay/k

ton/

bin

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

4.8.2 D-N S V

The day and night fluxes can also be extracted from the SK-II final data set by

recognizing those events which happened when the Sun was above the local hori-

zon and when it was below. Unlike the total flux, the day and night fluxes are

quoted using a threshold of 7.5 MeV due to low signal to noise ratio and increased

deviation from a non-flat background for the 7.0-7.5 MeV bin in the solar direction

101

7.0- 7.5MeV 7.5- 8.0MeV 8.0- 8.5MeV

8.5- 9.0MeV 9.0- 9.5MeV 9.5-10.0MeV

10.0-10.5MeV 10.5-11.0MeV 11.0-11.5MeV

11.5-12.0MeV 12.0-12.5MeV 12.5-13.0MeV

13.0-13.5MeV 13.5-14.0MeV 14.0-15.0MeV

15.0-16.0MeV 16.0-20.0MeV

Figure 4.22: The cosθSun distribution of data for each energy bin in the recoilelectron spectrum.

102

after the data set is divided. Their values are

Φday = (2.31 ± 0.07(stat.) ± 0.15(sys.)) × 106 cm−2sec−1,

Φnight = (2.46 ± 0.07(stat.) ± 0.16(sys.)) × 106 cm−2sec−1.

To quantify the possible indication of regeneration back into νe as mentioned in

Section 2.2.2, an asymmetry value between the day and night fluxes can be defined

asA = (Φday −Φnight)/(12 (Φday + Φnight)). The SK-II day-night difference yields

A = −0.063 ± 0.042(stat.) ± 0.037(sys.).

Unfortunately, while the night flux reflects an increase in the νe rate over the

day flux, the combined statistical and systematic errors make the measurement

consistent with zero.

The total flux variation as a function of time, or seasonal variation, for both SK-I

and SK-II solar data is shown in Figure 4.23. Each bin represents 1.5 months and is

seen to follow a sinusoidal trend consistent with the expected 1/r2 flux variations

due to the eccentricity of the earth’s orbit around the sun. SK-II has excellent

agreement with SK-I data, thus showing the continuation of the SK solar neutrino

measurement through two phases of the detector.

F C S A

With the completion of SK-II, the solar neutrino flux measurement of the Super-

Kamiokande experiment spans an interval of 9.5 years. This closely coincides with

the full period of solar cycle 23. To address any possible correlation of solar neutrino

flux with sun spot number, the SK-I and II flux time variation data are compiled

in 1-year bins between 1996 and 2006. The SK-I data set (from 1996 to 2001) is

taken from a 5.0 MeV threshold while SK-II is from 7.0 MeV. Errors are statistical

103

1

2

3

4

1998 2000 2002 2004 2006YEAR

Flu

x (x

106 /c

m2 /s

)

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

only. From 1996 to the end of the SK-II phase in October 2005, the solar neutrino

flux is stable and shows no pattern of correlation with the minima and maximum

of solar cycle 23. This is consistent with (and a continuation of) the Kamiokande

measurement and comparison with solar cycle 22 [33], albeit with a greater level

of precision for Super-Kamiokande.

4.8.3 E S

The recoil electron energy spectrum is obtained by dividing the total flux into 17

energy bins ranging from 7.0 to 20.0 MeV. The bin boundaries and flux values are

listed in Table II. Figure 4.25 shows the observed energy spectrum divided by the

expected spectrum without oscillation determined from the BP2004 SSM. The line

through the spectrum represents the average of all bins. This spectrum is valuable

for constraining the parameters governing solar neutrino flavor change.

104

Table 4.1: SK-II observed energy spectra expressed in units of event/kton/year.The errors in the observed rates are statistical only. The 7.0-7.5 MeV energy binis excluded from the day-night analysis. Correction is made for the reductionefficiencies in Figure 4.14. The expected rates neglecting oscillation are for theBP2004 SSM flux values. θz is the angle between the z-axis of the detector and thevector from the Sun to the detector.

Energy Observed rate Expected r ate(MeV) ALL DAY NIGHT 8B hep

−1 ≤ cosθz ≤ 1 −1 ≤ cosθz ≤ 0 0 < cosθz ≤ 17.0 − 7.5 43.7+5.2

−5.1 − − 112.4 0.2577.5 − 8.0 40.0+3.6

−3.5 36.4+5.1−4.9 43.6+5.2

−5.0 99.1 0.2458.0 − 8.5 34.9+2.5

−2.4 34.4+3.5−3.4 35.5+3.5

−3.4 85.9 0.2318.5 − 9.0 30.1+2.0

−1.9 27.0+2.8−2.7 33.0+2.8

−2.7 73.5 0.2159.0 − 9.5 24.5+1.6

−1.6 23.9+2.3−2.2 25.0+2.3

−2.2 61.4 0.1989.5 − 10.0 22.0+1.4

−1.4 20.7+2.0−1.9 23.3+2.0

−1.9 50.3 0.18110.0 − 10.5 16.6+1.2

−1.1 15.4+1.7−1.6 17.6+1.7

−1.6 40.7 0.16310.5 − 11.0 13.9+1.0

−1.0 13.5+1.5−1.4 14.2+1.5

−1.4 32.1 0.14511.0 − 11.5 10.3+0.9

−0.8 11.3+1.3−1.2 9.4+1.2

−1.1 25.3 0.12911.5 − 12.0 8.06+0.71

−0.66 7.11+1.00−0.90 8.96+1.03

−0.94 19.51 0.11312.0 − 12.5 6.28+0.62

−0.58 6.82+0.94−0.84 5.79+0.86

−0.77 14.67 0.09812.5 − 13.0 4.07+0.50

−0.45 4.18+0.73−0.63 3.97+0.70

−0.61 10.96 0.08413.0 − 13.5 3.32+0.43

−0.38 2.95+0.62−0.53 3.66+0.61

−0.53 7.91 0.07113.5 − 14.0 2.23+0.35

−0.30 2.95+0.57−0.48 1.59+0.44

−0.35 5.74 0.06014.0 − 15.0 2.77+0.39

−0.35 2.99+0.60−0.51 2.58+0.53

−0.45 6.90 0.09115.0 − 16.0 1.75+0.30

−0.26 1.37+0.42−0.32 2.08+0.45

−0.37 3.41 0.06316.0 − 20.0 1.37+0.27

−0.22 1.11+0.37−0.28 1.60+0.40

−0.31 2.52 0.089

105

0

1

2

3

4

1998 2000 2002 2004 2006YEAR

Flu

x (x

106 /c

m2 /s

)

0

50

100

150

200

Nu

mb

er o

f su

nsp

ots

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

4.9 S S E

A listing of all assigned systematic errors is given in Table 4.2. The second column

is the error when applied to the total flux measurement and the third column

(day-night) is the error applied to the day-night difference.

Table 4.3 is the list of energy correlated systematic errors (energy scale, energy

resolution, 8B spectrum) assigned to each bin in the recoil electron energy spectrum.

106

0.2

0.4

0.6

0.8

5 10 15 20Energy(MeV)

Dat

a/S

SM

BP

2004

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

107

Table 4.2: SK-II systematic error of each item in %. Numbers in parentheses are thevalues obtained from calibration data before application to the neutrino flux.

flux day-nightEnergy scale (absolute ±1.4%) +4.2 − 3.9Energy scale (relative ±0.5%) ±1.5Energy resolution (2.5 %) ±0.38B spectrum ±1.9Trigger efficiency ±0.51st reduction ±1.02nd reduction ±3.0Spallation dead time ±0.4Gamma cut ±1.0Vertex shift ±1.1Non-flat background ±0.4 ±3.4Angular resolution (6.0%) ±3.0Cross section ±0.5Live time ±0.1 ±0.1Total +6.7 − 6.4 ±3.7

Table 4.3: SK-II energy correlated systematic errors for each energy bin in %.Energy 8B Shape Energy Scale Energy Res.7.0 − 7.5 +0.47,−0.54 +1.07,−1.02 +0.42,−0.417.5 − 8.0 +0.52,−0.65 +1.71,−1.54 +0.36,−0.368.0 − 8.5 +0.64,−0.77 +2.34,−2.08 +0.27,−0.288.5 − 9.0 +0.82,−0.89 +2.96,−2.62 +0.15,−0.179.0 − 9.5 +1.05,−1.03 +3.57,−3.18 0,−0.029.5 − 10.0 +1.30,−1.181 +4.17,−3.75 +0.15,−0.18

10.0 − 10.5 +1.58,−1.35 +4.77,−4.32 +0.36,−0.3710.5 − 11.0 +1.88,−1.53 +5.36,−4.90 +0.60,−0.5711.0 − 11.5 +2.19,−1.73 +5.95,−5.48 +0.87,−0.8011.5 − 12.0 +2.51,−1.95 +6.55,−6.06 +1.18,−1.0412.0 − 12.5 +2.84,−2.19 +7.16,−6.63 +1.52,−1.3012.5 − 13.0 +3.18,−2.44 +7.78,−7.20 +1.89,−1.5713.0 − 13.5 +3.53,−2.71 +8.43,−7.76 +2.30,−1.8713.5 − 14.0 +3.89,−2.98 +9.09,−8.31 +2.74,−2.2014.0 − 15.0 +4.44,−3.41 +10.15,−9.12 +3.47,−2.7615.0 − 16.0 +5.17,−3.98 +11.68,−10.16 +4.59,−3.6816.0 − 20.0 +6.46,−5.19 +15.95,−12.83 +8.18,−7.40

108

C 5

O A

Since the announcement by Super-Kamiokande of the discovery of neutrino oscil-

lation in 1998, tremendous progress has been made in neutrino physics. Through

the efforts of Super-Kamiokande, SNO and KamLAND, the solar neutrino problem

appears to be understood, for the first time since its discovery over 30 years ago.

There is now strong evidence that neutrinos have mass and their flavor states mix

with each other. The evidence exists from atmospheric neutrinos, solar neutrinos,

reactor experiments, and long baseline oscillation experiments.

By the methods presented in the previous chapter, solar neutrinos events in

the initial SK-II data set are extracted and the corresponding flux determined.

Although the results from SK-II does not significantly change the experimentally

determined precision of the solar neutrino mixing angle and mass difference, it does

give overwhelmingly consistent results which further solidifies the hypothesis of

oscillating neutrinos. Since this is the second phase of the Super-Kamiokande

experiment, the SK-II oscillation analysis is very similar to that performed earlier.

It is the goal of this chapter to present this analysis and its results in detail.

5.1 A SK-II χ2

The comparison of observed data rates and SSM rates is done by fitting their energy

spectra in a χ2 analysis. The finite spectra boundaries were chosen based on SK-II’s

energy threshold and the limited statistics and energy resolution in the high energy

region. The binning of the spectra is the same as presented in Table 4.1. To begin,

109

a simplified version1 of the χ2 is given as

χ2(β, η) =17∑i=1

(di − βbi(tan2 θ,∆m2) − ηhi(tan2 θ,∆m2)

)2

σ2i

. (5.1)

The di is the data spectrum term with statistical error σi. The β and η are free

parameters scaling the SSM-derived 8B (bi) and hep (hi) rates, respectively, allowing

for arbitrary total neutrino rates to be chosen to minimize the χ2. The SSM rate is

modified for a number of neutrino oscillation parameter sets (tan2 θ, ∆m2), each of

which has χ2-minimizing values of β and η. The final step will be to find which

set (or sets) best favors the data spectrum by utilizing the distinctive signatures

different parameter sets produce.

To further clarify, the terms in equation 5.1, di, bi, and hi are in fact ratios of the

SK-II data and oscillated SSM MC rates over the unoscillated SSM MC rate. This is

shown as

di =Di

Bi +Hi(5.2)

bi =Bosc

i

Bi +Hi(5.3)

hi =Hosc

i

Bi +Hi. (5.4)

The di equation is the spectrum shown in Figure 4.25 and Di is listed in Table 4.1

as the observed total rate. For MC, whereas Bi and Hi are the expected rates in

Table 4.1, Bosci and Hosc

i must be determined from neutrino survival probabilities

and are hence known as oscillated rate predictions.

1Additional considerations relating to systematic errors as well as time variation effects will bediscussed later.

110

5.1.1 O R P

The SSM MC rate was obtained in the previous chapter to show how a SSM solar

neutrino flux would appear in SK-II and to compare with actual observation. This

MC rate is transformed by including the probability of electron neutrino survival.

The electron neutrino survival probability is given by Pνe(Eν) and incorporated into

Equation 4.14:

r(E) = ΦSSM Ne

∫∞

0

∫∞

0dEνdEeφ(Eν)R(Ee,E)

(dσνe

dEePνe +

dσνµ,τdEe

(1 − Pνe)). (5.5)

The dEν and dEe integration is taken over all true neutrino and recoil electron

energies. E is the recoil electron energy in the SK-II tank. The no-oscillation limit

occurs when when Pνe(Eν) = 1. For other probabilities, the σνµ,τ cross section must

be considered due to νµ and ντ interacting in SK-II (albeit with a much lower cross

section than νe).

The explicit form of Pνe(Eν) depends on whether or not matter effects play a

significant role in an electron neutrino’s journey from the core of the Sun to the

SK-II tank. Since this analysis considers neutrinos of mass differences equal to and

greater than 10−9eV2. Vacuum oscillations can be ignored. Oscillation must then

adjust for matter effects prompting P(Eν) to have a matter effective probability (see

Section 1.3.3).

However, the effective mass squared difference and mixing angle assume con-

stant mass densities. From the profiles in Figure 5.2, it is seen that densities in the

Sun and Earth vary greatly with the core-surface distance. This must be taken into

account and is done so numerically by adjusting the value A to correspond to the

correct solar density at r. Since it is assumed that the density in the Sun varies only

radially and is otherwise isotropic when viewed from the neutrino-producing core,

the probability of flavor change does not vary with time. The path through the

111

Earth is accounted for similarly but is slightly more complicated due to the Earth

axis rotation. The path of the neutrino through the Earth to SK transverses different

chords within the Earth depending on the time of day. A continuous model of the

myriad of different paths a neutrino might take is not developed for this analysis.

Instead, a system of binning based on the SK-II detector’s zenith angle with the

Sun is adopted. 1000 bins are utilized for propagation through the Earth (night

with 0 < Θz < 1) and 1 bin is used when Earth matter effects do not occur (day with

−1 < Θz < 0). P(Eν) becomes P(Eν, cosΘz) and can be determined based on which

bin the neutrino would be traveling.

Since Di represents the total day and night combined data rate for the i-th energy

bin, Bosci and Hosc

i need to be expressed as a 24-hour rate as well. Taking the newly

defined P(Eν, z), where z = cosΘz, and placing it in the oscillated rate prediction of

equation 5.5, Bosci and Hosc

i can be obtained for SK-II observed energy E:

Bosc(E) =1001∑j=1

τ(z j)τtot· r

8Bi (E, z j), Hosc(E) =

1001∑j=1

τ(z j)τtot· rhep

i (E, z j), (5.6)

where τ(z j) equals the live time in the jth zenith angle bin and is obtained from LE

and SLE trigger live time data. Figure 5.1 shows the 1000-bin night live time. τtot is

the total live time of the SK-II experiment (791 days). The 8B and hep superscripts

on the un-normalized rates distinguish which SSM flux φ(Eν) is used to compute

the rates. The oscillated SSM rates in Equation 5.6 are then binned for the SK-II

energy spectrum by

Bosci =

∫ Ehigh,i

Elow,i

dE · Bosc(E), Hosci =

∫ Ehigh,i

Elow,i

dE ·Hosc(E) (5.7)

for the i-th energy bin with boundaries Elow,i and Ehigh,i.

112

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1cos(ΘZ)

Day

s

0.45

0.46

0.47

0.48

0.49

-1 -0.5 0 0.5 1

Osc

illat

ed M

C/M

C

7-7.5 MeV

7.5-8 MeV8-8.5 MeV8.5-9 MeV9-9.5 MeV

9.5-10 MeV10-10.5 MeV10.5-11 MeV11-11.5 MeV11.5-12 MeV12-12.5 MeV12.5-13 MeV13-13.5 MeV13.5-14 MeV

-0.5 0 0.5 1

cos(ΘZ)

14-15 MeV15-16 MeV16-20 MeV

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

distance/sun radius

e- den

sity

in m

ol/c

m3

10-2

10-1

1

10

10 2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

02468

101214

0 0.2 0.4 0.6 0.8

Core Mantle

distance/earth radius

dens

ity in

g/c

m3

cos Θz

CoreMantle

0 0.2 0.4 0.6 0.8 1

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

113

5.1.2 SK-II S E T

The χ2 of equation 5.1 only includes statistical errors. Therefore, it is necessary to

add consideration for systematic uncertainties. These come in two types: energy

correlated and energy uncorrelated.

E C S E

The energy correlated (EC) systematic uncertainties consist of errors on the energy

scale, energy resolution, and the theoretical 8B neutrino spectrum (the assignment

of these errors is given in Chapter 4). Since these uncertainties directly impact

the shape of the SK-II energy spectrum, the χ2 must reflect these possible changes

when comparing Monte Carlo to data.

The assigned EC systematic uncertainties in Table 4.3 are applied to the spectrum

as multiplicative energy shape factors to reflect an actual one sigma shift. However,

since the data may better agree with a lesser or greater shift from the one sigma

assigned value, the factors are varied until the χ2 gives a minimum value.

Written out explicitly, the energy shape factors are

fx(Ei, δx) = (1 + δx ∗ ε±

x (Ei))−1

where x denotes the type of EC error (B for 8B spectrum shape, S for energy scale,

and R for energy resolution), δx is a unit-less parameter that is used to vary the

assigned error, and ε±x (Ei) is the assigned percent error from Table 4.3. When δx

is less than 0, the absolute value of the negative error |ε−| is used. These form a

combined energy shape factor

f (Ei, δB, δS, δR) = fB(Ei, δB) × fS(Ei, δS) × fR(Ei, δR),

114

which is used to modify the predicted combined rate ratio as

(β · bi + η · hi

)× f (Ei, δB, δS, δR). (5.8)

The χ2 becomes

χ2(β, η, δB, δS, δR) =17∑i=1

(di − (βbi + ηhi) × f (Ei, ~δ)

)2

σ2i

+ (δB)2 + (δS)2 + (δR)2. (5.9)

The δx factors are constrained to 0 to assure sensible sigma-level values when the

χ2 is minimized. The σi remain statistical uncertainties only. See Figure 5.3 for the

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

0.3

0.35

0.4

0.45

0.5

0.55

10 15 20Energy(MeV)

Osc

illat

ed/U

nosc

illat

ed

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

The f energy shape factor is essentially a correction to reflect possible uncertain-

ties in the data spectrum di. However, rather than modify the data according to the

115

assigned EC systematic errors, the predicted combined rate is inversely modified

thereby retaining the χ2 formulation and treating the δxs as minimizing parameters.

The minimization itself is difficult to perform analytically in full so the β and η pa-

rameters are calculated analytically at each point of a three-tiered iteration of the δx

parameters. This is done coarsely at first (each δx is independently incremented by

0.5 sigma) and then repeated in a smaller, minimally localized area using a simplex

search. The minimum χ2 is then identified for a unique combination of β, η, and ~δ.

E U S E

The energy uncorrelated (EU) systematic uncertainties are those which are not

reflected by explicit energy related errors. Mainly, they are comprised of uncer-

tainties in event reconstruction, extraction, and simulation and must be accounted

for in the oscillation analysis.

Table 4.2 list the EU systematic errors for the total flux measurement in SK-

II. A complete treatment of these systematics would include not only an error

estimate for the total flux but also an estimate for each individual energy bin as

is done with the EC systematic uncertainties. For use in the χ2, they would then

be combined in quadrature with the energy spectrum’s statistical uncertainties

and σi in Equation 5.9 would refer to the combined statistical-systematic errors

of the SK-II spectral bins. This is the method applied to the SK-I analysis but is

omitted for SK-II. This is due to the EU errors having negligible contributions when

compared to the large uncertainty in the SK-II energy scale (an EC error which is

notably small for SK-I). In the observed spectrum, the energy scale systematic error

increases with increasing energy leaving SK-II greatly affected not only due to the

large 1.4% sigma total value (SK-I: 0.64%), but also due to the analysis threshold

of 7.0 MeV (SK-I: 5.0 MeV). Below 7.0 MeV, the energy scale error would not have

such a dominating effect and thus EU systematics would need to be included. But

116

in SK-II, energies below 7.0 MeV are not analyzed.

However, the total flux EU systematic uncertainty is applied. This is due to the

SK-II spectrum constraining the combined rate (βφ8B + ηφhep) in the χ2. To account

for the total flux EU systematic error, it is helpful to start by Taylor expanding2

equation 5.1 (the energy shape factors will be added again shortly) to second order

around the minimum, βm and ηm. Written in terms of the curvature matrix, it

becomes3

χ2(β, η) = χ2m(βm, ηm) +

(β − βm

η − ηm

)T

C0

(β − βm

η − ηm

), (5.10)

where the curvature matrix C0 is defined as

C0 =

17∑i=1

b2

iσ2

i

bi·hiσ2

i

bi·hiσ2

i

h2iσ2

i

.It is known in this formulation that the curvature matrix is proportional to the

inverse of the errors of the fitting parameters β and η. Therefore it is possible to

scale the curvature matrix by αsys to allow the addition of the total EU systematic

error on the total flux scale factors. αsys is defined as

αsys =σ2

0

σ20 + σ

2sys, where σ2

0 =

17∑i=1

1σ2

stat,i

and χ2α(β, η) = χ

2m(βm, ηm) +

(β − βm

η − ηm

)T

αsys · C0

(β − βm

η − ηm

). (5.11)

It can be seen in the above equation that the minimum does not change (the second2The Taylor series of a quadratic function is exact and not an approximation due to the third and

higher derivatives vanishing.3Since setting the partial first derivatives of χ2 with respect to β and η to zero is required to

calculate the parameters’ minimizing values, the first order term in the Taylor expansion, wheresuch first derivatives would appear, is omitted.

117

order term vanishes at β = βm) but β and η are allowed a greater range that directly

reflects the increased error due to the total flux systematic uncertainty. Adding the

energy shape factors, the total χ2 for the SK-II energy spectrum shape is

χ2spec(β, η) =Min

(χ2α

(β, η, δB, δS, δR

)+ (δB)2 + (δS)2 + (δR)2

). (5.12)

This treatment can also be found in [34].

5.1.3 T-V A

So far, the χ2 comparison of data and MC has been confined to the energy spectrum

shape. While this is sufficient to exclude some parameter set candidates, it is

excluding the phenomenon of matter effects which can provide a more powerful

tool when investigating flavor change.

The likelihood function that was used to extract the solar signal from background

(Equation 4.15) can be modified with a scaling function that reflects the deviation

from the average number of events arising from time variations. Using a suitable

substitution for the zenith angle to reflect its time dependence, t ≡ Θz, the oscillated

rate predictions of Section 5.1.1 can be used to form the ratio ri(t j)/ravei where t j is

the time that corresponds to the j-th event and the denominator is the average over

all zenith bins. This then scales the probability density function of solar neutrino

events si, j giving a likelihood of

L = e−Σi(Bi+S)Nbin∏i=1

ni∏j=1

(Bi · bi j + S · Yi · si j ·

ri(t j)rave

i

). (5.13)

With a likelihood function describing solar neutrinos with a 24-hour period time

variation, it is possible to test the increase in likelihood when compared to neutrinos

with the time variation averaged (ri/ravei = 1, the original likelihood). This is done

118

by a likelihood ratio test and then casting it in terms of a change in χ2 values:

−2(

logL

Lave

)= −2

(log L − log Lave

)= χ2

tv − χ2ave = ∆χ

2tv

Before combining with the SK-II spectrum χ2spec, it is noted that the oscillated

rate predictions are altered by the flux and energy shape factors coming from the

χ2spec fit. This results in the fitted rate predictions (βbi + ηhi) × f (Ei, ~δ) in χ2

spec and

(βr8B(t j)+ηrhep(t j))× f (Ei, ~δ) in the likelihood differing only by the exclusion of energy

binning and a sum of all zenith angle bins in the time-variation rate predictions.

The full SK-II χ2 becomes:

χ2SK-II = χ

2spec + ∆χ

2tv. (5.14)

5.2 R

In presenting the SK-II χ2 fit results, two initial distinctions will be made: the recoil

electron spectrum fit of equation 5.12 and the time variation fit of equation 5.14. The

purpose for differentiating the fits is to show how the recoil spectrum shape alone

can favor certain oscillation parameters. This is purely the effect of the energy

dependence on neutrino survival probability and is heavily dependent on the

mixing angle and mass-squared difference. With the time variation fit, disfavored

regions will be seen based on Earth’s matter effects prompting νe regeneration in

the Earth.

To remind the reader, the χ2 fits in the equations referenced above have no

constraints on the β and η parameters. After minimizing these parameters for a

specified oscillation parameter set, tan2 θ and ∆m2 are systematically incremented

in value until 90,601 sets have been minimized. The χ2 maps are therefore two

dimensional depending on tan2 θ and ∆m2 and contours are plotted for minimum

χ2 values corresponding up to a 95% confidence level (c.l.).

119

Lastly, each contour χ2 plot contains three fitted maps: SK-II (violet), SK-I (thin

black line) and SK-I and SK-II combined fit (light blue). In the SK-Combined fit,

two separate χ2’s are minimized together sharing the flux factors β and η. Also,

the 8B spectrum shape systematic error is common between the two. All other

systematics, data, and MCs are not correlated.

5.2.1 SK-II S & T-V F

In the spectrum fit, SK gives exclusion areas where the oscillation parameters are

not favored at 95% c.l. (Figure 5.4, left side). While the significance of the SK-II

exclusion is smaller than that of SK-I, it does follow the same trend and can be

seen on the right side of Figure 5.4 when SK-II’s exclusion probability is lowered

to the 68.3% c.l. The loss of sensitivity is due to both the combined effects of larger

statistical errors, larger EC errors, and the absence of the four low-energy bins

between 5.0 and 7.0 MeV. This is evident in the values of the χ2 itself where less

data and larger denominators bring the SK-II value down relative to SK-I. At their

respective minimum χ2 values, SK-II is almost 17 units less than SK-I.

SK-II, while excluding fewer oscillation candidates, does offer slight increases

in exclusion power in the SK-Combined fit, most notably along tan2 θ at ∆m2≈

5 × 10−4eV2 and around tan2 θ ≈ 0.1 and ∆m2≈ 10−6eV2.

By adding the time variation term ∆χ2tv to the χ2, greater exclusion is obtained.

Also, the addition of SK-II data to SK-I continues to show improvement. Contours

at 95% c.l. are shown in Figure 5.5. These results, along with the spectrum fit, are

independent of third party data.

The SK-II flux factors β and η have minimizing values that correspond to 0.37 ×

ΦSSM and 50.78 × ΦhepSSM for the 8B and hep fluxes respectively. The large value

of hep is due to η compensating the low β and for the higher 16-20 MeV bin in

the Data/SSM spectrum. The energy shape factors give best fit sigma scalings of

120

νe→νµ/τ (95%C.L.)

∆m2 in

eV

2

10-9

10-8

10-7

10-6

10-5

10-4

10-3

tan2(Θ)10-4 10-3 10-2 10-1 1 10 102

νe→νµ/τ (68.3%C.L.)

∆m2 in

eV

2

10-9

10-8

10-7

10-6

10-5

10-4

10-3

tan2(Θ)10-4 10-3 10-2 10-1 1 10 102

Figure 5.4: The minimumχ2 map for various oscillation parameters tan2 θ and∆m2.The left side shows at 95% c.l. SK-II (violet), SK-I (black line) and SK-Combined(light blue) exclusion areas. The right side shows SK-II only at 68.3% c.l. to illustratethe consistency, albeit less sensitivity, of the SK-II exclusion when compared to SK-Iat 95% c.l.

121

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

resolution systematic errors.

νe→νµ/τ (95%C.L.)

∆m2 in

eV

2

10-9

10-8

10-7

10-6

10-5

10-4

10-3

tan2(Θ)10-4 10-3 10-2 10-1 1 10 102

νe→νµ/τ (68.3%C.L.)

∆m2 in

eV

2

10-9

10-8

10-7

10-6

10-5

10-4

10-3

tan2(Θ)10-4 10-3 10-2 10-1 1 10 102

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

5.2.2 SK-II R C F

To render regions in the χ2 plane where data favors various oscillation scenarios,

it is necessary to constrain the total rate of 8B neutrinos during fitting. Previously

when plotting for excluded regions, the 8B rate factor β was free to take any (pos-

itive) value in order to minimize the difference between data and MC oscillated

predictions. For the rate constrained fit, the rate factor β is tied to the total rate of

all active flavors of neutrinos measured via neutral current (NC) interactions on

Earth .

122

First, to constrain β, a term of (φcon. − β)2/σ2φ is added to the χ2 of equation 5.14.

The value of φcon. and its accompanying error is the NC value coming from the

measurement of SNO [17]. It is again divided by the BP2004 flux value and is

0.85 ± 0.07.

Figure 5.6 shows contours for both the SSM and NC constrained fits. It favors

the LMA and LOW regions in the SK-I and SK-Combined contours. This is due

to the relative flatness of the LMA and LOW spectral distortions. SK-II alone is,

again, lacking the sensitivity of SK-I.

It is important to mention that the comparison between the SK-I and SK-

Combined fits. Unlike with the spectral and day-night exclusion fits, the rate

constrained χ2 of SK-Combined does not offer greater precision despite the inclu-

sion of SK-II data. The LMA and low regions essentially remain unchanged.

The SK-Combined best fit parameters are tan2 θ = 0.55 and∆m2 = 6.6×10−5 eV2.

This is in close agreement with the SK-I result [18] of tan2 θ = 0.52 and ∆m2 =

6.3 × 10−5 eV2. The fitted 8B flux is 0.84 ×ΦSSM.

5.3 C O S E

5.3.1 S N O

SNO’s published flux and day-night asymmetry values are used to further con-

strain the allowed oscillation parameter region. The data include the NC and CC

flux measurements from both the 306-day D2O phase [16] and the 391-day salt

phase [17]. Their values are listed in Table 2.2. The SNO analysis presented here is

a total rate comparison of data and prediction and does not include any information

of the SNO spectra.

MC oscillated rate predictions that reflect the SNO energy resolution and unique

scattering target, deuterium, are needed for comparison with the data. The oscil-

123

νe→νµ/τ (95%C.L.)

∆m2 in

eV

2

10-9

10-8

10-7

10-6

10-5

10-4

10-3

tan2(Θ)10-4 10-3 10-2 10-1 1 10 102

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

124

lated rate predictions are formed in the same manner as SK-II (Equation 5.5) but

with the substitution of the neutrino-electron differential cross section with the

neutrino-deuteron differential cross section. Also, the resolution function R(Ee,E)

is given analytically for SNO and is

R(Te,T) =1

√2πσT

exp[−

(Te − T)2

2σ2T

], (5.15)

where a the change from total energy E to kinetic energy T is taken into account

when integrating with the differential cross section. The value σT differs between

Phase I and Phase II and is

σT(Te) = −0.0684 + 0.331√

Te + 0.0425Te Phase I,

σT(Te) = −0.131 + 0.383√

Te + 0.03731Te Phase II.

The rate predictions are analogous to the rates in 5.7 but are integrated over all

energies to form a total rate with no spectral information. They are then divided by

their unoscillated counterparts and are labels BCC and HCC. They are differentiated

from the SK-II spectrum predictions bi and hi by the uppercase lettering. The di

ratio has as its SNO equivalent value

DCC = φCC/φSSM,

where φSSM is the Standard Solar Model flux of 5.79×106 cm−2 s−1 and φCC is the

value listed in Table 2.2. No MC predictions are made to compare with the NC flux

since the ratio DNC = φNC/φSSM is directly reflected by the flux factor β.

With these terms, a SNO CC-NC χ2 can be formed using the flux factors β and

125

η for minimization. Written in matrix form,

χ2SNO(β, η) =

−→φTV−1−→φ (5.16)

where

−→φ =

(DCC − βBCC − ηHCC

DNC − βb − ηh

)and V−1 =

σ2CC σ2

ct

σ2ct σ2

NC

.The constants b and h are the SSM ratios of 8B and hep fluxes over the total flux and

are b = 0.9986 and h = 0.0014. The σct matrix element is the cross term correlation

of the CC and NC fluxes and can be found from

σ2r = σ

2CC

(∂r∂φCC

)2

+ σ2NC

(∂r∂φNC

)2

+ 2σ2ct

(∂r∂φCC

)(∂r∂φNC

), (5.17)

where r ± σr is the published CC/NC ratio with uncertainties and σCC and σNC are

the published uncertainties for the CC and NC fluxes.

The final SNO χ2 includes martix terms for both phases of SNO. Also, the Phase

I day-night asymmetry (7.0 ± 5.1%) is added as a constraint. The SNO χ2 is

χ2SNO(β, η) = χ2

SNO,I(β, η) + χ2SNO,II(β, η) +

(ADNCC − ADNpred(β, η))2

σ2ADN

, (5.18)

and is added to the SK-I and SK-II χ2. The resulting contour is shown in Figure 5.7.

It shows that with the addition of the CC fluxes and day-night asymmetry of SNO,

the LOW solution is no longer favored at 95% c.l. and only LMA remains.

5.3.2 R E & KLAND

The Homestake, GALLEX/GNO, and SAGE [36] rates can also be added to the χ2 to

form a ”global” contour encompassing 5 different solar neutrino experiments. The

126

νe→νµ/τ (95%C.L.)

∆m2 in

eV

2

10-9

10-8

10-7

10-6

10-5

10-4

10-3

tan2(Θ)10-4 10-3 10-2 10-1 1 10 102

Figure 5.7: The SK-I, SK-II, and SNO combined contour at 95% c.l.

127

radiochemical rates are constrained to the best fit β and η factors of the SK-SNO fit.

The parameter set that corresponds to a minimum χ2 is found to be tan2 θ = 0.40

and ∆m2 = 6.03 × 10−5eV2 and is in the LMA region of solutions.

The KamLAND [37] experiment, a liquid scintillator experiment measuring the

inverse beta decay process νe + p → e+ + n, was initially constructed to verify

the solar measurements’ favored LMA solution. Detecting antineutrinos from

surrounding nuclear reactors in Japan, KamLAND is able to measure ∆m2 with

greatly improved precision. The best fit contour is shown overlaid with the global

solar contour in Figure 5.8 and does confirm the LMA as the most likely parameter

region for quantifying νe → νµ,τ oscillation.

5.4 C

The conclusion of Super-Kamiokande-II has brought measurements consistent with

previous solar experiments. The 8B flux in SK-II is (2.38 ± 0.05(stat.)+0.16−0.15(sys.)) ×

106 cm−2sec−1 and a day-night asymmetry value was observed to be −0.063 ±

0.042(stat.) ± 0.037(sys.) which is consistent with zero. SK-II has brought the total

SK time-dependent flux measurement to a length of 9.5 years and this measurement

is compared with solar activity in solar cycle 23 resulting in no strong correlation.

In the combined SK-I and SK-II global oscillation analysis, the best fit is found

to favor the LMA region at tan2 θ = 0.40 and ∆m2 = 6.03 × 10−5eV2, in excellent

agreement with previous solar neutrino oscillation measurements. Although no

additional precision in the measurements was gained, reconstruction tools and

reduction cuts that were developed to compensate for SK-II’s loss of photocathode

coverage are now being used to lower the threshold of the SK-III analysis. SK-II, in

short, was the catalyst for improved research techniques at the Super-Kamiokande

experiment.

128

νe→νµ/τ (95%C.L.)

∆m2 in

eV

2 x10-4

0

1

2

sin2(Θ)0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Figure 5.8: The SK-I, SK-II, SNO, and radiochemical rate combined contour at 95%c.l. The green overlay is KamLAND.

129

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