Study of capture and annihilation of a few GeV
WIMPs inside the Sun by using an undergroundneutrino detector
Koun Choi
Division of Particle and Astrophysical Science,
Nagoya University
January 18, 2015

Abstract
Dark matter has an enigmatic invisible existence contributing most of the gravity
in the universe. Weakly interacting massive particles (WIMPs) are favored as particle candidates for non-baryonic cold dark matter, since their interaction strength can
explain the thermal relic abundance of dark matter. Recent event excesses or annual
modulation signals reported by direct detection experiments such as DAMA/LIBRA,
CoGeNT, CRESST and CDMS II Si would suggest possible evidence for WIMPs withlight masses of a few to a few tens of GeV/c2 . However, they conflict with null resultsfrom other experiments such as XENON100, LUX and SuperCDMS. A promising way
to identify a WIMP particle is to search for excess of a high-energy neutrino flux coming from the Sun, often referred to as the “solar WIMP search”. WIMPs in the dark
matter halo could scatter off nuclei inside the Sun and be captured, then pair-annihilate
producing neutrinos from decays of the annihilation products which propagate throughthe Sun and may finally be detected in neutrino detectors on the earth. Thanks to thehydrogen-rich composition and huge gravity of the Sun, tight limits on the scatteringcross section of WIMPs to protons by spin-dependent interaction have been placed byneutrino telescopes such as Super-Kamiokande, IceCube, Baksan and ANTARES.
For a critical interpretation of indirect searches by neutrino detectors and comparisons to direct detection experiments, astrophysical uncertainties are key issues. Thisis especially true for the light WIMP region, where the velocity distribution of WIMPs
in the dark matter halo are known to be a large uncertainty source for direct detection
searches. The effect of the uncertainties from the velocity distribution of WIMPs in
the solar WIMP search are precisely evaluated. We examine various sources of uncertainties including local circular speed of the Sun, high-velocity cut of halo, halo shapes
from cosmological simulations and the possible existence of the dark disc. We conclude
that the impacts of the uncertainties on the velocity distribution on the solar WIMP
search are not significant. The estimated size of the total uncertainty is found to beless than 24% for 20 GeV/c2 WIMPs for scenarios without the dark disc. For heavier
WIMPs, the uncertainties are larger but still below about 50%.
Super-Kamiokande (SK), a water Cherenkov neutrino observatory located in Kamioka
mine in Japan, is used to search for light WIMPs. The signal acceptances for a few
GeV/c2 WIMPs are improved by one to two orders of magnitude by using vertexcontained type of neutrino events. To discriminate signal from the overwhelming atmospheric neutrino background events, we perform a least-squares fit on event bins
finely defined by using angle, energy and flavor information. By fitting 3,903-days of1chapter 0

2

SK I-IV data, we examine the best-fit contribution of WIMP-induced neutrino events
added to back-ground events.
As a result, much better sensitivity for spin-dependently coupling WIMPs than any
other direct detection experiment is achieved. For the tested 4–200 GeV/c2 WIMP
masses and several annihilation channels, the data is in agreement with a no-WIMP
contribution and new constraints on WIMP-nucleon cross-sections are obtained. We
set the current best limit on the spin-dependent (SD) WIMP-proton cross section
for WIMP masses below 200 GeV/c2 (at 10 GeV/c2 , 1.49×10−39 cm2 for χχ → bb
and 1.31×10−40 cm2 for χχ → τ + τ − annihilation channel), also eliminating τ + τ −dominantly annihilating WIMPs with spin-independent (SI) coupling as a candidate
for DAMA and CoGENT claimed signals. We also rule out some new parameter spacefor SI coupling WIMPs below 6 GeV/c2 by τ + τ − annihilation channel.
In this thesis, astrophysical uncertainties, including uncertainties from the velocity
distribution, are quantified and reflected on the new limits from the SK for the firsttime. The upper limits on SD or SI WIMP-nucleon scattering cross sections obtainedin this work carry an uncertainty of about 30% for 4 GeV/c2 WIMP, which increases
for higher WIMP masses and resulting up to 60% (100%) for SD (SI) coupling WIMPs
of 200 GeV/c2 mass. We also find that the contradistinctive responses of direct andindirect detections to the velocity distribution tell us that a solar WIMP search can bea good complementary method to entangle uncertainties afflicting direct detection. In
terms of not only good sensitivity for signal but also unique sensitivities and insensitivities to astrophysical properties, the usefulness of this result and the searches fromfuture neutrino detectors are highlighted.

Acknowledgements
Firstly I’d like to thank my supervisor, Prof. Yoshitaka Itow without whose effort
this dissertation would be in very awful shape. I appreciate his advising in the past
four years of my Ph.D studies, for being a big wallet of ideas to which I could always
access and take as much as I could invest, for always (and I select this word withenough statistics) considering my future as an experimentalist, and for almost always
listening to what I want to do in advance to judging my capability.
I also appreciate Prof. Carsten Rott for having suggested me a great experience towork under his guidance, for his patience on the project and warm supports in general.
Many thanks to Piotr Mijakowski, Roger Wendell and Takayuki Tanaka for their greathelps when I first started my research. Thanks for taking your time to answer my naive
questions, and also for being good friends.
Many thanks to my colleagues and friends in the Super-K. Thanks to people whoare always there and maintaining the detector; Hayato san, Miura san, Moriyama san,Hide san, Tomura san, Kameda san, Sekiya san, Nakano kun, Nakahata san, Suzuki
san, and many people whom I can’t recall now, and the officers, security guards andthe observatory and dormitory labourers. I especially appreciate the solar WIMP
paper committee members’ effort in the revision of the analysis. And thanks to Ed forencouraging me to finish and move on.
I also thank Bish san, Lluis, Okumura san, SKanadians and especially Hiro foralways being a good drinking guru and for his wise advices. I also thank my father,
Goya and Stefan, Ssal, Cris, and my muse Jia. Special thanks to my mother for always
supporting me. Also thanks to my defence committee - especially Shiozawa san, whois going to come to Nagoya for my defence on Christmas day!
Finally, thanks to Nagoya university GCOE program for giving me a lot of opportunities.

3

Contents
1 Dark matter and WIMP
9
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.2 The gravitational existence of dark matter . . . . . . . . . . . . . . . .
91.2.1 The history of dark matter discovery . . . . . . . . . . . . . . .
91.2.2 The non-baryonic dark matter . . . . . . . . . . . . . . . . . . . 11
1.2.3 The cold dark matter . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3 Dark matter distribution in the Milky way . . . . . . . . . . . . . . . . 15
1.3.1 Modeling of dark matter distribution, and the standard halo model 15
1.3.2 N-body simulations . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4 WIMP, a particle candidate of cold dark matter . . . . . . . . . . . . . 18
1.5 Current experimental searches for WIMPs . . . . . . . . . . . . . . . . 20
1.5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5.2 WIMP-nucleon scatterings . . . . . . . . . . . . . . . . . . . . . 21
1.5.3 Direct detection searches on WIMP-nucleon scatterings . . . . . 23
1.5.4 Constraints by indirect detection searches on WIMP-nucleon scatterings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.6 Prediction of sensitivity of the Super-Kamiokande for light WIMPs . . 29
2 Capture of WIMPs in the Sun and the search for it in a neutrino
detector
34
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2 WIMP capture in the Sun . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 WIMP thermalization inside the Sun . . . . . . . . . . . . . . . . . . . 37
2.4 WIMP density evolution inside the Sun and the equilibrium condition . 40
2.5 Production of neutrinos inside the Sun . . . . . . . . . . . . . . . . . . 40
2.6 Utilization of DarkSUSY . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3 The velocity distribution and the capture rate of WIMPs
3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Uncorrelated and anti-correlated uncertainties in direct and indirect detections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 The dark matter velocity distribution . . . . . . . . . . . . . . . . . . .
3.4 The effect of WIMP velocity distribution to the capture rate of WIMPs
4

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48
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56

chapter 05

3.5

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

4 The Super-Kamiokande detector
4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Cherenkov radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 The detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 The photomultiplier tubes . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Electronics and data acquisition system in SK-IV period . . . . . . . .
4.6 Detector calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.1 PMT calibration . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.2 Water quality and light reflection . . . . . . . . . . . . . . . . .
4.6.3 Energy scale calibration . . . . . . . . . . . . . . . . . . . . . .

63
63
65
66
68
70
73
73
78
79

5 Simulation of WIMP neutrinos and atmospheric neutrinos at SK
5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 The simulation of WIMP-neutrino flux by WIMPsim . . . . . . . . . .
5.3 The WIMP-induced neutrino events at Super-Kamiokande . . . . . . .
5.4 atmospheric neutrino flux . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Atmospheric neutrino oscillation . . . . . . . . . . . . . . . . . . . . . .
5.5.1 Two-flavor oscillation in atmospheric neutrinos . . . . . . . . . .
5.5.2 Full three-flavor neutrino oscillation in atmospheric neutrinos . .
5.6 Neutrino interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.1 Elastic and quasi-elastic scatterings . . . . . . . . . . . . . . . .
5.6.2 Resonance single meson production . . . . . . . . . . . . . . . .
5.6.3 Deep inelastic scattering . . . . . . . . . . . . . . . . . . . . . .
5.6.4 Coherent pion production . . . . . . . . . . . . . . . . . . . . .
5.6.5 Nuclear effects . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7 Detector simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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88
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91
93
96
96
99
101
102
104
104
105
107
108

6 Description of Super-Kamiokande data used
112
6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.2 Reduction for fully contained events . . . . . . . . . . . . . . . . . . . . 113
6.2.1 First reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.2.2 Second reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.2.3 Third reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.2.4 Fourth reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.2.5 Fifth reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.2.6 FC reduction summary . . . . . . . . . . . . . . . . . . . . . . . 119
6.3 Reduction for partially contained events . . . . . . . . . . . . . . . . . 120
6.3.1 First reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.3.2 Second reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.3.3 Third reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.3.4 Fourth reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 122

chapter 0

6.4

6.5

6.6
6.7
6.8

6.3.5 Fifth reduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.6 PC reduction summary . . . . . . . . . . . . . . . . . . . . . . .
Reduction for upward-going muon events . . . . . . . . . . . . . . . . .
6.4.1 First reduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.2 Second reduction . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.3 Scanning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.4 Background estimation for upward-going muon . . . . . . . . .
6.4.5 Up-µ reduction summary . . . . . . . . . . . . . . . . . . . . . .
Event reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.1 Vertex fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.2 Ring counting . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.3 Particle identification . . . . . . . . . . . . . . . . . . . . . . . .
6.5.4 Momentum reconstruction . . . . . . . . . . . . . . . . . . . . .
6.5.5 Direction reconstruction . . . . . . . . . . . . . . . . . . . . . .
6.5.6 Decay electron search . . . . . . . . . . . . . . . . . . . . . . . .
FC categorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
PC OD stop/through separation . . . . . . . . . . . . . . . . . . . . . .
Up-µ separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6
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123
123
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125
125
125
126
126
127
128
128
133
138
138
142
146
147

7 Search for WIMP neutrinos in Super-Kamiokande149
7.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.2 Analysis strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.3 Analysis method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.4 Sensitivity study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
7.4.1 Sensitivity study for null hypothesis . . . . . . . . . . . . . . . 160
7.4.2 Statistical fluctuation in the signal MC . . . . . . . . . . . . . . 161
7.5 Limits on WIMP-induced neutrino events and flux . . . . . . . . . . . . 164
7.6 Limits on WIMP-nucleon elastic scattering cross-sections . . . . . . . . 168
7.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
8 Discussions: On the uncertainties in solar WIMP search
174
8.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
8.2 Uncertainties related to detector response and background . . . . . . . 175
8.2.1 Uncertainties in background atmospheric neutrino flux . . . . . 175
8.2.2 Uncertainties in detector response: Neutrino interactions . . . . 178
8.2.3 Uncertainties in detector response: event selections . . . . . . . 183
8.2.4 Uncertainties in detector response: Event reconstructions . . . . 188
8.3 Uncertainties in WIMP neutrino propagation . . . . . . . . . . . . . . . 190
8.4 Uncertainties in nuclear physics and astrophysics . . . . . . . . . . . . 191
8.4.1 Form factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
8.4.2 Solar model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
8.4.3 WIMP phase distribution . . . . . . . . . . . . . . . . . . . . . 195
8.4.4 Dynamics of solar system . . . . . . . . . . . . . . . . . . . . . 195

chapter 0

8.5

8.6

8.4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Uncertainties in particle physics . . . . . . . . . . . . . . . . . . . . . .
8.5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.5.2 Isospin violation . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.5.3 Equilibrium condition . . . . . . . . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7
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200
200
201
202
202

9 Conclusion

204

Bibliography207

To Lily, Ola and Oksu

8

Chapter 1
Dark matter and WIMP
1.1

Overview

The existence of cold non-baryonic dark matter is suggested from its gravitational
effects. Weakly interacting massive particles (WIMPs) are prominent candidates to
explain the cold non-baryonic dark matter particle. They naturally arise in Super
Symmetry (1) or models with large extra dimensions (2). By crossing symmetry, they
could be searched through the scattering cross section with ordinary matters in theStandard Model.
In Chap. 1.2, the observation history of dark matter is briefly summarised. In
Chap. 1.3, the distribution of the dark matter in the Galaxy will be described. In
Chap. 1.4, a short introduction of WIMP will be given. Finally in Chap. 1.5, methods
to detect the WIMPs and the current achievements will be briefly described.

1.2

The gravitational existence of dark matter

1.2.1

The history of dark matter discovery

Dark matter was first postulated due to the discovery of the discrepancy between
the mass deduced from gravitational potential and the mass estimated from luminous
matter. In 1933, Fritz Zwicky observed the radial velocities of eight galaxies in the
Coma galaxy cluster, and found that the observed velocity dispersion was much larger
than that expected from the gravitational field due to the visible matter. His explanation for this observation was that some “dark” matter is present with a much greatermass than luminous matter (3), and it might be regarded as the first reference to dark
matter (4).
Subsequently, another evidence of dark matter arose in 1939 by Horace Babcock (5).
He measured the Doppler shifts of star lights at various places around the outer edgeof the galaxy M31 and found they showed unexpected behaviors: the speed of stars
orbiting around the galactic center did not decrease following the expected Keplerian

9Chapter 1

10

dynamics, but instead remained roughly constant to far distances from the galactic
center. Today Babcock’s result is interpreted as evidence of dark matter, whose massive
gravitational field holds the stars more strongly than the visible field would do. But
at that time, he suggested either the possible foreground for the light propagation or
the modification of the Keplerian dynamics as an explanation.
Despite several related results obtained by others, until the 1970’s the scenario of
the existing dark matter was not accepted as a standard idea. In the 1970’s Rubin
and Ford measured the Doppler shifts of star lights in the external parts of the spiralgalaxies (6). The observation was very clear and straightforward that it was enoughto convince the community with the idea that the luminous stars are probably only
the visible tracers of a invisible massive volume in the galaxy. This observation was
extended to larger radius by Roberts and Whitehurst (7) using 21-cm line observations
made with the 300-ft radio telescope, that reached to 30 kpc as shown in Fig. 1.1.

Figure 1.1: The rotation curve of M31. The triangles beyond 12 kpc show the opticalobservation by Rubin & Ford (1970), the circles at the outer edge show the 21-cm line
observations by Roberts & Whitehurst. Note that the rotational curve appears to beflat beyond ∼ 15 kpc. Image is taken from (7).
A good visual and direct evidence for dark matter was the observation of the Bullet
cluster by both gravitational lensing and X-ray. The idea of directly estimating themass of an object by observing gravitational lensing effect by foreground galaxy clusters
was first suggested by Zwicky (3). The first observation of the gravitational lensing wasdone by the Hubble Space Telescope in the 1990’s. The distorted galaxy images taken
were used to calculate approximately how much mass should present behind the “lens”.
On the other hand, the observation of X-rays emitted by very hot gas in the cluster

Chapter 1

11

could be used to derive the mass profile of the baryonic matter. In the Bullet cluster,
the mass which is in charge of gravitational lensing, and the baryonic mass which isin charge of X-ray emission, were observed to be in separated locations, in a shape ofa bullet. It was interpreted that the two clusters of galaxies were colliding and thepassing gas particles belonging to each cluster were slowed down by electromagnetic
interactions, where much of the mass of the two clusters passed through each otherwithout interactions. The observation of the Bullet cluster by Clowe et al. (8) is shownin the Fig. 1.2.

Figure 1.2: The ‘Bullet cluster’, also known as 1E 0657-56. The pink-colored region
represents hot gas observed by the Chandra X-ray space telescope, the blue region
represents dark matter seen through its gravitational effects. Image is taken from (8).

1.2.2

The non-baryonic dark matter

By the end of 1970’s, the idea of the existence of the dark matter had been largely
accepted. But still its property was hidden in a veil until its shady property first started
manifesting in the Big Bang nucleosynthesis (BBN).
When the atomic nuclei heavier than hydrogen were primordially formed in the early
universe, the expansion rate of the universe which is related to the total matter density
would have determined the population of each light element. The BBN prediction
matches the observed populations only when the baryonic matter accounts for 4∼5%of the critical density of the universe as shown in Fig. 1.3. This was the first noticethat we are clueless about the left 95% of the energy budget of the universe.
The next strong evidence of the existence of non-baryonic dark matter came fromthe observed discrepancy between the total cosmological density of matter and thedensity of baryonic matter.

Chapter 1

12

Figure 1.3: The BBN prediction of the light elements. Predicted abundances of eachelement, with the width of the curve indicating the 2σ theoretical uncertainties, are inagreement with the measurements in the consistency interval (vertical band, Copi etal (9)) of baryon density. The blue consistency interval band corresponds to Tytler etal (10)’s determination of the deuterium abundance. Image is taken from (11) and thecourtesy of K. Nollett.
In 1964, Cosmic Microwave Background (CMB) was detected and brought a doubt
to the presumption of baryonic dark matter. To form the currently observed structures
as galaxies, clusters and superclusters from the density evolution of the baryonic matters, the theoretical calculation said that the initial density fluctuation of the universeat the epoch of decoupling (“recombination”) of the CMB was required to be above∼ 10−3 . And it was expected that we could find the footprint of it on the CMB, as
fluctuations of the temperature. The early observations of the CMB couldn’t find thisbut the upper limits were set to be much lower than the predicted limit of 10−3 . When
the COBE satellite finally measured the temperature fluctuation of the CMB, it appeared to be two orders of magnitude lower than expected; only about 30 µK where
the average temperature of the CMB was 2.7K. The COBE result showed a need fora electrically neutral (which is non-baryonic) mass which could avoid the radiationpressure slowing down the gravitational growth of density fluctuations, so that it could
already initialise the structure formation in advance to the recombination.
Later on, more precise analyses on the anisotropy of CMB were carried on and gave

Chapter 1

13

us stronger constraints on the abundance of non-baryonic matter. The typical angularscales of the oscillations of radiation in the early universe were measured as a angularpower spectrum of the CMB anisotropies, which revealed the effects of baryonic matterand dark matter. By measuring the first three peaks with good precision, the analysisof the WMAP nine-years results (12) estimated that 4.6% of the total energy of thecurrent universe is made up with baryonic matter, and the total matter density is about
28.6% of the total energy density, which indicated that about 24% of the energy wascomposed of non-baryonic dark matter. Figure 1.4 shows the all-sky picture and a pie
chart of the energy budget of the current universe, obtained from nine years of WMAP
data.

Figure 1.4: Left: The all-sky picture from nine-years WMAP data. The colors correspond to temperature fluctuations. Right: The contents of the energy source of thecurrent universe revealed by nine-years WMAP observation (13).
The evidence for that the majority of dark matter can not consist of baryons further
emerged from lensing observations. Gravitational lensing surveys excluded nearly all
the plausible baryonic objects such as the Massive Compact Halo Object (MaCHO)shiding in our galactic halo as candidate of dark matter.
Baryonic acoustic oscillation (BAO) provided stricter constraint on the total matter density and baryonic matter density of the universe. BAO characterizes acoustic
density perturbations in the early universe. In early stage of the universe, photons andbaryons were bound together until the universe expanded and cooled enough to formhydrogen. In this recombination epoch, photons decoupled from baryons and quickly
diffused away leaving the baryon wave crests stalled. The remaining over-dense baryon
shells are predicted to be observed at approximately 100h−1 Mpc, where h is defined by
the Hubble constant H0 as h = H0 /100 km s−1 Mpc−1 . As baryonic matter and dark
matter might have affected the oscillations by their gravities in different effects, theamout of total matter and barionic matter could be derived from the power spectrumof galaxies. The final study of the 2dFGRS found Ωm = 0.231 ± 0.021 and Ωb / Ωm
= 0.185 ± 0.046 (14); a study based on the SDSS data with the combination of Union
supernova result (15) yielded the total matter content Ωm = 0.286 ± 0.018 and theChapter 1

14baryonic matter content Ωb h2 = 0.02267 ± 0.00058 at 68% confidence limit (16), whereΩi = ρi /ρc with ρc , the critical density which was defined as 3H 2 /8πG.

1.2.3

The cold dark matter

It is natural to consider the neutrino as a candidate of dark matter due to its
nature of being non-baryonic and electrically neutral. However, a simple calculationshows that their total relic density is predicted to be2Ων h =

3Xmνi.
νi =1 93eV

(1.1)

The best laboratory constraint on neutrino mass comes from tritium β-decay experiments at Troitsk and Mainz (17) asmν < 2.05 eV(1.2)

with 95% C.L.. This implies an upper bound on the total neutrino relic density ofΩν h2 < 0.07.

(1.3)

A more stringent constraint on the neutrino relic density comes from the analysisof CMB anisotropies, combined with large-scale structure data, suggesting Ων h2 <0.0067 (95% C.L.). Thus the abundance of neutrinos is insufficient to be the dominantcomponent of dark matter.
Furthermore, observations imply that the majority of the dark matter can not becomposed of relativistic (“hot”) matter. In the middle of 1970’s, the first red-shiftobservation of bright galaxies showed that the galaxies are not distributed randomly as
supposed in early days, but they formed filaments and voids in a scale of up to several
tens of Mpc. N-body simulations of large scale cosmic structure found that without
dark matter, it could not form the large structures with filaments and voids observed
by SDSS and other surveys in the proper time scale. Additionally, in the scenarios withrelativistic dark matter, the small-scale, i.e. galaxy-sized initial density fluctuations
would have been washed out by the free-streaming of the dark matter (18; 19; 20). As
a result, relativistic neutrinos are ruled out as a main component of dark matter.
When photons decoupled from baryons at the recombination epoch, the distancethat a photon could diffuse became much greater. The photons would free-stream
from high to low density regions while dragging protons and electrons with them. This
process, known as Silk damping or diffusion damping is also known to have wipedout fluctuations; it would have erased all the perturbations existed under ∼ 1013 M .
However, in the scenarios with non-relativistic (“cold”) dark matter, the problem can besolved. If the dark matters were already cold during the period of structure formation,
as they were pressure-less, they might not have experienced Silk damping and stayedwith the original size of the perturbations. At the end of recombination where the smallscale anisotropies in the baryons had been all removed away by damping photons, theChapter 1

15

baryons rapidly fell into the potential wells of dark matter perturbations. As a result,in the scenario of the universe where the cold dark matter is the dominant form of
the matter, the Silk damping observed in CMB data is irrelevant for galaxy formation(21).
As a result, non-baryonic cold dark matter is needed to take the role of the seed
for perturbation in the evolution of the universe. This “bottom up” model of structureformation with cold dark matter has been proved by observations; the structure formation in the universe started from the smallest structures and then the gravitational
merging processes have formed the clusters of galaxies and so on, hierarchically.

1.3

Dark matter distribution in the Milky way

As summarised above, there are plenty of evidences that most of the matter of theuniverse is in the form of non-baryonic cold dark matter. During the structure formations of cosmological history, massive halos of dark matter are expected to have been
formed within which the visible part of the Galaxies (disk and bulge) are immersed.
During gravitational collapse and subsequent Virialization (“violent relaxation”), thecollisionless dark matter would have formed a halo that is roughly spherical in shape(22).
The fact that the Sun rotating around the galactic center has the speed of ∼220
km/s allows us to infer the depth of the potential well of the Milky way. The local
dark matter halo density near the Sun, ρ0 , is estimated to be 0.3 GeV/cm3 (23).

1.3.1

Modeling of dark matter distribution, and the standard
halo model

For a phase space distribution function of collisionless particles:f (xi , vi , t),(1.4)

the steady-state solution is given by the collisionless Boltzmann equation (24):
3
∂f
∂Φ ∂f
∂f X
+ (vi
−
)=0
∂t i=1 ∂xi ∂xi ∂vi(1.5)

where the density distribution generates the potential:∇2 Φ = 4πGZ

f d3 v.

(1.6)

For a spherical isotropic system, there’s a unique relationship between ρ(r) andf (v) given by Eddington’s equation (25):
1 Z E d2 ρ dΦ√f (E) = √ 2,
8π 0 dΦ2 E − Φ(1.7)

Chapter 1

16

where Φ is the relative potential as a function of radius, and E is the relative energy,
E = Φ − mv 2 /2.
The flat rotational curve of the Milky way implies the choice of density profile
ρ(R) = ρ0 (

R0 2
),
R

(1.8)

in which case the resultant velocity distribution function is the Maxwell-Boltzmann
distribution:22e−v /vrms 3
3f (v)d v = 3/2 3 d v.
(1.9)
π vrmsThe value of vrms can be obtained from the observation of rotational speed. For
example, using the condition of hydrostatic equilibrium for an isothermal sphere:−4πR2 dp = G

M dM
R2

(1.10)22= GM
and vrot, one can get the relation between vrms and vrot ,with p = 31 ρvrms
R2
vrms
d ln R
3
= −3
= .
2
vrot
d ln ρ
2(1.11)The observed value of the circular speed in the solar position v is about q
220 km/s,
which leads the local velocity dispersion of the dark matter halo vrms to be ' 3/2v '270 km/s.
High-velocity dark matters might escape from the Milky Way leaving a cut off inthe Velocity Distribution Function (VDF) at the galactic escape speed. The equation
formally derived extends to infinity, therefore velocity distribution is usually truncatedat escape velocity by hand as:22e−v /vrms
f (v)d v = 3/2 3 Θ(v − vesc )d3 v.
π vrms3(1.12)

The above description of the dark matter distribution is called the Standard (Maxwellian)Halo Model or SHM, and is commonly used as a canonical model in the dark mattersearches.
However, some important departures from the SHM are observed, and supported
by analytic calculations and numerical simulations. For example Hansen et al. (26)shows that the velocity distribution function (VDF) derived numerically through the
Eddington’s formula with power-low density profiles matches well to the analyticalderivation of non-extensive statistical mechanics (27; 28). The authors suggest that
self gravitating collisionless particles don’t obey the rule of extensivity, so they can’t
be described by the statistical mechanics derived from the Boltzmann entropy but the
non-extensive statistical mechanics from the generalized entropy should be applied.

Chapter 1

17

Among many choices of analytically derivable dark matter profiles, the NFW profile
is a suitable choice to the spherical halos simulated in N-body simulations:ρ0,(1 + rrs )2
rs

ρ(r) = r(1.13)

where ρ0 is the normalization parameter for the density and rs is the scale radius of
the halo. When r  rs , ρ(r) ∝ r−1 therefore the profile is cuspy at the core.
Recent high-resolution N-body simulations show that three-parameter profiles, forexample Einasto profile, fits better to the wide range of simulated dark matter halos.
It has a form of:
ρ(r) ∝ exp(−Arα ).
(1.14)The parameter α defines the steepness in a power law, and by choice of it, the Einasto
profile gets a flat or slow central slope, which doesn’t diverge at the center of the core.

1.3.2

N-body simulations

Cosmological N-body integration is a promising way to understand the dark matter
properties within a framework of the standard ΛCDM model. N-body simulations havebeen successful in showing large scale structure, growth of structure and dark matter
halo properties, evolution of the distribution and clustering of halos, accretion historyof halos and other halo statistics (29) in agreement with observations.
Despite their successes, there have been claimed several critical discrepancies between the observations and the N-body simulations. The “cuspy halo problem” arises
from cosmological simulations that seem to indicate sharply peaked (cuspy) density
distribution at the center of the galaxies, whereas the observations likely have no cusps
there. The discussions have been done on the possibility of that high-resolution CDMonly simulations may be being misinterpreted and maybe baryons strongly modify the
structure of subhalos, therefore it needs better understanding of the effcts of baryonicphysics.
Hydrodynamic galaxy formation simulations involving baryons, with adequate resolution and realistic feedback mechanisms, appear to be able to reproduce the evolutionof galaxies, formation of galactic spheroids via mergers, and galaxy images over wide
ranges in wavelength including stellar evolution and dust. So far, adding baryons ingalaxy formation simulations has been resulting in problems such as “angular momentum catastrophe” in galaxy formation, overcooling therefore disks becoming too small
and resulting in too high baryon-to-dark matter ratio, no bulge-less galaxies, too manystars overall, and so on. But by recent progresses at different mass scales with better
resolution and more realistic feedback mechanisms, the angular momentum catastrophe
appears to be resolved (30).
Recently, N-Body simulations of smoothed-particle hydro-dynamics including explicit H2 and metal cooling, star formation and supernovae driven gas outflows (31)show that the cusp can be removed by starbursts blowing out central gas; Starting at

Chapter 1

18high redshift, rapid and repeated gas outflows following bursty star formation transfer energy to the dark matter component and significantly flatten the originally cuspy
central dark matter mass profile of galaxies with present day stellar masses. “Themissing satellite problem” which means that much smaller number of dwarf galaxies
are observed than expected from simulation, and the “too big to fail (32) problem”
which means that the observed dwarf spheroidal galaxy masses are much smaller than
expected for the most massive subhalos from simulations, are also possibly solved by
future N-body simulations including baryons (33). Figure 1.5 shows an example of a
N-body simulation including baryon (34).
By several cosmological simulations of hierarchical structure formation including
baryons, it is suggested that the merger history of the Milky Way probably allows
the existence of a co-rotating structure made from materials accreted from merged
satellites, known as the dark disc (34; 35; 36; 37; 38). In those simulations the preexisting baryonic disc perturbs the merger process of the host galaxy with satellites, as
it draws satellites to the disc plane by dynamical friction and disrupts them by tidal
force. As a result, the accreted materials settle around the galactic disc of the hosthalo and produce a co-rotating equilibrium structure of dark matter, as well as the hot
stellar disc.

1.4

WIMP, a particle candidate of cold dark matterTo be a good dark matter candidate in terms of explaining gravitational observations, the abundance in each epoch of the universe, speed in early universe, decay
life time and annihilation cross-section etc. would be considered. However, the nonbaryonic nature of dark matter forces us to consider it, not only as a phenomenon
solely appearing in cosmology, but as a possible evidence of new physics beyond theStandard Model. In that vein, most attempts of modeling and detecting dark matter
have been made for target particles which are motivated by existing physics problems.
There are numbers of candidates for non-baryonic cold dark matter particle beyond
the Standard Model. Among the candidates, WIMPs, Weakly Interacting Massive Particles, are most favored as they arise from Super Symmetry (SUSY) or extra dimension
scenarios. Axions are also favored as a candidate of cold dark matter as they are apossible solution for the strong-CP problem.
WIMPs do not interact via electromagnetism, nor do they interact with strong force,but only with weak force. When the universe was at very high temperature, WIMPs
were in thermal equilibrium with the radiation. After the temperature dropped below
the mass of the WIMP as the universe expanded, its number density froze out, anda substantial number of WIMPs have been left. The remarkable fact is that thermal
relics with electroweak interactions can give us relic abundance of WIMPs in agreementwith concordance model, thus it is called the “WIMP miracle”.
In some extension of the Standard Model, there can be found particle models forChapter 1

19

Figure 1.5: The 20 Mpc/h-sized box of a N-body simulation including baryons (34)(top), for zoomed view on the central region (middle) and the galaxy in the center ofit (bottom). In the bottom plot, the red circle shows its Virial radius (264 kpc) andthe halo is concentrated in the center and has many subhalos, as expected. Figures aretaken from (34).

Chapter 1

20

WIMP such as a fourth generation of Dirac neutrino, a Majorana neutrino or a scalarcoupled WIMP. But what made WIMP most famous is that it naturally arises fromSUSY theory which is highly motivated to explain the hierarchy problem in the Standard Model. The most popular candidate, neutralino, is the LSP (lightest SymmetricParticle) in SUSY models. In the Minimal Supersymmetric extension of the Standard
Model (MSSM), the R-parity is conserved and sparticles (superpartners of SM particles) can only decay into an odd number of sparticles plus Standard Model particles.
The LSP is therefore stable and can only be destroyed via pair annihilation, making
it an excellent dark matter candidate (39; 40). They are predicted to have masses
ranging from a few GeV to a few TeV (41).

1.5

Current experimental searches for WIMPs

1.5.1

Overview

Figure 1.6: WIMP interactions with ordinary matter and corresponding detection
strategies.
If WIMPs are to have a cosmological density of order unity, they must have somesmall but finite coupling to ordinary matter; otherwise, they would not have annihilated
in the early universe and would be unacceptably overabundant today. By crossing
symmetry as shown in Fig. 1.6, the amplitude for WIMP annihilation to quarks isrelated to the amplitude for scattering of WIMPs off quarks. Therefore, WIMPs are
generically expected to have small but nonzero coupling to nuclei through the coupling
to quarks as well as self-annihilation.
Dark-matter searches mostly rely on the non-gravitational interactions with theStandard Model particles - weak interactions in the majority of the searches. By thetype of the interaction between WIMPs and the Standard Model particles used in asearch, the WIMP searches can be categorised in big three methods as illustrated inChapter 1

21

Fig. 1.6. The Feynman diagram following the arrow from right to left shows the production of dark-matter particles from the Standard-Model particles. This strategy isused in the WIMP searches in hadron colliders by using mono-jet and mono-photon signatures. The WIMP-nucleon scattering cross section can be deduced by these searches,
however, these searches employ effective field theories and depend on masses of mediating particles. The other Feynman diagram following the arrow from left to rightindicates WIMP annihilation. The annihilation cross-section allows us to search for
WIMPs by its annihilation products. “Indirect detection searches” aim at detecting
the final products as Standard-Model particles among which, gamma rays, neutrinos
and anti-particles are particularly targeted because of its strong back-ground discriminating power or directional information. Target objects are in various places of theuniverse wherever it is expected have WIMP population; Galactic Center, Milky Way
halo, dwarf galaxies and the Sun and the Earth. Dense regions are favored as theannihilation rate goes as the square of the density, but diffuse components from all
directions are also promising because of their competing abundances. Note that thegeneral definition of “indirect” comes from the fact that they search for secondary
products. In this sense, the search for decay products which doesn’t appear in the
diagram also falls into indirect detection category.
Finally, the Feynman diagram directing from bottom to top shows the scatteringinteractions of dark-matter particles with the Standard-Model particles, which are
strategies widely used in commonly called “direct detection”.
Detection of WIMPs in the Sun using neutrino detectors has sensitivity to bothannihilation and scattering cross sections. It is categorised in “indirect detection” inthe sense that it searches for neutrinos from those secondary annihilation products,
however, it is sensitive to WIMP interaction with Standard-Model particles same asdirect detections, by the mean that will be explained in the next chapter (Chap. 2).
In this chapter, methods and current status of the WIMP searches by direct detection and for the neutrinos coming from the WIMP annihilation in the Sun are
discussed.

1.5.2

WIMP-nucleon scatterings

As the elastic scattering of WIMPs off nuclei occurs in the non-relativistic limit inboth direct detection and solar WIMP detection using neutrinos, the simplifications
can be made: the axial-vector current becomes a “spin-spin interaction”, while thevector and tensor currents take the same form as the scalar interaction (42; 43). As aresult, two types of interactions are considered for WIMP search; spin-spin interaction
and scalar interaction. In the case of the spin-spin interaction, WIMPs couple to thespin of the nucleus with spin-dependent (SD) scattering cross-section; in the case of
the scalar interaction, WIMP couples to the mass of the nucleus with spin-independent(SI) scattering cross-section.
The WIMP-nucleon spin-dependent scattering cross-section at zero momentum
transfer can be written as

Chapter 1

22

J +1
G2F µ2N(ap < Sp(N ) > +an < Sn(N ) >)2,(1.15)
πJ
where GF is the Fermi constant, µN is the reduced mass mχ mN /(mχ + mN ) of themass of WIMP mχ and the mass of target nucleus mN . ap (an ) is the coupling constantfor proton(neutron), < Sp(i) >(< Sn(i) >) is the expectation value of the spin content of
the proton (neutron) group in the nucleus and J is the total angular momentum. The
SD scattering can be efficient when a nucleus has large number of unpaired protons or
neutrons.
Figure 1.7 shows possible tree level SD scattering diagrams.
σ SD = 32

Figure 1.7: Feynman diagrams contributing to spin-dependent elastic scattering of
neutralinos off quarks. Squark exchanges are allowed for SUSY neutralinos. Figure istaken from (42).
In the spin-independent interaction, the WIMP-nucleon scattering cross-section canbe written as
4µ2N(Zfp + (A − Z)fn )2 ,(1.16)
π
where fp and fn are coupling constants to the proton and neutron, respectively, A is
the mass number and Z is the atomic number of the target nucleus. The cross sections
for each nucleus is calculated with fp and fn defined for individual model of WIMP
candidate. However, a simplification can be made: assuming fp ' fn , the cross section
can be written as
σ SI =σSI

4m2N µ2N fp2=πm2p(1.17)µ2N,µ2p(1.18)= σ0SI A2

Chapter 1

23

where mp is proton mass and µp is the WIMP-proton reduced mass. Note that all theinformation about a specific model is encoded in fp inside σ0SI , where σ0SI is the SI
WIMP-nucleon (equally to proton and neutron) scattering cross-section.
Figure 1.8 shows possible tree level diagrams contributing to SI scattering. For SI
scattering, one-loop amplitudes for interactions with gluons are also important (42).

Figure 1.8: Feynman diagrams contributing to spin-independent elastic scattering of
neutralinos off quarks. Squark exchanges are allowed for SUSY neutralinos. Figure istaken from (42).

1.5.3

Direct detection searches on WIMP-nucleon scatterings

The basic idea of direct detection is waiting for dark matter scattering off a material
which is buried in the deep underground to reduce background for a very weak signal.
When WIMPs pass through the material, they may experience weak scattering andtransfer a small amount of kinetic energy to a nucleus. The recoil energy of the nucleus
is:4mχ mN
1 + cosθ
1
×.
(1.19)E R = mχ v 2 ×2
2(mχ + mN )2
The first part is the kinetic energy of incoming WIMP, the middle part is the “massmatching factor” between the WIMP mass mχ and the target nucleus mass mN and thelast part is the angular factor, with the θ being a scattering angle in the center-of-massframe. Maximal energy transfer happens for head-on recoil (when θ=0). For the caseof the WIMP and the target have the same mass,
1
ERmax ∼ mχ v 2 .
2(1.20)

It indicates that it is a good choice to select a material to have similar mass with thetarget WIMP. The WIMPs distributed in the dark matter halo in the Galaxy will getboosted in the coordinate of the detector as the solar system moves in the Galaxy

Chapter 1

24with the speed ∼220km/s. The typical incoming velocity of the WIMPs thus will bev ∼ 10−3 c, as a result the expected maximal recoil energy is about:
1 mχ
)keV,
ERmax ∼ (2 GeV

(1.21)

resulting tens to hundreds of keV signal for tens to hundreds GeV mass of WIMPs.
Observing this tiny amount of deposited energy is the challenge of direct detection
experiments.
The expected event rate is:
dR
ρχ Z
dσ 3
= NT
d ~v
f (v)vdERmχ vmindER(1.22)

with NT as the number of nuclear targets.
Typically a very low event rate is expected; for σ ∼ 10−46 cm2 , the event rate of∼1 count/day/kg is expected for the most effective target nucleus, thus usually tonscale detectors are demanded. Typically detectors have a finite energy threshold set
at around a few to few tens of keV which cut off sufficient amount of signal, therefore
have more sensitivity to energetic WIMPs. Many sources of backgrounds, some ofthem having a similar exponential spectrum to the signal, are also a challenge fordirect detection experiments.
To observe the recoil energy signal, various signal channels are used such as scintillation lights, ionization yields, thermal energy and bubbles. Most of the current
detectors use one of these channels or a combination of them to efficiently reduce backgrounds of for instance γ’s, electrons and nucleons. In Fig. 1.9, the various direct
detection techniques and experiments are shown.
As for target materials; heavy nuclei are widely used to enhance the SI cross section. Liquid noble gases, such as Xe or Ar, form a dense and homogeneous target with
self-shielding and allow large detector masses at moderate costs. They also yield highlight (40 photons/keV) and charge. Noble gases usually can produce signals in two
ways; scintillation and ionization processes. Single phase liquid noble gas detectors
which are sensitive to scintillation yield such as XMASS, CLEAN and DEEP generally
have advantage that they can be built larger without high-voltage eletrodes, resulting
less radioactive background. Double phase detectors using the time projection chamber
technique can further discriminate background events reading both charge and scintillation signals. XENON100, LUX, PandaX, ArDM, DarkSide are on data-taking or in
the R&D process.
Another type of detectors use cryogenic materials such as Ge (CDMS, EDELWEISS), CaWO4 (CRESST). They typically detect phonons and have a high yield.
CRESST uses a calcium tungstenate crystal as the target, and detects both a phonon
signal and the scintillation lights. By 730 kg-days of data taking, they observed event
excess over known backgrounds at more than 4 σ significance in 2011 (45). CDMS is asemiconductor detector measuring signal by ionization and phonons. With Si target of
low atomic mass, they have claimed the observation of three WIMP-candidate events

Chapter 1

25

Figure 1.9: The direct detection techniques and the experiments categorised by thetechniques being used. Image is taken from M. Schumann’s slides in ICRC 2013 (44).
in the 140.2 kg-days of data taken, which is consistent with low mass WIMP signal at
3 σ C.L. (46).
Because of the relative velocity of the Earth to the Sun added to the motion of
the Sun in the Galactic frame, the event rate is expected to have annual modulation.
The modulation amplitude in the region of maximal sensitivity is expected to be '7% for halo distributions which are commonly used (47). So far DAMA (DAMA/NaI
and DAMA/LIBRA) and CoGENT have reported annual modulation signals observed.
DAMA has been taking data for 13 years using NaI crystal measuring scintillation
lights, and observed an annual modulation signal interpreted as WIMP interactions
with 8.9 σ significance. By a year-data taking of CoGENT, a P-type Point Contact
Germanium Detector measuring ionization, it is showed 2.8 σ annual modulation (48).
However, their allowed signal regions are completely rejected by recent results fromXENON100 (49), LUX (50) and SuperCDMS (51). The current best constraints are
made by two phase Xe detectors, i.e., XENON100 (49) and LUX (50), and are rapidly
rejecting the theoretically profound WIMP parameter space around the scattering crosssection of 10−45 cm2 . Figure 1.10 shows the current best limits from direct detection
searches.
Recently, direct detection experiments are making steep gains in sensitivity andachieving nearly zero-background searches. In the near future, the experiments willreach the irreducible solar-neutrino background at around the sensitivity of SI WIMPnucleon cross-section ∼ 10−48 cm2 .
Some detectors have a sensitivity for SD interactions as well. Natural isotopes with

Chapter 1

26

some unpaired protons or neutrons are used as a target. Si and Ge are almost composedof spin-less isotopes, however, 73 Ge (7.8% of natural Ge) and 29 Si (4.67% of natural Si)
contribute to the sensitivity to spin-dependent coupling WIMPs in spite of their smallabundance. Superheated fluids are also used; COUPP is a bubble chamber using CF3 I
and PICASSO, SIMPLE are superheated droplets made of C4 F10 .
Figure 1.10 shows the current limits from WIMP detection experiments on spinindependent WIMP-nucleon cross section and spin-dependent WIMP-proton cross section.

1.5.4

Constraints by indirect detection searches on WIMPnucleon scatterings

Indirect solar WIMP searches have strong sensitivity to spin-dependent coupling
WIMPs as the Sun is mainly composed of protons. Tight limits on the SD scatteringcross section of WIMPs on protons have been placed by neutrino telescopes such as
Super-Kamiokande (56), AMANDA (57), IceCube (54), BAIKAL (58), Baksan (59)and ANTARES (60).
Super-Kamiokande (SK) is a 50 kton Water Cherenkov neutrino observatory locatedin Kamioka mine in Japan. The Super-Kamiokande collaboration has analysed theiraccumulated data and reported the result in 2004 (61) and 2011 (56). The analysis donein 2004 used a dataset of 1679.6 days of SK-I muons which pass through the detector.
The updated analysis in 2011 increased the dataset to SK I-III (3109.6 days). In
addition, it introduced a new event categorization of muons to use the neutrino energy
information to reduce background, in order to improve the limits especially for lower
mass WIMPs. As muon events dominate the observed events in SK above 10 GeV,
the signal acceptance for WIMPs with mass above 10 GeV was good, also the goodangular resolution of muon events allowed an efficient way to distinguish signal frombackground. As a result, the search resulted the best limit for WIMPs in the mass
range from 10 GeV to a few hundreds of GeV at the moment.
The IceCube is a large Cherenkov detector situated in the ice at the South Pole.
Spanning a surface area of roughly 1 km2 and depths between 1450 and 2450 m, IceCube
has strong sensitivity for heavy WIMPs. Current report (54) provides the world best
limit for spin-dependent coupling cross section for WIMP masses above 20 GeV, which
eliminates the cross sections of about 10−40 cm2 and the MSSM parameter space which
is not reached by direct detection experiments.
Figure 1.11 shows the current best limits from searches for the neutrinos comingfrom the WIMP annihilation in the Sun.

Chapter 1

27

Figure 1.10: Top: Constraints and signal claims on spin-independent WIMP-nucleon
cross sections as a function of WIMP mass (52; 49; 53). Bottom: Constraints on spindependent WIMP-proton cross sections as function of WIMP mass for direct detection
experiments (IceCube (54) result is shown in comparison). Figure is taken from (55).

Chapter 1

28

Figure 1.11: 90% C.L. upper limits on spin-independent scattering cross-section (top)and spin-dependent one (bottom) as function of mass from solar WIMP searches areshown. The shaded regions indicate allowed MSSM parameter spaces. Results fromdirect detection experiments are shown for comparison. Figure is taken from (54).

Chapter 1

1.6

29

Prediction of sensitivity of the Super-Kamiokande
for light WIMPs

As mentioned above, recently, there have been possible signals claimed by direct
detection experiments such as DAMA/LIBRA (62), CoGeNT (48), CRESST (45) and
CDMS (46), mostly for light WIMP mass below 30 GeV, but these claims conflict withmany other experimental results (52; 49; 50; 51). Investigation with an independent
detection method and uncertainties on this region will be therefore appreciated.
As typical neutrino energy resulting from WIMP annihilation in the Sun is roughly
1/3 to 1/2 of the WIMP mass (42), the SK’s sensitivity to few-GeV neutrinos makes
it suitable for this search. The huge gravity and hydrogen-rich composition of the Sun
combined with high sensitivity of SK allow promising sensitivity to light WIMPs (63;64; 65; 66), especially for spin-dependent coupling case.
Recent theoretical works (64; 65; 66) have calculated the expected power of SK data
to test light WIMPs. Kappl et al.’s work shows that using SK-I fully-contained type ofneutrino events, it can be proved that spin-dependently coupling WIMPs annihilatingeven subdominantly into neutrinos, taus, bottoms or charms will fail to explain the
DAMA signal region (65). Fig. 1.12 shows their work on SD WIMP-nucleon scatteringcross section and the result of a sensitivity study using 200 kton contained type of
events done by Carsten et al. (66).
The sensitivity of Super-Kamiokande for SI WIMP-nucleon cross section for light
WIMPs is also expected to be competitive with direct detection. The huge size of
the Sun allows sizable contribution even from less abundant elements than hydrogen,and heavier elements scatter energetic WIMPs more efficiently. Kappl et al.’s study(65) shows that spin-independently coupling WIMPs which annihilate dominantly into
neutrino or tau pairs will be inconsistent with the DAMA and CoGENT favored regions.
The upper limit on SI WIMP-nucleon scattering cross section in case of using SK I
fully-contained events studied by Kappl et al. is shown in Fig. 1.13.
Motivation for considering isospin violating dark matter (IVDM) arises from recent
CDMS (46) result which can be “reconciled” with conflicting results by introducing destructive interference fn /fp < 0 (67; 68). The IVDM model is also motivated by current
accelerator searches that low mass (< 20 GeV) WIMPs have been essentially ruled out
in the context of the MSSM. But we can find theoretically well-motivated light WIMPs
for example in nMSSM (next-to-Minimal Supersymmetric Standard Model) scenariosand generally light WIMP candidates found in those scenarios do not necessarily always
conserve isospin (68).
As the Sun is composed of relatively light elements, the Super-Kamiokande is expected to have strong sensitivity to IVDM with destructive interference (69; 70). Figure
1.14 shows the sensitivity study done by Chen et al. (69) assuming 1679.6 live-days
data of the SK. Most of the DAMA and CoGeNT signal regions are expected be excluded assuming bb̄, τ + τ − or cc annihilation channels.
Previous Super-Kamiokande solar WIMP search (56) used upward-going muons
produced in the surrounding rock. The new sensitivity for the light WIMPs can beChapter 1

30

achieved by increasing signal acceptance to a few-GeV neutrinos, whose interaction
vertices are expected to be contained inside the detector. However, the atmospheric
neutrino background events increase as ∝ E −2.7 at low energy. Also angular correlation
between the parent neutrino and daughter lepton becomes worse for low-energy events.
Therefore for light WIMP search, introducing large background in the solar WIMP
search is inevitable. Developing an analysis tool to increase signal acceptance and to
improve the sensitivity to signal against the increased background is a goal of thiswork.
Another challenge the solar WIMP search faces, while confirming results of directdetection searches, is to understand the effect of astrophysical uncertainties. The phasespace distribution of WIMPs in the Galaxy is a large source of uncertainty for WIMP
searches. Uncertainty in the local WIMP density will make a similar overall shift of thelimits for direct detection and solar WIMP searches. However, the velocity distributionof WIMPs will affect the direct detection and the solar WIMP search in different ways.
In the existing literature there have been extensive discussions about the impact of thevelocity distribution on direct detection (see e.g. (71; 72) and references therein), but
for the solar WIMP search, there have been no overall quantitative investigation on
the size of the uncertainties except for specific scenarios (73; 74; 75; 76; 77; 66; 78; 79).
In Chap. 2, details of scattering and annihilation processes inside the Sun, neutrino
production and the propagation to the detector will be described. In Chap. 3, theuncertainties in the WIMP velocity distribution and its effect on the solar WIMP search
will be discussed. In Chap. 4, we briefly introduce the SK detector and its calibration
procedure. In Chap. 5 and 6, MC simulations and event reduction, reconstruction andcategorization of data and MC will be summarised. In Chap. 7, the analysis methodof the SK solar WIMP search and results will be shown. In Chap. 8, the uncertainties
in the analysis will be summarised. Finally, the conclusion and the future prospect are
given in Chap. 9.

Chapter 1

31

Figure 1.12: Left: Sensitivity study done by Rott et al. (66) of 200 kton-years neutrino
detector data on SD WIMP-proton scattering cross section for annihilations into τ + τ −(blue) and bb̄ (green). Here FC νe events with three different direction cuts are used.
Figure is taken from (66). Right: Sensitivity study (65) of the SK-I data on SD WIMPproton cross section for various annihilation channels. The limits are obtained by using
fully contained events at low mass region. Also shown are results from direct detection
experiments. Figure is taken from (65).

Chapter 1

32

Figure 1.13: Sensitivity study done by kappl et al. (65) of the SK-I data on SI WIMPnucleon cross section for various annihilation channels. At lower WIMP mass region
FC events contribute to the sensitivity. Also shown allowed regions and upper limits
claimed by various direct detecation expriments. Figure is taken from (65).

Chapter 1

33

Figure 1.14: Sensitivity study done by Chen et al. (69) of the SK solar WIMP analysis(solid curves) in isospin conserving case (upper panel) and violating (lower panel) casewith fn /fp = 0.7. DAMA (orange circle) and CoGeNT (purple circle) signal regions
and the limits from other direct detection experiments (dashed line) are shown together.
Figure is taken from (69).)

Chapter 2
Capture of WIMPs in the Sun andthe search for it in a neutrino
detector
2.1

Overview

As the solar system travels on the Milky Way arm, WIMPs in the dark matter halo
could occasionally become gravitationally captured to the Sun after losing energy byscattering off nuclei within the Sun. Once captured, WIMPs may lose energy in subsequent scatters and settle at the center of the gravitational well and thermalize. The
WIMPs would then pair-annihilate in the deep solar core, after which produced neutrinos propagate outward through the Sun and may be detected in terrestrial neutrino
detectors.
The first WIMP detection attempts using neutrinos were motivated by the work
done in 1985 by Press and Spergel, who calculated the capture rate of heavy particles
by the Sun (80). In 1988, Gould refined the calculations of Press and Spergel andderived exact formulae for the capture rates (81; 82) which have been used in many
subsequent studies including this work.
In section 2.2, we will review the WIMP capture process in the Sun. In Chap. 2.3,
the WIMP number density evolution in the core of the Sun will be briefly reviewed andin Chap. 2.4, it will be linked to the assumption on equilibrium between the capture
rate and the annihilation rate. In Chap. 2.5, the annihilation of WIMPs and the
production of neutrinos will be introduced. Throughout the dissertation, calculationsof the capture of WIMPs are done by using the numerical package for neutralino dark
matter calculations, DarkSUSY (83). In Chap. 2.6, brief explaination of the usage of
the DarkSUSY in this work will be given.

34

Chapter 2

2.2

35

WIMP capture in the Sun

WIMPs that are abundant in the dark matter halo of the Milky Way can scatter off
a nucleus in the Sun as it revolves around the Galactic center. The WIMPs may loseenough energy by elastic scattering off a nucleus to fall below the escape velocity of the
Sun at the point of the scatter, and then be gravitationally captured. The differential
WIMP capture rate on nucleus i per unit shell volume dV at distance r from the center
takes the following form :ρH Z ∞ fη (u)dCi=duΩi (q),
dV
mχ 0
u

(2.1)

where ρH is the local dark matter halo density set to be 0.3 GeV/cm3 (23; 84) andmχ is the mass of the WIMP. fη (u) is the Velocity Distribution Function (VDF) seenin the reference frame of the Sun. For isotropic VDF, fη (u) can be simply calculated
from the original VDF in the Galactic frame, fo (v), asfη (u) =

Z 1
−1

q

fo ( v 2 + v 2 + 2vv cosθ) dcosθ,(2.2)

where ~u = ~v + v~ and θ is the angle between ~v , WIMP velocity in the Galactic frame,and ~v , the speed of the Sun in the Galactic frame. The bound velocity of a WIMP in
the gravitational field of the Sun, w, and the escape velocity at the scattered position,
uesc (r), are related via w2 = u2 + uesc (r)2 . A WIMP can be captured when w drops
below uesc (r) after losing kinetic energy through scattering off a nuclei. The capture
probability Ωi (q) can be written as (82)
mi Z qmax 2
F (q)dq,(2.3)Ωi (q) = σi ni 22µi qminwhere σi is the WIMP-nucleus i elastic-scattering cross-section at zero-momentum
transfer, ni is the number density of nucleus i in the Sun, mi is the mass of nucleus i,µi is the reduced mass of nucleus i and the WIMP, q is the recoil energy where1qmin = mχ u2
2

(2.4)

is the minimum recoil energy required to be captured,1
qmax = β+ mχ w2 ,
2

(2.5)

is the kinematically determined maximal recoil energy withβ± =4mi mχ
,(mi ± mχ )2(2.6)

and F 2 (q) is a nuclear form factor taken to have exponential form from (82). The σiSD
and σiSI shown in Chap. 1 are not the real total cross sections used in the calculation

Chapter 2

36

Figure 2.1: Density distribution inside the Sun as a function of radius.
over dq 2of the capture rate. The actual cross section is obtained by integrating dσ(q)dq 2considering nuclear form factor F 2 (q). Where actual differential cross sections are usedin the analysis, we define the ‘zero-momentum transfer cross section’ that are denotedby σiSD and σiSI and can be regarded to be total cross sections assuming F 2 (q) = 1. This
allow us to reduce them into σ0SD and σ0SI , which are the fiducial variables compared
between models and various detection methods.
For SD WIMP capture, we neglect contributions from elements heavier than hydrogen. For the SI coupling case, we assume isospin-conserving scattering and use the
SI zero-momentum transfer cross section as shown in Eq. 1.5.2.
To calculate the total capture rate, Eq. 2.1 must be summed over all relevant
nuclear species inside the Sun and integrated over the distance r from the center as,
CC =

Z R
0

4πr2 dr

X dCi
i

dV

,

(2.7)

where R is the radius of the Sun.
Depending on the WIMP scattering nature with ordinary matter which is describedin the previous chapter, the total capture rate is decided. The sun is mostly composedof hydrogen (∼74% of total mass) and helium (∼25% of total mass), then oxygen,carbon, iron, and so on. The capture rate also depends on the solar composition.
The radial distribution of each element inside the Sun is adopted from the BS2005-OPmodel in the DarkSUSY. An example of radial distribution of total mass is shown in
Fig. 2.1.
In the case when the strength of the interaction cross sections are the same, spinindependent coupling is more efficient to capture WIMPs than spin-dependent couplingChapter 2

37

Figure 2.2: Capture rates of WIMPs in the Sun divided by interaction strength [pb]for spin-independent and spin-dependent coupling cases, as a function of WIMP mass.
Figure is taken from (65).
for most of the WIMP masses. An example of the capture rates from both scatterings
are shown in Fig. 2.2.
For the spin-independent coupling case, the contributions come from several elements. In the case of heavy elements with A1 such as iron, besides coherent enhancement ∝A2 , the mass-matching factor becomes approximately ∝A2 , resulting in
the capture rate being boosted ∝A4 . Therefore, despite being the fifth most abundantelement, iron could capture the WIMPs most efficiently. However, because of the formfactor suppression, the capture from scattering off iron can be suppressed by several
orders of magnitude for WIMPs with mass of several hundred GeV (42). Therefore thecapture of heavy WIMPs through spin-independent coupling is dominated by scattering off oxygen (82). Figure 2.3 visualises the capture rates from several elements forspin-independent coupling WIMPs.

2.3

WIMP thermalization inside the Sun

Once captured, WIMPs bound in the gravitational potential well of the Sun will
experience further rescatterings by solar nuclei through which they could lose moreenergy. As they lose energy, they can move to smaller orbits and finally settle ata dense core of the Sun and thermalize. The thermalization time depends on the
WIMP-nucleon cross section which governs the rescattering rate. It also depends on
the WIMP mass, as the mass governs the number of scatters required for the WIMP
to be completely embedded in the Sun (86). Once they are embedded in the solar
radius, they will quickly reach thermal equilibrium with the solar medium. Figure 2.4
shows the mass dependence of the number of scatters required to bring the WIMP
orbit within the two times solar radius, calculated by A. Peter (86).

Chapter 2

38

Figure 2.3: Capture rate of WIMPs in the Sun with assumed interaction strength ofσSI = 10−43 cm2 . The total capture rate (Ctotal ) and the contribution from several mostimportant elements are shown together, as a function of WIMP mass: He (CHe ), thesum of C, O and N (CCN O ), the sum of Fe and Ni (CF e+N i ), and Ne (CN e ). Figure istaken from (85).

Figure 2.4: The number of scatters X, required to bring a WIMP orbiting with a semimajor axis a after first scattering down to orbiting with af = 2R , as a function of the
WIMP-nucleus mass ratio mχ /mA . Figure is taken from (86).

Chapter 2

39

Considering those effects, the estimated lower limit on the elastic scattering crosssection for which the median thermalization time of captured WIMPs is less than theage of the solar system is calculated (86). In the spin-dependent (spin-independent)
scattering case, the thermalization time exceeds the age of the solar system if σSD (σSI )< 10−48 cm2 (10−51 cm2 ) for mχ = 100 GeV. These limits are low enough comparedto the expected sensitivity of indirect detection with existing neutrino detectors, so
it is safe to assume that WIMPs “instantly” achieve thermal equilibrium with solar
medium.
The WIMPs that are in thermal equilibrium will spatial distribution described by
the global temperature and the local gravitational potential. At the radial distancefrom the center of the Sun r, the equilibrium number density, ne (r) will be describedas:ne (r) = n0 e−mχ Φ(r)/kB T ,(2.8)
where n0 is a normalization
factor, mχ is the WIMP mass, Φ(r) is the gravitational
Rrpotential of the Sun, i.e. 0 drGM (r)/r2 , kB is the Bolzmann factor and T is the temperature to which WIMPs are thermalized. Typically the temperature of the WIMPs
are assumed to be constant at the temperature at the center of the Sun, T .
Assuming that the density of highly condensed ordinary matter within a sphere ofradius r is constant and equal to the central density, ρ for simplicity, then with thegravitational potential Φ(r) = 2πGρr2 /3 with Newton’s constant G, one can find that
the WIMPs are concentrated in the center of the Sun with the typical radius of rχ , as
(87):
2

2

nχ (r) = n0 e−r /rχ , with
3T
)1/2 ∼ 0.01Rrχ = (
2πGρ mχs(2.9)
100GeV.
mχ

(2.10)The thermalized WIMPs are predicted to distribute within 1% of the radius of theSun. Thus it is safe to assume that the annihilations of WIMPs happen at the centerof the Sun.
WIMPs can eventually be accumulated in the center of the Sun in this fashion
and be ready to annihilate. The WIMP annihilation rate is proportional to N 2 as apair of WIMP annihilate together. Also it is proportional to the thermally averaged
annihilation cross section < σv >. The annihilation rate is described as:
1
ΓA = CA N 2
2(2.11)

where

1 Z
CA = 2 4πr2 drn2e (r) < σv >Nwith N being the total number of WIMPs.

(2.12)

Chapter 2

40

Using the equations 2.9 and 2.10, CA is calculated as a function of < σv >, WIMP
mass and the density and the temperature of the solar core (87):CA =< σv > (Gmχ ρ 3/2
) ,3T

(2.13)

with the solar core density ρ of about 151 g/cm3 and the solar core temperature Tof about 15.5 E6 K around the center of the Sun.

2.4

WIMP density evolution inside the Sun and theequilibrium condition

Considering capture and annihilation described above together, the evolution of thetotal number of WIMPs in the Sun, N , can be described by the following differential
equationdN= CC − CA N 2 − CE N,(2.14)
dt
where CC is the capture rate, CA is linked to the annihilation rate ΓA as described in theprevious chapter, and CE is the evaporation rate. WIMPs could also become unboundfrom the Sun in scatters, in a process known as ‘evaporation’. The evaporation process
is not considered and the discussion on it will be given in Chap. 8.
Neglecting the evaporation effect, the equation can be solved and the current annihilation rate can be written as
ΓA =CC
tanh2 (t /τ ),2(2.15)

where t is the age of the Sun, τ is (CC CA )−1/2 . In the literature, τ is often regarded
as “equilibrium time” because when t  τ , or when equilibrium between capture andannihilation has been achieved, the time dependence of the annihilation rate will be
removed as:
1
Γequi
= CC
A
2

(2.16)and the capture rate, CC , regulates the annihilation rate, Γequi
A . Some discussion about
the equilibrium condition will be given in Chap. 8.

2.5

Production of neutrinos inside the Sun

The captured WIMPs in the core of the Sun can self-annihilate in a pair andcreate the Standard Model particles. By hadronization and decays of the annihilation
products, neutrinos and anti-neutrinos can be produced as final products and escape
the Sun, then be detected by the detectors on the Earth.

Chapter 2

41

The three-flavor neutrino and anti-neutrino fluxes at production in the core of the
Sun are shown in Fig. 2.5. For the bb̄ channel, WIMPs hadronize and produce B
mesons, which interact in the solar medium before decay. This channel is considered
to represent the “softest” spectrum among the channels considered in this and othertypical analyses. For the W + W − channel, the high-energy and the low-energy peaks
correspond to the prompt decay of the W boson and decay products of hadrons, respectively. For larger WIMP masses, i.e. for larger boosts of the primary W, this featureis smeared into a smooth spectrum as seen in Fig. 2.6. The τ + τ − channel produces
far more tau neutrinos than electron neutrinos and muon neutrinos. Annihilation toν ν̄ pair is helicity-suppressed for Majorana particle (39) whereas it is not the case forgeneral dark matter candidates such as those found in extra dimension scenarios. This
channel is not studied in this analysis, therefore the τ + τ − channel represents the mostenergetic (“hardest”) spectrum. The figures are generated using WIMPsim (88) andmore plots of the neutrino fluxes from WIMP annihilation at various distances can befound in (89).
The propagation of the neutrinos experiences both vacuum and matter oscillations
with three-flavor mixing. Also neutrino interactions in the solar medium are considered.
A charged current (CC) interaction would reduce the amount of the flux, and a neutral
current (NC) interaction will reduce the energy of the neutrino. When a muon orelectron (anti-) neutrino interacts through charged current, produced electrons arestable and muons are stopped before decay. But when a tau neutrino interacts, theproduced tau lepton can decay and produce secondary neutrinos which can propagate
to the detector. Figure 2.7 shows the three flavor neutrino and anti-neutrino fluxes atthe surface of the Sun.
After passing through the Sun, neutrinos propagate in the vacuum as it is oscillating.
The electron neutrinos, which remained almost without mixing throughout the solar
medium with high electron number density, now is efficiently mixed with the muon andtau neutrinos. As a result all three flavors of neutrino and anti-neutrinos end up having
similar spectrum at 1AU. Figure 2.8 shows the three flavor neutrino and anti-neutrino
fluxes at 1AU.

42

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Neutrino fluxes at creation (Sun)
Mass: 6
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νµ
ντ

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-1

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dNν/dz (ann )-1dNν/dz (ann )Chapter 2

Antineutrino fluxes at creation (Sun)
Mass: 6
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5
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ν_ µ
ντ

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-1

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10

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
z = Eν / mχ

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Neutrino fluxes at creation (Sun)
Mass: 80.3
Channel: 8
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νµ
ντ

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dNν/dz (ann )-1dNν/dz (ann )z = Eν / mχ

Antineutrino fluxes at creation (Sun)
Mass: 80.3
Channel:
8
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ν_ µ
ντ

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-1

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Neutrino fluxes at creation (Sun)
Mass: 200
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νe
νµ
ντ

1

-1

10

dNν/dz (ann )-1dNν/dz (ann )z = Eν / mχ

-1

10

-2

10

-3

10

Antineutrino fluxes at creation (Sun)
Mass: 200
Channel:
11
_
ν_ e
ν_ µ
ντ

1

-1

10

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10

-2

-3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
z = Eν / mχ10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
z = Eν / mχ

Figure 2.5: The WIMP-induced neutrino fluxes at the production at the center of
the Sun for mχ = 6 GeV bb̄ channel, 80.3 GeV W + W − channel, and 200 GeV τ + τ −channel. Fluxes are shown as a function of neutrino energy normalized to WIMP mass(z) for neutrino (left column) and anti-neutrino fluxes (right column).

43

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Neutrino fluxes at creation (Sun)
Mass: 80.3
Channel: 8
νe
νµ
ντ

1

-1

10

dNν/dz (ann )-1dNν/dz (ann )Chapter 2

-1

10

-2

10

-3

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Neutrino fluxes at creation (Sun)
Mass: 100
Channel: 8
νe
νµ
ντ

1

-1

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10

-2

-3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
z = Eν / mχ10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
z = Eν / mχ

Figure 2.6: The WIMP-induced left-handed neutrino fluxes at the production at thecenter of the Sun for W + W − channel for mχ = 80.3 GeV (left) and mχ = 100 GeV(right) are compared. The flux is shown as a function of neutrino energy normalized
to WIMP mass (z).

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1

10

10

-1

Neutrino fluxes at Rsun
Mass: 6
Channel: 5
νe (total)νµ (total)ντ (total)νe (secondary)νµ (secondary)ντ (secondary)

-1

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44

dNν/dz (ann )-1dNν/dz (ann )Chapter 2

1

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-1

-1

Neutrino fluxes at Rsun
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10z = Eν / mχ

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1

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-2

Antineutrino fluxes at Rsun
Mass: 80.3
Channel:8_ν_ e (total)
ν_ µ (total)
ν_ τ (total)
ν_ e (secondary)ν_ µ (secondary)ντ (secondary)-2z = Eν / mχ

10

Antineutrino fluxes at Rsun
Mass: 6
Channel:5_ν_ e (total)
ν_ µ (total)
ν_ τ (total)
ν_ e (secondary)ν_ µ (secondary)ντ (secondary)

10-1Antineutrino fluxes at Rsun
Mass: 200
Channel:11_
ν_ e (total)
ν_ µ (total)
ν_ τ (total)
ν_ e (secondary)ν_ µ (secondary)ντ (secondary)-2-3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
z = Eν / mχ

Figure 2.7: The WIMP-induced neutrino fluxes at the surface of the Sun for mχ =6 GeV bb̄ channel, 80.3 GeV W + W − channel, and 200 GeV τ + τ − channel. The flux
is shown as a function of neutrino energy normalized to WIMP mass (z) for neutrino(left column) and anti-neutrino flux (right column). Three-flavor oscillation with parameters: θ12 = 33.461◦ , θ13 = 9.097◦ , θ23 = 40.686◦ , δ = 300, ∆m221 = 0.766 × 10−4
eV2 , ∆m232 = 0.274 × 10−2 eV2 are used assuming normal mass hierarchy.

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1

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Mass: 6
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45

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-1

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1

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10

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Mass: 6
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ν_ µ (total)
ν_ τ (total)
ν_ e (secondary)ν_ µ (secondary)ντ (secondary)

10-1Antineutrino fluxes at 1 AU
Mass: 200
Channel:11_
ν_ e (total)
ν_ µ (total)
ν_ τ (total)
ν_ e (secondary)ν_ µ (secondary)ντ (secondary)-2-3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
z = Eν / mχ

Figure 2.8: The WIMP-induced neutrinos flux at 1AU for mχ = 6 GeV bb̄ channel, 80.3 GeV W + W − channel, and 200 GeV τ + τ − channel. The flux is shown as
a function of neutrino energy normalized to WIMP mass (z) for neutrino (left column) and anti-neutrino flux (right column). Three-flavor oscillation with parameters: θ12 = 33.461◦ , θ13 = 9.097◦ , θ23 = 40.686◦ , δ = 300, ∆m221 = 0.766 × 10−4
eV2 , ∆m232 = 0.274 × 10−2 eV2 are used assuming normal mass hierarchy.

Chapter 2

2.6

46

Utilization of DarkSUSY

DarkSUSY (83) is a simulator written in Fortran language which can be utilized
for calculations of expected WIMP signal in direct detections and indirect detectionsof various annihilation products such as gamma rays, antimatter, solar and Earthneutrinos, and so on. DarkSUSY version 5.0.6 is used for this work. It is a voluminous
program, but for the use in solar WIMP neutrino detection, the functionalities that
DarkSUSY provides can be categorized as follows:
1. Scans parameter space of SUSY scenarios and predict WIMP candidate properties
2. For a given WIMP model, calculates the capture rate
3. Calculates the annihilation rate accordingly
4. For a determined annihilation rate, read WIMPsim output files and delivers theneutrino flux to the user.
Here is a brief sketch of the usage of the DarkSUSY in this work for each process.
1) In DarkSUSY, we can define the model space that we want to scan by setting
ranges of SUSY parameters directly, and indirectly introducing bounds from recent
accelerators and CMB measurements. For this analysis, we set the rough range of
SUSY parameters for CMSSM and released the other constraints as much as possible.
When one runs DarkSUSY in ‘random model mode’, DarkSUSY will automaticallyscan possible WIMP scenarios which are randomly chosen within the model space weset. We ran it for about 5,000,000 scenarios first, and among the results we collected themodels which have WIMP masses close to our region of interest (4, 6, ... GeV). For theselected models, we made the ‘mode’ files which are a summary of SUSY parameters
for each WIMP hypothesis.
2) The next step is running DarkSUSY in ‘read model mode’ for each WIMP
hypothesis. In running in the read model mode, according to the SUSY parameters
read from mode file, DarkSUSY analytically calculates the capture rate as described in
the previous chapter. We let DarkSUSY calculate the capture rate for spin-dependent
coupling WIMPs and spin-independent coupling WIMPs separately. In calculationof the capture rate for SD coupling WIMP, SI scattering inside the Sun is ignored
by setting it to be zero, and vice versa. With setting the local dark matter density,
the velocity distribution of dark matter in the halo and other nominal parameters,
DarkSUSY allows the precise calculation of the capture rate for a selected WIMP massand scattering cross section. For inputting the velocity distribution of dark matter,both analytic equations and numerical tables can be used.
3) DarkSUSY also decides how many annihilations are allowed to happen for the
scenario. In this analysis, to achieve equilibrium for any chosen model, we manipulated
the age of the Sun to be very old. In this way, the model-dependent annihilation rate
calculated by DarkSUSY is ignored and the result is equivalent to solving Eq. 2.16.

Chapter 2

474) If we don’t choose a particular annihilation channel, DarkSUSY will run for
multiple channels with branching ratio decided by the set of SUSY parameters in themode file. Otherwise, it can force the annihilation into a specified channel. We used
this function to run DarkSUSY for a single channel.
We can also feed the type of yield of interest, which are differential neutrino/antineutrino fluxes in this analysis. Then DarkSUSY reads the selected type of yieldfrom the tabulated output of WIMPsim, and interpolates or extrapolates the flux andmagnifies it by the ratio between the calculated annihilation rate and the WIMPsim
nominal annihilation rate. After that, DarkSUSY adjusts it in the finer format thatuser specified; it applies the energy threshold, cuts the angular size from the Sun
and chooses the flavor. In this analysis, the energy threshold is set to be zero. Also
the angular size of the flux is turned off as the source is regarded to be point-like. Insummary, the resultant neutrino flux we get from DarkSUSY is the neutrino flux whose
differential spectrum and flavor ratio are calculated by WIMPsim, and for which theabsolute normalization is calculated by DarkSUSY for a certain mass and scatteringcross section.
In summary, we get a neutrino flux corresponding to a certain WIMP mass, anniSD/SI
using DarkSUSY.
hilation channel and WIMP-nucleus scattering cross section σ0

Chapter 3
The velocity distribution and thecapture rate of WIMPs
3.1

Overview

Under the assumption of equilibrium between the capture rate and the annihilation
rate, the uncertainty on the annihilation is directly given by the uncertainty on the
WIMP capture in the Sun. This capture mechanism is theoretically well understood sothat we can study the impacts of the various effects that contribute to its uncertainty
and is the subject of this chapter.
In this chapter, we will give a comprehensive and general treatment of uncertainties from the dark matter velocity distribution in indirect solar WIMP search. We will
discuss the general response of solar WIMP search to the change of the velocity distribution which will be helpful to understand the effect on capture rate from individual
source of uncertainties. The uncertainties will be quantified and given as a functionof WIMP mass, making it easy to incorporate in future experimental and theoreticalworks. We study the effect of the velocity distribution of dark matter on capture ratein the Sun considering four sources of uncertainties: circular speed of the Sun, galactic
escape speed of dark matter, departures from a Maxwellian distribution and existenceof the dark disc.
In chapter 3.2 we show how solar WIMP search responds to the velocity distribution
in comparison to direct detection. In Chap. 3.3, we introduce the velocity distributionsof dark matter halo, for which we discuss the impact on the capture rate in Chap. 3.4.
In the final chapter 3.5, we conclude and summarise the chapter.

3.2

Uncorrelated and anti-correlated uncertainties
in direct and indirect detections

The local velocity distribution of dark matter impacts on direct and indirect searches
very differently. First, WIMPs in the Sun have been accumulating over a very long time48

Chapter 3

49

period. Even though there is no evidence for very large density fluctuations or changes
in the velocity distribution, we point out that direct detection rates are only sensitive
to the present local dark matter density and velocity distribution, while indirect solar
WIMP search rates are a result of the dark matter halo density and velocity distribution that has been sampled by the Sun over the equilibration time scale. Hence,indirect detection rates are related to the average dark matter density and velocity
distribution at the solar orbit.
Secondly, as opposed to direct detections looking for energetic scattering of WIMPs,
the less energetic ones are easier to be captured by the Sun. Thus direct detections are
sensitive to WIMPs with higher velocity, by contrast, indirect detections are sensitive
to the lower part of the velocity distribution. This contrast in the responses of direct
and indirect detections to the velocity distribution will manifest in anti-correlated sizeand sign of the uncertainties; i.e. whereas the direct detection event rate is largely
affected by any source of uncertainty, the size of the effect on capture rates can be
marginal; or whereas direct detection event rate is boosted, the capture rate can be
reduced, and so on.
The maximum velocity of a WIMP to be captured by nucleus i at distance r fromthe centre of the Sun is given when qmin (Eq. 2.4) equals to qmax (Eq. 2.5), whereumax (i) =qβ− uesc (r).

(3.1)

√ The escape velocity at the surface of the Sun is ∼ 618 km/s. In the case ofβ− ∼ 1, umax (i) is large enough compared to the mean velocity of the WIMPs in thegalactic halo being ∼ 300 km/s, so to allow an efficient capture process. However,when the mass-matching between the WIMP and the nucleus fails then β− becomes
smaller than 1, the capture of high-velocity WIMPs will be suppressed.
To estimate umax for the total capture rate, we have to consider contributions frommultiple elements with their radial distribution inside the Sun. Figures 3.1 and 3.2
visualise the relative capture rate for several WIMP masses as a function of the dark
matter velocity in the local reference frame at the Sun, whereas the Sun rotates aroundthe galactic center with v = 220 km/s, and in the Galactic frame which is definedsuch that the plane of our Galaxy has a latitude of zero, respectively. Capture rates arecalculated using delta function for VDF and are relatively normalised to the highest
capture rate for the mass. Because the capture in the Sun is dominated by the mostabundant hydrogen and other light elements, for the light WIMPs good mass-matching
will allow the capture to be efficient up until the high-velocity tail, where for heavy
WIMPs it starts losing its efficiency in the lower-velocity region. Compared to capture
by SD interactions, capture by SI interaction keeps high efficiency longer until ∼50
GeV, which is the ‘resonance range’ (82; 42), helped by mass-matching scatterings
with heavier elements such as He, O, Si and Fe.
Meanwhile, the umin , the minimum velocity detectable in direct detections, is determined when the maximal recoil energy qmax = β+ 21 mχ u2 is the same as the energy

Capture rate (SD)

Chapter 3

50

5 / 10GeV
50 / 100GeV
200 / 500GeV
1000 / 10000GeV

1

10

-1

100

200

300

400

500

600

700

800

Capture rate (SI)u[km/s]5 / 10GeV
50 / 100GeV
200 / 500GeV
1000 / 10000GeV

1

10

-1

100

200

300

400

500

600

700

800

u[km/s]Figure 3.1: The capture rate for sample WIMP masses as a function of the relative
velocity to the Sun. The rates are normalised to the rate when u = 0. We show the SDcase (top) compared to the SI case (bottom) for eight sample masses of 5 (red solid)/ 10GeV (red dashed), 50 (green solid) / 100GeV (green dashed), 200 (blue solid) /500GeV (blue dashed), 1000 (magenta solid) /10000 GeV (magenta dashed).

51

Capture rate (SD)

Chapter 3

5 / 10GeV
50 / 100GeV
200 / 500GeV
1000 / 10000GeV

1

10

-1

100

200

300

400

500

600

Capture rate (SI)v[km/s]5 / 10GeV
50 / 100GeV
200 / 500GeV
1000 / 10000GeV

1

10

-1

100

200

300

400

500

600

v[km/s]Figure 3.2: The capture rate for various WIMP mass as a function of the velocity inthe galactic frame is given relative to the capture rate for v = 220 km/s. We show theSD case (top) compared to the SI case (bottom) for eight sample masses of 5 (red solid)/ 10GeV (red dashed), 50 (green solid) / 100GeV (green dashed), 200 (blue solid) /500GeV (blue dashed), 1000 (magenta solid) /10000 GeV (magenta dashed).

Chapter 3

52

threshold for the detector, Eth ,sumin =

2Eth
.
β+ mχ

(3.2)

Below umax and above umin , the capture efficiency or detection efficiency are not100% because typically WIMPs cannot transfer qmax to nucleus. As a result, indirect
detection will more efficiently capture WIMPs at low-velocity, while direct detections
will have strong favor to the high-velocity region. Helped by Fig. 3.1 to estimate umax ,and keeping in mind that umin is dependent on the energy threshold and nuclear used
in individual direct detectors, we will follow the simple notation of ‘high-velocity’ and‘low-velocity’ hereafter.
Figure 3.1 shows that the heavier the WIMP is, the smaller umax will be due tomass-matching with light nuclei inside the Sun. In the calculation of umin , there isan additional mχ besides the mass-matching factor in the denominator of Eq. 3.2,
therefore for a type of direct detection experiments, the umin for heavier WIMPs will
be smaller until mχ  mN .
Taking the above into account; light WIMP direct detection is sensitive to highvelocity tail, while light WIMP capture comes from wide range of VDF (starting fromthe low-velocity tail). In contrast, heavy WIMP indirect detection can only see thelow-velocity part of the VDF, while direct detection for heavy WIMPs can see a wide
range (starting from the high-velocity tail). Be aware that umin and umax are definedin different ways, so the regions to which direct and indirect detection are sensitive are
not always separated. For example, for a liquid Xe detector with an energy thresholdof 10 keV, the typical umin required to produce a nuclear recoil is of the order of ∼300
km/s for 100 GeV WIMPs, while in indirect detection for 400 km/s WIMPs are still
possible to be captured with spin independent coupling.
The contrasts in the responses of direct and indirect detections to VDF would
manifest clearly when the uncertainty only affects the high or the low velocity tail.
In this case, when either indirect detection or direct detection is strongly affected byan uncertainty in VDF, the other detection method would possibly be blind to thechange or affected to a minor degree. As a result, the effects of the uncertainties
in both detection methods might be uncorrelated. It is also interesting to interpret
some type of uncertainties as a ‘shift of WIMP population from one (low or high)region to another’, which makes the changes in low and high velocity regions anticorrelated. Because of the contrasts in the responses of direct and indirect detections
to the velocity distribution, the sign of the effect in direct and indirect detections might
be anti-correlated; i.e. whereas capture rate will be decreased, direct detection event
rate will be increased (76). With the situations above, we will be able to interpretthe response of capture rates to the uncertainties in velocity distribution which will be
discussed in following chapters.

Chapter 3

3.3

53

The dark matter velocity distribution

In this section we survey the dark matter velocity distribution in the galactic dark
matter halo. We introduce VDFs obtained from various simulations. We also shortly
discuss about the dark disc. These VDFs are later used as benchmark scenarios forwhich we compute capture rates of WIMPs in the Sun.
The SHM serves as a useful benchmark model, but recent cold dark matter N-bodysimulations indicate the significant deviation of the VDF of dark matter halos from theSHM (90; 34; 91; 92). We choose three benchmark VDFs from recent works as shownin the left plot of Fig. 3.3 and show them in the local reference frame at the Sun as
shown in the right plot of Fig. 3.3. VDFs in the local reference frame at the Sun are
shown in the form of f (u)/u as this term is inserted in the capture rate as shown in
Eq. 2.1.
Our benchmark distributions are taken from three recent N-body hydrodynamical
simulations: the Aquarius (93) project; an N-body simulation with Baryons (34) carried
with the cosmological Adaptive Mesh Refinement (AMR) code RAMSES (94); theRhapsody cluster re-simulation project (95).

Name& Reference
Vogelsberger etal. (90)

Ling et al. (34)

Mao et al. (92)

Descriptionof simulation
Largest dark matter
only simulation of a
Milky Way-sized dark
matter halo

Descriptionof VDFMedian of the velocity modulus distributions for 2 kpc
boxes centred between 7
and 9 kpc from the galactic
centre, with a broad bumpat v ' 250 km/s
A Milky Way-sized Velocity
modgalaxyfrom ulus in a spherical shell 7 <high-resolution
N- R < 9 kpc around the galacbody simulation with tic centre, with platykurbaryons
tic shape (K = 2.39) andhigh velocity dispersion (v0= 267.2 km/s)
Cluster re-simulation Stacked velocity distribuprojecttion for 96 halos at r/rs =0.15, peaks at low-velocity

SourceFig.2

Fig.4(d)Fig.1

Table 3.1: The sources and descriptions of the benchmark VDFs used in this chapter
are summarised.
The green line in Fig. 3.3 shows the VDF from the result of Aquarius project, takenfrom Fig.2 in Vogelsberger et al. (90). It shows the median of the velocity modulus

54

0.5

f(u)/uf(v)Chapter 3

SHM
Vogelsberger et al.
Ling et al.
Mao et al.

0.45
0.4

0.2

0.16

0.35

0.14

0.3

0.12

0.25

0.1

0.2

0.08

0.15

0.06

0.1

0.04

0.05

0.02

0

100

200

300

400

500

600

SHM
Vogelsberger et al.
Ling et al.
Mao et al.

0.18

700

v[km/s]

0

100

200

300

400

500

600

700

u[km/s]Figure 3.3: The VDFs of dark matter halo from simulations are shown together withthe standard Maxwellian halo (SHM). Left panel shows the normalised VDF f (v)s in
the galactic frame and right panel shows the VDFs in the local reference frame at the
Sun in the form of f (u)/u in eq. 2.1. Detailed descriptions for Vogelsberger et al. (90)(green), Ling et al. (34) (red), Mao et al. (92) (blue) simulated halos are given in the
text.
distributions for 2 kpc boxes centred between 7 and 9 kpc from the centre of simulated
halo, Aq-A-1 (93). The broad bump above 250 km/s originates from bumps in eachsingle box at approximately the same velocity, with similar amplitude. They foundthat these features appear in all six of simulated halos, which suggests that they do
not reflect local structures but rather a global property of the inner halo, possibly aconsequence of real dynamical structure. We take this VDF as a representative VDF
with bumpy structures.
The velocity modulus in a spherical shell 7 < R < 9 kpc around the galactic centershown in Fig.4(d) of Ling et al. (34) is taken in the red line in Fig. 3.3. The VDF is
platykurtic, i.e. fatter than a Gaussian distribution with the same standard deviation.
Equilibrated self-gravitating collisionless structures can be described by non-extensive
statistical mechanics, which gives theoretical support to the use of the Tsallis distribution for describing gravitational structures such as galactic halos (26). For the particles
in this shell they showed the VDF is indeed well fit by the Tsallis distribution and thebest fit kurtosis parameter K = 2.44 agrees well with the observational value K = 2.39.
This agreement is better than the one from the best-fit generalised Maxwellian, which
is K = 2.71. The best-fit Tsallis distribution gives the velocity dispersion parameterv0 = 267.2 km/s. This VDF will serve as a representative of a realistic description ofa galaxy, considering interactions between the baryonic and the dark components.
The third benchmark velocity distribution is taken from Fig.1 of Mao et al. (92),which is the stacked VDF for ninety-six halos from the Rhapsody simulations (95).
They argue that the largest current theoretical uncertainty in the VDF arises fromthe unknown radial position of the solar system relative to the dark matter halo scaleradius, r/rs . In the range of current observations ((92) and its references) r/rs isChapter 3

-3

x 10

0.14

SHM

0.12

ldd/lH=0.25

0.1

ldd/lH=1.00

f(u)/uf(u)/u

x 10

55

0.14
0.12
0.1

0.08

0.08

0.06

0.06

0.04

0.04

0.02

0.02

0

100 200 300 400 500 600 700 800

u[km/s]0-3

ldd/lH=0.25,m=120km/s,vlag=50km/s
ldd/lH=0.25,
m=50km/s,vlag=0km/s
ldd/lH=0.25,
m=50km/s,vlag=150km/s

100 200 300 400 500 600 700 800

u[km/s]Figure 3.4: The dark disc velocity distributions added to dark matter halo in the local
reference frame at the Sun are shown in the form of f (u)/u. red: with ρdd /ρH = 0.25,
σ = 50 km/s, vlag = 50 km/s, green: ρdd /ρH = 1, blue: σ = 120 km/s, magenta:
vlag = 0 km/s, cyan: vlag = 150 km/s. Unless otherwise specified in the legend,parameters are fixed to ρdd /ρH = 0.25, σ = 50 km/s, vlag = 50 km/s and ρH =0.3 GeV/cm3 . Note that the total dark matter density is increased to ρdd + ρH . The
SHM with no dark disc is shown for comparison (black).
suggested to be within 0.15 ∼ 6. Here we picked the marginal value of r/rs = 0.15.
This halo will represent the halo model peaking at low-velocity.
Descriptions of our benchmark VDFs are summarised in Tab. 3.1. Note that VDFs
are regarded to be isotropic throughout our discussion to simplify the argument.
Simulations show that the local density of the dark disc ρdd could range from a few
percent (38) up to ∼ 1.5 times (36) of the local dark matter halo, ρH . We consider
scenarios with the mild dark disc (ρdd =0.25ρH ) and the strong dark disc (ρ√dd = ρH )assuming Maxwellian VDF with fiducial parameters for VDF; vrms = 50 3 km/s
and the relative circular velocity to the Sun, vlag = 50 km/s following (96) and (75).
However, the VDF of the dark disc also contains large uncertainties. The velocity
dispersion is likely to be substantially larger (37) and more realistic values can beprovided by full cosmological hydrodynamics simulations (75). The uncertainty of vlag
is discussed in (36). In√ order to test properties of the dark disc velocity distribution,
we varied vrms to 120 3 km/s, and vlag to 0 and to 150 km/s fixing the density toρdd = 0.25ρH . Figure 3.4 shows the distributions of combined dark matter halo anddark disc in the local reference frame at the Sun. Note that for the dark matter halo
we fix the total local density to the standard value ρH = 0.3 GeV/cm3 and allow theadditional contribution from the dark disc.

Chapter 3

3.4

56

The effect of WIMP velocity distribution to thecapture rate of WIMPs

In this section we discuss uncertainties associated with WIMP capture in the Sun.
We evaluate the impact of the local circular speed of the Sun, the halo escape speed,
various VDFs for the dark matter halo and the dark disc introduced in the previous
section on capture rate, respectively.
For our evaluation we use DarkSUSY (83) with which we can precisely compute
capture rates as explained in Chap. 2. We modify VDFs parameters or introduce new
VDFs in the DarkSUSY and calculate the capture rate for various WIMP masses for
SD or SI coupling cases.

f(u)/u

x 10

-4

0.14

SHM (vsun=220km/s)
vsun=200km/s
vsun=240km/s
vsun=280km/s

0.12
0.1
0.08
0.06
0.04
0.02
0

100

200

300

400

500

600

700

u[km/s]Figure 3.5: The VDFs in the local reference frame at the Sun with varied v are shown
together with the standard Maxwellian halo (SHM) in the form of f (u)/u, for v =200 km/s (red), 240 km/s (green) and 280 km/s(blue).
We first investigate how the WIMP capture rate in the Sun changes with respect
to the local circular speed of the Sun v . We took the range 200 ∼ 280 km/s from
McMillan et al. (97) and fixed vrms to 270 km/s. In Fig. 3.5 we show the VDFs withvaried v in the local reference frame at the Sun and the effect on capture rate in
Fig. 3.6 as a function of WIMP mass. The size of the uncertainty in capture rate canbe well visualised by showing the relative change compared to the SHM with fiducial
value v = 220 km/s. For this purpose we introduce a boost factor, which is definedby the ratio of the capture rate of the assumed velocity distribution (C) divided by
the capture rate of SHM (CSHM ) as C/CSHM . Results are separately shown for WIMPs
that undergo a SD or SI interaction only as each process experiences different effective
capture from the multiple nuclei inside the Sun. It appears that higher vrms reduces

57

2.2
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4

vsun=200km/s, vrms=270km/s
vsun=200km/s, vrms=245km/s
vsun=240km/s, vrms=270km/s
vsun=240km/s, vrms=294km/s
vsun=280km/s, vrms=270km/s
vsun=280km/s, vrms=343km/s

1

10

2

3

4

10
10
10
WIMP mass[GeV]

Capture boost (SD)

Capture boost (SD)

Chapter 3

2.2
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4

vsun=200km/s, vrms=270km/s
vsun=200km/s, vrms=245km/s
vsun=240km/s, vrms=270km/s
vsun=240km/s, vrms=294km/s
vsun=280km/s, vrms=270km/s
vsun=280km/s, vrms=343km/s

1

10

2

3

4

10
10
10
WIMP mass[GeV]

Figure 3.6: The capture boosts defined as C/CSHM are shown as a function of WIMP
mass for the SD (left) and SI (right) couplings assuming an orbital speed of the Sun ofv = 200 km/s (red solid), 240 km/s (green solid)qand 280 km/s(blue solid) with fixed
vrms = 270 km/s, dashed lines for varied vrms = 3/2v with same color scheme.
the capture rate and the size of the uncertainty appears larger for heavy WIMPs. For
example, for a 20 GeV WIMP with SD (SI) coupling the capture rate changes as muchas −24 % (−15 %) when v = 280 km/s, whereas for a 1000 GeV WIMP with SD (SI)
coupling the change will be −45 % (−38 %).
This result, in agreement with previous studies (66) for SD coupling, is easly understood; the higher the relative speed between the WIMPs in the halo and the Sun, thestronger the WIMPs in the local reference frame at the Sun get boosted and become
harder to be captured. This is especially true for the capture of heavy WIMPs that are
sensitive to the low-velocity tail and thus experience more significant diminution. On
the other hand, the shift of WIMPs to the high-velocity region will increase the signal
rate in direct detection of light WIMPs (71). As a result direct and indirect detections
will get opposite signs in the change, thus they are ‘anti-correlated’. However, Fig. 3.5
gives us a hint that this can be reversed for the high mass region, because the capture
rate and the scattering rate are calculated by integrating over f (u)/u (see also (76)).
In Fig. 3.7, we show the relative changes in capture rate for the set of our benchmark
velocity distributions. Several comments are in order here.
The small structures in Vogelsberger et al. (90) dark matter halo seen in the galactic
frame (Fig. 3.3 right) are washed out in the local reference frame at the Sun (Fig. 3.3left). Because the circular speed of the Sun is almost 1/3 of the size of the typical VDF
of dark matter halo, we expect that any structure of the size smaller than this will beeasily spread out. The SD (SI) coupling capture of 10 GeV WIMPs that come from theentire range of the VDF, including the bumpy shape above 250 km/s in Vogelsberger et

58

1.7
1.6
1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7

Capture boost (SI)

Capture boost (SD)

Chapter 3

Vogelsberger et al.
Ling et al.
Mao et al.

1

10

2

3

10
10
10
WIMP mass[GeV]

4

1.7
1.6
1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7

Vogelsberger et al.
Ling et al.
Mao et al.

1

10

2

3

10
10
10
WIMP mass[GeV]

4

Figure 3.7: The capture boosts for SD (left), SI (right) coupling WIMPs as a function of
WIMP mass for VDFs of the dark matter halo in Vogelsberger et al. (90) (green), Ling
et al. (34) (red) and Mao et al. (92) (blue) compared to SHM. Detailed descriptions of
VDFs can be found in Chap. 3.3.
al. (90), gives ∼2 (1)% change compared to the SHM. In the same simulation (90), theyfound that the contributions from either unbound streams or sub-halos are likely subdominant. For ultra-fine structures beyond the highest resolution of the current N-bodysimulations, it is studied to have negligible impact on direct detections (98; 99; 100).
Note that the effect of these properties on the capture rate in the Sun is expected tobe more suppressed because of the larger size and the longer time scale of the capture.
The transformation from the halo frame to the local reference frame at the Sun alsochanges the overall characteristics of VDF. Because the capture rate is not sensitive
to specific changes in the structure of the VDF, we can simply interpret the overall
effect as another source of anti-correlation between the low-velocity and high-velocity
populations. Simply, either a high velocity dispersion or high kurtosis will result ina tendency to enrich the high-velocity region at the local reference frame of the Sun.
In Fig. 3.1, we can see that for 50 (500) GeV WIMPs the SD (SI) capture process isefficient for velocities below 300 km/s. The highly peaked f (u)/u in this region of Mao
et al. (92) halo seen in the right plot of Fig. 3.3 brings 23 (19)% increase of SD (SI)
capture rate for 50 (500) GeV WIMP and keeps more efficient capture than SHM forlighter WIMPs. As opposed to it, in Fig.6 in Mao et al.(92), the blue line shows thenegative boost for our adopted halo benchmark model of Mao et al. (92) for 10 GeV
above 300 km/s (umin /uesc > 0.55), which manifests the anti-correlated responses of
direct and indirect detections discussed previously. We also found that Ling et al. (34)
dark matter halo with broad shape and high velocity dispersion gives relatively smallabundance in this velocity range, resulting in ∼8 (8)% decrease of the SD (SI) capture
rate for 50 (500) GeV WIMP and keeps the negative boost for lighter WIMPs.
By investigating three characteristic halos, we conclude that the capture rate is not

59

1.05
1.04
1.03
1.02
1.01
1
0.99
0.98
0.97
0.96
0.95

Capture boost (SI)

Capture boost (SD)

Chapter 3

vesc=498km/s
vesc='

1

10

2

3

4

10
10
10
WIMP mass[GeV]

1.05
1.04
1.03
1.02
1.01
1
0.99
0.98
0.97
0.96
0.95

vesc=498km/s
vesc='

1

10

2

3

4

10
10
10
WIMP mass[GeV]

Figure 3.8: The capture boosts for SD (left), SI (right) coupling WIMPs with the escapespeed of the Milky Way of red : 498 km/s, green : infinity, compared to 544 km/s in
SHM as a function of WIMP mass. The normalisation of the VDFs are kept to unity
for different truncations.
significantly affected by the VDF itself. This is also found to be the case in direct
detections (101), generally the size of uncertainty from VDF is discussed to be within
10 ∼ 20% (90; 102).
Some changes in the halo models or astrophysical assumptions will give rise to an
overall shift of the WIMP population from the high-velocity to the low-velocity region
or vice versa. In this case, the expected change in event rates for both direct andindirect searches are rather moderate. For the orbital speed of the Sun, the uncertainty
is found to be less than ∼40% in this work and ∼20% for various direct detector set-ups
examined (76). The expected change in capture rate due to the halo VDF is found to
be less than ∼20% in this work and a similar magnitude is expected for direct detection
experiments in general (90; 101; 76; 102). The effect is smaller for direct detection in
the heavy WIMP scenarios and for indirect detection in light WIMP scenarios, as for
those cases a wide range of the velocity distribution contributes. For example, even
if the WIMP distribution has shifted from the low-velocity to the high-velocity region
which is harder to be captured, they can still contribute to the indirect neutrino signal
in a light WIMP scenario, although not in a heavy WIMP scenario. While the different
response to these type of uncertainties is a limiting factor in the comparison between
direct and indirect detection (76) it also stresses the complementarity between the two
methods.
N-body simulations of galactic halos show a steeper fall-off at high-velocity tail thanwhat is expected for a Maxwell-Boltzmann distribution (103; 104). Figure 3.8 examines
the cases with different velocity cuts varied to 498 km/s (105) or infinity (106; 107),
compared to the SHM with the fiducial value vesc =544 km/s. Since the normalization of
the VDF is kept to be 1 for different truncations, the low and high parts of VDF are notcompletely uncorrelated, but it can be regarded to be approximately uncorrelated. The
indirect detection of heavy WIMPs is blind to the reduction of the WIMP population

10

60

ldd/lh=1.00,m=50km/s,vlag=50km/s
ldd/lh=0.25,m=50km/s,vlag=0km/s
ldd/lh=0.25,m=50km/s,vlag=50km/s
ldd/lh=0.25,m=120km/s,vlag=50km/s
ldd/lh=0.25,m=50km/s,vlag=150km/s

2

10

1
1

10

2

3

4

10
10
10
WIMP mass[GeV]

Capture boost (SI)

Capture boost (SD)

Chapter 3

10

ldd/lh=1.00,m=50km/s,vlag=50km/s
ldd/lh=0.25,m=50km/s,vlag=0km/s
ldd/lh=0.25,m=50km/s,vlag=50km/s
ldd/lh=0.25,m=120km/s,vlag=50km/s
ldd/lh=0.25,m=50km/s,vlag=150km/s

2

10

1
1

10

2

3

4

10
10
10
WIMP mass[GeV]

Figure 3.9: The capture boosts for SD (left), SI (right) coupling WIMPs for dark discscenarios. green: with ρdd /ρH = 1, magenta: vlag = 0 km/s, red: ρdd /ρH = 0.25,
σ = 50 km/s, vlag = 50 km/s, blue: σ = 120 km/s, cyan: vlag = 150 km/s. Unlessotherwise specified in the legend, parameters are fixed to ρdd /ρH = 0.25, σ = 50 km/s,
vlag = 50 km/s and ρH = 0.3 GeV/cm3 (Note that the total dark matter density is
increased to ρdd + ρH ). The boosts in capture rate as a function of WIMP mass are
given relative to a scenario without the dark disc.
in the high-velocity tail, while the marginal changes at the high mass come from therenormalization of the low-velocity part. Also the changes in the low mass region are
mild, because the WIMP population in the high-velocity tail is originally small andkinematically hard to capture. As a result the uncertainty from a high-velocity cut isless than 1% for indirect detection for any WIMP masses.
In light WIMP scenarios the uncertainty on the high-velocity tail requires a carefulinterpretation of direct detection experiments (75; 71; 102; 103; 108), while the size ofthe effect on capture rate in the Sun as studied here will be marginal. Note that several
studies have pointed out the large uncertainty in modeling of the high-velocity tail ofthe dark matter halo (103; 102; 77; 72), therefore it requires a careful interpretation of
direct detection results in the low mass region (75; 71; 102; 103; 108). This merits thecomplementarity of direct and indirect searches.
In a heavy WIMP scenario the opposite effect will occur; indirect detection results
need to be interpreted in a careful manner as the capture will be governed by the
low-velocity part of the VDF. Luckily, sources of uncertainties in the low-velocity tailof simulated halos are expected to be small (79).
Another example of uncertainty, which would change the VDF in an uncorrelatedmanner, is the existence of the dark disc. Figure 3.9 shows the capture boosts for thepreviously discussed dark disc scenarios. The mean relative velocity of co-rotating dark
disc with respect to the Sun is smaller than that of the non-rotating dark matter halo.
Hence, the dark disc primarily populates the low-velocity region. We consider SD andChapter 3

61

SI couplings exclusively, which was not done previously.
The first thing we notice is that the boost effect appears more dramatically in the
spin-dependent coupling case. This is because SD capture is more sensitive to the lowvelocity region. Figure 3.4 shows that the dark disc contribution is mainly contained
below 200 km/s. In Fig. 3.1, we can see that for WIMP masses above 100 GeV(1000 GeV) for SD (SI) coupling, efficient captures are allowed for WIMPs with relative
velocity to the Sun below 200 km/s. Taking these facts into account, the boost factor
is as much as 15 (6) for SD (SI) coupling at 100 GeV for the strong (ρdd /ρH = 1) dark
disc model, and larger in the higher mass region.
Secondly, our result for the strong dark disc on SI coupling is in agreement with theprevious study done by Ling (75). A study by Bruch et al. (73) has discussed CMSSM
models that define the ratio between SI and SD couplings for specific scenarios andfound boost factors as much as an order of magnitude for 100 GeV WIMPs. By running
DarkSUSY (83), we confirm that some 100 GeV neutralinos in CMSSM can result inan order of magnitude boost for the strong dark disc. We point out that this highboost can only be realised in scenarios with large SD contributions.
Lastly, we confirm that the capture boost can be limited with larger values ofrelative velocity or velocity dispersion used for modeling dark disc (75).
In manner contrary to the uncertainty of escape speed, enhancement of low-velocity
region due to the dark disc will significantly affect heavy WIMP indirect detection,
whereas direct detections also get a sizeable effect, with a factor 2 ∼ 3 for WIMP
masses above 50 GeV (96) for recoil energies of 5 − 20 keV. Below this mass direct
detections are hardly altered, for example the so called channeling region of the DAMA
annual modulation signal is not changed (75). In summary, the uncertainty comingfrom the dark disc on capture rates can be orders of magnitude depending on the WIMP
mass and coupling type. But for the reason that the dark disc only adds an extra
contribution to which indirect detection is more sensitive, it makes the interpretationof null results from indirect detections more conservative.
Possible sources of uncertainty in the low-velocity tail have been extensively discussed in the literature (109; 110; 111; 112; 113; 86; 114; 115). The phase space of
low-velocity WIMPs could be influenced by the gravitational fields of the Sun and theplanets. Recently, Sivertsson et el. (78) concluded that the loss of the weakly captured
population by the Sun due to Jupiter depletion, which has been known to substantiallyreduce the capture of WIMPs heavier than a TeV, can be compensated with gravitational diffuse components. As a result, the simple free-space approximation in (109)
turned out to be accurate (78).

3.5

Summary

We conclude that the uncertainties for light WIMPs are mild; for 20 GeV WIMP,
they are 18 (14)% from local circular speed, 1 (1)% from high-velocity cut for thetested range for SD (SI) couplings. We also found that even though the form of theChapter 3

62dark matter halo VDF is unknown the impact on the capture rate of light WIMP
is suppressed because the effect is entirely integrated, which is consistent with thediscussion in (116); it is up to 16 (10)% for the examined dark matter halos fromcosmological simulations. In summary, the estimated sizes of the uncertainties frompossible sources in VDF are found to be less then 24% (17%) for 20 GeV SD (SI)
coupling WIMPs, if there was no dark disc.
For heavier WIMPs, the uncertainties are larger because of their sensitivity to the
low-velocity tail, but it still is in the level of 50%; they are 45 (32)% from local circular
speed, 1 (1)% from high-velocity cut for the tested range, up to 11 (16)% from thedeviation of the VDF from Maxwellian for SD (SI) coupling 500 GeV WIMP.
The results are summarised in Tab. 3.2.
WIMP mass
(GeV)
20
500

v
(%)
18 (14)45 (32)

vesc
(%)
1 (1)
1 (1)

VDF
(%)
16 (10)11 (16)Total(%)
24 (17)46 (36)

Table 3.2: Summary of the size of the uncertainty for each source; v is the local circular
speed of the Sun, vesc is the high-velocity cut, VDF is model of velocity distributionfunctions. Also total sum of them are shown for two cases of WIMP mass, for SD
coupling (SI coupling) cases.

By quantitative research of the uncertainties, we concluded that the size of uncertainties in the solar WIMP search are moderate, which ensure the interpretation of
their results within 50% error. This result will be taken into account for the resultof the Super-Kamiokande data analysis together with other uncertainties, in Chap. 8.
The existence of the dark disc enriching the low-velocity population could result in asignificant increase in capture rate especially for heavy WIMPs, but the interpretationof the indirect searches compared to direct searches will become more conservative with
its consideration.

Chapter 4
The Super-Kamiokande detector
4.1

Overview

The Super-Kamiokande (SK) is a cylindrical water Cherenkov detector located in
the Kamioka mine in Japan. The Super-Kamiokande observes Cherenkov lights which
are emitted by the relativistic charged particles travelling in the water by photomultiplier tube (PMT)s. Figure 4.1 shows a display of a neutrino event observed in the
Super-Kamiokande detector. The Cherenkov light produces characteristic ring patternon the finely aligned PMTs on the wall of the water tank.
The Super-Kamiokande, whose name stands for “Super-Kamioka Nucleon Decay
Experiment”, was first proposed to search for nucleon decays and various studies ofneutrinos. The Super-Kamiokande sets the world best limits on lifetimes of various
nucleon decay modes. The studies of neutrino in the Super-Kamiokande target various
sources such as the Sun, atmosphere and the human-maid neutrino beams; it has
been used as a far detector for long-baseline neutrino oscillation experiments suchas K2K and T2K. The Super-Kamiokande has been contributing significantly to ourunderstanding of neutrino physics including the first evidence of neutrino oscillation
by analysing atmospheric neutrinos, the confirmation of the solar neutrino oscillationand the high-precision measurement of the mixing angle θ13 using the neutrino beam.
Various astrophysical sources, such as supernovae, gamma ray bursts and dark matterare studied using neutrinos as well.
Super-Kamiokande was commissioned after a great success of the Kamiokande experiment in neutrino physics and astrophysics. The construction was started in 1991
and finished in 1995, and the operation was started in April 1996. After about five
years (1678 days) of operation which is called “SK-I”, it was shut down for maintenanceand upgrade in July 2001. On November 12, 2001, a terrible accident occurred during
refilling water after maintenance work is done. An implosion of a PMT destroyed morethan a half of PMTs by a chain reaction. The operation was resumed (called “SK-II”)
from October 2002 with temporary PMT coverage which was about a half of that of
SK-I. SK-II continued measurement for three years and paused in October 2005 torecover the photo coverage. After recovering, it resumed operation (called “SK-III”) in
63

Chapter 4

64

NUM
4
RUN
9000
SUBRUN
396
EVENT
64753459
DATE **-Jul-15
TIME
13:41:42
TOT PE:
3858.0MAX PE:
24.0
NMHIT : 1456
ANT-PE:
25.4
ANT-MX:
4.2
NMHITA:30

90/00/00:NoYet:NoYet
90/00/00:NoYet:NoYet
90/00/00:NoYet:NoYet
90/00/00:NoYet:NoYet
90/00/00:NoYet:NoYet
90/00/00:NoYet:NoYet
90/00/00:;R= 0:NoYet
R :Z : PHI : GOOD
0.00: 0.00: 0.00:0.000
CANG : RTOT : AMOM : MS

RunMODE:NORMALTRG ID :00000111
T diff.:0.264E+04us
FEVSK :81002803
nOD YK/LW: 2/ 1
SUB EV : 0/ 1
Dec-e: 0( 0/ 0/ 0)*NO KEKGPS RED*

Comnt;

Figure 4.1: The development view of the Super-Kamiokande detector taken with anobserved neutrino event. The size of the small circles correspond to the Cherenkov
photons detected in each PMT, and the larger circle indicates the Cherenkov ringpattern.

Chapter 4

65

June 2006 with similar PMT coverage to the original design and that of SK-I period.
In September 2008 new electronics modules were developed and installed. After the
upgrade of electronics, the running period is changed once again (called “SK-IV”) andbeing continued until now.
This chapter will start with a brief introduction of Cherenkov radiation in Chap.
4.2. In Chap. 4.3, the detector site and facilities will be introduced. In Chap. 4.4,
photomultiplier tubes installed in the SK is explained roughly. Chap. 4.5 summarises
the electronics system of the SK-IV. In Chap. 4.6, short summary of the SK calibration
system will be given.

4.2

Cherenkov radiation

When a charged particles travel with velocity v in medium with refraction index
n, if v  c/n, the Cherenkov photons are emitted. Passing through the medium, theelectromagnetic field of the charged particle excites the atoms and molecules whichquickly return to ground state by emitting Cherenkov photons. The geometry of theemitted photon with speed of c/n, being slower than the charged particle with speedof v = βc, results in a cone-shaped shock wave front as shown in Fig. 4.2.

Figure 4.2: Image explaining Cherenkov photon propagation.
The characteristic half opening angle θC (“Cherenkov angle”) between the directionof the traveling particle and the Cherenkov photon is determined as:
c/n
c
=
.
(4.1)
βc
nv
For a particle traveling with v ∼ c, the Cherenkov angle has a maximum angle being
about 42◦ .
cos θC =Chapter 4

66

A momentum threshold to generate Cherenkov radiation is:
p = mv ≥

mc
.
n

(4.2)

With the refractive index of water being ∼1.34, the thresholds for e, µ, π ± and p are
0.57, 118, 156 and 1051 MeV/c respectively.
The number of photons emitted by a charged particle of charge ze per unit path
length x per wavelength λ is written as a function of λ:2παz 2
d2 N
c2
=
(1
−
),
dxdλ
λ2
n(λ)2 v 2

(4.3)

where α is the fine structure constant. Around 340 photons/cm are emitted for λranging 300 ∼ 600 nm.

4.3

The detector

Super-Kamiokande is located in the Mozumi mine of the Kamioka Mining andSmelting Company near the village of HigashiMozumi, Gifu, Japan. Geographic coordinate is 36◦ 25’N and 137◦ 18’E, with the altitude of about 370 m. The detector lies
under the peak of Mt. Ikenoyama so it is beneath 1000 m of rock, which corresponds
to 2700 m-water-equivalent overburden. The reduced cosmic ray muon even rate is ∼2.7 kHz in the detector. Figure 4.3 shows the schematic view of the detector and its
location. The coordinate system of the detector used throughout this work and foundother SK publications is drawn in Fig. 4.4.
The Super-Kamiokande is a cylindrical tank with a diameter of 39.3 m and a heightof 41.4 m filled with 50 kilo ton of purified water.
Clean and abundant water supplied by the Kamioka mine is continuously circulated
in the SK tank through water purification system in the flow rate of 35 ton/hour. To
maximize water transparency and to remove radioactive materials, the water used tofill the SK tank is highly purified by a multi-step system including filtration, reverse
osmosis (RO) and degasification.
The radon background present in the mine is in relatively high level. In orderto reduce back-ground radioactive events which is a background source for the solar
neutrino observation in the MeV energy region, also to keep low radon level aroundthe experimental area, fresh air pumped in from a site outside and well away fromthe mine entrance is purified and is continuously pumped to inside the SK tank at a
positive pressure with flow rate about 18 m3 /hour (119).
Within the tank, stainless-steel frameworks are spaced approximately 2 ∼ 2.5 m
inside the tank wall to support PMTs. The remaining PMT glass envelopes, rest
spaces are covered with black opaque polyethylene sheet which optically separates
the tank to two concentric cylindrical regions. The inner and the outer part of theframes are called the inner detector (ID) and the outer detector (OD) respectively.

Chapter 4

67

Figure 4.3: A sketch of the Super-Kamiokande detector and its location. Figure istaken from (117).

Chapter 4

68

Figure 4.4: A sketch of the definition of detector coordinate system. Figure is takenfrom (118).
The inner detector is instrumented with approximately 11,100 (5200 for SK-II) 20-inch
Hamamatsu photomultiplier tubes (PMTs). The outer detector is instrumented with
∼ 1,880 8-inch PMTs for use as veto counter for incoming and outgoing muons. The
OD acts as a passive shield for low energy particles such as neutrons and γ rays fromthe surrounding rocks as well. The drawing of ID and OD is shown in Fig. 4.5.
PMTs are connected to high voltage supplies and electronics via cables which arebrought up to the tank top and distributed to four huts which contain electronics racks
and front-end DAQ computers. Another electronics hut is built at the center of thefour huts to contain electronics and associated computers for triggering, housekeeping,and Global Positioning System (GPS) time synchronization systems. The details of
the detector can be found elsewhere (117).

4.4

The photomultiplier tubes

The ID PMTs were mounted approximately every 70 cm on the side wall, top andbottom of the tank. The photocathode coverage of the ID surface is 40% for SK-I,III
and IV and about 47% of it for SK-II. 50 cm PMTs (Hamamatsu R3600) are developed
by Hamamatsu Photonics in collaboration with Super-Kamiokande (120) and used for
ID. The bialkali (Sb-K-Cs) coated photo-cathode has peak quantum efficiency of about
21% at 360 ∼ 400 nm wavelength as shown in Fig. 4.7. The neck of each PMT was

Chapter 4

69

Figure 4.5: A sketch of the ID, OD regions in the SK detector.
coated with a silver reflector to block external light. Figure 4.6 shows a schematic viewof the ID PMT. The collection efficiency at the first dynode is over 70%. The dynamic
range covers from 1 to 300 photoelectron (p.e.). The transit time spread for a single
p.e. signal is about 2.2 ns. The average dark noise rate at the 0.25 p.e. threshold isabout 3 kHz. The ID PMTs were operated with a gain of 107 at a supply high voltage
ranging from 1700 to 2500 V. Fig. 4.8 shows the charge distribution for single p.e.,obtained from the calibration measurement with nickel source (121). A clear single
p.e. peak can be seen.

Figure 4.6: Schematic view of a 50 cm PMT used for ID. Figure is taken from (117).
After the accident in 2001, the PMTs have been instrumented with acrylic covers toChapter 4

70

Figure 4.7: quantum efficiency of the bialkali (Sb-K-Cs) coated photocathode as afunction of wavelength. Figure is taken from (117).
avoid chain reaction of implosions since SK-II period. A 12 mm thick UV-transparent
acrylic dome is used to cover the photocathode of each PMT, and the side of the PMT
is protected with Fiber-Reinforced Plastic (FRP) shield with holes letting water flows
through inside the case. Transparency of the acrylic cover in water is more than 96%
above wavelength 350 nm.
To prevent the geomagnetic field affecting the PMT response, 26 horizontal and vertical Helmholtz coils are implemented around the detector, which reduce the magnetic
field from 450 mG to 50 mG.
The OD is implemented with 20 cm PMTs (Hamamatsu R1408) facing outward.
The photocathode of the OD PMT is covered with a 60 cm × 60 cm × 1.3 cm wavelength shifter plate, in order to increase light collection efficiency.

4.5

Electronics and data acquisition system in SKIV period

After 12 years of operation without a major upgrade, the detector data acquisition(DAQ) system was upgraded in 2008. In the designing of the new system, the main goal
was to implement the system able to use software trigger instead of hardware trigger
used for SK-I,II and III. The new system is developed that all the data of PMT hits
can be sent to the online PCs, and then various software trigger can be applied in orderto pick up the events for each science goal. In this way, more advanced science goals
can be examined such as: more precise determination of oscillation parameters and theChapter 4

71

Figure 4.8: The single p.e. distributions in pC unit from the calibration data withnickel source in SK-III. The black line shows the data with normal set up of SK PMTs,
the green line shows the data with double gain and half threshold and the red line islinear extrapolation. Figure is taken from (121).

Chapter 472

observation of energy dependence of neutrino oscillation in the low energy region using
solar neutrinos by the decreased energy threshold, fully detecting nearby supernova
burst by the increased size of the storing data, detecting relic supernova neutrinos bytagging 2.2 MeV gamma ray from neutrons.
The newly developed front-end electronics is called QBEE, which stands for “QTC
(charge (Q) to Time Converter) Based Electronics with Ethernet”. The QBEE has 24
analog input channels - a board is equipped with eight QTC chips which have three
input channels each - which can collect signals from 24 PMTs. A custom applicationSpecific Integrated Circuit (ASIC) is developed as a high-speed QTC. It integrates the
input charge and issues an output pulse whose width is proportional to the amount ofcharge. Figure 4.9 shows the block diagram of the QTC.

Figure 4.9: Block diagram of a QTC. Each channel has three gain ranges: small,medium and large. Figure is taken from (122).
The output of QTC is sent to the multi-hit Time-to-Digital converter (TDC) called
AMT-3. The TDC measures the width of QTC output pulse by detecting the timeof leading and falling edges. Then the data of the width and arrival timing of theleading edge is sent to a Field-Programmable Gate Array (FPGA) and then finally tothe front-end PC. The front-end PC passes the data to 10 merger PCs in which thesoftware triggers for physics analyses are applied to select events. One front-end PC
receives the data from 30 QBEE boards. The data transfer is greatly improved byadopting Ethernet for transferring data to online PCs; 11.8 MB/s per QBEE board
has been achieved.
Figure 4.10 shows the block diagram of the new DAQ system. Detailed descriptions are found in (123). In the merged data, the software trigger searches the events
satisfying the trigger conditions for physics analyses. The basic trigger is applied by
calculating the number of PMT hits in a 200ns sliding window, which is the same as
the hardware trigger of the previous system for SK I-III. In SK-IV, there are usedfour types of basic trigger - Super Low Energy (SLE), Low Energy (LE), High Energy(HE) and special high energy (SHE) triggers. The different event widths are storedChapter 4

73

Figure 4.10: The schematic view of the data acquisition system in SK-IV. In the leftpart, newly developed front-end electronics boards connected with PMTs are shown.
In the right part, components of the new online system are shown. Figure is takenfrom (123).
for different trigger types; SLE trigger can be set to save 1.5µs of data and the muchless frequently triggering LE and HE triggers can save data for like 40µs. SHE has
been introduced with a threshold of 58 hits since September 2011. The thresholds andevent widths for these triggers are shown in Tab. 4.1. In addition to the basic trigger,
special triggers, for example for neutrino tagging, for the detector calibration with an
external source and for the T2K data are implemented.
More information of the new electronics and online system can be found in (123).

4.6

Detector calibration

4.6.1

PMT calibration

High-voltage setting
Setting appropriate HV for each PMT was performed in order to achieve same
output charge for the same incident light intensity for all PMTs. At the center ofthe detector, an isotropic light source, the “scintillator ball”, which is a 5 cm diameter
acrylic ball containing a diffuser is deployed. Lights from a Xe lamp are passed througha UV filter and then injected to the scintillator ball. To control the incident lightyields to each PMT at different locations, 420 pre-calibrated “standard PMTs” areChapter 4

74

TriggerThreshold (hits)

OD
SLE
LE
HE
SHE
AFT

22
34
47
50
70 → 58 (from Sep, 2011)
SHE, no OD

Event Width (µs)−0.5 ∼ 1.0−5 ∼ 35−5 ∼ 35−5 ∼ 35
35 ∼ 535

Table 4.1: The thresholds and event widths for OD and various ID triggers.

mounted in the ID and served as references for other neighbor PMTs. The HV valuesof the standard PMTs are initially adjusted so that they give the same ADC values
for incident light of ∼ 30 p.e.’s. The HV settings of other PMTs are adjusted so thatthe observed charges match those for the neighbor standard PMTs. After determining
the HV setting for all PMTs, the reproducibility is checked and confirmed to be within
1.3% RMS as seen in Fig. 4.11.

Figure 4.11: The percent differences in observed charge from the respective reference
value are accumulated for all PMTs. Figure is taken from (121).

Chapter 4

75

Relative gain
To measure the relative gain difference for each PMT, a nitrogen-laser-driven dyelaser is used as a light source. The measurements are done in two ways: first, the lightsource flashes at high-intensity, Iw , from which every PMT gets a sufficient numberof photons and create an average charge Qobs (i) in repeated events. Secondly, themeasurement uses flashes at low-intensity, Is , in which only a few PMTs are hit by the
number of times, Nobs (i), in repeated events, and each of them can be safely assumed
to be a single-p.e. hit. Qobs (i) and Nobs (i) have following factors involved:
Qobs (i) ∝ Is × a(i) × qe (i) × G(i),Nobs (i) ∝ Iw × a(i) × qe (i),

(4.4)
(4.5)

where a(i) is the acceptance at the i-th PMT, qe (i) is the quantum efficiency of thei-th PMT and G(i) is the gain of the i-th PMT.
By comparing two measurements, G(i) can then be derived as:
G(i) ∝

Qobs (i).
Nobs (i)(4.6)

By normalizing with the average gain over all PMTs, G = Qobs /N obs , the relative
gain of each PMT can be obtained. Figure 4.12 shows the accumulated relative gain
for all PMTs. The RMS of the distribution was found to be 5.9%. After adjusting HV
to have the same gain for all PMTs, this deviation is inferred to come from different
QEs among PMTs. The relative gain differences among PMTs are made as table andare used as fine corrections.
Absolute gain
For a uniform and stable source of single-p.e. level light, the “nickel source” that
isotropically emits ∼9 MeV γ-rays from thermal neutron capture by Ni is used. The
nickel source consists of Ni wires stored in a container with Cf-252 neutron sources. The
nickel source is deployed at the center of the tank and generates about 100 Cherenkov
photons per capture, assuring more than 99% of observed signals are from single p.e..
Applying the relative gain corrections obtained in the previous chapter, the cumulative single-p.e. distribution are measured for all PMTs. The resulting charge distribution is shown in Fig. 4.8. The sharp peak near zero is caused by electrons which
punch-through the first dynode, and the second round peak corresponds to single p.e.
signal. The averaged pC over the whole distribution was defined as the conversion
factor from pC to single p.e.; the obtained value of this conversion factor for SK-I,
SK-II, SK-III and SK-IV are 2.044 pC/p.e., 2.297 pC/p.e., 2.243 pC/p.e. and 2.658
pC/p.e., respectively.

Chapter 4

76

Figure 4.12: Accumulated relative gains of all PMTs. Figure is taken from (121).
Relative differences in QEs
We used the nickel source data to be compared with MC simulation which predicts
the number of photons arriving at each PMT. After correcting the distance from thesource position to the PMT position, the acceptance as a function of incident angle
and light propagation in the water, the remaining differences in observed number ofhits are attributed to come from the QE differences of PMTs. Figure 4.13 shows thedifference between number of hits from the calibration data and MC, after correcting
the distance and the angle-dependent acceptance. The obtained relative QEs of the
PMTs are tabulated and used in the MC simulations.
Relative timing
The timing response of a PMT depends on length of the cable, time walk of PMT
pulses at the discriminator, and processing time of electronics. The overall effect on
timing can be calibrated by injecting fast light pulses at various intensities. The laser
light from a fast Nitrogen laser is injected to a diffuser ball located at around the centerof the tank.
The TQ-map, timing as a function of the pulse height is fitted with a polynomial
function. Each PMT has its own TQ-map. Figure 4.14 shows an example of TQ
distribution for a PMT. The parameters resulting from the fitting are used to correct
the time response for each PMT as a function of charge.
The timing resolution is evaluated for each Qbin by accumulating all PMT timing
distributions after correction by their TQ-maps. The timing resolution is measured

Chapter 4

77

Figure 4.13: Hit probability as a function of PMT position, after normalising by averagevalue of all the PMTs for data (red line) and MC (blue line). For the barrel PMTs
shown in top figure, the horizontal axis denotes the z (cm) position of PMTs. The lower
figures show top (left) and bottom (right) PMTs, where the horizontal axis shows thesquare of the distance from the center (cm2 ). Figure is taken from (121).

Chapter 4

78

Figure 4.14: Typical scatter plot of TQ distribution for a single PMT for charge versus
hit timing after time-of-flight-correction. Larger T corresponds to earlier hits. Figure
is taken from (121).
to be 2.1 ns at single p.e. level and 0.5 ns at 100 p.e. level. Figure 4.15 shows thetiming distribution together with the fitted function resulting from the fit for Qbin =14, which corresponds to ∼1 p.e..
The stability of the time-response of PMTs during SK detector operation is monitored in real-time and correction is applied to the TQ-map. As a result, it is checked
to be stable within ± 0.1 ns for a few years.

4.6.2

Water quality and light reflection

By injecting a collimated laser beam with the wavelength at 337, 375 405, 445
and 473 nm vertically through the SK tank, the PMT hit timing distributions are
obtained for PMTs mounted at top and five barrel parts of the SK tank as shownin Fig. 4.16. While the sharp peak in later timing (right region of plots) represents
photons reflected by the bottom PMTs or black sheets, the hits in the left region arefrom photons scattered by the water. The total number of scattered photons and theshape of the time distribution are used to tune scattering and absorption properties
for the water.
The intensity of the light traveling in water exponentially decreases as:−1

I(λ) = I0 (λ)e L(λ) ,(4.7)

Chapter 4

79

Figure 4.15: Accumulated timing distribution of all the PMTs in Qbin=14. The fitted
asymmetric Gaussian distribution is shown by the green solid curve. Figure is takenfrom (121).
where L(λ) is the attenuation length. Attenuation length in water is described as L =(αabs + αscat.sym + αscat.asym )−1 in the SK MC, where αabs , αscat.sym and αscat.asym areabsorption, symmetric scattering and asymmetric scattering coefficients respectively.
Physically, the symmetric scattering may consist of Rayleigh and symmetric Mie scattering and the asymmetric scattering may consist of forward Mie scattering. Though
in the MC, forms of these parameters are empirically determined and the coefficientsare tuned by fitting the laser beam data with the least-χ2 method. The results of the
fit are shown in Fig. 4.17.
The laser-beam data for water transparency is continuously measured and the time
variations are monitored at several wavelengths. Figure 4.18 shows time variations of
the water parameters in SK-IV from October 2008 to November 2012. The symmetricscattering coefficient is relatively stable, being about 3% for August 2008 to November
2012. The absorption and asymmetric scattering coefficients have relatively larger timedependence, being about 20 ∼ 40% and 20 ∼ 60% in the same time period, respectively.

4.6.3

Energy scale calibration

The energy scale is determined by vertical cosmic-ray through-going muons whichstop in the bottom OD layer. The accuracy of the absolute energy scale is checkedwith four sources covering wide momentum range from a few tens of MeV/c to about
10 GeV/c.

Chapter 4

80

Figure 4.16: Typical hit PMT timing distributions of after time-of-flight correction for
the wavelength at 405 nm, for data (black circle) and MC (red solid line) normalized bytotal p.e.’s. The top plot is for the PMTs mounted at the tank top, and the following
five plots are for the PMTs on the divided regions on the barrel, from the top to thebottom. The hits within the first two blue lines are used to estimate the absorption
and scattering properties, and the hits between the last two blue lines are used for themeasurement of reflection, for which readers are referred to see (121). Figure is takenfrom (121).

Chapter 4

81

Figure 4.17: The water coefficient functions (lines) fitted to data obtained in April
2009 (circles). The asymmetric (pink) and symmetric (blue) scattering coefficients,
the absorption (red) coefficients are shown together with the all fitted functions added(black). Figure is taken from (121).

Chapter 482

Figure 4.18: Time variation of the asymmetric (black) and symmetric scattering (purple) parameters, and the absorption (blue) parameter in SK-IV, for several wavelengths.
Figure is taken from (121).

Chapter 4

83

Track lengths of high energy stopping muons
The first source of energy scale calibration covering 1 ∼ 10 GeV/c of momentum
range is high energy muons. Muons entering the detector from the top, traveling
downward and stopping inside the detector are selected to be used. With the known
energy loss being about 2.3 MeV per cm, the reconstructed muon track length iscompared with total p.e.’s. The track of stopping muon is reconstructed by connecting
the entering position at the detector and the vertex position of the subsequent decay
electron. The accuracy of the absolute energy scale is checked by comparing calibration
data and Monte Carlo simulation. As a result, data and MC agree within 0.7%, 1.1%,
2.0% and 2.2% for SK-I, SK-II, SK-III and SK-IV, respectively. Figure 4.19 shows theagreement between data and MC for SK-I.

Figure 4.19: The averaged Pp.e. /Pmom where Pmom is the momentum estimated withmuon track for SK-I data (solid) and MC (dashed) (left), and the ratio between MC
and data (right). Figure is taken from (124).

Cherenkov angles of low energy stopping muonsCherenkov angles of low energy cosmic ray muons are used for lower momentum (<400 MeV/c) calibration. The Cherenkov angle has momentum dependence as following:
s

1
m2
1
cosθC ==
1+ 2 .
nβ
n
p

(4.8)

Chapter 484

Selection of sample muons have same criteria as above, and total p.e.s are required tobe less than 1500 (750 for SK-II) which correspond to pc< 380 MeV/c in addition tothem. The data and MC agree within 0.7%, 1.3%, 2.1% and 2.1% for SK-I, SK-II,
SK-III and SK-IV respectively. Figure 4.20 shows the agreement between data andMC for SK-IV.

SK-IV

SK-IV

1.4
1.04
1.02

MC/DataP(pe)/P(θ)

1.2

1

0.98

0.8

Data
MC

0.6
200

1

250

300

350

400

0.96
450

Reconstructed Momentum in P(θ)

200

250

300

350

400

450

Reconstructed Momentum in P(θ)

Figure 4.20: The averaged Pp.e. /Pθ where Pθ is the momentum estimated with θC for
SK-IV data (solid) and MC (dashed) (left), and the ratio between MC and data (right).
Figure is taken from (125).

Invariant mass of pions
Pions produced by neutrino interactions can also be a source of calibration as theinvariant mass of π 0 can be reconstructed from two γ rays. The distribution of theinvariant mass peaking around 135 MeV is compared to MC as shown in Fig. 4.21.
The peak positions obtained by Gaussian fitting of data and MC agree within 0.7%,
1.3%, 0.3% and 1.7% for SK-I, SK-II, SK-III and SK-IV respectively.
Momentum spectra of decay electrons
Decay electrons produced from decay of cosmic-ray stopping muons are abundant
and their momentum spectra are well understood. To reject ∼6 MeV γ rays produced
from µ− capture on nucleons, events whose NHIT50 is greater than 60 (30 for SK-II) areChapter 4

85

Super Kamiokande IV 1096.7 days : Monitoring

100

80

60

40

20

0
0

50

100

150

200

250

300

Mass(2 e-like rings) MeV/c2

Figure 4.21: Invariant mass distribution of neutrino induced π 0 events (black dot) incomparison to MC (red box). Figure is taken from (125).

SK-IV

Number of events

4000

Mean Values
Data: 37.36 ± 0.05 MeV
MC: 37.11 ± 0.04 MeV

3000

2000

1000

0
0

20

40

60

Reconstructed momentum

80

Figure 4.22: Momentum spectrum of decay electrons (black dot) in comparison to MC(blue solid line) for SK-IV. Figure is taken from (125).

Chapter 486

selected. Figure 4.22 shows the momentum spectra of decay electrons in comparison
to the MC. The mean value of the spectrum agrees with MC within 0.6%, 1.6%, 0.8%
and 1.6% for SK-I, SK-II, SK-III and SK-IV respectively.
The stopping muons and decay electrons are used to check long-term variationof energy scale. The RMS of the mean reconstructed momentum of stopping muonsdivided by muon track length, and the mean reconstructed momentum of decay electron
are calculated. The size of the fluctuations are estimated to be 0.74%, 1.60%, 2.08%
and 0.39% for SK-I, SK-II, SK-III and SK-IV, respectively.
Combining the size of the uncertainties in energy scale calibration with the time
variation of them in quadrature, the total uncertainty in the energy scale is estimatedto be 1.1%, 1.7%, 2.7% and 2.3% for SK-I, SK-II, SK-III and SK-IV, respectively.
Figure 4.23 shows the estimated uncertainty of each source for SK I-IV.
As decay electron events are expected to uniformly occur, the averaged momentumof decay electrons are compared to the MC for five zenith angles. The detector gains
are uniform over zenith angles within ±0.6%, ±0.6%, ±1.3% and ±1.3% for SK-I, SKII, SK-III and SK-IV respectively. They are also uniform over azimuthal angles within
±1%.

87

(MC - DATA) / DATA (%)

Chapter 4

SK-I
8
6
4
2
0
-2
-4
-6
-8

(MC - DATA) / DATA (%)

10

2

3

4

10
10
10
momentum range (MeV/c)

10

5

SK-II
8
6
4
2
0
-2
-4
-6
-8
10

(MC - DATA) / DATA (%)µ rangeµ angle0π massdecay eµ rangeµ angle0π massdecay e
2

3

4

10
10
10
momentum range (MeV/c)

10

5

SK-III
8
6
4
2
0
-2
-4
-6
-8
10

µ rangeµ angle0π massdecay e
2

3

4

10
10
10
momentum range (MeV/c)

105Figure 4.23: Estimated energy scale uncertainties for four sources in their momenta
ranges, for SK I-IV are shown. Figure is taken from (125).

Chapter 5
Simulation of WIMP neutrinos andatmospheric neutrinos at SK
5.1

Overview

Neutrino events in the Super-Kamiokande are simulated by Monte-Carlo method.
The simulation is composed of three parts: expectation of neutrino flux, neutrino
interactions and particle tracking including Cherenkov photons in the detector.
For the calculation of the WIMP-neutrino flux at SK, the WIMPsim (88) simulator
is used. The atmospheric neutrinos originated from the cosmic ray interactions arebackground of the WIMP search. For the atmospheric neutrino flux, calculation by M.
Honda et al. (Honda flux) (126; 127; 128; 129) is mainly adopted to predict the flux
at SK.
By the neutrino interaction simulation code “NEUT” (130) (131), secondary particles are produced at the SK detector and surroundings considering several types ofneutrino interactions. With the detector simulation, Cherenkov photons are produced,
tracked and then finally recorded in electronics simulation.
In Chap. 5.2, the simulation of the WIMP-neutrino flux at SK site by WIMPsim isdescribed. Following chapter 5.3 will show the resulting WIMP-induced events detectedin SK. In Chap. 5.4, the atmospheric neutrino flux will be described and in Chap. 5.5
its oscillated feature will be explained. The neutrino interactions in the simulation
is summarised in Chap. 5.6. After that, events are propagated through the detector
response simulations, about which it is briefly described in Chap. 5.7. The informationof simulated events are recorded in the same structure as the observed data that canbe later processed by common reconstruction and reduction software programs.

88Chapter 5

5.2

89

The simulation of WIMP-neutrino flux by WIMPsim

For expecting the WIMP-neutrino spectrum at SK, the WIMPsim (88) version
3.01 is used. WIMPsim calculates the annihilation process of WIMPs inside the Sunthen tracks produced neutrinos all the way from the center of the Sun to a detector
in an event-based scenario. The propagation of the neutrinos includes vacuum andmatter oscillations with the full three-flavor mixing and neutrino interactions on theway through the Sun such as absorption, energy losses and regeneration from tau
decays.
We ran WIMPsim for WIMP mass 4, 6, 10, 20, 50, 80.3 (only for W + W − channel),
100 and 200 GeV for χχ → bb̄, χχ → W + W − and χχ → τ + τ − annihilation channels
assuming 100% brunching fraction of annihilation. For bb̄ and W + W − channels they
started from 6 GeV and 80.3 GeV, respectively instead of 4 GeV due to kinematical
restrictions. For each mass and annihilation channel, we simulated 106 annihilations.
WIMPsim calls the simulator Pythia to simulate the hadronization and decay of theannihilation products at the center of the Sun. And then it collects the produced
neutrinos and anti-neutrinos. WIMPs are predicted to be thermalized before annihilation and the initial momenta are neglected. As a result, a neutrino properties at thecreation simply are regulated by WIMP mass and annihilation channel.
After that, WIMPsim let all the emerged neutrinos propagate outwards throughthe solar medium treating neutrino oscillations and interactions simultaneously. Using neutrino-nucleon interaction cross-section, WIMPsim randomizes the frequency of
the interaction in the solar medium, which decides the interaction point or let theneutrino pass through without any interaction. On the way to the interaction point,
neutrinos are oscillated with total Hamiltonian including MSW potential term, whichis calculated for each density layer from the center along the radial direction sliced inreasonable size: a width of 0.3 % of the radius of the Sun. Three-flavor oscillation withparameters: θ12 = 33.461◦ , θ13 = 9.097◦ , θ23 = 40.686◦ , δ = 300, ∆m221 = 0.766 × 10−4
eV2 , ∆m232 = 0.274 × 10−2 eV2 are used assuming normal mass hierarchy. These values are consistently applied in the treatment of the oscillation effect on both WIMP
neutrinos and background (BG) atmospheric neutrinos.
The tracking of neutrinos is once stopped at 1AU from the Sun, then with respect
to the time and location of the detector, WIMPsim let the flux propagate forward or
backward to the exact distance to the detector, considering the matter effect in the
Earth. We generated the flux at the SK location averaging from the spring equinox tothe autumn equinox. The final neutrino spectrum delivered at the SK site is shown in
Fig. 5.1. Note that the fast oscillating pattern seen in Fig. 2.8 is washed out due toeccentricity of the Earth’s orbit, i.e. the variation of the Sun-Earth distance.

90

10 6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
z = Eν / mχ

10 7

-1

Generated with WimpSim, www.physto.se/˜edsjo/wimpsim/

Antineutrino fluxes at detector
Mass: 80.3
Channel:
8
_
ν_ e
ν_ µ
ντ

10 6

10 5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

z = Eν / mχz = Eν / mχ

10 6

10 5

-1

10 8

-2
-35

Neutrino fluxes at detector
Mass: 200
Channel: 11
νe
νµ
ντ

cm ann )Generated with WimpSim, www.physto.se/˜edsjo/wimpsim/

dNν/dz (10-2-1

cm ann )-35

dNν/dz (10

10 8

-2

Neutrino fluxes at detector
Mass: 80.3
Channel: 8
νe
νµ
ντ

cm ann )Generated with WimpSim, www.physto.se/˜edsjo/wimpsim/

10 5

10 7

10 6

z = Eν / mχ

10 6

10 8

Antineutrino fluxes at detector
Mass: 6
Channel:
5
_
ν_ e
ν_ µ
ντ

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-35

10 7

10 7

Generated with WimpSim, www.physto.se/˜edsjo/wimpsim/

10 5

dNν/dz (10dNν/dz (10

-35

-2

-1

cm ann )

10 5

10 8

10 8

-2

-1

cm ann )Neutrino fluxes at detector
Mass: 6
Channel: 5
νe
νµ
ντ

-35

-35

dNν/dz (1010 7

Generated with WimpSim, www.physto.se/˜edsjo/wimpsim/

dNν/dz (10

10 8

-2

-1

cm ann )Chapter 5

10 7

Generated with WimpSim, www.physto.se/˜edsjo/wimpsim/

Antineutrino fluxes at detector
Mass: 200
Channel:
11
_
ν_ e
ν_ µ
ντ

10 6

10 5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

z = Eν / mχz = Eν / mχ

Figure 5.1: The WIMP-neutrino fluxes at the SK site for mχ = 6 GeV bb̄ channel, 80.3
GeV W + W − channel, and 200 GeV τ + τ − channel. The flux is shown as a functionof neutrino energy normalized to WIMP mass (z) for neutrino (left column) and antineutrino flux (right column).

Chapter 5

91

In this analysis, the neutrino flux before their interactions at the detector or at thesurrounding rocks was used to simulate SK events with NEUT and detector simulation.
In summary, the input parameters we used for WIMPsim are:• Number of annihilation simulated for each hypothesis of WIMP mass and channel:
10,000,000,• Oscillation parameters: Three-flavor oscillation with parameters: θ12 = 33.461◦ , θ13 =9.097◦ , θ23 = 40.686◦ , δ = 300, ∆m221 = 0.766 × 10−4 eV2 , ∆m232 = 0.274 × 10−2eV2 are used assuming normal mass hierarchy, and• Location of the detector: at SK site (36◦ 25’N, 137◦ 18’E).
Figure 5.1 is an example of neutrino fluxes used in this analysis; the time-averaged
differential neutrino flux for six neutrino and anti-neutrino flavors at the SK location.
The information of angular distribution is not used because the WIMP-neutrino flux
can be regarded to be point-like.

5.3

The WIMP-induced neutrino events at SuperKamiokande

When the neutrinos from WIMP annihilations arrive at SK, it will produce leptons
and hadrons by interaction inside the detector and in the surrounding rocks and water,which can be detected by Cherenkov radiation reaching to PMTs in SK. The existing
atmospheric neutrino Monte Carlo simulation (MC) samples which have been alreadyproduced in large statistics (500 years for each SK era) to predict atmospheric neutrino
induced events in SK were used to simulate WIMP events by reweighting method. We
randomly divided the MC samples into two sets which equally have 250 years statistics
for each SK era and used for BG and signal MC, respectively.
To have information of the direction of the Sun at event time, date and time information are stored for data. For MC events, date and time are randomly distributed.
In the MC set to be used for signal MC, events whose angle between the true neutrino direction and the direction from the Sun at the event time are < 10◦ and whose
true neutrino energies is in the range of 0 ∼ WIMP mass (GeV) are used to mimic
WIMP-induced neutrino events. This choice of the angle (< 10◦ ) does not introduce
any bias to produce WIMP neutrino MC sample, because the collection of the events
inside 10◦ cone doesn’t have a preference of certain direction in the detector coordinate
system, also the averaged location of the Sun is almost uniformly distributed in the
SK coordinate. We also checked it by MC simulation by making a signal MC withdifference choices of angular selection such as 5◦ and 15◦ . The result of the analysis
shows consistency within the MC statistical limit.
We used all νµ , ν µ , νe , ν e , ντ and ν τ flavors. Table 5.1 shows the number of SK
I-IV events used for signal MC compared to the total number of events in atmospheric
neutrino MC (500 years, SK I-IV) for each mass and annihilation channel of WIMPs.

Chapter 5

92

4 GeV
6 GeV
10 GeV
20 GeV
50 GeV80 GeV
100 GeV
200 GeV

WIMP event fraction to total event(%)
19228
0.29
20797
0.31
22075
0.33
23112
0.35
23951
0.36
24234
0.36
24336
0.37
24605
0.37

Table 5.1: Number of SK I-IV events used for signal MC, and the fraction to total event
number of atmospheric neutrino MC sample (6643287), for each mass and annihilation
channel of WIMP.

In the atmospheric neutrino MC, there’s a variable “hondaflx” assigned to each MC
event, which represents absolute atmospheric neutrino flux defined for each flavor indimension of [1/Sr/GeV/m2 /s]. This value is interpolated from original Honda flux
calculation which has been updated since 2011 by Honda et al. (129). To produce theneutrino spectrum from WIMP annihilations, we reweight the hondaflx value eventby event with the weighting factors calculated from the ratio of original hondaflx to
the WIMP neutrino flux. For WIMP neutrino flux, the output fluxes from WIMPsim
for six neutrino flavors are tabulated in 1000 energy bins whose sizes are mχ /1000.
This procedure was repeated for each WIMP mass and annihilation channel. For theabsolute normalization of the total WIMP νµ flux set to be 1/km2 /yr, the weight factor
is defined to be1/(πsin2 (10◦ ))/(mχ /1000)/106 /(365 × 24 × 3600),(5.1)

which is multiplied by ‘WIMP neutrino flux’ in a bin size
[1/10◦ /(mχ /1000GeV)/km2 /yr]

(5.2)

divided by hondaflx[1/Sr/GeV/m2 /s]. The normalization factor for other five flavor
fluxes are calculated to hold the relative normalization to νµ flux by the additional
normalizing factor.
The energy spectrum of original atmospheric neutrino flux and WIMP neutrino
fluxes are shown in Fig. 5.2. The atmospheric neutrino fluxes originally have large
variance in 3D angular distribution, therefore they are shown after averaging in eachenergy bin. The WIMP neutrino fluxes are shown for representative samples whichis expected to have the softest (6 GeV bb̄) and the harder (200 GeV τ + τ − ) spectrum
among tested hypotheses.
The reconstructed momentum distributions for the signal MC sample before and

93

10

4

10

3

10

2

WIMP neutrino flux[10-31/m2/Γ]

atmospheric neutrino flux[/m2/s]

Chapter 5

10
1
10
10

-1

-2

1

10

10

10

8

10

7

2

1

Energy[GeV]

10

10

2

Energy[GeV]

Figure 5.2: Left: True energy spectrum of unoscillated atmospheric muon-neutrino (νµ )(black), ν µ (red), νe (green) and ν e (blue). Right: True energy spectrum of WIMP
neutrinos per annihilation for 6-GeV bb channel, and 200-GeV τ + τ − channel shown
together (colors are the same as on the left, and ντ (magenta) and ν τ (cyan)).
after weighting are shown in Fig. 5.3 for 6 GeV bb̄ and 200 GeV τ + τ − hypotheses,
respectively.

5.4

atmospheric neutrino flux

The flux model calculated by M. Honda et al. (126) (127) (128) (129)(the Honda
flux) is mainly used to predict the atmospheric neutrino flux coming in SK. For thehigher energy neutrinos which the Honda model doesn’t predict, the Volkova flux (132)
is applied for energy above 10 TeV.
Figure 5.4 shows the primary cosmic ray flux model used in the Honda model (solidline) together with experimental measurements.
Primary protons and nuclei may interact with molecules in the Earth’s atmosphereand produce secondary particles such as pions and kaons through hadronic interactions.
In the Honda model, the US standard atmosphere model (145) is used to reproduce
the density structure of the atmosphere.
Atmospheric neutrinos are produced by decays of pions and kaons:
π+ →

µ+
-

−

π →

µ−

+νµ

(5.3)

e+ + νe + ν¯µ ,
+ν¯µ

(5.4)
(5.5)

Chapter 5

94

Figure 5.3: The reconstructed momentum distributions of signal MC before (black)and after (red) weighting for 6 GeV bb̄ channel (left) and for 200 GeV τ + τ − (right)
channel are shown. Y axis shows the number of WIMP events whose normalization isarbitrary.

Figure 5.4: Primary cosmic ray flux used in Honda model (black solid). Also shown
are the results from observations: Webber (133) (crosses), LEAP (134) (upward triangles), MASS1 (135) (open circles), CAPRICE (136) (vertical diamonds), IMAX (137)(downward triangles), BESS98 (138) (circles), AMS (139) (squares), Ryan (140) (horizontal diamonds), JACEE (141) (downward open triangles), Ivanenko (142) (upward
open triangles), Kawamura (143) (open squares) and Runjob (144) (open diamonds).
Figure is taken from (126).

Chapter 595-

e− + ν¯e + νµ(5.6)

The flavor ratio between νµ + ν¯µ and νe + ν¯e is about 2 for neutrino energies below a
few GeV and becomes larger as neutrino energy increases, as more secondary muonsreach the ground before decaying.
The energy spectrum of atmospheric neutrinos at the SK site is shown in Fig. 5.5
for the Honda model in comparison to other models as well as previous version of the
Honda model. The flux is calculated in the 3-dimensional way and shown in averageover the 3-dimensional directions.

Figure 5.5: The energy spectrum of atmospheric neutrinos at the SK site for the Honda
model (left) and the flavor ratio and neutrino/anti-neutrino ratios (right). Also shown
are the calculations from other models: the Bartol model (146) (dashed line) and the
FLUKA model (147) (dotted line). The previous version of Honda model (126) isshown as well (dot-dashed line). Figure is taken from (129).

Chapter 5

5.5

96

Atmospheric neutrino oscillation

As quarks mix themselves in weak interaction, the leptons would do the same;differently from quarks, the definition of neutrinos is set on weak eigen states. The
phenomena of neutrino oscillations were first proposed by Maki, Nakagawa, Sakata(148) and Pontecorvo (149). It was proposed for three lepton flavors, and the full
three-flavor mixing is observed by the recent experiments.
A flavor eigenstate of neutrino can be written as a superposition of three mass
eigenstates:| να i =3X

Uαi | νi i

(5.7)

i=1

where | να i is the flavor eigenstate with the α corresponds to e, µ or τ , and | νi i is
the mass eigenstate, where i corresponds to 1,2 and 3. The unitary mixing matrix U
is generally parameterized by three mixing angles (θ12 , θ23 and θ13 ) and a single phaseδ relating to charge-parity (CP) violation as:cos θ130 sin θ13 e−iδ
1
0
0


 
0
1
0
U =  0 cos θ23 sin θ23 ×iδ− sin θ13 e 0
cos θ130 − sin θ23 cos θ23


 







cos θ12 sin θ12 0× − sin θ12 cos θ12 0  .
0
0
1
(5.8)

The time evolution of the flavor eigenstate is represented as:| να (t)i =X

Uαi exp(−iEi t) | νi (0)i

(5.9)

i

=

XX
i

Uαi Uα∗0 i exp(−iEi t) | να0 (0)i,(5.10)

α0

where | ν(t)i is a neutrino state at time t, Ei is the energy of the mass eigenstate | νi i.
The probability of the initial state να to oscillate to state νβ at t is:
P (να → νβ ) = | hνβ (t) | να (0)i |2X∗= |Uβi Uαiexp(−iEi t) |2(5.11)(5.12)

i

=

X
i

5.5.1

| Uαi Uβi |2 +X∗∗
Uαi UβiUαjUβj exp(−i(Ei − Ej )t).

(5.13)

i6=j

Two-flavor oscillation in atmospheric neutrinos

As mentioned above, atmospheric neutrinos are produced by decays of mesons which
are generated through the primary cosmic rays interactions in the atmosphere.
νµ → ντ 2-flavor oscillation is good approximation for predicting atmospheric neutrino flux detected on the ground base. The mixing matrix U can be simplified in
two-flavor mixing case:!
cosθ sinθU=,(5.14)−sinθ cosθ

Chapter 5

97

and the transition probability in equation (5.13) can be simplified to:
P (να → νβ ) = sin2 (2θ)sin2 ((Ei − Ej )t).
2(5.15)In the relativistic limit, |p|  m, the following approximation is allowed:
Ei − Ej =q(m2i + p2i ) −≈ (p +≈q(m2j + p2j )m2m2i) − (p + j )
2p
2p

∆m2ij
,
2E

(5.16)(5.17)(5.18)

where ∆m2ij ≡ m2i − m2j is the difference in squared masses.
Combining equations 5.18 and 5.15, the survival probability of να can be writtenas:∆m2ij Lν)
4Eν
1.27∆m2ij (eV2 )Lν (km)= 1 − sin2 2θsin2 ().
Eν (GeV)P (να → να ) = 1 − sin2 2θsin2 ((5.19)(5.20)

Hence the oscillation probability depends on the mixing angle θ, the mass squaredifference ∆m2ij , neutrino flight length Lν and neutrino energy Eν .
The atmospheric neutrino flux peaks at around 40 MeV and extends to above TeV.
Atmospheric neutrino experiments usually have good sensitivity for neutrino energies
above a few hundreds MeV. Neutrino flight length ranges from about 15 km to thediameter of the Earth (∼13000 km, produced at the other side of the Earth). Therefore
observation of atmospheric neutrinos can confine the oscillation parameter ∆m2 , if it islied within 10−1 ∼ 10−4 eV2 as well as the mixing angle θ. As the prediction of the flavor
ratio (νµ + ν¯µ )/(νe + ν¯e ) is about 2% accurate for neutrino energy below 10 GeV (126),
the measured flavor ratio which was roughly one half of what was expected, led thediscovery of neutrino oscillation in atmospheric neutrinos, and the precise measurementof the ∆m2 and θ values (150; 151; 152; 153; 154).
Super-Kamiokande (SK) reported in 1998 the zenith angle dependent deficit ofneutrinos (155) (Figure 5.6). It was reported that the observation was consistent withthe νµ ↔ ντ oscillation hypothesis with sin2 (2θ) > 0.82 and 5 ×10−4 < ∆m2 < 6×10−3
eV2 at 90% confidence level. Later SK provided a L/E analysis(156) as shown in Fig.
5.7.

Chapter 5

98

Figure 5.6: Zenith angle distributions of sub-GeV events (left) and multi-GeV events(right). The hatched regions show the MC expectation for no oscillations with statistical errors and the solid lines show the best-fit expectation for νµ ↔ ντ oscillations.
MCs are normalized to the data livetime. Figure is taken from (155).

Chapter 5

5.5.2

99

Full three-flavor neutrino oscillation in atmospheric neutrinos

The discovery of non-zero neutrino mixing angle θ13 by the T2K and reactor experiments (157) excited the world by opening the possibility to measure CP violationin lepton sector. Equation 5.8 shows that, in order to access the leptonic CP violation
phase δ, the mixing angle θ13 must be nonzero.
The SK atmospheric neutrino data doesn’t have a good sensitivity to constrain
θ13 solely, but as it covers a wide range of L/Eν and can detect νe , it is possible toanalysis the data in the three-flavor mixing framework in one mass scale dominanceapproximation.
The Earth contains a large number of electrons and νe interacts with them via
CC while traversing the Earth. Therefore in three-flavor oscillation, matter effect(158; 159) plays an important role. In order to describe the oscillation effect due tothese effects and the interference of them, full oscillation parameters are taken into
account, including the CP violation phase δ.
When considering all oscillation parameters, νµ ↔ νe transition probability in constant matter density is:
P (νµ ↔ νe ) = cos2 θe13 cos2 θ23 Pe2+ sin2 θe13 cos2 θe13 sin2 θ23 (2 − Pe2 )−2 sin2 θe13 cos2 θe13 sin θ23 cos θ23 (cos δR2 − sin δI2 ),(5.21)

where
Pe2 = sin2 2θ12,M sin2

φm
2

R2 = − sin 2θ12,M cos 2θ12,M sin2

(5.22)
φm2

1
I2 = − sin 2θ12,M sin φm .
2

(5.23)(5.24)

φm is the phase of oscillation in matter, and θe13 is the mixing angle in matter with
θe13 ≈ θ13,M (160).
Figure 5.8 shows P(νµ → νe ) as a function of zenith angle and neutrino energy. The
ratio between the expected electron events (Ne ) and the non-oscillation electron events(Ne0 ) is:Ne= (Pee − 1) + rPeµNe0
' ∆1 (θ13 ) + ∆2 (θ12 , ∆m221 ) + ∆3 (θ13 , θ12 , ∆m221 , δ),(5.25)(5.26)

where r = Nµ /Ne , and r ∼ 2 to 3.5 in ∼100 MeV to ∼10 GeV. The phenomenon ofelectron neutrino oscillation appears in wide range of energy as shown in Fig. 5.8.

Chapter 5

100

Figure 5.7: The ratios of data to unoscillated MC as a function of the L/E withstatistical error (points) and the ratio of oscillated MC with the best-fit expectationfor νµ ↔ ντ oscillations to unoscillated MC (solid line). Also shown are the bestfit expectation for other scenarios such as neutrino decay (dashed line) and neutrinodecoherence (dotted line). Figure is taken from (156).

1

P(νµ → νe)

1

0.7

0.8

νe flux ratio Φosc / Φ0

1.4

0.8
0.6

0.6

1.3

0.6
1.2

0.5

0.2

0.4

0

-0.2

0.3

-0.4

0.2

-0.6

0.2

1.1

0

1

-0.2

0.9

-0.4
0.8
-0.6

0.1

-0.8
-1 -1
10

0.4
cosΘν

cosΘν

0.4

1

10
Eν (GeV)

10

20

0.7

-0.8
-1 -1
10

1

10

10

2 0.6

Eν (GeV)Figure 5.8: Left: Probability of νµ → νe oscillation as a function of zenith angle andneutrino energy, in case of (∆m221 , ∆m232 , sin2 θ12 , sin2 θ23 , sin2 θ13 , δ) = (7.7 × 10−5 ,2.1 × 10−3 , 0.3, 0.5, 0.04, 0). Right: The νe flux ratio Φosc /Φ0 assuming the same
oscillation parameters as the left figure. Figures are taken from (124).

Chapter 5

101

The global best fit results of neutrino oscillation parameters used to oscillate atmospheric neutrino MC events assuming normal hierarcy in this analysis are summarizedin Tab. 5.2 assuming normal hierarchy.
Parameterused value
θ12 ◦33.461
◦
θ13
9.097
◦
θ23
40.6862∆m32
0.274 × 10−2 eV2∆m221
0.766 × 10−4 eV2
δ300
Table 5.2: Summary of global fit values of neutrino oscillation parameters used in theanalysis.

5.6

Neutrino interaction

The simulation library called “NEUT” (130) (131) is used to simulate the neutrino
interactions with proton, oxygen and other nuclei. When a neutrino scatters off anucleus, it actually interacts with a quark bound in a nucleon, which is bound in a
nucleus by strong force. As a neutrino only interacts via weak force, charged current(CC) and neutral current (NC) weak interactions are to be considered. Figure 5.9
shows the Feynman diagrams of NC and CC interactions.

Figure 5.9: Feynman diagrams of neutrino-quark NC (left) and CC (right) interactions.

In the NC interaction, the neutrino scatters elastically off the quark where theChapter 5

102

outgoing neutrino cannot be detected in a water Cherenkov detector. In the CC interaction, the out-going particle is a charged lepton, which can be detected by Cherenkov
radiation.
NEUT covers the neutrino interactions as following:• CC/NC (quasi-)elastic scattering: ν + N → l + N 0• CC/NC resonance single meson production: ν + N → l + N ∗ + meson• CC/NC deep inelastic interaction: ν + N → l + N 0 + hadrons• CC/NC coherent pion production: ν +16 O → l +16 O + π,
where N and N 0 are the nucleons (proton or neutron), N ∗ is baryon resonance and l
is a lepton.

5.6.1

Elastic and quasi-elastic scatterings

The differential cross section dσ/dq 2 for charged current quasi-elastic (CCQE) interaction of anti-neutrino off free proton can be written as (161):
2m2p G2F cos2 θC
dσ
2
2 s−u
2 (s − u)
=
[A(q
)
+
B(q
)
+
C(q
)
],
dq 2
8πEν2
m2p
m4p

(5.27)

where mp is the proton mass, GF is the Fermi coupling constant, θC is the Cabibo
angle, Eν is the neutrino energy, q is the four-momentum transferred to the lepton, sand u are Mandelstam variables (161) and the form factors A, B and C are:A(q 2 ) =q2
q2
m2l − q 2
2
[(4
−
)|F
|
−
(4
+
)|F 1 |2
A
4m2p
m2p
m2p V

q2
q2
4q 2 FV1 ξFV2
2 2
|ξF
)
−
|
(1
+
V
m2p
4m2p
m2p
m
− ((FV1 + ξFV2 )2 + |FA |2 )],mp
q2
B(q 2 ) =(FA (FV1 + ξFV2 )),2
mp−

C(q 2 ) =1
q2
|ξFV2 |2 ),(|FA |2 + |FV1 |2 −44m2p

(5.28)(5.29)(5.30)

where ml is the outgoing lepton mass, FV1 (q 2 ) and FV2 (q 2 ) are vector form factors,
FA (q 2 ) is axial vector form factor and ξ is the anomalous magnetic moment µp − µnwhich is set to be 3.71. The form factors are determined from the experiments.

Chapter 5

103

σ (10

-38

2

cm )

σ (10

-38

2

cm )

In the calculation of the cross sections of neutral current elastic scattering, cosθCin the Eq. 5.27 is replaced with unity, and the nucleon form factors are replaced bycorresponding factors (162).
For the scattering off 16 O, as the protons are not free but are bound in nucleus,
Fermi motion of the nucleon and Pauli principle are taken into account. The calculation
is done based on the Smith and Moniz Fermi-gas model (163).
Figure 5.10 shows the total cross section of CCQE scattering as comparing the
NEUT calculation with experimental data.
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

-

(a) νµn→µ p

0

2

4

–

ANL
BNL
SKAT

6
8
10
Eν (GeV)

2

4

12

14

GGM (1977)
GGM (1979)
Serpukhov
SKAT+(b) νµp→µ n0

Serpukhov
GGM (1977)
GGM (1979)

6
8
10
Eν (GeV)12

14

Figure 5.10: The CCQE scattering cross section of (a) νµ off neutron and (b) ν¯µ off
proton are shown as a function of neutrino energy. Solid line indicates the scattering
off a free proton, the dashed line indicates scattering off a bound nucleon in 16 O.
The experimental data from ANL (164), Gargamelle (165) (166; 167), BNL (168),
Serpukhov (169) and SKAT (170) are shown together. Figure is taken from (125).

Chapter 5

5.6.2

104

Resonance single meson production

The simulation of the resonance productions of single π, η and K is adopted from
Rein and Sehgals model (171). The interaction process is separated into two parts as:
ν + N → l + N∗
N ∗ → meson + N 0 .

(5.31)(5.32)< 2 GeV, resonance
Where the invariant mass of the intermediate resonance W is ∼production of single meson often (∼ 80%) occurs. In order to avoid double countingof the interactions, the W is restricted to be less than 2 GeV. As a result, eighteen
resonances below 2 GeV are considered in the calculation. The angular distributionof the π in the final state for the P33 (1232) resonance followed the Rein and Sehgal’s
method, and the angular distributions of the π in the other resonances are set to be
isotropic in the resonance rest frame. Pauli-blocking effect for nucleons from ∆ decays
is taken into account by applying minimum momentum cut. Pion-less ∆ decay is also
considered which turns out to compose 20% of the ∆ decay events.
Figure 5.11 and 5.12 show charged-current cross section of single meson productionfor νµ and ν¯µ respectively.

5.6.3

Deep inelastic scattering

The cross section for charged-current deep inelastic scattering of neutrino or antineutrino off nucleon is given by integrating the following equation for the invariant
mass W > 1.3 GeV (172):
G2 mp Eν
y2
y
d2 σ
= F
((1 − y ++ C1 )F2 (x, q 2 ) ± y(1 − + C2 )xF3 (x, q 2 )),
dxdy
π
2
2

(5.33)

where Eν and El are the energy of incoming neutrino and outgoing lepton in thelaboratory frame, respectively, x = −q 2 /(2M (Eν − El )) and y = (Eν − El )/Eν are the
Bjorken scaling parameters and the factors C1 , C2 are:(5.34)
C1 =

m2l
xymp
m2
−
− l2 −
4mp Eν x
2Eν
4Eν
2mp Eν x
2ml

C2 = −yMl2

4mp Eν x.

(5.35)

F2 and xF3 are the nucleon structure functions taken from GRV98 (173) and fromthe A. Bodek and U.K. Yang’s correction for the lower q 2 region (174).
In the region 1.3 < W < 2.0 GeV, only pions are considered as outgoing mesons
and in order to not overlap with the single pion production previously described, only

105

−38

cm2)

νµ p → µ− p π+

1.2σ(ν µ )(10σ(ν µ )(10

−38

cm2)

Chapter 5

1
0.8

1
0.9

νµ n → µ− n π+

0.8
0.7
0.6
0.5

0.6

0.4

0.4

0.3
0.2

0.2
0.1
0

0

σ(ν µ )(10

−38

cm2)

1
1
0.9

10

Eν (GeV)νµ n → µ− p π0

1

10Eν (GeV)ANL Radecky, Phys.Rev.D 25, 1161 (1982)
ANL Campbell, Phys.Rev.Lett. 30, 225 (1973)

0.8
0.7

ANL Barish, Phys.Rev.D 19, 2521 (1979)

0.6
BEBC Allen Nucl.Phys.B 264, 221 (1986)

0.5
0.4

BEBC Allen Nucl.Phys.B 176, 269 (1980)

0.3
BEBC Allasia Nucl.Phys.B 343, 285 (1990)

0.2
0.1
0

BNL Kitagaki Phys.Rev.D 34, 2554 (1986)

1

10

Eν (GeV)FNAL Bell Phys.Rev.Lett. 41, 1008 (1978)

Figure 5.11: The charged-current single pion production cross sections for νµ as afunction of neutrino energy. The solid lines show the NEUT calculations and theexperimental data are shown together with error bars. Figure taken from (125).
multi-π production is considered. The mean multiplicity of produced charged pions
is taken from the Fermilab 15-foot hydrogen bubble chamber experiment (175) result
which was:
< nπ >= 0.09 + 1.83lnW 2 .
(5.36)In the region W > 2.0 GeV, the PHYTIA/JETSET (176) simulations are used totreat π, K, η, ρ productions and so on.
The charged-current DIS cross section for νµ and ν¯µ are shown in Fig. 5.13.

5.6.4

Coherent pion production

When the four-momentum transfer from neutrino to the oxygen nucleus is small, thenucleus recoils as a whole, unfragmented, which is called “coherent scattering”. When

cm2)

0.8
0.7

1
0.9

0.7
0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

cm2)
σ(ν µ )(10

1
1

0.9

10

0
Eν (GeV)νµ p → µ+ n π0νµ n → µ+ n π−

0.8

0.6

0

−38

νµ p → µ+ p π−

−38

1
0.9

106

σ(ν µ )(10σ(ν µ )(10

−38

cm2)

Chapter 51

10Eν (GeV)

BEBC Allen Nucl.Phys.B 264, 221 (1986)

0.8

BEBC Allasia Nucl.Phys.B 343, 285 (1990)

0.7
0.6
0.5

P. Schreiner, GGM, Proceedings of 1979 symposium at Fermilab

0.4
0.3

FNAL Barish Phys.Lett.B 91, 161 (1980)

0.2
0.1
0

1

10

Eν (GeV)

BEBC Allisia, Z.Phys.C 20, 95 (1983)

Figure 5.12: The charged-current single pion production cross sections for ν¯µ as a function of neutrino energy. The solid lines are the NEUT calculations and the experimental
data are shown together with error bars. Figure taken from (125).
neutrino energy is enough high, the coherent scattering results in the production of anadditional particle such as a pion.
The simulation of the coherent pion production is based on the modified Rein and
Sehgal model (177). The differential cross section is expressed as follows:

G2 mp 2 2
1
d3 σ=
fπ A Eν (1 − y) ×[σ πN ]22
2
dq dydt
2π
16π totm2
×(1 + r2 )( 2 A 2 )2 e−b|t| Fabs ,mA + qwith
r = RefπN (0)/ImfπN (0),(5.37)(5.38)(5.39)(5.40)

where t is the square of the four-momentum transferred to the nucleus, mA is theaxial-vector mass, fπ = 0.93mπ , b = 80GeV−2 , G is the weak coupling constant, andσ/Eν (10-38cm2 GeV-1)

Chapter 5

107

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

GRV94LOGRV94LO (w/ B-Y corr.)
GRV98LO
GRV98LO (w/ B-Y corr.)

CCFR 90
CDHSW 87
GGM-PS 79

0

10

CHARM 88
BNL 80
CRS 8020

30 50

BEBC-WBB 79
IHEP-JINR 96
IHEP-ITEP 79

CCFRR 84
SKAT

100

200

150

250

Eν (GeV)Figure 5.13: The charged-current DIS cross sections for νµ (upper lines and points) andν¯µ (lower lines and points) are shown as a function of neutrino energy.The current used
one is the GRV98 model with the B-Y correction (dot-dashed lines). The experimental
data are shown together with error bars. Figure taken from (125).
A = 16 is the atomic number of oxygen respectively. Fabs , which accounts for theabsorption of pion in the nucleus, is as taken to be following form:
Fabs = e−<x>/λ ,
with
−1
πN
λ = σinel ρ,(5.41)(5.42)(5.43)πN
πN
where ρ is the nuclear density. σtot
and σinel
are the averaged total and inelastic crosssections respectively, taken from the experimental results.
Figure 5.14 shows the charged-current coherent pion cross-section used in NEUT
together with experimental limits.

5.6.5

Nuclear effects

When secondary particles are produced through the neutrino interaction with a
O nucleus or other nuclei in the rock, the subsequent interactions of them within the
nucleus are simulated. All of the mesons are tracked from the production points toeither the exiting points or the absorption points in the nuclei. For pion interaction, inelastic scattering, charge exchange and absorption are considered. When pion scatters,
the Fermi motion and Pauli blocking effect are taken into account. The interaction
probabilities for pion momenta pπ ≤ 500 MeV/c is calculated based on the model of16

Chapter 5

108

Figure 5.14: The flux-averaged νµ charged-current coherent pion cross-section from
Rein-Sehgal model, used in NEUT is shown with measurements in SciBooNE (178)
and in T2K (179) using the NEUT Rein-Sehgal model. Figure is taken from (179).
Salcedo et al. (180). For the case of pπ > 500 MeV/c, scattering cross section of π ±off free proton is used. For kaon interaction, elastic scattering and charge exchange
are considered by using the cross sections measured by K ± − N scattering experiments(181) (182) (183). For η interactions, the absorption interaction ηN → N ∗ → π(π)N is
considered (184). The secondary interactions of nucleons generated inside the oxygen
are also treated. The nucleon-nucleon elastic scattering cross section bases on the measurements by Bertini et al. (185). A single or two delta(s) production is also considered
and the pion production from decay of produced ∆s is also taken into account using theisobar production model by Lindenbaum et al. (186). Total cross section calculated byusing NEUT including quasi-elastic scattering, resonant single pion production, coherent pion production and deep inelastic scattering, together with individual componentsfor νµ interaction with a nucleon bound in oxygen nucleus is shown in Fig. 5.15.

5.7

Detector simulation

The detector simulation, “SKdetsim” has been developed based on the GEANT3
program (188). In the SKdetsim, propagation of particles considering further interactions and production of secondary particles, generation and propagation of Cherenkov
photons, and the response of PMTs and the detector electronics are considered. The
simulation is tuned using calibration measurements.
Hadronic interactions in water are simulated with the CALOR package (189). In
this package, pion interactions are well simulated until low energy region (about 1

Chapter 5

109

Figure 5.15: The NEUT total interaction cross section and individual components forνµ interaction with a nucleon bound in 16 O. The colored region shows the predicted νµflux spectrum at SK without oscillation for comparison. Figure is taken from (187).

Chapter 5110

GeV/c). For lower energy region (pπ ≤ 500 MeV/c), the NEUT cascade model is used.
GEANT3 controls the production and propagation of Cherenkov light, as well asprocessing any particle decays and interactions through the water that are required.
For Cherenkov photons, Rayleigh scattering, Mie scattering and absorption are sumilated during propagation in the water. Also reflection and absorption at the detector
surface are properly considered. The PMTs collect generated photons according to
the quantum efficiency obtained by the calibration measurement. The detected chargeof each hit PMT is decided according to the random number which are distributed
following the measured average single-p.e. distribution as shown in Fig. 5.16.
Finally, the electronic response of the detector is generated considering the timing
response of PMTs. The relation between timing and charge is measured using laser
calibration. The timing of each PMT is smeared by a random variable distributed
following Gaussian distribution. After HITSUM signal is calculated, the same software
trigger which is applied to data is applied with dark noise being set at 5.8kHz.

Figure 5.16: The single-p.e. distributions for the MC simulation (solid line). The blueline is obtained from the nickel data in SK-IV. The difference between them is due tothe threshold function of the QBEE, and the ratio is also put into the MC simulation.
Figure is taken from (121).
As up-going muon (up-µ) events are produced outside the Super-Kamiokande detector, they are independently simulated. The neutrino interactions occurring in thesurrounding rocks and in the OD water were separately treated. After simulating neutrino interactions in the rocks, the propagation of produced muons through the rocks

Chapter 5111

was simulated by considering energy loss at very high energy including radioactive process and multiple Coulomb scattering. Only the events in which muons can reach thedetector are selected and considered for further detector simulation steps. The muon
propagation range Ref f in the rock is shown in Fig. 5.17.
The neutrino interaction and the propagations are simulated in “standard rock”
which has: ρ = 2.65 g/cm3 , Z = 11 and A = 22 ((190)).

Figure 5.17: The muon propagation range in the in “standard rock”. Figure is takenfrom (191).

The simulation covers the rocks within 4 km from the detector, which is the distancethat muon produced from a few TeV neutrino event can travel.
As a result of the simulation, 85% of the total up-µ events turns out to be producedin the rocks and the remaining in the OD water.

Chapter 6
Description of Super-Kamiokande
data used
6.1

Overview

High-energy neutrino events detected in the Super-Kamiokande are categorized into
three types as shown in Fig. 6.1:
• Fully contained (FC)• Partially contained (PC)• Upward-going muon (Up-µ).
When the neutrino interacts inside the detector, it is called “contained event type”.
They are discriminated one to another by the OD activity; “fully-contained (FC)events” are in which all observed Cherenkov light is contained in the ID, “partiallycontained (PC) events” additionally have an exiting particle that deposits energy in
the OD. “Upward-going muons (up-µ)” are high-energy muons produced by neutrino
interactions in the rocks and water surrounding the detector. To discriminate themfrom cosmic rays, muons going upward direction are only selected.

Fully contained

Partially contained

Upward stoppingmuon

Upward through-going
muon

Figure 6.1: Schematic view of three types of high-energy neutrino events in SK.
112

Chapter 6

113

This analysis uses data accumulated during the SK I-IV run periods. The SK-I
(1996 to 2001) FC and PC samples correspond to 1489 live-days, and the up-µ sample
corresponds to 1646 days of livetime. SK-II (2002 to 2005) had 799 live-days of FC
and PC samples and 828 live-days of up-µ sample. SK-III (2006 to 2008) contains 518
live-days of FC and PC with 636 live-days of up-µ events. SK-IV (2008 to present)
data-taking is ongoing; in this analysis we used the data collected until March 2012,
containing 1096.7 live-days of FC, PC and up-µ events.
Events falling into each event category will experience independent back-ground
events reduction process which will be explained in the following former three chapters(Chap. 6.2 to 6.4). In Chap. 6.5, the event reconstruction methods are explained.
After reducing the back-grounds, data are further distributed into 18 event categories
which are 13, 3 and 2 for FC, PC and up-µ respectively. The FC events are distributed
in seven sub-GeV (visible energy < 1.33 GeV) and six multi-GeV (visible energy > 1.33
GeV) sub-categories, and then further divided based on particle identification (e-like,µ-like and π 0 -like), number of reconstructed Cherenkov rings and number of decay
electrons. PC events are sub-divided into “OD stopping” and “OD through-going”
accoroding to their energy depoit in OD. Up-going muons either stop in the detector(stopping up-µ) or pass through the detector (through-going up-µ). Through-going upµ events are further divided into “showering” and “non-showering” categories (192).
The criteria for the event categorization will be given in the latter three chapters (Chap.
6.6 to 6.8).

6.2

Reduction for fully contained events

The FC reduction consists of five steps. Between third and fourth reduction steps,
reconstruction fitters are applied. Since the number of ID PMTs was about half for
SK-II period, reduction cut values differ from other period while sharing the conceptsof the criteria.

6.2.1

First reduction

The first reduction applies rough cuts to reduce low energy and cosmic ray muon
background events. An event is rejected, if one of these criteria is not satisfied:
1. PEtot > 200 p.e.s (100 p.e.s for SK-II),
where PEtot is number of total p.e.s observed by ID PMTs. This cut is to reject
radioactive low energy background events. 200 p.e.s (100 p.e.s for SK-II) corresponds to 22 MeV/c of electron momentum, which is safely lower than the 30
MeV energy cut used for high energy neutrino analyses.
2. NHITA800 ≤ 50 or OD trigger is off,
where NHITA800 is number of hit OD PMTs detected in a 800 nsec fixed time
window. This criterion removes cosmic ray muon events.

Chapter 6

114

If there was an event entered within 30 µsec window after a selected event by theabove criteria, it is kept as attached to the previous events as ‘sub-event’s in order tosearch decay electrons. After the first reduction, the number of events decreases from106 events/day to ∼1500 events/day.

6.2.2

Second reduction

The second reduction is composed of two cuts to give additional rejection of cosmic
ray muons, and electric noise events respectively. An event is rejected if one of thesecriteria is not satisfied:
1. NHITA800 ≤ 25 (30 for SK-IV), if PEtot < 100,000 p.e.s (50,000 p.e.s for SK-II)
or OD trigger is off, to reject more cosmic ray muons.
2. PEmax /PE300 < 0.5,
where PEmax is the maximum number of p.e.s observed by an ID PMT, PE300 is
the maximum number of p.e.s observed by total ID PMTs in a 300 nsec sliding
time window. This criterion rejects low energy electric noise events, to whose
signal a single PMT largely contribute.
Event rate further reduces to ∼200 events/day after the FC 2nd reduction.

6.2.3

Third reduction

Third reduction is composed of several cuts optimised to reduce various types ofmuons, flasher events and low energy radioactive and electric noise events.
Through-going muon cut
A specified fitter for the through-going muon is applied for the events who havemore than 1000 PMTs which individually detect more than 230 p.e.s. The fitter first
selects the entrance point by searching the earliest hit PMT with some neighboring hit
PMTs, and selects the exit point by searching for the center of the hit ID PMTs. The
cut criteria are following:
1. goodness of through-going muon fit > 0.75
2. NHITAin ≥ 10 or NHITAout ≥ 10
where NHITAin (NHITAout ) is number of OD hit PMTs located within 8 m fromthe entrance (exit) point in a 800 nsec fixed time window.
Stopping muon cut
A specified fitter for the stopping muon cut is applied to select the entrance point.
The cut criteria are following:• NHITAin ≥ 10 (or if goodness of fit ≥ 0.5, NHITAin ≥ 5 only for SK-I)

Chapter 6

115

Cable hole muons
To pass the PMT cables through to the electronics huts, there are cable holes on
the top of the tank. OD PMTs are not installed at the positions of these holes which
occurs the missing information of the OD activities for the events occurring aroundthese holes. To compensate the defect, 2 m × 2.5 m plastic scintillation counters are
set as veto for four big cable holes (For eight smaller holes, back-ground rejection using
software will be applied in 5th reduction).
The cut criteria related to the veto counters are:
1. One veto counter hit
2. Lveto < 4 m,
where Lveto is the distance from the cable hole to the event vertex.
Flasher event cut
Flasher events are background events caused by PMTs emitting light from internal
corona discharges. Typical flasher events would have broader hit timing distributions
than neutrino events. Using this feature, following cut criteria is applied:
1. NMIN100 ≥ 15 (or NMIN100 ≥ 10 if the number of hit ID PMTs ≤ 800 (ForSK-I),
where NMIN100 is the minimum number of ID PMTs hit in a 100 nsec sliding
time window
2. NMIN100 ≥ 20 (For SK-II to SK-IV)
Accidental coincidence events cut
Sometimes a cosmic ray muon event enters in the trigger gate which is activated bya low energy event. These events are removed by following cuts:
1. NHITAof f ≥ 20,
where NHITAof f is the number of hit OD PMTs in a 500 nsec fixed off-time
window (the timing window in which we do not expect a signal).
2. PEof f > 5000 p.e.s (2500 p.e.s for SK-II),
where PEof f is the number of p.e.s observed ID PMTs in a 500 nsec fixed offtiming window.
Low energy events cut
The remaining low energy radioactive and electric noise events are removed byfollowing criterion:

Chapter 6

116• NHIT50 < 50 (25 for SK-II),
where NHIT50 is the number of hit ID PMTs in a 50 nsec sliding time window.
Event rate becomes ∼35 events/day after the third reduction.

6.2.4

Fourth reduction

A pattern matching algorithm is used to remove the remaining flasher events because they usually show repeated hit patterns. The correlation factor of two events (Aand B) is computed as follows:
r=

N
A
B
B
(QA
1 X
i − hQ i) × (Qi − hQ i),
N i
σA × σB(6.1)

where N is the number of 2m × 2m patches of ID wall, hQA(B) i and σA(B) are theaverage charge and its standard deviation for event A and event B. If the distance
between the PMTs which have maximum pulse heights in the two events is smaller
than 75 cm, an offset value 0.15 is add to r. Another matching parameter dks is
introduced by KS (Kolmogorov-Smirnov) test. The normalised accumulated charge
distributions of two events are compared and the maximum distance dks is picked up
which is expected to be small, when two events have similar hit patterns.
A likelihood method is used to set a cut criterion:





top10
X
X
1
1 1  top10
i−2
ln (Pr(ri , Qi )) − 2ln( ) + Cr +Prob = (Probr + Probdks ) =
2
2 10
Nα
i=1
i=1





top10
X
X
1  top10i−2
ln (Pr(dksi , Qi )) − 2ln( ) + Cdks (6.2)
10
Nβ
i=1
i=1

!

Pr is a probability density function (PDF), i is count for compared events up to tenof them, Qi is average total charge of two compared events, N is number of combinationand tunning parameters are set to be α=0.909, β=0.674, Cr =−0.31, Cdks =−3.39.
Probabilities of getting r and dks are estimated from PDF made from MC, comparedwith expectation assuming neutrino event.
If Prob exceeds 3.0, events A and B are determined to be matched events. Figure
6.2 shows the Prob distribution for neutrino data and MC against flasher events.
The event rate after FC fourth reduction is ∼32 events/day.

6.2.5

Fifth reduction

The fifth (final) reduction step is composed of dedicated cuts for each backgroundsource.

8

Chapter 6

117

Cut parameterNo. of Events

Inside Fiducial volume

Neutrino data
Neutrino MC

SK-III

Cut
Flasher

Prob
Figure 6.2: Prob distributions, regarding FC 4th reduction to reject flasher events for
SK-III data and MC against flasher events inside fiducial volume. Figure is taken froma SK internal document generated by K. Okumura.
Invisible muon cut
If a cosmic-ray muon has momentum too low to generate Cherenkov radiation(“invisible”) and accompanies a decay electron which is successfully tagged, the event
may be misidentified as a neutrino event. Events fulfilling following cuts are recognized
as invisible muon and are rejected:
1. PEtot < 1000 p.e.s (500 p.e.s for SK-II)
2. NHITACearly > 9,
where NHITACearly is the maximum number of hit PMTs in the OD hit cluster
in a 200 nsec sliding time window in the range of 8900 ns. If there’s a OD hit
cluster within 500 cm from the early muon cluster in a 500 nsec fixed time window
coming after the scanning for the early muon cluster, that number of hit PMTs
(NHITAC500 ) is added to NHITACearly .
Coincidence muon cut
The coincidence muon events which are remaining after the third reduction are
further removed by the following cuts:
1. PE500 < 300 p.e.s (150 p.e.s for SK-II),
where PE500 is the total number of p.e.s observed in the ID in a 500 nsec fixed
time windowChapter 6

118

2. PElate ≥ 20 p.e.s,
where PElate is the maximum number of hit OD PMTs in a 200 nsec sliding time
window in the range of 1200 nsec starting after the earlier fixed time window.
Sub-cable hole muon cut
Besides the cable hole muon cut in the third reduction, another cut is introduced
from SK-IV to remove cosmic ray muon events coming through eight sub-cable holes
where veto counters are not set. The cut criteria are as following:
1. Goodness of stopping muon fit ≥ 0.4
2. PEtot > 4000
3. downward direction (cosθ < −0.6)
4. Lveto < 2.5 m.
By this cut, the cosmic ray back-ground contamination in the final FC multi-GeV (willbe defined soon after) data set decreased to 0.1% level.
Long-tail flasher cut
The additional flasher cut tighter than in third reduction is applied. Events satisfying the following criteria are regarded as flasher events and removed:
1. NMIN100 ≥ 6 if goodness of Point-fit < 0.4
Point-fit will be explained in Chap. 6.5.
From SK-II:
2. NTMIN100 ≤ 5 if the goodness of Point-fit < 0.3.
Stopping muon cut
The remaining stopping muons are rejected by the similar criterion with that in FC
third reduction step:• NHITAinF C5 ≥ 10 (or if goodness of fit ≥ 0.5, NHITAinF C5 ≥ 5).
Here NHITAinF C5 is counted by applying 800 ns time window without TDC correction for event which has 8 or more sub event, which is made for third reduction.
After FC fifth reduction, FC event rate becomes ∼30 events/day.

Chapter 6

6.2.6

119

FC reduction summary

In the final FC data set after passing through five reduction steps, neutrino events
which satisfy following criteria are finally selected to be used for this analysis:
1. Vertex of neutrino interaction locates inside the fiducial volume (> 2m from the
ID wall),
2. Number of hit PMTs in the OD cluster (NHITAC) < 16 (10 for SK-II) and3. Evis > 30 MeV.
The second criteria (“fiducial volume cut”) largely rejects the remaining background
events such as cosmic-ray muons and flashers.
Detection efficiencies in each reduction step are estimated by atmospheric neutrino
MC. The number of events per day in the final sample are 8.21±0.07, 8.24±0.10,
8.41±0.12 and 8.09±0.09 for SK-I, SK-II, SK-III and SK-IV respectively. Event rates
are stable through all SK periods. The detection efficiencies, the number of events inthe final sample and the event rates are summarised in Tab. 6.1.
FC
reduction step
1st
2nd
3rd
4th
5th
5th (FV)MC
Data
event rate/day

SK-I

SK-II
SK-III
SK-IV
Detection efficiency (%)
100.0
99.97
100.0
100.0
100.0
99.92
99.98
99.99
99.93
99.78
99.81
99.82
99.29
99.38
99.30
99.00
99.26
99.30
99.24
98.95
99.25
99.95
99.62
99.19
Number of events
13461.3
7222.54753.610669.5
12232
6584
4356
8929
Event rate
8.21±0.07 8.24±0.10 8.41±0.12 8.09±0.09

Table 6.1: Detection efficiencies, number of events and event rates of the FC neutrino
events used in the analysis are summarised. Number of events for MC are normalized
to data livetime without oscillation effect.
Background events remaining in the final FC data set are mostly cosmic-ray muonsand flasher events. Flasher events is a background for e-like sample and cosmic-ray
muons are a background for µ-like sample. After applying fiducial volume cut, remaining background events are checked by eye-scanning with a visual event display inreal-time weakly base. The studied contamination rates of the final FC data set aresummarized in Tab. 6.2.

Chapter 6

120

SK-I
SK-II
SK-III
SK-IV
BG
cosmic ray flasher cosmic ray flasher cosmic ray flasher cosmic ray flasher
Sub-GeV
0.2%
0.1%
0.2%
0.4%
0.3%
0.1%
0.1%
0.1%
Multi-GeV
0.3%
0.5%
0.2%
0.2%
0.3%
0.2%
0.1%
0.1%
Table 6.2: Summary of background contamination in FC final data set, estimated byeye-scanning.

6.3

Reduction for partially contained events

Partially contained events are distinguished from FC events mainly by looking at
OD activities. In the elimination of cosmic ray backgrounds, cuts based on straightforward OD information used in the FC reduction are not as useful for PC events.
Therefore the PC reduction is composed of more dedicated reduction means.
During the reconstruction between SK-II and SK-III, the OD segmentation was
installed. As this change allows much efficient selection of PC events, the cut criteria
from SK-III largely differ from the previous ones. Some cuts were removed in thesecond and third reduction, and the fourth reduction was completely modified. In thefifth reduction, several new cuts were added.
Here PC reduction corresponding to from SK-III is summarised. For the reductions
applied for SK-I and SK-II, refer (193) for example.

6.3.1

First reduction

Events satisfying the following criteria are selected:
1. PEtot ≥ 1000 p.e.s.
Particles in PC events must travel at minimum 2m in the ID, which requires
muons to have a momentum ≥ 500 MeV/c. A cut of 1000 (500 for SK-II) p.e.s in
the ID is applied to reject events unable to reach the OD. 1000 p.e.s correspondto 310 MeV/c for muons, conservatively.
2. NHITAtop < 10 or NHITAbottom < 10,where NHITAtop (NHITAbottom ) is number of OD hits at the top (bottom) of OD.
This rejects muons through-going from top (bottom) to bottom (top).
3. NHITAendcap < 25 or NHITAside < 70,
where NHITAside is number of OD hits on the side region of the OD, and “endcap”
refers to either top or bottom. This rejects corner clipping muons which leave
hit PMTs in both endcap and side region of the detector.
4. ODRmean < 2100 cm, if OD hits < 20 in 500 nsec time window,
where ODRmean is the average distance between all hit pairs which is defined as

Chapter 6

121

follows:
ODRmean =

1

N
−1
X

Npair i=1−−|→
xi − →
xj | .

(6.3)

This criterion is to reject through-going muons as their ODRmean s are expected
to be large.
Many through-going muons are rejected at this stage. The first reduction reduces
the event rate to ∼ 9000 events/day.

6.3.2

Second reduction

The second reduction removes more corner clipping muons, and events with a large
number of hits in the second largest cluster. From SK-III, events satisfying the following
criteria are selected:
1. NHITAendcap < 20 or NHITAendcap < MAX(NHITAside ),where MAX(NHITAside ) is defined as:e(α−β×NHITAside ) with α = 5.8, β = 0.023 if NHITAside < 75, α = 4.675, β =0.008 otherwise.
2. Nouter2 ≤ 10,
where “Nouter2” is the number of hits in the second largest hit cluster. The
clusters are formed based on the charge gradients between OD patches, which
are composed of 11 × 11 PMTs.
Event rate after second reduction is ∼270 events/day.
After the second reduction, precise fit is applied and the event rate is reduced to∼100 events/day.

6.3.3

Third reduction

At the 3rd reduction flashers are rejected using the same cut as the FC third
reduction that used the flasher event’s feature of broad timing distribution:• NMIN100 > 14 (if the number of hit ID PMTs < 800, > 9)
Event rate after PC third reduction is ∼2.6 events/day.

Chapter 6

6.3.4

122

Fourth reduction

Because of OD segmentation introduced from SK-III, fourth reduction is completely
rewritten based on the muon fitter “muboy” to powerfully identify remaining cosmicray muons which have survived so far due to limited OD activity. Muboy classifies
each event as a stopping muon, a through-going muon, a multiple muon or a cornerclipping muon.
Six variables and corresponding selection cuts which use the muboy information are
defined as below, and if an event passes any of these selection cut for muon events, it
is rejected from the PC sample.
Five selection criteria based on the results of the fitter are:
1. Muboy Angle < 90◦ ,where Muboy Angle is the angle between the fitted direction and the direction
between vertex fitted by the Point-fit and the center of the largest OD chargecluster. As for the Point-fit, see Chap. 6.5.
2. Muboy Dotprod > -0.8,
where Muboy Dotprod is the angle between the fitted direction and the direction
between the Point-fit vertex and the earliest saturated ID PMT.
3. Muboy Track length < 1750 cm,
where Muboy Track length is the length of the fitted muon track.
4. Muboy Goodness < 0.52,
where Muboy Goodness is the goodness of fit by muon fitter.
5. Muboy Corner ≥ 300 cm,
where Muboy Corner is the distance between fitted entrance point and the cornerof the tank.
For stopping muons which are required to fulfill at least four of these criteria including the 2nd one, additional selection cut is applied:• Muboy Goodness < 0.5 or Muboy Ehit8m < 10,where Muboy Ehit8m is number of OD hits within 8 m from the fitted entrance
point in a 500 nsec fixed time window.
An additional cut is applied to all events to reduce more low energy back-ground
events:• PEtot < 2900 p.e.
2900 p.e. corresponds to ∼ 500 MeV/c, which is less than the minimum momentum
which PC event needs to have to reach the OD, ∼ 700 MeV/c.
The event rate after the PC fourth reduction is ∼1.8 events/day.

Chapter 6

6.3.5

123

Fifth reduction

Fifth reduction is composed of a series of dedicated cuts for each difficult remaining
background source. There are two types of cuts: hard cuts and soft cuts. PC events
are required to pass all hard cuts and allowed to fail soft cuts once. Hard cuts include
through-going muon cuts, a stopping muon cut, a cable hole muon cut and a cornerclipping muon cut. Soft cuts also include through-going muon cuts, stopping muoncuts, a corner clipping muon cut in a milder way, and a decay electron cut. The
criteria of each cut is described elsewhere (193). The event rate after the PC fifth
reduction is ∼1.2 events/day.

6.3.6

PC reduction summary

The final data set of PC events are selected by the following criteria:
1. Vertex of neutrino interaction locates inside the fiducial volume (> 2m from the
ID wall),
2. NHITAC ≥ 16 (10 for SK-II),
3. Evis > 350 MeV(Total observed charge in ID > 3000 p.e.s (1500 p.e.s for SK-II).
Detection efficiencies in each reduction step are estimated by atmospheric neutrino
MC. The detection efficiencies, the number of events in the final sample and the event
rates are summarised in Tab. 6.3.
Remained back-ground events, which are mainly due to cosmic ray muons, arechecked by eye-scanning after applying fiducial volume cut. Contamination of background events is estimated by extrapolating vertex distribution of the OD-tagged muon
sample into fiducial area. Table 6.4 summarizes the estimated contamination for each
SK period.

6.4

Reduction for upward-going muon events

For the up-µ reduction, most background is cosmic ray muons and some radioactive
radiations. Most tricky background comes from the cosmic rays coming from aroundhorizontal direction, for which even-base selection is not able. The reduction composesof three reduction steps, eye-scanning and background estimation.

6.4.1

First reduction

Low-energy and very high energy events are rejected by requiring:
6000 p.e.s (3000 p.e.s for SK-II) < PEtot < 1,750,000 p.e.s (800,000 p.e.s for SK-II).
6000 p.e.s corresponds to the muon momentum of 1 GeV/c and to the track length ofChapter 6

124

PC
reduction step
1st
2nd
3rd
4th
5th
5th (FV)MC
Data
event rate/day

SK-I

SK-II
SK-III
SK-IV
Detection efficiency (%)
98.98
98.58
99.09
99.63
96.74
93.43
98.52
98.73
95.69
92.32
98.51
98.68
89.86
84.60
97.87
97.42
88.66
74.80
96.61
96.15
80.98
74.80
88.80
86.30
Number of events
913.8
448.6
356.0
744.9
902
427
344
735
Event rate
0.66 ± 0.02 0.65 ± 0.04 0.62 ± 0.03 0.66 ± 0.02

Table 6.3: Detection efficiencies, number of events and event rates of the PC neutrino
events used in the analysis are summarised. Number of events for MC are normalized
to data livetime with two-flavor oscillation effect.

cosmic ray BG

SK-I SK-II
0.2% 0.7%

SK-III
1.8%

SK-IV
0.6%

Table 6.4: Estimated contamination in the PC final data set by cosmic ray muons.

Chapter 6125

3.5 m. Comparing to the requirement for final sample (tracklength > 7 m), it is rather
conservative cut applied at this stage. After that, an upmu fitter is applied and relect
events whose goodness of fit is less than 0.3 or down-going. After first reduction, theevent rate is ∼9000 events/day.

6.4.2

Second reduction

To reject down-going cosmic-ray muons, dedicated fitting algorithms are used asfollowing;
1. First muon fitter is applied to a event.
2. If the event is up-going and the goodness of fit is above the threshold, the event
is save.
3. If the event is down-going and the goodness of fit is above the threshold, theevent is rejected.
4. If the event is travelling horizontally and the goodness of fit is above the threshold,or if the goodness of fit is below the threshold, the judgement is postponed.
5. The event is brought to the first step for the next muon fitter.
The algorithm continues until the event is judged by criterion 2 or 3, or passes
all fitters. Detailed description of the muon fitters and definition of goodness of fit
can be found in (194). After second reduction, the event rate is reduced to be ∼250
events/day.

6.4.3

Scanning

The remaining background corner clipper and cosmic ray muons are rejected byeye scanning (Note that FC and PC eye scannings are used for systematic study, notfor reductions themselves). All the remaining events are checked one by one by two
experts and efficiency of scanning is estimated to be ∼100%. About remaining half ofthe events are rejected in this stage.

6.4.4

Background estimation for upward-going muon

Background cosmic-ray muons mostly come from around horizontal direction (−0.1< cosθ < 0). Background contamination is estimated by extrapolating the distributionof downward events in 0 < cosθ with the exponential function with constant term.
The uncertainties on the background estimation is evaluated the maximum changeon the number of background events fit parameters deduced from the fitting results.
They are summarized in Tab. 6.5.

Chapter 6

126

stop µthrough-going µ (non-shower)
through-going µ (shower)

SK-I
16%
11%
18%

SK-II
21%
15$
14%

SK-III
20%
19%
24%

SK-IV
17%
14%
20%

Table 6.5: Uncertainties of up-µ background estimation in the zenith angle −0.1 < cos θ
< 0.

6.4.5

Up-µ reduction summary

In the final up-µ data set after passing through three reduction steps, events which
satisfy following criteria are finally selected to be used for this analysis:
For stopping muons;
1. Fitted momentum ≥ 1.6 GeV/c which corresponds to track length = 7 m,
2. NHITEX < 10 (16 for SK-II),
where NHITEX is number of hit OD PMTs within 8 m from the exit point.
For through-going muons;
1. Travelling in ID ≥ 7 m,
2. NHITEX ≥ 10 (16 for SK-II).
The detection efficiencies, the number of events in the final sample and the event
rates are summarised in Tab. 6.6. Detection efficiencies in each reduction step are
estimated by atmospheric neutrino MC.

6.5

Event reconstruction

A ring imaging Cherenkov detector Super-Kamiokande can derive various event
information by reconstruction of the Cherenkov light cones detected at PMTs. Fromreconstructed ring shape, radius of the ring, i.e., the Cherenkov emission angle, thenumber of rings, the vertex position of the Cherenkov emission, and the directions
and energies of particles can be obtained. Also the hit patterns allow to discriminate
particle type and momentum.
For FC and PC events, the reconstruction process starts from fitting the vertex
position of an event. With the vertex position, the ring fitting algorithm identifies otherpossible rings. After that, the particle type for each ring is identified. For single ring
events, this particle identification (PID) is used to improve the vertex reconstruction.
Finally, the momentum for each ring is determined and decay electrons are searched.
Up-µ events are also reconstructed by using some of the above processes. PC andup-µ events are basically treated as µ-like events.

Chapter 6

127

SK-I
stop
thru

SK-IV
stop
thru

80.98

SK-II
SK-III
stop
thru
stop
thru
detection efficiency (%)
74.80 88.80 86.30 80.98 74.80
number of events
1892.4 364.6 926.2 286.0 725.1

88.80

86.30

474.0

1257.6

1866

final
729.6
sample
(MC)
final
462
sample(data)
BG
429
subtracted
event
0.26
rate (per ±0.01
day)

223

1835.6 210

871

210

735

306

1345

833.4

193

722.9

284

1330.9

1.12
0.25
1.01
0.30
1.14
0.26
1.21
±0.03 ±0.02 ±0.03 ±0.02 ±0.04 ±0.02 ±0.03

Table 6.6: Detection efficiencies, number of events and event rates of the up-µ neutrino
events used in the analysis are summarized. “thru” means ’through-going’. Number of
events for MC are normalized to data livetime with two-flavor oscillation effect. The
number of events for “final sample” and “BG subtracted” are shown for before andafter BG subtraction, respectively, where the event rate is calculated for the latter.

6.5.1

Vertex fitting

Vertex position is reconstructed by the hit timing information of PMTs. Assumingthat all the PMT hits are created by photons coming from a single point, a fitter
called “Point-fit” selects a point which gives the maximal goodness-of-fit of the timing
residual, i.e., photon arrival time subtracted by the time of flight from a tested vertex,
distribution of hit PMTs as a vertex of the event.
Using the vertex information reconstructed, the Cherenkov emission angle of thedominant ring is determined as an open angle giving the maximal goodness-of fit.
After that, using another fitter called “TDC-fit”, much precise vertex position isdetermined. This fitter assumes that photons are emitted along the particle track for
the PMTs hit inside the opening angle. Fixing the Cherenkov ring, various vertex
point is tested and the point with maximal goodness-of-fit is selected.
For single ring events, a more precise fitter called “MS-fit” is used. This fitter
is applied after particle identification, that the fitter can use the information of the
expected light pattern for each particle flavor. Performance of MS-fit is checked forthe estimated resolution studied using MC. Table 6.7 summarises the vertex resolution,
which is the difference between the true vertex and the reconstructed vertex, of FC andChapter 6

128

PC samples for each SK period. Note that MS-fit is also used for up-µ reconstruction,
assuming that the vertex locates at the ID surface.
Vertex resolution (cm)
SK-I SK-II SK-III
FC sub-GeV
single-ringe-like31.2 35.6
31.1µ-like23.8 30.3
23.9
FC multi-GeV
multi-ringµ-like67.4 111.8
73.8
PC
53.6 62.8
52.6

SK-IV

31.5
23.5

87.8
67

Table 6.7: Vertex resolutions of FC/PC samples for each SK period.
Figure 6.3 shows the vertex distributions of FC and PC events comparing the data
and MC. Figure 6.4 shows the vertex distributions of FC and PC events comparing thedata and MC.

6.5.2

Ring counting

Possibly existing other rings are searched for by a pattern recognition technique,“Hough transform method” (195) and likelihood technique. Figure 6.5 illustrates themethod. For every hit PMT, a Cherenkov emission cone from the fitted vertex with
opening angle 42◦ are drawn weighted with detected p.e.s after subtracting p.e.’s thedominant Cherenkov ring. The most probable direction of the ring is deduced fromthe maximum of the distribution are taken as ring candidates.
Ring candidates are tested whether they are true rings or not by likelihood techniqueto compare the (N + 1)-ring hypothesis to the N-ring hypothesis up to N = 5.
Figure 6.6 and Fig. 6.7 show the ring-counting likelihood distributions comparing
the data and MC. Difference between the data and MC is taken as systematic uncertainty for ring counting. Among the reconstructed rings, rings with low momentum
and overlapping with other energetic rings are discarded.

6.5.3

Particle identification

Each ring is classified as two types: ”e-like” events which are supposed to be produced from e± , γs have diffused ring pattern Electrons and positrons propagate in
the water as generating electromagnetic shower through Bremsstrahlung (e− → e− γ)
and following photon pair production (γ → e− e+ ), and multiple scatterings of low
energy electrons. High-energy gamma rays also generate electromagnetic showers andChapter 6

129

Figure 6.3: r2 distributions in dimension of [cm2 ] for SK-IV (left) FC and PC (right)
events. For FC sample, Data (black dotted) is shown in comparison to MC (red solid)
assuming two-flavor oscillation effect.

Figure 6.4: z distributions in dimension of [cm] for SK-IV (left) FC and PC (right)
events. For FC sample, Data (black dotted) is shown in comparison to MC (red solid)
assuming two-flavor oscillation effect.

Chapter 6

130

42 deg. ringhit PMT(possible center)

center(most probable)

Cherenkov ring

Figure 6.5: A drawing of basic idea of “Hough transform method”. By drawing rings
around the hit PMTs, center of the unknown Cherenkov ring can be revealed.
are indistinguishable from electrons. “µ-like” events have rings ring with sharper edge
produced from µ± , π ± s. Figure 6.8 shows an example for event displays sample of νeand νµ MC events. A likelihood using Cherenkov ring pattern is defined and used forthe particle identification.
Together with likelihood by hit pattern, the opening angle information is also used in
PID algorithm. The Cherenkov rings from muons and charged pions would have smaller
opening angle as they are less energetic than light electrons for the same neutrinoenergy.
Figures 6.9 to 6.12 show the PID likelihood distributions comparing the data and
MC. Discrimination between multi-ring e-like and µ-like events suffers difficulty due toring overlapping. Difference between the data and MC is taken as systematic uncertainty for ring counting.
π 0 s can be identified due to its short lifetime (of 8.4×10−17 s) with 99% probability
to two γs. By searching for two e-like rings, applying a cut on the invariant mass (to bewithin 85 MeV ∼ 215 MeV) and requiring no decay electron and sub-GeV momentum,
NC single π 0 production event can be identified.
However, in case that one of the two rings is missed, the event can be misidentified
as a single-ring e-like event. To compensate it, an algorithm called “POLfit” is applied
for single-ring e-like events (196). Using the initially tagged ring, the best second ringcandidate is searched for which maximizes the likelihood difference between two ringsand single-ring assumptions. With the second ring candidate, the invariant π 0 mass isrecalculated and if it exceeds 100 MeV and the likelihood passes an additional cut, theChapter 6

131

Super Kamiokande I 1489.2 days
450

single ring

Super Kamiokande II 798.6 days
250

multi ring400

single ring

multi ring

CCQE

CCQE

CC Single π

350

CC Single π

200

CC DISCC DIS

CC Coherent π

300

CC Coherent π

NC

NC

150

250
200

100

150
100

50

50
0
-15

-10

-5

0

5

10

15

Ring counting likelihood sub-GeV(FC), stack0-15

Super Kamiokande III 518.1 days
220

-10

-5

0

5

10

15

Ring counting likelihood sub-GeV(FC), stack

Super Kamiokande IV 1096.7 days : Monitoring

450
single ring

multi ring200

single ring

multi ring

400

CCQE

CCQE

CC Single π

180

CC Single π

350

CC DIS

160

CC DIS

CC Coherent π

CC Coherent π

300

NC

NC

140
250

120
100

200

80

150

60

100

40

50

20
0
-15

-10

-5

0

5

10

15

Ring counting likelihood sub-GeV(FC), stack

0
-15

-10

-5

0

5

10

15

Ring counting likelihood sub-GeV(FC), stack

Figure 6.6: Ring-counting likelihood distributions for SK-I (top left), SK-II (top right),
SKIII (bottom left) and SK-IV (bottom left) FC sub-GeV events. Data (black dotted)
is shown in comparison to MC (shaded) for various interaction mode with two-flavor
oscillation effect. The hatched regions indicate various neutrino interaction modes.

Chapter 6

132

Super Kamiokande I 1489.2 days

Super Kamiokande II 798.6 days

400

220
single ring

multi ring

200

350
CCQE

CC DIS

CCQE
CC Single π
CC DIS

160

CC Coherent πNC

250

multi ring180

CC Single π

300

single ring

CC Coherent π
NC

140
120

200

100
150

80

100

60
40

50
0
-15

20
-10

-5

0

5

10

15

Ring counting likelihood multi-GeV(FC), stack0-15

Super Kamiokande III 518.1 days
140

single ring

-5

0

5

350

single ring

300

CCQE

CCQE
CC Single π

CC DISCC DIS

250

CC Coherent π

CC Coherent π

NC

NC

80

200

60

150

40

100

20

50

0
-15

-10

-5

0

5

15

multi ring

CC Single π

100

10

Ring counting likelihood multi-GeV(FC), stack

Super Kamiokande IV 1096.7 days : Monitoring

multi ring

120

-10

10

15

Ring counting likelihood multi-GeV(FC), stack

0
-15

-10

-5

0

5

10

15

Ring counting likelihood multi-GeV(FC), stack

Figure 6.7: Ring-counting likelihood distributions for SK-I (top left), SK-II (top right),
SK-III (bottom left) and SK-IV (bottom left) FC multi-GeV events. Data (black
dotted) is shown in comparison to MC (shaded) for various interaction mode with
two-flavor oscillation effect. The hatched regions indicate various neutrino interaction
modes.

Chapter 6

133

Figure 6.8: MC event displays of an electron produced from νe interaction (right) anda muon produced from νµ (left).
event is classified as an NC single π 0 event.

6.5.4

Momentum reconstruction

Number of p.e.s in each hit PMT is distributed to each ring according to an expected
p.e. distribution. The qiobs , the observed p.e.s in the i-th PMT, is divided into the n-th
ring as:
exp
qi,n
obs
obs
qi,n = qi × P exp ,n0 qi,n0

(6.4)

exp
obs
where qi,nis the fractional p.e.s from the n-th ring in the i-th PMT, qi,nis the expected
p.e. for the n-th ring in the i-th PMT.
The expectation is based on observation, estimated as a function of opening angle,
assuming azimuthal angular symmetry.
For a momentum of a particle, the number of assigned fractional p.e.’s from all hit
PMTs inside the Cherenkov cone with opening angle 70◦ in 300 nsec timing window,
which is defined around the peak of the residual timing distribution, are summed.
In the summation, each hit PMT is corrected with the acceptance of PMTs, also
considering light attenuation, scattering, reflection effects as:

Chapter 6

134

Super Kamiokande I 1489.2 days

Super Kamiokande II 798.6 days

300
e-like 3469

180

muon-like 3184

e-like 1856

muon-like 1684

160

250

140
200

120
CCQE muon

CCQE muon
100

150

80
100

60
40

50
20
0
-10

-8

-6

-4

-2

0

2

4

6

8

10

PID likelihood sub-GeV 1ring (FC)0
-10Super Kamiokande III 518.1 days
120

e-like 1268

-8

-6

-4

-2

0

2

4

6

8

10

PID likelihood sub-GeV 1ring (FC)Super Kamiokande IV 1096.7 days : Monitoring200

muon-like 1139

e-like 2457

muon-like 2412

180
100

160
140

80
CCQE muon

CCQE muon

120
100

60

80
40

60
40

20

20
0
-10

-8

-6

-4

-2

0

2

4

6

8

10

PID likelihood sub-GeV 1ring (FC)

0
-10

-8

-6

-4

-2

0

2

4

6

8

10

PID likelihood sub-GeV 1ring (FC)

Figure 6.9: PID likelihood distributions for SK-I (top left), SK-II (top right), SKIII(bottom left) and SK-IV (bottom left) FC sub-GeV 1-ring events. Data (black dotted)
is shown in comparison to MC (red solid) with two-flavor oscillation effect. The hatchedhistograms show CCQE νµ components.

Chapter 6

135Super Kamiokande I 1489.2 days
e-like 856

Super Kamiokande II 798.6 days

muon-like 712

100

120
100

e-like 396

muon-like 400

80
CCQE muon

80

CCQE muon

60

60
40
40
20

20
0
-30

-20

-10

0

10

20

30

PID likelihood multi-GeV 1ring (FC)0-30Super Kamiokande III 518.1 days

-20

-10

0

10

20

30

PID likelihood multi-GeV 1ring (FC)Super Kamiokande IV 1096.7 days : Monitoring

120

45
e-like 274

muon-like 238

e-like 579

40

muon-like 520

100

35
80

30
CCQE muon

25

CCQE muon
60

20
40

15
10

20
5
0
-30

-20

-10

0

10

20

30

PID likelihood multi-GeV 1ring (FC)

0
-30

-20

-10

0

10

20

30

PID likelihood multi-GeV 1ring (FC)

Figure 6.10: PID likelihood distributions for SK-I (top left), SK-II (top right), SKIII(bottom left) and SK-IV (bottom left) FC multi-GeV 1-ring events. Data (black dotted) is shown in comparison to MC (red solid) with two-flavor oscillation effect. The
hatched histograms show CCQE νµ components.

Chapter 6

136

Super Kamiokande I 1489.2 days
e-likeSuper Kamiokande II 798.6 days
100

muon-likee-likemuon-like

100
80
80
nu_mu CC

nu_mu CC

60

60

40

40

20

20

0
-300

-200

-100

0

100

200

300

PID likelihood for Pmax ring in sub-GeV multi-ring (FC)0-300Super Kamiokande III 518.1 days
e-like

40

-200

-100

0

100

200

300

PID likelihood for Pmax ring in sub-GeV multi-ring (FC)Super Kamiokande IV 1096.7 days : Monitoringmuon-likee-like60

muon-like

35
50
30
40

nu_mu CC25
20

nu_mu CC

30

15

20

10
10

5
0
-300

-200

-100

0

100

200

300

PID likelihood for Pmax ring in sub-GeV multi-ring (FC)

0
-300

-200

-100

0

100

200

300

PID likelihood for Pmax ring in sub-GeV multi-ring (FC)

Figure 6.11: PID likelihood distributions for SK-I (top left), SK-II (top right), SKIII(bottom left) and SK-IV (bottom left) FC sub-GeV multi-ring events. Data (black
dotted) is shown in comparison to MC (red solid) with two-flavor oscillation effect.
The hatched histograms show CCQE νµ components.

Chapter 6

137

Super Kamiokande I 1489.2 days

Super Kamiokande II 798.6 days

120
e-like180

muon-likee-likemuon-like

160

100

140
80

120
nu_mu CC

nu_mu CC
100

60

80
40

60
40

20
20
0
-600

-400

-200

0

200

400

600

PID likelihood for Pmax ring in multi-GeV multi-ring (FC)

0
-600

-400

-200

0

200

400

600

PID likelihood for Pmax ring in multi-GeV multi-ring (FC)Super Kamiokande III 518.1 days

Super Kamiokande IV 1096.7 days : Monitoring

45

100
e-likemuon-likee-likemuon-like

40
80

35
30
nu_mu CC25

nu_mu CC

60

20
40
15
10

20

5
0
-600

-400

-200

0

200

400

600

PID likelihood for Pmax ring in multi-GeV multi-ring (FC)

0
-600

-400

-200

0

200

400

600

PID likelihood for Pmax ring in multi-GeV multi-ring (FC)

Figure 6.12: PID likelihood distributions for SK-I (top left), SK-II (top right), SKIII(bottom left) and SK-IV (bottom left) FC multi-GeV multi-ring events. Data (black
dotted) is shown in comparison to MC (red solid) with two-flavor oscillation effect.
The hatched histograms show CCQE νµ components.

Chapter 6138

RT OTn =

X
X
X
cosθi
GM C
ri
[α × (qiobs × exp( ) ×
)−
Si −
Ri ],
Gdata
L
f (θi )
i
i
i

(6.5)

where
α = normalization factor,
Gdata , GM C = relative PMT gain for data and MC,
L = light attenuation length in water, measured by using cosmic ray through-going
muons,
ri = distance between the vertex position and the i-th PMT,f (θi ) = correction function for the PMT acceptance as a function of photon incidence
angle θi ,Si = expected p.e.s for i-th PMT from scattered photons, andRi = expected p.e.s for i-th PMT from reflected photons on black sheets.
Then RT OTn is converted to a momentum using the MC simulation.
q As a result, thereconstructed momentum resolutions are estimated to be 1.7 + 0.7/ P (GeV/c) % andq

0.6 + 2.6/ P (GeV/c) % for single-ring electron and single-ring muons, respectively.
Figure 6.13 show the momentum distributions of SK-IV events comparing the data
and MC.

6.5.5

Direction reconstruction

The direction of single-ring event is decided by MS-fit as described before. Figures
6.14 and 6.15 show the angular resolution for single ring events from CCQE interactions, studied using MC single-ring events. The angular resolution is defined tobe the difference between the true lepton direction and the reconstructed Cherenkov
ring direction. Figure 6.16 shows the angular resolution for muon events. Table 6.8
summarises the angular resolution of FC and PC samples for each SK period. For
multi-ring events, the directions are decided as the direction of the vector sum of thereconstructed momentum of each Cherenkov ring.
Figure 6.17 shows the angular correlation studied for all the categories using MC for
SK-I events (197). The angular correlation is defined as the angle between the parent
neutrino and the reconstructed direction for which 68 % of the events are included.

6.5.6

Decay electron searchMuons stopped in the detector may produce decay electrons which can be also
observed. Decay-electrons (“decay-e”) may be tagged within the timing window of
the primarily triggered muon event (“in-gate”), or searched in a afterward-triggered‘sub-event’ attached to the previous event (“sub-event”). Thanks to the improvementof electronics, the search criteria related to dead time have been removed, also tagging
efficiency has increased for SK-IV.

Chapter 6

139

Figure 6.13: Momentum distributions for SK-IV FC 1-ring events for sub-GeV e-like(top left), sub-GeV µ-like (top right), multi-GeV e-like (bottom left) and multi-GeVµ-like (bottom right). Data (black dotted) is shown in comparison to MC (red solid)
with two-flavor oscillation effect.

Chapter 6

140

Figure 6.14: Angular difference between the true direction of out-going leptons andreconstructed direction of CCQE events in FC MC samples for SK-IV. Left: Sub-GeV
1-ring e-like. Right: Sub-GeV 1-ring µ-like. Figures are taken from (125).

Figure 6.15: Angular difference between the true direction of out-going leptons andreconstructed direction for CCQE events in FC MC samples for SK-IV. Left: MultGeV 1-ring e-like. Right: Multi-GeV 1-ring µ-like. Figures are taken from (125).

Chapter 6

141

Figure 6.16: Angular difference between the true direction of out-going muon and thereconstructed direction for up-µ stopping event, up-µ non-showering events and up-µ
showering events for SK-I. Figure is taken from (198).

Figure 6.17: The angular correlation between neutrino direction and reconstructed ring
direction as a function of the outgoing lepton momentum. Figure is taken from (197).

Chapter 6142

Angular resolution (degree)
SK-I SK-II SK-III SK-IV
FC sub-GeV
single-ringe-like
3.1
3.2
3.0
3.0
µ-like
1.9
2.1
1.9
1.9
FC multi-GeV
single-ringe-like
1.2
1.3
1.2
1.2
µ-like
0.8
1.0
0.8
0.8
PC
1.0
1.2
0.9
0.9
Table 6.8: The angular resolutions of FC/PC samples for each SK period.
For “in-gate” type, another peak in the timing distribution after the primary event
is searched and any peaks with more than 50 (25 for SK-II) hits in the 30 nsec window
above the background level are identified as decay electron.
For “sub-event” type, the sub-events which fulfill the following criteria are identified
as decay electron:
1. NHITtot > 50 (25 for SK-II)
where NHITtot is total number of hit PMTs (This cut is removed for SK-IV),
2. The goodness of vertex fit > 0.5,
3. NHIT50 > 30 (16 for SK-II),
4. PEtot < 2000 (1000 for SK-II).
Tagging efficiency of decay electron is 80% for µ+ and 63% for µ− decays for SKI-III,since it is lower for µ− as ∼20% of µ− s are captured by 16 O nuclei (199). For SK-IV,
the tagging efficiency has been improved to be 96% for µ+ and 83% for µ− decays.

6.6

FC categorization

FC data is categorized into 13 sub samples - 7 sub-GeV ( Evis < 1.3 GeV) and 6
multi-GeV ( Evis > 1.3 GeV) samples.
Seven sub-GeV samples are as follows:
1. FC sub-GeV single-ring e-like 0 decay-e electron sample

Chapter 6

143

2. FC sub-GeV single-ring e-like 1 decay-e electron sample
with one or more decay electrons
3. π 0 -like sample
fitted as π 0 -like by π 0 -fitter and π 0 likelihood
4. FC sub-GeV single-ring µ-like 0 decay-e electron sample
5. FC sub-GeV single-ring µ-like 1 decay-e electron sample
6. FC sub-GeV single-ring µ-like 2 decay-e electron sample
with two or more decay electrons
7. FC sub-GeV 2-ring π 0 -like sample
Two rings are recognized and both of them are identified as e-likeNo decay electrons‘Single-ring e-like (µ-like)’ events are events who have a single ring identified as
e-like (µ-like). e-like events has to satisfy the momentum cut pe > 100 MeV/c, and forµ-like events it is pµ > 200 MeV/c.
FC sub-GeV single-ring e-like sample has events from νe CCQE interactions andalso contains NC π 0 background events. A high fraction of CCQE events are contained
in in the 0 decay-e sample. In case one of the two gamma rays produced in NC π 0
events is missed, the NC π 0 event could be misidentified as a single-ring e-like event.
A specialized π 0 fitter is used to reconstruct the second gamma-ray ring based on thefitted first ring, a mass cut, as well as a likelihood method is used to separate π 0 -likeevents.
After applying the π 0 likelihood method, remaining e-like events are divided into
two categories by the detected number of decay electrons: 0 decay-e sample with nodecay electrons and 1 decay-e sample with one or more decay electrons.
As νµ CCQE events are expected to produce one decay electron from a muon, thefraction of CCQE events is high in 1 decay-e single-ring µ-like sample. FC sub-GeV
single-ring µ-like events are divided to three categories according to the number ofdecay electrons reconstructed: 0 decay-e, 1 decay-e and 2 decay-e (2 or more decay
electrons).
Table 6.9 shows the summary of purities of FC sub-GeV single-ring samples.
Left sub-GeV events are π 0 -like multi-ring events, which have two or more e-likerings recognized.
Six multi-GeV samples are as follows:
1. FC multi-GeV single-ring νe -like sample
Number of decay electrons > 0

Chapter 6

MC events
Q.E.
CC
single mesonνe + ν emulti πcoherent π
CC νµ + ν µNCMC events
Q.E.
CC
single mesonνµ + ν µmulti πcoherent πCC νe + ν e
NC

144

FC sub-GeV single-ring e-like
FC sub-GeV
0
0 decay-e 1 decay-e
π -likesingle-ring e-like
1932.9196.1
95.92224.9
81.5%
3.7%
13.0%
72.2%
12.5%
56.0%
6.0%
16.2%
0.9%
10.4%
1.0%
1.0%
1.0%
8.4%
0.4%
1.6%
0.2%
12.1%
1.3%
1.3%
3.9%
9.3%
78.2%
7.6%
FC sub-GeV single-ring µ-like
FC sub-GeV
0 decay-e 1 decay-e 2 decay-e single-ring µ-like
365.8
1651.9130.2
2147.9
63%
79.6%
5.4%
72.3%
11.7%
15.1%
69.3%
17.8%
0.9%
1.3%
14.8%
2.0%
0.4%
1.1%
7.5%
1.4%
5.4%
0.1%
0.05%
1.0%
18.5%
2.8%
3.0%
5.5%

Table 6.9: Number of FC sub-GeV single-ring events and the compositions of interaction modes in them, for SK-IV MC normalised to the SK-IV livetime assuming
two-flavor oscillation.

2. FC multi-GeV single-ring ν e -like sample
Number of decay electrons = 0
3. FC multi-GeV single-ring µ-like sample
4. FC multi-GeV multi-ring νe -like sample
Events that passed through the MME likelihood cut
Events that have νe and ν e separation likelihood > 0
5. FC multi-GeV multi-ring ν e -like sample
Events that passed through the MME likelihood cut
Events that have νe and ν e separation likelihood < 0
Most energetic ring identified as µ-like
Momentum pµ > 600 MeV/c
Evis > 600 MeV

Chapter 6145

Figure 6.18 shows the schematic diagram which summarizes the steps for divisionof multi-GeV samples.

Figure 6.18: The schematic diagram of division steps for the multi-GeV samples. Figure
is taken from (125).
Multi-GeV single-ring e-like sample is divided into νe -like and ν̄e -like samples by the
number of decay electrons. Table 6.10 shows the composition of interaction modes in
the multi-GeV single-ring νe -like sample and ν̄e -like sample, together with the case ofwithout separation. By the νe /ν̄e separation, the composition of CC interaction modein νe -like (ν̄e -like) sample has been increased by 5.2% (6.1%).
‘Multi-ring νe /ν e -like (µ-like)’ events are events whose most energetic ring is identified as e-like (µ-like). In order to make CC νe + ν̄e -enriched multi-GeV multi-ring e-like(MME) sample, a selection is applied based on likelihood method using variables suchas PID likelihood, momentum fraction of the most energetic ring, number of decay
electrons and the distance between the neutrino vertex and any muon decay electrons.
This MME likelihood selection increases CC νe + ν̄e purity significantly, as shown in
Tab. 6.11.
Multi-GeV multi-ring e-like sample is sent through another cut to separate into
νe -like and ν̄e -like samples using a likelihood method using variables such as number ofdecay electrons, number of rings and transverse momentum of the most energetic ring,according to the fact that the outgoing electrons from CCνe interaction with water
nuclei tend to have smaller energy. Table 6.10 shows the composition of interaction

Chapter 6

146Interaction modeCC νeCC ν eCCνµ + ν µ
NC
Total

Without separation
57.6%
30.6%
3.2%
8.8%
568.7

νe -like ν e -like
62.8% 55.9%
10.8% 36.7%
10.6% 8.1%
15.7% 6.6%
135.9 432.8

Table 6.10: Interaction modes of multi-GeV single-ring sample for the cases without
and with the νe /ν̄e separation. The numbers are for SK-IV MC, normalised to data
livetime with two-flavor oscillation effect.
Interaction mode composition Without LM M E
CC νe + ν e
53.8%
CC νµ + ν µ
21.5%
NC
24.7%
Total number of events
654.3

With LM M E
75.5%
7.2%
17.3%
341.2

Table 6.11: Interaction modes of multi-GeV multi-ring sample for the cases without
and with the MME likelihood. The numbers are for SK-IV MC, normalised to data
livetime with two-flavor oscillation effect.

modes in the multi-GeV multi-ring νe -like sample and ν̄e -like sample, together with thecase of without separation. By the νe /ν̄e separation, the composition of CC interaction
mode in νe -like (ν̄e -like) sample has been increased by 3.3% (1.5%).

6.7

PC OD stop/through separation

PC events are categorised to two samples - OD stopping and through-going events.
OD stopping events deposit less charge in the OD compared to OD through-going
events which travel through the OD into the rocks surrounding the detector. Selection
criteria for OD stopping events are based on the energy deposited into the OD region.
1. PC OD stopping sample
Maximum number of p.e.s observed in OD in a 500 nsec sliding time window(PEanti ) is less than PEexp /1.5
where PEexp is the expected number of p.e.s in the OD from the track length in
the OD.
The most energetic or the second most energetic ring is identified to be µ-like.
2. PC OD through-going sample

Chapter 6

147

Interaction modeCC νeCC ν eCCνµ + ν µ
NC
Total

Without separation
56.1%
19.5%
7.2%
17.3%
341

νe -like ν e -like
59.4% 53.1%
17.9% 21.0%
8.3%
6.1%
14.4% 19.8%
161.9 168.1

Table 6.12: Interaction modes of multi-GeV multi-ring sample for the cases without
and with the νe /ν̄e separation. The numbers are for SK-IV MC, normalised to data
livetime with two-flavor oscillation effect.
Remaining PC events after the OD stopping events are selected.

6.8Up-µ separation

Up-µ events are further divided into stopping and through-going muon samples as
described before. After stopping / through-going muons are separated by fitter and
OD activities, the through-going up-µ data are further separated into showering muon
sample and non-showering muon sample. Energetic muons produce electromagnetic
shower while losing energy through processes such as Bremstrahlung, e+ e− pair production and photonuclear reactions. Selection of showering muons uses χ2 techniquebased on expected charge (194). The through-goin muon events are categorized to be
showering muons if they pass following criteria:
1. Difference between average corrected charge and the expected charge for nonshowering muon (∆(Q) = <Q> − ql ) > 0.5 if χ2 valuable for showering selection≥ 50,
2. or ∆(Q) > 4.5 - 0.08χ2 if χ2 < 50,
otherwise they are categorised as non-showering through-going muons. Table 6.13
summarises the number of data and MC events in each eighteen samples used in thisanalysis, for each SK period.

Chapter 6

FC sub-GeV
single-ringe-like0-decay
1-decay
π 0 -likeµ-like0-decay
1-decay
2-decay
2-ring π 0 -likeFC multi-GeV
single-ringνe -likeν e -likeµ-likemulti-ringνe -likeν e -likeµ-likePC
OD stopping
OD through-goingUp-µstopping
non-showering
showering

148

SK-I
Data
MC

SK-II
Data
MC

SK-III
Data MC

SK-IV
Data
MC

2992
301
176

2705.4
248.1
160.0

1573
172
111

1445.4
138.9
96.3

1092
118
58

945.3
85.3
53.8

2098
243
116

1934.9
198.4
96.2

1025
2012
147
524

893.7
1883.0
130.4
492.8

561
1037
86
266

501.9
1006.7
71.3
259.8

336
742
61
182

311.8
664.1
46.6
172.2

405
1833
174
380

366.3
1654.1
132.2
355.9

191
665
712

152.8
656.2
775.3

79
317
400

78.4
349.5
415.7

68
206
238

54.9
231.6
266.4

156
423
420

135.9
432.8
554.8

216
227
603

224.7
219.7
640.1

143
134
337

121.9
121.1
337.0

65
80
228

81.8
72.4
231.4

175
212
479

161.9
179.1
499.0

143
759

141.4
772.4

77
350

71.3
377.3

54
290

52.0
304.0

109
626

115.0
629.9

435.9
1564.4271.6

388.3 241.2
1306.3 725.3
285.1 108.1

210.1
687.7
106.9

193.7
612.9
110

210
286.7
486.3 1047.6
123.2 282.9

248.7
847.6
f 209.2

Table 6.13: Number of events for data and MC, in each SK run period. Number of
events for MC are normalized to data livetime with two-flavor oscillation effect.

Chapter 7
Search for WIMP neutrinos inSuper-Kamiokande
7.1

Overview

Light WIMPs whose masses are up to 200 GeV could produce neutrinos with energyof few MeV ∼ about 200 GeV, by pair-annihilations inside the Sun. For this energy
range, the Super-Kamiokande (SK) data are dominated by atmospheric neutrino events
produced by cosmic ray interactions with the atmosphere. To test the contribution of
the WIMP-induced neutrino signal to the atmospheric neutrino BG, SK I-IV data are
fitted against signal MC plus BG MC by pulled χ2 method, using the information of
direction, momentum and flavor of signal and BG. First we will introduce the strategyof the analysis in Chap. 7.2. After explaining analysis method in Chap. 7.3, theresults of the sensitivity study will be given in Chap. 7.4. Finally, the result of theanalysis using SK data will be given in Chap. 7.5 and interpreted to the WIMPnucleon scattering cross-sections in Chap. 7.6. The resultant numbers will be shortly
summarised in Chap. 7.7 before moving to discussions in the next Chapter.

7.2

Analysis strategy

The search for the light WIMPs using the Super-Kamiokande is motivated by recent
observations as mentioned in Chap. 2. For scenarios with light WIMPs, most of thesignal will end up in the contained neutrino event (FC+PC) categories. Tables 7.1, 7.2
and 7.3 show the expected fraction of WIMP-induced events in four event categories
which span the energy range from the lowest to the highest, normalized to the total
number of WIMP-induced events for candidate WIMP masses and for bb̄, W + W − andτ + τ − annihilation channels, respectively. Fig. 7.1 visualizes them. It can be seen thatfor 10 GeV bb̄ WIMP, 98% of the signal goes into FC+PC categories.
Table 7.4 shows the fraction of the number of events from each flavor compared tothe total number of events for signal MC, for 6 GeV bb̄, 80.3 GeV W + W − and 200
149

Chapter 7

150

GeV τ + τ − , respectively. Each flavor is shown for combination of ν and ν̄, and the ratioof the number of events produced through CC/NC type of interactions are separately
shown. To show the statistical fluctuation in the signal MC, the mean values are shown
together with 90% upper and lower values for 100 fluctuated MC samples. Note thatelectron neutrino and tau neutrino induced events make up about 44% of the signal
for 6 GeV bb̄ WIMP.
To increase signal acceptance, our analysis is updated to use FC and PC events
as well. We also started using e-like events which only contribute to FC events. As aresult, all e-like and µ-like high-energy events collected in SK are used. Compared tothe up-µ only analysis, the signal acceptance is increased by 47 times for the 10 GeV
bb̄ channel WIMP hypothesis.
The main source of background is atmospheric neutrinos coming from all directions.
The solar atmospheric neutrinos, originating from cosmic ray interactions in the Sun’satmosphere, is expected to be a source of background coming from the Sun. The
calculation by Hettlage et al. shows that for a Cherenkov detectors with an effective
area of 104 m2 and a thickness of 500 m, the expected muon neutrino event rate withenergies greater than 10 GeV is about 0.1 ∼ 0.3 events per year (200), therefore canbe ignored. Figure 7.2 shows the expected event fraction of atmospheric neutrinos inSK event categories.
To reduce the overwhelming atmospheric neutrino BG events which increase as∝ E −2.7 , the angular cut approach used in the previous analysis is not as efficient,because the correlation of lepton direction and parent neutrino direction is worse forlow-energy events.

6 GeV
10 GeV
20 GeV
50 GeV
100 GeV
200 GeVFC sub GeV
70
42
20
7.4
3.3
1.4

FC multi GeV
28
49
55
44
33
24

PC up-µ1.8 0.17
6.4 2.5
14
11
21
28
22
42
19
56

Table 7.1: The fraction(%)s of WIMP-induced events in four categories compared tothe total WIMP-induced events in SK high energy data are shown for tested WIMP
masses, for bb̄ channel.
In this analysis, we adopt a fitting approach whose format was developed for SK
atmospheric neutrino oscillation analyses. The different energy spectra of the atmospheric neutrino BG and the signal, the distinctive angular distribution of the signal
peaking toward the direction of the Sun and the flavor composition information are
fully used in the fitting approach. The details of the analysis will be described in next
chapter.

Chapter 7

151

80.3GeV
100GeV
200 GeVFC sub GeV
0.32
0.21
0.09

FC multi GeV
25
18
10

PC up-µ
19
56
14
68
9.8
80

Table 7.2: Fraction(%)s of WIMP-induced events in four categories compared to thetotal WIMP-induced events in the SK high energy data are shown for tested WIMP
masses, for W + W − channel.

4GeV
6 GeV
10GeV
20GeV
50GeV
100GeV
200 GeVFC sub GeV
46
26
13
5.3
1.1
0.26
0.09

FC multi GeV
49
60
58
44
27
17
10

PC up-µ4.1 0.97
9.4 4.3
17
13
22
29
22
50
15
68
11
79

Table 7.3: Fraction(%)s of WIMP-induced events in four categories compared to thetotal WIMP-induced events in the SK high energy data are shown for tested WIMP
masses, for τ + τ − channel.

Figure 7.1: Visualized fractions of WIMP-induced events in four categories of SK highenergy data are shown for assumed WIMP masses and annihilation channels; blue :FC sub GeV, red : FC multi GeV, yellow : PC, green : up-µ events.

Chapter 7

152

1000

FC νe
FC νµ
PC

800

Events / 1000 days

600
400
200
0
160
140
120
100
80
60
40
20
0 -1
10

Upward stopping µ
Upward through-going µ

1

10
10
Eν (GeV)

2

10

3

10

4

Figure 7.2: The expected event rates for FC νe , FC νµ , PC (top) and upward stoppingµ, upward through-going µ (bottom).

mχ
(GeV)
6 GeV

80.3 GeV

200 GeVannihilation
νechannel
CC
NC
bb̄
35.0
3.37
(34.2
(-3.02+35.7) +3.66)
+
−
W W
12.6
2.30
(-9.05(-0.645+16.9) +5.41)
+ −
τ τ
7.32
1.39
(-4.48(-0.368+10.2) +3.50)

νµ
CC
50.9
(-50.2+51.6)
68.4(-63.8+73.7)
78.7(-75.9+82.0)

ντ
NC
5.04(-4.76+5.26)
3.27(-2.20+4.44)
1.88(-1.43+2.52)

CC
4.37E-4(-1.63E-4+7.25E-4)
8.95(-7.48+10.73)8.35(-7.18+9.75)

NC
5.75(-5.41+6.01)
3.60(-2.44+4.85)
2.00(-1.54+2.67)

Table 7.4: Fraction (%) of the number of signal events from each flavor compared tothe total number of events for three representative WIMP masses and channels. Eachflavor is for combination of ν and ν̄, and further shown are the ratio of the numberof events produced through CC / NC type of interactions. The mean values of 100
fluctuated signal MC sets are shown together with 90% lower and upper values in theparentheses.

Chapter 7

153

-1

-0.5

0

0.5

1

50

-1

-0.5

0

0.5

1

100

45

90

Up stop µ

40

80

-1

45
40

70

35

30

60

30

25

50

25

20

40

20

15

30

15

10

20

10

5

10

0

0

0.8
0.5

1
1

50

90

Up stop µ

40

80

1
1

0

45
40
35

30

60

30

25

50

25

20

40

20

15

30

15

10

20

10

5

10

0.6

0.8

1

0

0.95
0.5

1
1

50

Non-Showering µ

70

0.4

1

Showering µ-1 0.85-0.5 0.90

35

0

0.5

5

0.97 0.98
0.99-10.96-0.5
0
0.5
100

45

0

50

Non-Showering µ35-1 0.4-0.5 0.60-0.5Showering µ50.96 0.97 0.98 0.99

1

0

0.85

0.9

0.95

1

Figure 7.3: Angular distributions for 50 GeV τ + τ − channel are shown in three up-µ
samples. Red dashed : WIMP-induced neutrino signal magnified, in ‘the cosine value
between the direction from the Sun and the reconstructed direction’ (top) and ‘cosine
value between the true direction of the parent neutrino and the reconstructed direction’(bottom, used in the analysis).

7.3

Analysis method

The data and BG/signal MC events in 18 event categories are distributed to finer
bins. The visible energy and reconstructed angle are used to define bins. For angular binning, the cosine value between the angle from the Sun and the reconstructed
direction, cosθsun is used, by reading solar position relative to the SK tank for each
event time from SK low energy solar oscillation analysis library. In the cosθsun distribution, we can check for example in Fig. 7.6 that BG event are almost evenly spread,because of the mixing of zenith angular bins into cosθsun bins due to the location of
the SK(36◦ 25’N), which smears the original anomaly in zenith bins introduced by atmospheric neutrino oscillation. Because of the sample size of 10◦ open angle we took,
the signal events are not perfectly coming from the direction of the Sun, therefore theangular distribution for the signal will be more spread than they are supposed to be.
To correct this, instead of the cosine value between the direction from the Sun andthe reconstructed direction, we used the angle between reconstructed direction and thetrue direction of the parent neutrino as the definition of cosθsun for signal MC events.
Figures 7.3 and 7.4 show the angular distributions of the three up-µ samples for thesetwo different definitions.
For PC, FC multi-GeV samples and sub-GeV CC samples, 10 equally-sized angular bins which span from cosθsun = −1 to 1 are used. For FC sub-GeV µ-like 2
decay electron, Sub GeV e-like 1 decay electron, Sub GeV π 0 -like 1-ring, Sub GeV

Chapter 7

154

-1

-0.5

0

0.5

1

50

-1

-0.5

0

0.5

1

100

45

90

Up stop µ

40

80

-1

45
40

70

35

30

60

30

25

50

25

20

40

20

15

30

15

10

20

10

5

10

0

0

0.8
0.5

1
1

50

90

Up stop µ

40

80

1
1

0

45
40
35

30

60

30

25

50

25

20

40

20

15

30

15

10

20

10

5

10

0.6

0.8

1

0

0.95
0.5

1
1

50

Non-Showering µ

70

0.4

1

Showering µ-1 0.85-0.5 0.90

35

0

0.5

5

0.97 0.98
0.99-10.96-0.5
0
0.5
100

45

0

50

Non-Showering µ35-1 0.4-0.5 0.60-0.5Showering µ50.96 0.97 0.98 0.99

1

0

0.85

0.9

0.95

1

Figure 7.4: Angular distributions for 200 GeV τ + τ − channel are shown in three up-µ
samples. Red dashed : WIMP-induced neutrino signal magnified, in ‘the cosine value
between the direction from the Sun and the reconstructed direction’ (top) and ‘cosine
value between the true direction of the parent neutrino and the reconstructed direction’(bottom, used in the analysis).

π 0 -like multi-ring samples, angular information are not used. For up-µ samples, signal
is expected to highly concentrate around cosθsun = 1. Referring T.Tanaka’s previous analysis (56) which calculated open angle containing 90% of WIMP-induced up-µ
events for stopping, non-showering and showering through-going samples separately,assuring at least 6 BG events to be remained in each bin, we optimized the bin sizes
to be {-1, 0.28, 0.36, 0.44, 0.52, 0.6, 0.68, 0.76, 0.84, 0.92, 1.0}, {-1, 0.955, 0.96, 0.965,
0.97, 0.975, 0.98, 0.985 , 0.99, 0.995, 1.0}, {-1, 0.82, 0.84, 0.86, 0.88, 0.9, 0.92, 0.94,
0.96, 0.98, 1.0} for stopping, non-showering and showering through-going up-µ samples,
respectively. As discussed in previous chapter, the contamination from the cosmic-ray
events left around horizontal direction are estimated and subtracted from the first andsecond closest bins to the horizon for 3 up-µ categories respectively. Assuming thiscontamination will spread evenly in cosθsun distribution, we take the fraction of contaminating events versus total up-µ events, and reduced the normalizations of all up-µbins according to it.
All sub-GeV samples are further divided into 5 momentum bins. For multi-GeV
samples in FC, PC, up-µ, 1 ∼ 6 energy bins are defined considering energy range ofeach sample. As a result, events will be distributed in 684 two-dimensional (angle& energy) bins for each SK run period respectively, in total 2736 bins for SK I-IV.
After that, bins whose number of BG events in a bin are less than 6 are merged with

Chapter 7

155

Upmu
shower
FC
FC
FC
FC
FC
(1x10 bins)
1ring
Multi-ring Multi-ring Multi-ring
1ring
1ring
PC
PC
Upmu
µ-likeνe-likeanti-νe-likeνe-likeanti-νe-likeµ-likestop
throughstop(4x10 bins) (4x10 bins) (2x10 bins) (3x10 bins) (3x10 bins) (4x10 bins)(2x10 bins) (4x10 bins) (3x10 bins)

Log10 P (MeV/c)

5.0

Upmu
non-shower(1x10 bins)

4.0
3.7
3.4
3.0

3.2
3.0
2.8
2.6
2.4

FC Sub-GeV
1-ring e-like

FC Sub-GeV
1-ring π0-like

0-decay
1-decay(5x10 bins) (5x1 bins)(5x1 bins)

FC Sub-GeV
2-ring π0-like

FC Sub-GeV 1-ring µ-like
0-decay
1-decay
2-decay(5x10 bins) (5x10 bins) (5x1 bins)
3.2
3.0
2.8
2.6
2.4
2.2

2.0

3.2

2-decay(5x1 bins)

2.8
2.6
2.4
2.2
2.0

Figure 7.5: Illustration of the binning used in this analysis : FC: 370 bins, PC: 60 bins,
up-µ: 50 bins are used. White boxes indicate that a momentum bin is divided into 10
angular bins between -1 < cosθsun < 1. Shaded boxes indicate that a momentum bin
is not further divided into angular bins (FC single-ring e-like 1-decay, π 0 -like, µ-like
2-decay samples and 2-ring π 0 samples).
neighboring bins, also SK-I, SK-II, SK-III, SK-IV bins are merged together as:
Nndata →X

fjn ˙j )X

data
Nm,SKmn(Nnexp )(1 +

X
j

→

m

exp
Nn,SKm
(1 +

X

n
fj,m
j,m )(7.1)

j

As a result, total 480 bins are used for fitting and they are illustrated in Fig. 7.5.
Figures 7.6 to 7.7 show the cos θsun and reconstructed momentum distributions, for
6 GeV bb̄ WIMP hypothesis as an example.

Chapter 7

156

450
400

SubGeV µ-like 0dcy e

350

1000

800

SubGeV µ-like 1dcy e

800

700

SubGeV µ-like 2dcy e

600

300
500

600

250

400

200
400

300

150

200

100
200

100

50
0

-1

-0.5

0

0.5

1

1400
1200

0

-1

-0.5

0

0.5

1

1200

-1

-0.5

0

0.5

1

800

1400

SubGeV e-like 0dcy e

0

SubGeV e-like 1dcy e

700

SubGeV π0-like 1-R

600
1000

1000

800

800

600

600

400

400

200

200

500
400
300

0

-1

-0.5

0

0.5

1

200
100

0

-1

-0.5

0

0.5

1

lalala

2200
2000
1800

0

-1

-0.5

0

0.5

1

300
350

SubGeV π0-like M-R

MultiGeV µ-like

300

250

1400

250

200

1200

200

MultiGeV e-like_ν e

1600

150
1000
150

800

100

600

100

400

50

50
200
0

-1

-0.5

0

0.5

1

0

-1

-0.5

0

0.5

1

0

-1

-0.5

0

0.5

1

350
120
500

MultiGeV e-like_ν eMultiRing µ-like

300

400

100

MultiRing e-like_ ν e

250
80
200

300
60
150
200

40
100

100

20

50
0

-1

-0.5

0

0.5

1

0

-1

-0.5

0

0.5

1

0

-1

-0.5

0

0.5

1

400
220
200
180

80

MultiRing e-like_ ν e

PC Stop

70

350

PC Through

300
160

60

140

50

250

120

200

40
100
80

30

150

60

20

100

40
10

20
0

-1

-0.5

0

0.5

1

50

0

50

-1

-0.5

0

0.5

1

100

45

90

Up stop µ

40

80

0

-1

45

Non-Showering µ

40

35

70

35

30

60

30

25

50

25

20

40

20

15

30

15

10

20

10

5

10

0

0

0.4

0.6

0.8

1

-0.5

0

0.5

1

50

Showering µ50.96 0.97 0.98 0.99

1

0

0.85

0.9

0.95

1

Figure 7.6: cos θsun distributions of signal MC for 6 GeV bb̄ are shown in the 18 event
categories. Black cross: SK I-IV data, blue solid: atmospheric neutrino backgroundnormalized to livetime of data), red dashed: signal MC events in arbitrary normalization.

Chapter 7

157

3000
1000

SubGeV µ-like 0dcy e

2500

800
600

1000

200

500

0 -1

10

3000

SubGeV µ-like 2dcy e

1500

400

3500

220
200
180
160
140
120
100
80
60
40
20
0 -1

SubGeV µ-like 1dcy e

2000

0 -1

10

1

10

1

400

SubGeV e-like 0dcy e

350

1

300

SubGeV e-like 1dcy e

250

SubGeV π0 -like 1-R

300
2500

200

250
2000

200

1500

150

1000

100

500

50

0 -1

10

2400
2200
2000
1800
1600
1400
1200
1000
800
600
400
200
0

SubGeV π0 -like M-R

600
500
400
300
200
100

10

50

10

1

0 -1

100

0 -1

800
700

150

1

0 -1

10

1

MultiGeV µ-like

1400

1000
800
600
400
200
0

1

10

1
500

MultiGeV e-like_ν e

1000

MultiRing µ-like
400

1200

MultiGeV e-like_ν e

1200

1600
1400

1

1600

10

MultiRing e-like_ ν e

800

1000

300

600

800
600

400

200

200

100

400
200
0

1

0 -1

0

10

10

1

10

1

400

10

1600

500

MultiRing e-like_ ν e

PC Stop

300

300

1400

250

1200

200

1000

PC Through

800
150
200

600
100

100

400

50

0

200

0

1

10-1

10

900

1

0

102

10

10-1

1

102

10

7000

Up stop µ

800

6000

Non-Showering µ

1200

700
5000

1000

500

4000

800

400

3000

600

600

300

Showering µ

400

2000
200

200

1000

100

3

0

10

102

3

10

104

5

10

0

20

40

60

80

× 10
100

3

0

20

40

60

80

× 10
100

Figure 7.7: Reconstructed momentum [GeV] distributions of signal MC for 6 GeV bb̄
are shown in the 18 event categories. Black cross: SK I-IV data, blue solid: atmospheric
neutrino background normalized to livetime of data, red dashed: signal MC events inarbitrary normalization.

Chapter 7

158

If the expected number of events in each bin is low, the number of events in eachbin follows Poisson distribution, therefore the natural logarithm of the joint probability
to obtain the observed data, i.e. the log-likelihood −2lnL is used for fitting. The
data is compared against signal plus BG using the maximum log-likelihood method,
particularly with “pulled technique”. In this approach, χ2 is defined where ∂χ2 /∂j = 0
for every j as :2χ= min 2
{j }

N
X

J
J
X
X
BG
n
χ
[Nn (1 +fj j ) + βNn (1 +gjn j ) − Nndata +
n=1
j
j



Nndata ln(

JXj 2 
Nndata
)]
+
(
) ,
P
P
χ
σj
NnBG (1 + Jj fjn j ) + βNn (1 + Jj gjn j )
j

(7.2)

where n is the index of the bins; Nndata is the number of events observed in each bin;
NnBG is the background expectation in the bin after oscillated and normalized; Nnχ
is the number of events of signal MC in the bin, where β, the global normalization
parameter, stands for the allowed fraction of WIMP-induced events. β is the only
parameter allowed to move free in the fitting, without prior regulation of the range.
j is the systematic uncertainty index; σj is the 1σ value of the j-th systematic error;j is the pull to it and fjn /σj (gjn /σj ) is the predicted fractional change of the numberof background (signal) events in the n-th bin due to 1σ change of the j-th systematic
error.
148 systematic errors on neutrino flux, neutrino interaction, detector response areconsidered. List of the sources and some information about computation of the systematic errors will be presented in the next chapter.
The data could be fitted either in covariance approach or pull approach. The two
approaches are equivalent for χ2 distribution (201). In this analysis we can argue thatour −2lnL follows χ2 distribution with sufficiently large degree of freedom (480 bins
and a free parameter) and we extend the argument of (201) to maximum log-likelihood
method and use the pull approach, because it is more practical than the covariance one
when J  N ; i.e. the minimization of the Eq. 7.2 leads to a J × J matrix inversion
rather than the N × N covariance matrix inversion (201).
Differentiating χ2 with respect to the systematic j yields
NX
N data Anj2j∂χ2= 2 [Anj − n] + 2 = 0,
∂j
Bnj
σj
n

(7.3)

Anj = NnBG fjn + βNnχ gjn

(7.4)

where
and
Bnj = NnBG (1 +JXjfjn j ) + βNnχ (1 +JXjgjn j ).

(7.5)

Chapter 7

159

After arranging it to have the linear terms in left-hand side, the minimization
problem is specified to solve J × J polynomial equations :X Nndata Anj 
1Nndata Anj Anl

+
δ

=
lj
l

2
Cnj
σl2
Cnj
n



!

X
l,n

PJ

1
PJ

1+

j

Anj j+j Anj j Cnj

 − Anj ,(7.6)

Cnj

where
Cnj = NnBG + βNnχ .

(7.7)

Inverting this matrix allows one to solve for the set of j that minimize χ2 . The χ2
at each tested β is calculated by varying j to minimize χ2 .
Again note that the merging is done such that it is predicted to have at least 6 BG
events in each bin after scaling to data livetime, also the bins with no BG events are
excluded from the fitting. Although our calculation is based on Poisson distribution,these will allow us to interpret the result with Gaussian error assumption and the χ2
test.
We first run the fit for a wide range of β with rough gradation to find the rangeof β spanning about −5σ ∼ 5σ, then run a fit again with more precise (about 0.1σ)
gradation for that narrower range of β. In the resultant χ2 distribution as a functionof β, the best-fit β, or βmin is defined at the point with minimum-χ2 .

Figure 7.8: Bayes’ theorem says that our new knowledge of the distribution of β, given
our measurement β̂, is given by the shaded function after appropriate renormalization.
(α in this figure corresponds to β in this analysis. Figure is taken from Particle Data
Group (PDG).)
In case where βmin is consistent with no WIMP hypothesis, we set 90% upper limit
on β by drawing ∆χ2 distribution; the definition of ∆χ2 is the difference of the χ2value for each β compared to the χ2 value at βmin , χ2min . Among the tested signal

Chapter 7

160

hypotheses, some of them results in negative value of βmin . To deal with these, we
reveal the Bayes’ theorem in the calculation of upper limit. The Bayes’ theorem canbe written as:f (β̂|β)π(β);(7.8)
g(β|β̂) = R
f (β̂|β)π(β)dβIn order to derive a confidence limit on β, one would like to know g(β|β̂)dβ, whichis the probability of β lies between β and β + dβ, given our measurement β̂. The
conditional probability density function f (β̂|β) about experimental information β̂ fora fixed and unknown value of β will be delivered from ∆χ2 distribution.
If we assume that β cannot lie in the unphysical region but can lie anywhere in thephysical region with constant probability, then a ‘prior’ π(β) is assumed to be 0 whenβ has negative value, flat when β is positive. This leads to the following formula:g(β|β̂) = f (β̂|β)/Z

f (β̂|β)dβ;(7.9)

if β is in the physical region,g(β|β̂) = 0;otherwise.
It is known that at best fit maximum likelihood −2lnL is asymptotically distributedas χ2 distribution with N − m degree of freedom, where m is the number of fit parameter which is 1 in our case. And χ2 distribution for large degree of freedom can beapproximated to follow normal distribution. We calculate the 90% upper limit on β,β90 above which the right tail of Gaussian distribution corresponds to 10% of physical region (colored) as in Fig. 7.8, in either case that βmin falls in physical region or
unphysical region.
Fig. 7.9 shows an example of ∆χ2 distribution versus β, for 6 GeV bb̄ channel
WIMP hypothesis.

7.4

Sensitivity study

7.4.1

Sensitivity study for null hypothesis

To understand the effect of the statistical fluctuation of the data in fitting, we
generate the fake SK data by setting the number of events in each bin according to a
Poisson distribution. We also make j ’s fluctuated according to a Gaussian distribution
and add fjn j to the number of events in j-th bin to take into account the systematicfluctuations. By generating 500 sets of fake data for each WIMP mass and channel
hypothesis and performing a pulled χ2 test to each, we perform a toy MC study forthe null hypothesis assuming data are composed of background only.
Fig.7.10 shows the distribution of βmin for the 10 GeV τ + τ − WIMP hypothesis as an
example. The two green lines show the 90% band within which 90% of the βmin values

Chapter 7

161

Figure 7.9: The ∆χ2 distribution for 6 GeV bb̄ channel. Blue line indicates ∆χ2 = 1.65σ
which corresponds to 90% upper limit without Bayesian prior. Green line indicates∆χ2 = 2.71σ, which corresponds to 90% upper limit with Bayesian prior, if minimum
was at β = 0. Finally red line indicates recalculated Bayesian 90% upper limit used inthis analysis. For this example, ∆χ2 =4.9. The βmin , ∆χ2 and β90 s for all test WIMP
hypotheses will be summarized in Chap. 7.5.

of 500 toy MC sets are contained, and the result of the fit to the data is marked with
red line. Figure 7.11 shows the distribution of β90 for 10 GeV τ + τ − WIMP hypothesis.
The guidelines indicate the mean value of β90 and the 90% upper and lower limits of
β90 s, together with β90 from the data result.
Figure 7.12 shows the resultant best fit βmin s of data and toy MC for 16 tested
WIMP hypotheses. Figure 7.13 shows the resultant 90% Bayesian upper limit β90 s ofdata and toy MC for 16 tested WIMP hypotheses.

7.4.2

Statistical fluctuation in the signal MC

The fluctuation of BG MC can be negligible as the statistics used is about 200
times that of data. However due to the statistical limitation of the events in the signal
MC as shown in Tab. 5.1, larger fluctuation is expected for the MC. By testing 100
sets of signal MCs for 6 GeV bb̄, 80.3 GeV W + W − and 200 GeV τ + τ − hypotheses, thedifferences resulted in 90% upper limit are found to be about 2.8% for 6 GeV bb̄, 9.6%
for 80.3 GeV W + W − and 6.6% for 200 GeV τ + τ − respectively. The 90% bands of theupper limit on flux of 100 sets of signal MC are shown in Tab. 7.5.
As the size of the difference coming from the signal MC is much smaller than thesize of it in the data, it is not considered as a source of uncertainty in the final result.

Chapter 7

162

Figure 7.10: The distribution of βmin for the 10 GeV τ + τ − WIMP hypothesis is shown.
The green lines mark the 90% bands of βmin s from 500 toy MC sets, and the red line
shows the βmin from the data result.

mχ
(GeV)
6 GeV
80.3 GeV
200 GeVannihilation νµ(meanvalue)
channel(km−2 y−1 )bb̄
4.28×1014
W +W −
8.59×1011
τ +τ −
1.68×1011

νµ (lower)νµ (upper)

fraction(km−2 y−1 )4.16×1014
8.00×1011
1.59×1011(km−2 y−1 )4.40×1014
9.41×1011
1.79×1011

(%)
2.80
9.55
6.55

Table 7.5: The 90% upper limit on νµ flux from WIMP annihilations for the sensitivity
study by 100 signal MCs are shown, for mean value (third column) and lower/upper
values (fourth/fifth columns) of 90% band of it. The last column shows the fractionalchange in the 90% upper value compared to the mean value.

Chapter 7

163

muon neutrino flux[/km2/y]Figure 7.11: The distribution of β90 for 10 GeV τ + τ − WIMP hypothesis is shown. The
blue line shows the mean value of β90 and two green lines mark the 90% upper andlower limits of β90 s. The red line shows the β90 from the data result.
1015

1014

10

102

WIMP mass[GeV/c]

Figure 7.12: Fitted βmin s for data (solid), the 90 % bands of βmin s from 500 toy MC(band), and median value of βmin s from 500 toy MC (dashed) are shown as a functionof WIMP mass for τ + τ − (red), bb̄ (green) and W + W − (blue) channels. We added
5×1014 to all βmin s in order to plot in log scale.

164muon neutrino flux[/km2/y]

Chapter 7

1015

1014

1013

1012

τ+τ10

bb

W+ W

-

102

WIMP mass[GeV/c 2]

Figure 7.13: The 90% upper limit on muon-neutrino flux from WIMP annihilations in
the Sun at SK for the τ + τ − channel in red solid; bb in green dashed; W + W − in blue
dot-dashed. The shaded regions show 1σ bands of the sensitivity study results (color
scheme is the same as for data).
For example, for 10 GeV τ + τ − hypothesis, the statistical fluctuation coming from thedata shown as a band in Fig. 7.13 is found to be 15.6 times larger than the fluctuation
coming from the signal MC.

7.5

Limits on WIMP-induced neutrino events andflux

When we perform the fit for only the point β = 0, it corresponds to the fittingof atmospheric neutrino three flavor oscillation without WIMP contribution in cosθsun
distribution. The goodness of fit for null hypothesis is checked by sensitivity study
explained in the previous chapter. But we also can quickly view the p-value obtained
from Pearson’s chi-squared test for no WIMP contribution. The χ2 value for null
hypothesis is 518.039, corresponding to χ2 /d.o.f = 518.039/479 = 1.08. The test pvalue is 0.106, which is consistent with the conclusion of the sensitivity study.
Table 7.6 lists the χ2 values at minimum or χ2min , ∆χ2 at 0, Bayesian ∆χ2 from theminimum, β90 (90% upper limit on WIMP-induced νµ flux in SK I-IV) for data andmean values of it for sensitivity study for all WIMP hypotheses tested.
The total number of WIMP-induced events in SK I-IV at the fitted minimum point,
Nmin and the number of total WIMP-induced events in SK I-IV at 90% upper limit,
N90 for tested WIMP hypotheses are listed in Tab.7.7. Figure 7.14 visualizes Nmin s

Chapter 7

mχ
(GeV)
4 GeV
6 GeV
10 GeV
20 GeV
50 GeV
80.3 GeV
100 GeV

200 GeV165

annihilation χ2minchannelτ +τ −
508.1
bb̄
507.8
τ +τ −
507.8
bb̄
507.4
+ −
τ τ
507.6
bb̄
507.4
τ +τ −
509.0
bb̄
509.0
τ +τ −
508.9
+
−
W W
508.8
bb̄
509.0
W +W −
508.9
+ −
τ τ
508.9
bb̄
508.9
W +W −
508.9
+ −
τ τ
508.9

∆χ2
at 0
0.87
1.24
1.25
1.65
1.40
1.65
0.06
0.04
0.14
0.17
0.06
0.14
0.16
0.12
0.07
0.08

Bayesian
∆χ2
4.4
4.9
4.9
5.4
5.1
5.4
2.4
3.0
2.3
2.3
2.4
2.3
2.3
2.3
2.4
2.4

νµ
(km−2 y −1 )1.50×1014
2.94×1014
7.08×1013
1.40×1014
3.10×1013
5.31×1013
1.32×1013
1.98×1013
2.67×1012
1.09×1012
7.54×1012
6.29×1011
6.98×1011
2.81×1012
1.67×1011
1.91×1011

νµ sensitivity(km−2 y −1 )2.43×1014
4.67×1014
1.19×1014
2.50×1014
5.30×1013
1.01×1014
1.63×1013
2.29×1013
2.97×1012
1.14×1012
8.37×1012
6.11×1011
6.41×1011
2.59×1012
1.68×1011
1.88×1011

Table 7.6: χ2 value at minimum (χ2min ), ∆χ2 at 0, Bayesian ∆χ2 from the minimum,
90% upper limit on WIMP-induced νµ flux in SK I-IV for data and median value of it
for 500 sets of sensitivity study are shown.

Chapter 7

166

and N90 s for bb̄, τ + τ − , W + W − channels, respectively.
mχ
(GeV)
4 GeV
6 GeV
10 GeV
20 GeV
50 GeV
80.3 GeV
100 GeV

200 GeVannihilation Nminchannel(3903 days)
+ −
τ τ
-41.0
bb̄
-55.2
τ +τ −
-44.1
bb̄
-56.1
+ −
τ τ
-33.8
bb̄
-43.9
+ −
τ τ
-4.66
bb̄
-5.01
τ +τ −
4.17
+
−
W W
4.98
bb̄
2.48
W +W −
2.38
+ −
τ τ
2.21
bb̄
2.74
W +W −
0.640
+ −
τ τ
0.629

N90
(3903 days)
58.1
74.2
45.2
54.6
35.3
40.9
32.8
35.5
23.4
21.1
27.5
15.3
15.0
19.4
10.4
10.9

Table 7.7: Number of total WIMP-induced events in SK I-IV at fitted minimum (Nmin )and 90% Bayesian upper limit of the number of total WIMP-induced events in SK I-IV
(N90 ) are listed.
Given no significant excess of WIMP-induced events, we can set upper limits on
the WIMP-induced neutrino flux. We chose νµ flux for a simple representation of theresults, but it can be also νe flux or total flux as it doesn’t matter because we fix therelative normalization between them. Because we weight atmospheric neutrino eventsby the weight factor wi and the fitting parameter β, the total number of WIMP-induced
events Nχ can be calculated as:X(dfatm /dE)i × β × wi × [(dσ/dE)i × NT × η] × 3903days

i

=

X

(dfχ /dE)i × β × [(dσ/dE)i × NT × η] × 3903days,(7.10)iwhere i is the index for energy bin whose size is mχ /1000 and wi = (dfχ /dE)i /(dfatm /dE)i ,NT is the number of target and η is the detection efficiency.
The differential signal flux in each energy bin is therefore (dfχ /dE)i × β and thePPtotal flux will be i ((dfχ /dE)i × β) = ( i (dfχ /dE)i ) × β = β because our initial signal
flux before fitting was normalized to 1. Therefore the 90 % upper limit on total νµ flux
from WIMP annihilations is simply β90 .

Chapter 7

167

best fitted number
90% upper limit

75
50
25
0
-25
-50
-75

-100
-125
-150
10

10

100

Total WIMP-induced events in SKI-IV

Total WIMP-induced events in SKI-IV

100

best fitted number
90% upper limit

80
60
40
20
0
-20
-40
-60
-80

-100

2

WIMP mass(GeV)

10

10

2

WIMP mass(GeV)

Total WIMP-induced events in SKI-IV

100

best fitted number
90% upper limit

80
60
40
20
0
-20
-40
-60
-80

-100
80

90

100

200

WIMP mass(GeV)

Figure 7.14: The best fitted number of total WIMP-induced events in SK I-IV for each
WIMP mass, Nmin for bb̄ (top left) / τ + τ − (top right) / W + W − (bottom) channel are
shown in green crosses. 90% Bayesian upper limit of total WIMP-induced events, N90
are shown in red crosses. Black bars correspond to fitted 1σ from the fitted minimum
as guidelines.

Chapter 7

168

Tab. 7.6 lists the 90% upper limits on νµ flux from WIMP annihilations (β90 s) fordata and mean value of it from toy MC study for all WIMP hypotheses tested. Figure
7.13 visualize the results.
The overall effect of the systematic errors is up to 10% on the WIMP sensitivity,
where the largest contributions come from neutrino interaction and atmospheric neutrino flux uncertainties. Comparing to the case in which all systematic errors are turnedoff, the study shows the size of the effect for each group of the source of uncertainties:• sensitivity wtih all systematic error ON: <10.4%,• w/ only 16 neutrino interaction errors ON: <9.8%,• w/ only 25 atmospheric flux errors ON: <7.1%,• w/ only 14 × 4 reduction errors ON: <5.6%,• w/ only 11 × 4 reconstruction errors ON: <1.5%,• w/ only 7 WIMP neutrino flux errors ON: <1.0%.
The list of each source of uncertainties will be given in Chap. 8.
Among individual source of errors, the θ23 uncertainty on the signal flux is expected
to affect the result up to ∼ 1%. For heavy (50 GeV∼ for τ + τ − , 100 GeV∼ for bb
channel) WIMP hypotheses, the Q2 spectrum in deep inelastic scattering for invariant
mass W >1.3 GeV is expected to be a source of ∼ 1% uncertainty.
In overall, the 1σ statistical uncertainties in the data, determined by 500 toy MCsets are about 74.7% for 6 GeV bb̄, 97.8% for 80.3 GeV W + W − and 89.0% for 200
GeV τ + τ − respectively. The systematic uncertainties in BG and signal flux, neutrino
interaction, data reduction and reconstruction are about 10.4% for 6 GeV bb̄, 2.2%
for 80.3 GeV W + W − and 1.4% for 200 GeV τ + τ − respectively. The 90% systematic
uncertainty in the generation of signal MC (statistical flucuation) is about 2.8% for 6
GeV bb̄, 9.6% for 80.3 GeV W + W − and 6.6% for 200 GeV τ + τ − respectively.

7.6

Limits on WIMP-nucleon elastic scattering crosssections

The amount of neutrino flux at SK is directly proportional to the WIMP annihilation rate inside the Sun, and to capture rate under the equilibrium condition, andthen to WIMP-nucleon scattering cross-section.
As discussed in Chap. 2, we used DarkSUSY to compute the neutrino flux for theassumed WIMP mass, annihilation channel and WIMP-nucleus scattering cross section
SD/SI
σ0. The ratio between the DarkSUSY-outcome neutrino flux and WIMP-nucleon
scattering cross sections is used to convert the 90% C.L. upper limit on the νµ flux
calculated in the previous chapter to the 90% C.L. upper limit on the WIMP-nucleon
scattering cross sections.

Chapter 7

169

Table 7.8 shows the conversion factor from the νµ flux to SD, SI and IVDM SI
scattering cross-sections used in the analysis. Figure 7.15 shows them graphically, for
SD, SI and IVDM SI scattering coupling WIMPs, respectively. The explanation of
IVDM will be given in the next chapter.
The 90% upper limit on SD WIMP-proton and SI WIMP-nucleon cross-section arelisted in Tab. 7.9 and shown in Figures 7.16 and 7.17 together with other experimental
limits.
mχ (GeV ) annihilation σSD,p (km2 ×σSI,p (km2 ×channelyr/cm2 )yr/cm2 )(fn/fp=1)
+ −
4 GeV
τ τ
1.44E-18
6.62E-20
6 GeV
bb̄
5.84E-18
1.79E-19
τ +τ −
2.25E-18
7.11E-20
10 GeVbb̄
1.06E-17
2.02E-19+ −τ τ
4.10E-18
8.06E-20
20 GeVbb̄
2.70E-17
2.72E-19+ −τ τ
1.05E-17
1.09E-19
50 GeVbb̄
1.19E-16
5.58E-19
τ +τ −
4.65E-17
2.26E-19+−
80 GeVW W
2.80E-16
9.45E-19
100 GeV bb̄
4.27E-16
1.18E-18
W +W −
4.30E-16
1.24E-18+ −τ τ
1.71E-16
4.91E-19
200 GeV bb̄
1.65E-15
2.86E-18
W +W −
1.74E-15
3.21E-18+ −τ τ
6.75E-16
1.24E-18

σSI,p(km2 ×yr/cm2 )(fn/fp=−0.7)
3.15E-19
8.53E-19
3.38E-19
9.63E-19
3.84E-19
2.27E-18
9.10E-19
1.05E-17
4.27E-18
3.05E-17
4.91E-17
5.16E-17
2.05E-17
2.97E-16
3.34E-16
1.30E-16

Table 7.8: Conversion factors from νµ flux to SD WIMP-proton scattering cross-section,
SI and SI (IVDM) WIMP-nucleon scattering cross-sections are listed.

7.7

Summary

In this analysis, we increased the signal acceptance by using fully-contained andpartially-contained neutrino events, and all three flavors of neutrinos added to upgoing muons in Super-Kamiokande. We also used minimum-χ2 method to use energy,
direction and flavor informations. We fitted Super-Kamiokande I-IV data to find theallowed contribution of WIMP-induced neutrino events added to large background of
atmospheric neutrino events.

170

2

SD cross section / muon neutrino flux [pb//km /y]

Chapter 7

10

10

10

-16

-17

bb channel
tautau channel
WW channel

-18

0

25

50

75

10

10

125

10

150

175

200

-15

2

IVDM SI cross section / muon neutrino flux [pb//km /y]

10

100

WIMP mass(GeV)

2

SI cross section / muon neutrino flux [pb//km /y]

10

-15

-18

10

10

-19

bb channel
tautau channel

10

WW channel
-20

0

25

50

75

100

125

150

175

200

WIMP mass(GeV)

10

-16

-17

-18

bb channel
tautau channel
WW channel

-19

0

25

50

75

100

125

150

175

200

WIMP mass(GeV)

Figure 7.15: Conversion factor from νµ flux to SD (top), SI (bottom left) andIVDM(bottom right) scattering cross-section.

Chapter 7

171

mχ
(GeV)
4 GeV
6 GeV
10 GeV
20 GeV
50 GeV
80.3
GeV
100 GeV

200 GeVannihilation σSD,p (cm2 )channelτ +τ −
2.22×10−40bb̄
1.72×10−39+ −τ τ
1.63×10−40bb̄
1.49×10−39τ +τ −
1.31×10−40bb̄
1.43×10−39+ −τ τ
1.42×10−40bb̄
2.34×10−39+ −τ τ
1.28×10−40
W +W −
3.13×10−40

σSI,p (cm2 )(fn/fp=1)
8.73×10−42
4.56×10−41
4.35×10−42
2.40×10−41
2.12×10−42
1.20×10−41
1.19×10−42
8.99×10−42
4.92×10−43
8.26×10−43

σSI,p (cm2 )(fn/fp=−0.7)
2.45×10−40
1.28×10−39
1.22×10−40
6.75×10−40
5.95×10−41
4.48×10−40
4.47×10−41
3.90×10−40
2.14×10−41
3.73×10−41

3.19×10−39
2.80×10−40
1.24×10−40
4.49×10−39
3.00×10−40
1.33×10−40

7.13×10−42
6.26×10−43
2.76×10−43
6.34×10−42
4.23×10−43
1.88×10−43

3.27×10−40
2.87×10−41
1.26×10−41
3.04×10−40
2.03×10−41
9.03×10−42bb̄
W +W −
τ +τ −bb̄
W +W −
τ +τ −

Table 7.9: The 90% upper limits on SD WIMP-proton scattering cross-section, SI and
SI (IVDM) WIMP-nucleon scattering cross-sections are listed.

As a result, we found no signal observed and the null result was interpreted as upper
limit on the SD and SI WIMP-proton elastic scattering cross-section for χχ → bb̄ andχχ → τ + τ − WIMP annihilation channels. The result of the study is consistent with nosignal from 10 GeV WIMP with SD coupling of 1.31 × 10−40 cm2 (1.49 × 10−39 cm2 ) to
proton where τ + τ − (bb̄) annihilation channels are assumed respectively, and consistent
with no signal from 10 GeV WIMP with SI coupling of 2.12×10−42 cm2 (2.40×10−41 cm2 )to proton where τ + τ − (bb̄) annihilation channels are assumed respectively. We set
current best limit on SD cross-section for WIMP mass below 200 GeV and ruled out
some fraction of signal-claimed region in SI parameter space.

172

WIMP-proton SD Cross section[pb]Chapter 7

10
1
10
10
10
10
10

-1

-2

-3

-4

τ+τ-

bb

W +W -

-5

10

10

2

WIMP mass[GeV]

Figure 7.16: 90% C. L. upper limits on SD WIMP-proton cross section calculated forthe DarkSUSY (83) default are shown for the τ + τ − channel in red solid; bb in red
dashed; W + W − in red dot-dashed. Also shown are limits from other experiments:
IceCube (54) bb (brown dashed) / W + W − (brown dot-dashed); BAKSAN (59) τ + τ −(pink solid) / bb (pink dashed) / W + W − (pink dot-dashed); PICASSO (202) (bluesolid); SIMPLE (203) (green solid). The black shaded regions are the 3σ C.L. signal
claimed by DAMA/LIBRA (62; 204).

173

WIMP-neutron SI Cross section[pb]Chapter 7

10

10

10

10

10

-3

-4

-5

-6

-7

τ+τ10

bb

W +W -

-8

10

10

2

WIMP mass[GeV]

Figure 7.17: 90% C. L. upper limits on the SI WIMP-nucleon cross section calculatedusing the DarkSUSY (83) default are shown for the τ + τ − channel in red solid; bb
in red dashed; W + W − in red dot-dashed. Also shown are signal claims from otherexperiments: DAMA/LIBRA (black shaded regions, 3σ C.L.); CoGENT (48) (magenta shaded region, 90% C.L.); CRESSTII (45) (violet shaded regions, 2σ C.L.);
CDMS II Si (46) (blue shaded region, 90% C.L.); and limits: SuperCDMS (51) (cyansolid); CDMSlite (205) (blue dashed); XENON10 S2-only (52) (dark green dashed);
XENON100 (49) (green solid); LUX (50) (brown solid).

Chapter 8
Discussions: On the uncertainties
in solar WIMP search
8.1

Overview

In order to set upper limits on cross sections appropriately from searches for theneutrinos coming from the WIMP annihilation in the Sun, we have to consider various
sources of uncertainties which can be grouped into three categories:• A) uncertainties in annihilation in the Sun,• B) uncertainties in neutrino propagation from the Sun to a detector,• C) uncertainties related to detector response and background.
The last category C is detector specific and hence need to be studied by the corresponding experiments. In this chapter, we focus on the Super-Kamiokande detector.
The errors in category B is location specific and the DarkSUSY (83) and WimpSim (88)packages can, for example, be utilised to obtain precise neutrino flux at the given detector site.
These uncertainties are treated with the “pulled” method and included in the result
shown in the previous chapter. Total 73 sources of systematic errors are consideredin the predictions of atmospheric neutrino flux, WIMP neutrino flux, neutrino interaction, event reduction, event selection and reconstruction. The best fit systematic
uncertainties, j ’s, and the estimated 1σ errors, σj ’s together with the brief descriptionof the estimation of the errors are summarized in the following chapters; for errors
in the background atmospheric neutrino flux and detector response, and in the signal
WIMP neutrino propagation in Chap. 8.2 and Chap. 8.3, respectively.
The category A includes astrophysical uncertainties as well as particle- and nuclearphysics uncertainties. These errors don’t alter the energy spectrum nor the flavor
composition of the neutrino flux but only alter the normalization of the flux. Therefore
they are common for searches at all neutrino detectors. These errors are considered174

in the conversion from the neutrino flux to the capture rate. To provide the reliableinterpretation of the result of solar WIMP neutrino searches in comparison to direct
detection experiments, these uncertainties are quantified and the new result of the SK
solar WIMP search is shown with the “uncertainty band” at the end of this chapter.
On the uncertainties which are not reflected in the result, a short comments will begiven.
Chapter 8.4 briefly explains and summarizes the uncertainties related to nuclear
physics and Chap. 8.5 discusses the uncertainties in particle physics assumptions made.
The final summary is given in Chap. 8.6.

8.2

Uncertainties related to detector response andbackgroundThe systematic error tables for 480 bins are generated for BG and signal, respectively. For simplicity, MC events are weighted for 2D atmospheric oscillation scenario
for BG, and to have 1:1:1:1:1:1 flavor fatio for three-flavor neutrino and anti-neutrinos
for signal.

8.2.1

Uncertainties in background atmospheric neutrino flux

Twenty-five systematic errors corresponding to the uncertainty in BG atmospheric
neutrino flux are summarized in Tab. 8.1, where j ’s at the best fit are shown forthree representative WIMP hypotheses. Most of the systematic errors considered inoscillation analysis are commonly used except those certainly related to zenith angle
distribution. Uncertainties in atmospheric neutrino oscillations are introduced as varying five neutrino oscillation parameters by 1σ deviation taken from PDG (206) andconsidering the matter effect in the Earth.
The uncertainty in the absolute atmospheric neutrino flux includes the uncertaintyof π production in the hadronic interaction model and the hadronic interaction crosssections. Calculations done by Honda et al. (126) are used:• Eν < 1 GeV: it linearly decreases with log Eν from 25% at 0.1 GeV to 7% at 1
GeV,
• Eν > 1 GeV: 7% in 1 ∼ 10 GeV, and it linearly increases with log Eν from 7%
at 10 GeV to 12% at 100 GeV, then it linearly increases with log Eν from 12%
at 100 GeV to 20% at 1 TeV.
As the differences in absolute normalization of flux models become large above ∼10
GeV, additional 5% is assigned for normalizations of FC multi-GeV, PC and UPMUstop samples.
The uncertainty in the flavor ratio (νµ + ν µ )/(νe + ν e ) of the atmospheric neutrino
flux is estimated from the difference in the flavor ratio of the FLUKA (147) and Bartol(146) flux models normalized to the Honda flux (126), as shown in Fig. 8.1. They are;

Chapter 8176

Systematic error related to BG neutrino flux
Absolute
Normalizationσ(%)

Flux Eν < 1 GeV

25

Fit value (/σ)6b80.3 W
0.89
0.90

Eν > 1 GeV

15a

0.75

0.72

0.73

Eν < 1 GeV
1 < Eν < 10 GeV
Eν > 10 GeV

2
3
5a

-0.90
-0.95
0.52

-0.90
-0.94
0.54

-0.90
-0.94
0.52

Eν < 1 GeV
1 < Eν < 10 GeV
Eν > 10 GeV

5
5
8a

1.19
0.59
-0.02

1.19
0.58
-0.03

1.19
0.58
-0.03

Eν < 1 GeV
1 < Eν < 10 GeV
Eν > 10 GeV

2
6
15a
10a
10
5

-0.15
0.31
0.07
-0.62
-0.09
-1.13

-0.14
0.30
0.08
-0.61
-0.10
-1.13

-0.14
0.30
0.07
-0.61
-0.10
-1.13

5
20
50
20
10
13.3
3.6
14.0
6.6
2.7
6.8

-0.17
0.00
0.62
0.08
-0.01
-0.23
0.01
-0.23
0.11
-0.08
-0.01

-0.16
0.00
0.62
0.08
-0.01
-0.23
0.01
-0.23
0.13
-0.08
-0.01

-0.16
0.00
0.62
0.08
-0.01
-0.22
0.01
-0.23
0.12
-0.08
-0.01

a

200 τ
0.90

(νµ + ν µ )/(νe + ν e ) ratioνe /νe ratioνµ /νµ ratio

K/π ratio
Neutrino path length
Sample-by-sample
normalization
Solar Activity

FC Multi-GeV
PC & stopping up-µ
SK-I
SK-II
SK-III
SK-IV

oscillation parameter θ13
oscillation parameter θ12
oscillation parameter θ23
oscillation parameter ∆m32
oscillation parameter ∆m21
Matter effectaThey are energy dependent. See text for the detail.

Table 8.1: Systematic errors related to atmospheric neutrino flux calculation. They
are common in all SK periods. The third column shows the estimated 1σ error size
compared to nominal value, and the 4-6th columns show the best fit values of the systematic error parameter  for three representative WIMP hypotheses; “6 b”, “80.3 W ”,
“200 τ ” indicate 6 GeV bb̄, 80.3 GeV W + W − , 200 GeV τ + τ − channels, respectively.

Chapter 8

177• 2% for Eν < 1 GeV,
• 3% for 1 < Eν < 10 GeV,(νµ+νµ)/(νe+νe) double ratio• 5% for 10 < Eν < 30 GeV, and then it linearly increases with logEν from 5% at
30 GeV to 30% at 1 TeV.

1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6 −1
10

1

10

102

103
Eν(GeV)

Figure 8.1: Double ratio of (νµ + ν µ )/(νe + ν e ) of the atmospheric neutrino flux for
FLUKA flux/Honda flux (black solid) and Bartol flux/Honda flux (red dashed). Figure
is taken from (125).
The uncertainty in the ν/ν ratio of the atmospheric neutrino flux is estimated fromthe difference in the flavor ratio of the FLUKA (147) and Bartol (146) flux models
normalized to the Honda flux (126) as shown in Fig. 8.2. They are:for ν e /νe ,• 5% for Eν < 1 GeV,
• 5% for 1 < Eν < 10 GeV,
• 8% for 10 < Eν < 100 GeV, and then it linearly increases with logEν from 8%
at 30 GeV to 30% at 1 TeV.
For ν µ /νµ ,• 2% for Eν < 1 GeV,
• 6% for 1 < Eν < 10 GeV,
• 6% for 10 < Eν < 50 GeV, and then it linearly increases with logEν from 6% at
50 GeV to 60% at 1 TeV.

1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5 −110178

νµ/ νµ double ratioνe/ νe double ratio

Chapter 8

1

10

102

103
Eν(GeV)

1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5 −1
10

1

10

102

103
Eν(GeV)

Figure 8.2: Double ratio of ν e /νe (left) and ν µ /νµ (right) of the atmospheric neutrino
flux for FLUKA flux/Honda flux (black solid) and Bartol flux/Honda flux (red dashed).
Figure is taken from (125).
Above a few tens of GeV, K’s contribute to the production of the neutrinos in
addition to π + and π − decays. The uncertainty of K/π production ratio in cosmic ray
interactions is estimated by using the SPY experiment measurement (207) as:• 5% for Eν < 100 GeV, and it linearly increases with Eν from 5% at 100 GeV to
20% at 1 TeV, then it is 20% for Eν > 1 TeV.
The uncertainty in the position of the neutrino production is related to the atmosphere density distribution. By varying the density distribution by 10 %, the differencein the resultant neutrino flux is taken as the systematic uncertainty.
For the uncertainty in solar activity, ±1 year is added on eleven-years solar activity
cycle and accordingly altered primary cosmic ray flux takes the uncertainty of 20% for
SK-I, 50% for SK-II, 20% for SK-III and 10% for SK-IV.
For the uncertainty in atmospheric neutrino oscillation, five neutrino oscillation
parameters are varied by 1σ uncertainty based on the experimental results (206), as
shown in Tab. 8.2.
The matter effect in the neutrino oscillation is affected by the electron density inthe Earth, which depends on the chemical composition of the Earth. As the Earth’s
core is composed of heavier elements, the electron density in the core is expected to beless than other regions by ∼ 6.8%. Reduction of 6.8% electron density is taken as theuncertainty in the matter effect.

8.2.2

Uncertainties in detector response: Neutrino interactions

Sixteen systematic errors corresponding to neutrino interactions are considered.
The results are summarized in Tab. 8.3 and briefly explained below.

Chapter 8

179

Parameter used value−1σ+1σ
sin(2θ12 )0.873−0.031+0.0312sin (2θ13 )0.098−0.013+0.0132sin (θ23 )
0.5
−0.07
+0.07
∆m232 [eV2 ]7.46e-5−0.2e-5 +0.2e-5∆m221 [eV2 ]2.44e-3−0.15e-3 +0.17e-3
Table 8.2: Summary of 1σ variations of neutrino oscillation parameters taken as uncertainties.

CCQE interaction cross section between neutrino and nucleon is calculated basedon the Smith and Moniz model as described in Chap. 5. The Smith and Moniz model iscompared to the Nieves et al. model (208) and the difference is taken as the uncertainty.
The uncertainties in total CCQE cross section, ν/ν ratio, and flavor ratio (νµ + νµ )/(νe + ν e ) are taken in this way. Figure 8.3 shows the calculated CCQE cross section
ratio of Nieves model to Smith and Monitz model for total cross section and double
ratios for ν/ν and flavor ratio.
Direction resolution becomes an issue in a search for point like signal; the uncertainty of axial mass MA is expected to be a large source of uncertainty in q 2 spectrum
therefore in direction reconstruction. The size of the effect of the uncertainties from
MA for CCQE and single pion production interactions are taken by comparing the differential cross section dσ/dq 2 with the MA = 1.4 and that with the fiducial value MA= 1.2. Figure 8.4 visualizes the change of angular resolution for two MA assumptions.
The Hernandez model (209) is compared to the Rein and Sehgal model for estimating the size of uncertainties relating to single meson production cross section. The crosssection ratios between two models are shown in the left panel of Fig. 8.5. The ratio
between the π 0 production interaction to the π ± production interactions is about 40%
and is taken as the systematic error. The double ratios of the ν/ν for two models aretaken as the systematic uncertainties of the ν/ν ratio in the single meson production
as shown in the right panel of Fig. 8.5 For the uncertainty in the cross section of single
meson production itself, available experimental data of νµ p → µ− pπ + is used. By taking the difference between the MC expectation and data, the systematic uncertainty is
estimated to be 20%.
The systematic uncertainty in deep inelastic scattering cross section is obtained bytaking the difference between the reference model and the CKMT model (210).
The uncertainty of q 2 spectrum in deep inelastic scattering is taken into accountby comparing the reference model with the original GRV98 model. It is separately
calculated for the case W < 1.3 GeV and W > 1.3 GeV. Figure 8.6 visualizes thechange of angular resolution for two models.
For NC and CC νe coherent pion production interactions, the Rein and Sehgal model
is compared to the SciBooNE experimental measurement (211) and the uncertainty isChapter 8

Systematic error in neutrino interaction
CCQE cross section
CCQE ν/ν ratio
CCQE (νµ + νµ )/ (νe + ν e ) ratio
MA in CCQE and single meson production
Single meson production cross section
Single π production, π 0 /π ± ratio
Single π production, ν/ν ratio
DIS cross section
Q2 spectrum in DIS (W < 1.3 GeV)
Q2 spectrum in DIS (W > 1.3 GeV)
Coherent π production
NC/CC ratioCC ντ cross section
NC in FC µ-like (hadron simulation)
Decay electron tagging (π + interaction)d
DIS model difference180

σ(σ/%)
a

1.0
1.0a
1.0a
20
20
40
1.0b
5
1.0c
1.0d
100
20
25
10
10
10

Fit value
6b
0.33
2.07
1.67
-0.98
0.71
-0.53
-0.60
0.71
0.00
0.51
-0.27
0.45
0.26
-0.47
-0.55
-0.42

(/σ)
80.3 W
0.34
2.05
1.67
-0.98
0.72
-0.53
-0.57
0.72
0.00
0.51
-0.27
0.44
0.26
-0.46
-0.54
-0.41

200 τ
0.35
2.05
1.67
-0.98
0.71
-0.53
-0.57
0.74
0.00
0.51
-0.28
0.45
0.26
-0.46
-0.54
-0.41

a

Difference from the Nieves model is set to 1.0.
Difference from the Hernandez model is set to 1.0.
c
Difference from the GRV98 model is set to 1.0.
dDifference from the GRV98 model is set to 1.0.
b

Table 8.3: Systematic errors related to neutrino interactions. They are common in all
SK periods. The second column shows the estimated 1σ error size compared to nominal
value, and the 3-5th columns show the best fit values of the systematic error parameter for three representative WIMP hypotheses; “6 b”, “80.3 W ”, “200 τ ” indicate 6 GeV
bb̄, 80.3 GeV W + W − , 200 GeV τ + τ − channels, respectively.

181

2

µ/e double ratio(Nieves/Smith-Moniz)

Ratio(Nieves/Smith-Moniz)

Chapter 8

1.8
1.6
1.4
1.2
1
0.8
0.6
0.4

1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6

0.2
50

100

150

200

250

! / ! double ratio(Nieves/Smith-Moniz)

0

1.5

300

350

400

450
500
E! (MeV)

0.5
0

50

100

150

200

250

300

350

400

450
500
E! (MeV)

1.4
1.2
1
0.8
0.6
0.4
0.2
0

50

100

150

200

250

300

350

400

450
500
E! (MeV)Figure 8.3: The CCQE cross section for νe +ν e (solid line) and for νµ +ν µ (dashed line)of Nieves model normalized by the Smith and Monitz model (top left), flavor ratio ((νµ + ν µ )/(νe + ν e ) ) of Nieves model normalized by the Smith and Monitz model (topright), ν/ν ratio of Nieves model normalized by the Smith and Monitz model (bottom)
are shown as a function of neutrino energy. Figure is taken from (125).

Chapter 8

182

Ma=1.2Ma=1.40.3

Ma=1.2Ma=1.4

1

0.25
0.8
0.2
0.6
0.15
0.4
0.1
0.2

0.05

0

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0

1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

CC ν/ ν double ratio

CC νµ cross section ratio

Figure 8.4: The angular distribution for two choices of MA ’s are compared for sub-GeV(left) and multi-GeV (right) signal MC events with arbitrary normalization, for 4 GeV
τ + τ − channel. The black line shows the angular distribution for default MA =1.2, andthe red line shows that for 20% increased MA (=1.4). Note that the systematic error isapplied only to the CCQE and single meson production events, but the plot is drawn
for entire signal MC to visualize overall effect on angular distribution of the signal.

0.5

1

1.5

2

2.5

3

3.5

4

4.5 5
Eν (GeV)

2.2
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0

0.5

1

1.5

2

2.5

3

3.5

4

4.5 5
Eν (GeV)Figure 8.5: The Hernandez model normalized by that of the Rein and Sehgal modelfor νµ p → µ− pπ + (solid line), νµ n → µ− pπ 0 (dashed line) and νµ n → µ− pπ + (dottedline) are shown for the predicted cross section (left) and the double ratio for the ν/ν
ratio (right). Figure is taken from (125).

Chapter 8

183

1
0.35

fiducial
GRV98 model

fiducial
GRV98 model

0.3

0.8

0.25
0.6
0.2
0.15

0.4

0.1
0.2
0.05
0

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Figure 8.6: The angular distribution for two DIS models are compared for sub-GeV(left) and multi-GeV (right) signal MC events with arbitrary normalization, for 4 GeV
τ + τ − channel. The black line shows the angular distribution for default, and the redline shows that for the GRV98 model. Note that the systematic error is applied only
to the DIS events, but the plot is drawn for entire signal MC to visualize overall effecton angular distribution of the signal.

taken to be 50%. For CC νµ coherent pion production interaction, the uncertainty istaken to be 100%.
The uncertainty in NC/CC ratio is taken to be 20%.
The uncertainty of the CC ντ cross section is estimated to be 25%, comparing to
the Hagiwara model (212).

8.2.3

Uncertainties in detector response: event selections

Fourteen systematic errors correspond to event selections are summarized in Tab. 8.4
to Tab. 8.7 respectively for SK I-IV.
The systematic uncertainty for FC/PC separation is estimated to be 0.6% for SKI, 0.5% for SK-II, 0.9% for SK-III and 0.02% for SK-IV from the difference of the
NHITAC distribution between the data and MC.
The uncertainty in FC reduction efficiency is estimated by comparing the distributions of each cut variable for data and MC. They are 0.2%, 0.2%, 0.8% and 0.3% for
SK I-IV, respectively.
The contamination of the background in the final FC data set are estimated to be
0.2%, 0.2%, 0.3% and 0.1% for sub-GeV µ-like sample for SK I-IV, respectively, 0.1%,
0.4%, 0.1% and 0.1% for sub-GeV e-like sample for SK I-IV, respectively, 0.3%, 0.2%,
0.3% and 0.1% for multi-GeV µ-like sample for SK I-IV, respectively, 0.5%, 0.2%, 0.2%

Chapter 8(SK-I) Systematic error
in event reduction
FC/PC separation
FC reduction
PC reduction
Non-ν BG (e-like)
Non-ν BG (µ-like)PC-stop/through separation

Fiducial volume
up-µ reduction
up-µ stop/through separation
up-µ
through-goingshowering
separation
Energy cut for up-µ stop
Path length cut for up-µ through-going

184

σ(%)
0.6
0.2
2.4
1.0
1.0
Top
46.1
Barrel
7.01
Bottom 22.7
2.0
1.0
0.4
3.4
-0.9
-1.5

Fit value
6b
-0.06
0.02
-0.44
0.16
-0.02
-0.21
-0.16
-0.08
-0.16
-0.16
0.00
0.00

( σ)
80.3 W
-0.07
0.02
-0.43
0.16
-0.02
-0.21
-0.15
-0.08
-0.17
-0.16
0.00
0.00

200 τ
-0.07
0.02
-0.43
0.16
-0.02
-0.21
-0.15
-0.08
-0.17
-0.16
0.00
0.00

0.00
0.01

0.00
0.01

0.00
0.01

Table 8.4: Systematic errors related to event reduction for SK-I period. The second
column shows the estimated 1σ error size and the 3-5th columns show the best fit valueof the systematic error parameter i for three representative WIMP hypotheses; “6 b”,“80.3 W ”, “200 τ ” indicate 6 GeV bb̄, 80.3 GeV W + W − , 200 GeV τ + τ − channels,
respectively.

Chapter 8(SK-2) Systematic error
in event reduction
FC/PC separation
FC reduction
PC reduction
Non-ν BG (e-like)
Non-ν BG (µ-like)PC-stop/through separation

Fiducial volume
up-µ reduction
up-µ stop/through separation
up-µ
through-goingshowering
separation
Energy cut for up-µ stop
Pathlength cut for up-µ through-going

185

σ(%)

Fit value
6b
0.5
0.16
0.2
0.04
4.8
-0.75
1.0
0.05
1.0
-0.13
Top
19.37 -0.13
Barrel9.44 -0.60
Bottom 12.9 -0.17
2.0
0.05
1.0
-0.23
0.6
0.00
4.4
0.00

( σ)
80.3 W
0.16
0.04
-0.74
0.05
-0.12
-0.13
-0.60
-0.17
0.05
-0.23
0.00
0.00

200 τ
0.16
0.04
-0.74
0.05
-0.12
-0.13
-0.60
-0.17
0.05
-0.23
0.00
0.00

-1.3
-2.3

0.00
-0.02

0.00
-0.02

0.00
-0.02

Table 8.5: Systematic errors related to event reduction for SK-II period. The second
column shows the estimated 1σ error size and the 3-5th columns show the best fit valueof the systematic error parameter i for three representative WIMP hypotheses; “6 b”,“80.3 W ”, “200 τ ” indicate 6 GeV bb̄, 80.3 GeV W + W − , 200 GeV τ + τ − channels,
respectively.

Chapter 8(SK-3) Systematic error
in event reduction
FC/PC separation
FC reduction
PC reduction
Non-ν BG (e-like)
Non-ν BG (µ-like)PC-stop/through separation

Fiducial volume
up-µ reduction
up-µ stop/through separation
up-µ
through-goingshowering
separation
Energy cut for up-µ stop
Pathlength cut for up-µ through-going

186

σ(%)
0.9
0.8
0.5
1.0
1.0
Top
86.6
Barrel
28.7
Bottom 12.1
2.0
1.0
0.4
2.4
2.0
-2.8

Fit value
6b
-0.02
0.06
-0.08
-0.02
-0.39
-0.17
-0.33
0.00
-0.19
0.16
0.00
0.00

( σ)
80.3 W
-0.01
0.05
-0.08
-0.02
-0.39
-0.16
-0.33
0.00
-0.19
0.16
0.00
0.00

200 τ
-0.02
0.05
-0.08
-0.02
-0.39
-0.16
-0.33
0.01
-0.19
0.16
0.00
0.00

0.00
0.07

0.00
0.07

0.00
0.07

Table 8.6: Systematic errors related to event reduction for SK-III period. The second
column shows the estimated 1σ error size and the 3-5th columns show the best fit valueof the systematic error parameter i for three representative WIMP hypotheses; “6 b”,“80.3 W ”, “200 τ ” indicate 6 GeV bb̄, 80.3 GeV W + W − , 200 GeV τ + τ − channels,
respectively.

Chapter 8(SK-4) Systematic error
in event reduction
FC/PC separation
FC reduction
PC reduction
Non-ν BG (e-like)
Non-ν BG (µ-like)PC-stop/through separation

Fiducial volume
up-µ reduction
up-µ stop/through separation
up-µ
through-goingshowering
separation
Energy cut for up-µ stop
Path length cut for up-µ through-going

187

σ(%)

Fit value
6b0.02 0.00
0.3
0.07
1.0
-0.26
1.0
-0.01
1.0
-0.16
Top
40.33 0.11
Barrel8.54 0.13
Bottom 6.8
0.02
2.0
-0.04
1.0
0.24
0.6
0.00
3.0
0.00

( σ)
80.3 W
0.00
0.07
-0.26
-0.01
-0.16
0.11
0.13
0.02
-0.04
0.24
0.00
0.00

200 τ
0.00
0.07
-0.25
-0.01
-0.16
0.11
0.13
0.02
-0.04
0.24
0.00
0.00

1.7
-1.5

0.00
-0.16

0.00
-0.16

0.00
-0.16

Table 8.7: Systematic errors related to event reduction for SK-IV period. The second
column shows the estimated 1σ error size and the 3-5th columns show the best fit valueof the systematic error parameter i for three representative WIMP hypotheses; “6 b”,“80.3 W ”, “200 τ ” indicate 6 GeV bb̄, 80.3 GeV W + W − , 200 GeV τ + τ − channels,
respectively.

Chapter 8

188

and 0.1% for multi-GeV e-like sample for SK I-IV, respectively, as discussed in Chap.
6.
The systematic uncertainty of PC stopping/through-going separation is estimatedby the difference of PEanti /PEexp distributions between the data and MC. The estimations are done separately for OD bottom, barrel and top.
The systematic uncertainty of up-µ stopping and through-going separation is estimated by the difference of NHITEX distributions between the data and MC. The
systematic uncertainty of up-µ non-showering and showering separation is estimatedby comparing the ∆(Q) distribution between the data and MC.

8.2.4

Uncertainties in detector response: Event reconstructions

Eleven systematic errors correspond to FC/PC/up-µ reconstructions are summarized in Tab. 8.8 to 8.11 respectively for SK I-IV.

(SK-I) Systematic errorσ(%)
in event reconstruction and selection
Ring separation
10
PID (single-ring)
1.0
PID (multi-ring)
10
Energy calibration
1.1
Multi-GeV single-ring e-like non CCνe BG
13.2
Multi-GeV multi-ring e-like non CCνe BG
12.1
7.16
Multi-GeV multi-ring νe and ν e separation
Sub-GeV 1-ring π 0 selection
100Sub-GeV 2-ring π selection
5.6
Decay-e tagging (µ-decay)
10
Multi-GeV multi-ring e-like other separation 10

Fit value
6b
0.51
-0.19
0.13
0.00
0.39
-0.10
-0.45
0.16
-0.15
-0.36
-0.14

( σ)
80.3 W
0.52
-0.19
0.13
0.00
0.39
-0.10
-0.45
0.16
-0.15
-0.36
-0.15

200 τ
0.52
-0.19
0.13
0.00
0.39
-0.10
-0.46
0.16
-0.15
-0.36
-0.14

Table 8.8: Systematic errors related to event reconstruction and selection for SK-I.
The second column shows the estimated 1σ error size and the 3-5th columns show
the best fit value of the systematic error parameter i for three representative WIMP
hypotheses; “6 b”, “80.3 W ”, “200 τ ” indicate 6 GeV bb̄, 80.3 GeV W + W − , 200 GeVτ + τ − channels, respectively.
For the ring seperation and the particle identification, the difference in the likelihood
distribution between the data and MC are taken as the uncertainty for each event
category.
The uncertainties for absolute energy scale are estimated to be 1.1%, 1.7%, 2.7%
and 2.3% for SK I-IV, respectively, as discussed in Chap. 4.6.3.

Chapter 8(SK-2) Systematic errorσ(%)
in event reconstruction and selection
Ring separation
10
PID (single-ring)
1.0
PID (multi-ring)
10
Energy calibration
1.7
Multi-GeV single-ring e-like non CCνe BG
38.1
Multi-GeV multi-ring e-like non CCνe BG
11.1
7.91
Multi-GeV multi-ring νe and ν e separation
Sub-GeV 1-ring π 0 selection
10
Sub-GeV 2-ring π 0 selection
4.4
Decay-e tagging (µ-decay)
10
Multi-GeV multi-ring e-like other separation 10.

189

Fit value
6b
-1.06
0.28
0.64
0.00
-0.37
0.12
0.07
0.19
-0.39
-0.44
0.22

( σ)
80.3 W
-1.05
0.28
0.64
0.00
-0.37
0.12
0.07
0.19
-0.39
-0.45
0.22

200 τ
-1.05
0.28
0.64
0.00
-0.37
0.12
0.07
0.19
-0.39
-0.44
0.22

Table 8.9: Systematic errors related to event reconstruction and selection for SK-II.
The second column shows the estimated 1σ error size and the 3-5th columns show
the best fit value of the systematic error parameter i for three representative WIMP
hypotheses; “6 b”, “80.3 W ”, “200 τ ” indicate 6 GeV bb̄, 80.3 GeV W + W − , 200 GeVτ + τ − channels, respectively.

(SK-3) Systematic errorσ(%)
in event reconstruction and selection
Ring separation
10
PID (single-ring)
1.0
PID (multi-ring)
10
Energy calibration
2.7
Multi-GeV single-ring e-like non CCνe BG
26.7
Multi-GeV multi-ring e-like non CCνe BG
11.4
7.68
Multi-GeV multi-ring νe and ν e separation
Sub-GeV 1-ring π 0 selection
100Sub-GeV 2-ring π selection
5.9
Decay-e tagging (µ-decay)
10.
Multi-GeV multi-ring e-like other separation 10.

Fit value
6b
0.29
-0.14
0.31
0.00
0.14
-0.29
-0.94
0.22
-0.12
0.22
-0.42

( σ)
80.3 W
0.29
-0.14
0.31
0.00
0.14
-0.29
-0.94
0.22
-0.12
0.21
-0.43

200 τ
0.29
-0.14
0.31
0.00
0.14
-0.29
-0.94
0.22
-0.12
0.21
-0.43

Table 8.10: Systematic errors related to event reconstruction and selection for SK-III.
The second column shows the estimated 1σ error size and the 3-5th columns show
the best fit value of the systematic error parameter i for three representative WIMP
hypotheses; “6 b”, “80.3 W ”, “200 τ ” indicate 6 GeV bb̄, 80.3 GeV W + W − , 200 GeVτ + τ − channels, respectively.

Chapter 8(SK-4) Systematic errorσ(%)
in event reconstruction and selection
Ring separation
10
PID (single-ring)
1.0
PID (multi-ring)
10
Energy calibration
2.3
Multi-GeV single-ring e-like non CCνe BG
17.6
Multi-GeV multi-ring e-like non CCνe BG
11.6
6.82
Multi-GeV multi-ring νe and ν e separation
Sub-GeV 1-ring π 0 selection
10
Sub-GeV 2-ring π 0 selection
5.6
Decay-e tagging (µ-decay)
10
Multi-GeV multi-ring e-like other separation 10

190

Fit value
6b
0.56
0.08
0.59
0.00
0.06
0.61
-0.28
0.41
-0.02
0.25
-0.55

( σ)
80.3 W
0.55
0.08
0.59
0.00
0.06
0.61
-0.28
0.41
-0.02
0.24
-0.55

200 τ
0.55
0.08
0.59
0.00
0.06
0.61
-0.28
0.41
-0.02
0.24
-0.54

Table 8.11: Systematic errors related to event reconstruction and selection for SK-IV.
The second column shows the estimated 1σ error size and the 3-5th columns show
the best fit value of the systematic error parameter i for three representative WIMP
hypotheses; “6 b”, “80.3 W ”, “200 τ ” indicate 6 GeV bb̄, 80.3 GeV W + W − , 200 GeVτ + τ − channels, respectively.
For the multi-GeV multi-ring e-like non CC νe BG and νe and ν e separation, thecross section of the final state interactions of pions and the pion multiplicity are main
expected source of the uncertainty (213). The cross sections of pions is compared to external experimental data, also the pion multiplicity is compared with the experimental
data from the CHORUS experiment (214).
The uncertainties in the selection efficiency of π 0 -like events in sub-GeV 1-ring
π 0 -like and 2-ring π 0 -like samples are estimated by comparing the data with MC foreach momentum region, and by shifting the absolute energy scale. The combined
uncertainties are summarized in Tab. 8.12.
The uncertainty in decay electron detection efficiency is estimated to be 1.5% for
SK I-III and improved to 0.8% for SK-IV, by using cosmic ray muons (215).

8.3

Uncertainties in WIMP neutrino propagation

For the uncertainties in the propagation of WIMP neutrino flux, seven systematic
errors are considered for oscillation parameters and matter effect inside the Sun andthe Earth. Five neutrino oscillation parameters are varied by 1σ deviation based on
the experimental results (206), as shown in Tab. 8.13. The electron density inside theSun and the Earth are varied by 10% in charge of the uncertainties in matter effects.
We simulate the neutrino flux using WIMPsim as varying each error source andChapter 8191

1σ (%)
SK-I SK-II SK-III
100 < Pe < 250 MeV/c
11.2 7.5
7.7
250 < Pe < 400 MeV/c
11.5 8.9
26.4
400 < Pe < 630 MeV/c
23.4 17.5
12.5
630 < Pe < 1000 MeV/c 19.1 10.7
26.7
1000 < Pe < 1330 MeV/c 13.0 11.1
26.7

SK-IV
10.8
11.6
10.1
15.1
14.5

Table 8.12: 1σ values for systematic uncertainties of single-ring π 0 -like events selection
for SK-I, SK-II, SK-III and SK-IV.
compare to the original flux used in the analysis.
Parameterused value◦
θ1233.461
θ13 ◦
9.097
◦
θ23
40.6862
2∆m32 [eV ] 0.274 × 10−2∆m221 [eV2 ] 0.766 × 10−4

1σ
36
9.75
49
0.289 × 10−20.726 × 10−4

Table 8.13: Summary of 1σ variations of neutrino oscillation parameters taken as
uncertainties.
The pulls for systematic errors related to signal flux are marginal except for θ23
which gives some contribution. Figure 8.7 shows fractional change in signal events due
to 1σ increase of θ23 . The fit values of all signal flux errors are summarized in Tab.
8.14.

8.4

Uncertainties in nuclear physics and astrophysics

8.4.1

Form factor

In the spin-dependent coupling case, only hydrogen nuclei contribute, therefore
form factor is not considered. However, the spin-independent coupling involves form
factor correction, especially when WIMPs scatter off heavy elements.
The Helm-Gould nuclear form factor (82) for the momentum transfer q on theelement i used in DarkSUSY has a form:|Fi (q 2 )|2 = exp(−∆E/Ei0 ),(8.1)

Chapter 8

192

Zenith Angle Distributions
osc_5_0specdt
Entries
11
Mean 640.3
RMS
365

20

osc_2_1
Entries3
Mean
0.1
RMS 0.5833

200Sub-GeV e-like 1-dcy e

0-20

Sub-GeV e-like 0-dcy e

0

20

20

Sub-GeV π 0-like 1-R

0-20

Multi-GeV e-like ν e

0MultiRing_elike_nue Zenith vs P W IMP

0

Sub-GeV µ -like 2-dcy e

0-20

Multi-Ring e-like ν e

0-20

Sub-GeV π 0-like M-R

500

1000

lepton momentum (MeV)

0

-20
-1

0

PC Stop

UpThruNonShower_mu Zenith vs P WIMP

0-20
osc_53_13
Entries9
Mean 0.7848
RMS 0.09883

20

0

PC Thru-20

osc_41_14
Entries
10
Mean
0
RMS
0

20

PCThru Zenith vs P WIMP

Multi-Ring e-like ν e

osc_44_9
Entries
10
Mean 0.9883
RMS 0.006118

20

UpThruShower_mu Zenith vs P WIMP

0-20

osc_38_17
Entries5
Mean 0.3367
RMS 0.3879

MultiRing_mulike Zenith vs P WIMP

Multi-GeV mu-likecos zenith

-20

-20
osc_35_12
Entries8
Mean 0.3979
RMS 0.5542

20

MultiGeV_mulike Zenith vs P WIMP

0

0

-20
osc_29_16
Entries3
Mean 0.7111
RMS 0.1496

200

osc_50_8
Entries9
Mean 0.7609
RMS
0.129

20

MultiRing_elike_nuebar Zenith vs P WIMP

-20
osc_20_15specdt
Entries
11
Mean 462.7
RMS
325.1

20

Multi-GeV e-like ν e

20

UpStop_mu Zenith vs P WIMP

PCStop Zenith vs P WIMP

-20
osc_32_11
Entries9
Mean 0.609
RMS 0.2339

20

Sub-GeV µ -like 1-dcy e-20

MultiGeV_elike_nuebar Zenith vs P WIMP

-20
osc_17_10specdt
Entries
11
Mean 597.8
RMS
352.1

20

0

osc_26_7
Entries3
Mean 0.6736
RMS 0.1733

20

MultiG eV _elike_nue Zenith vs P W IMP

0

Sub-GeV µ -like 0-dcy e

osc_47_4
Entries
10
Mean 0.8147
RMS
0.112620

SubGeV_mulike_1dcy Zenith vs P WIMP

-20
osc_23_6
Entries9
Mean 0.7934
RMS 0.09978

osc_14_3
Entries
4
Mean -0.04957
RMS
0.549820

SubGeV_mulike_0dcy Zenith vs P WIMP

-20
osc_8_5specdt
Entries
11
Mean 615.9
RMS
350.6

osc_11_2
Entries
7
Mean-0.005517
RMS 0.5594

20

S ubG eV_elike_0dcy Zenith vs P W IMP

0

1

-20
-1

Multi-Ring mu-like0

cos zenith

1

cos zenith

cos zenith

Figure 8.7: signal events with 1σ varied θ23 subtracted by signal events for nominal
spectrum divided by signal events for nominal spectrum in percentage. They are shownfor eighteen event categories used in the analysis.

Chapter 8

Systematic error related tosignal neutrino flux
oscillation parameter θ13
oscillation parameter θ12
oscillation parameter θ23
oscillation parameter ∆m32
oscillation parameter ∆m21
Neutrino interaction in the Sun
Matter effect in the Earth

193

σ(%)
13.3
3.6
14.0
6.6
2.7
10.0
10.0

Fit value
6b
-0.01
0.03
0.05
0.00
0.00
0.00
0.00

(/σ)
80.3 W
0.00
0.08
-0.02
0.00
0.00
0.00
0.00

200 τ
0.00
0.00
0.00
0.00
0.00
0.00
0.00

Table 8.14: Systematic errors related to WIMP neutrino flux calculation. They arecommon in all SK periods. The second column shows the estimated 1σ error size
compared to nominal value, and the 3-5th columns show the best fit values of the systematic error parameter  for three representative WIMP hypotheses; “6 b”, “80.3 W ”,
“200 τ ” indicate 6 GeV bb̄, 80.3 GeV W + W − , 200 GeV τ + τ − channels, respectively.
where ∆E = q 2 /2Mχ , Ei0 = 3h̄2 /2Mχ Ri2 with nuclear radius Ri = [0.91(mi /GeV)1/3 +0.3] × 10−15 m are the used values.
The choice of parameters are compared to the choices in other works to estimate
the size of the uncertainty. The effective radius Ref f,i = [1.23 × exp(log(mi )/3) − 0.6] ×
10−15 m as in Helm-Lewin-Smith form factor in (116) and (216; 217; 218) was usedinstead of nuclear radius used in (82) and DarkSUSY, and the difference was taken as
upper error. The lower limit is taken by changing the factor 3/2 to 5/2 (219) as well as
using the effective radius instead of the nuclear radius. It is a reasonable approximation
to the more accurate Helm form factor for ∆E <∼ 2Ei0 (216; 217; 218). The sizes ofthe effects on the capture rate of WIMPs as a function of WIMP mass are shown in
Tab. 8.15.

8.4.2

Solar model

The uncertainty on the composition of the Sun is calculated by comparing the DarkSUSY default choice, the BS2005-OP model, with the BS2005-AGS,OP model (220)with lower heavy element abundances. Table 8.16 shows the resultant numbers ofchange in the capture rate.
The effect is small for the SD-coupling case, about −3% lower capture rate isresulted. In the SI-coupling where heavier elements than hydrogen contribute, uncertainties are larger: 21% higher capture rate for 20 GeV and 26% of it for 500 GeV are
resulted.

Chapter 8

194

mχ
upper
(GeV)
4
SI 0.3%
6
SI 1.0%10
SI 3.1%20
SI 8.8%50
SI 16%
100
SI 19%
200
SI 22%

lower
−1.4%
−2.8%
−6.6%
−17%
−33%
−41%
−47%

Table 8.15: Relative magnitudes of changes in the capture rate of WIMPs through SI
coupling due to the form factor uncertainty are shown for several WIMP masses.

mχ
(GeV)
4
6
10
20
50
100
200

SD
SI
SD
SI
SD
SI
SD
SI
SD
SI
SD
SI
SD
SI

upper

lower

14%
16%
18%
21%
23%
24%
26%

−2.7%
−2.8%
−3.1%
−2.8%
−3.1%
−3.5%
−3.2%
-

Table 8.16: Relative magnitudes of changes in the capture rate of WIMPs due to theuncertainty of the solar model, are shown for several WIMP masses, respectively for
SD and SI coupling cases.

Chapter 8

8.4.3

195

WIMP phase distribution

Large uncertainties on WIMP search comes from the understanding of the local
dark matter phase space.
The uncertainty in local dark matter density is expected to be a factor of two (221;222; 223; 224; 225), so do the uncertainties in the capture rate in the Sun and signal
rate of direct detection searches, as scattering rate simply increases with the local
density. Therefore it will make a similar overall shift of the limits for all direct andindirect detection experiments and doesn’t effect the relative interpretation of them.
For the effect of the uncertainties in the velocity distribution of WIMPs, numbers
determined in Chap. 3 are used. The uncertainties in the choice of the velocity distribution function, the orbital speed of the Sun and the high speed tail of the dark matter
halo are summed in quadrature; They are ∼ O(20)% for 20 GeV WIMP and larger
for heavier WIMP; ∼ O(40)% at 200 GeV. The possibility of co-rotating structurewith the Sun can largely boost the signal by order of magnitude (73; 75; 116; 226).
However, it is not considered as it makes the interpretation of solar WIMP search only
conservative compared to direct detection searches.

8.4.4

Dynamics of solar system

The simplified dynamical history of WIMPs in Eq. 2.15 has several possible modifications.
As discussed in Chap. 3, the effects of the planets in the solar system on the capturerate, known as Jupiter depletion and solar gravitational diffusion, are canceled eachother and determined to be negligible (109; 78) so not considered.
Solar evaporation is the process where captured WIMPs gain energy by rescatterings
inside the Sun enough to escape out of the gravitational field of the Sun (227; 81; 228).
This process can be important for the calculation of the density evolution of the light
WIMPs inside the Sun, and is expected to drop rapidly for a WIMP mass of above
4 GeV (227; 81; 63; 229). Figure 8.8 shows the evaporation rate, E , as a functionof WIMP mass (229). In the new SK analysis, WIMPs with masses above 4 GeV are
studied in order to avoid the evaporation effect and to assume the equilibrium between
the capture and the annihilation rates.
The evaporation mass, mevap , which is defined as the mass with which the inverseof the evaporation rate, E , is equal to the age of the Sun t = 4.7 × 109 yrs (228),
can be approximated as (65):mevap = m0 + 0.32log10 (σ0)GeV
10−40 cm2

(8.2)

with m0 = 3.02 (m0 = 3.5) for SD (SI) coupling case. As a result, for the cross-sections
around new SK limits, which are about 10−40 cm2 (10−41 cm2 ) for SD (SI) couplings,
the limits could be extended down to mevap ∼ 3.02 GeV for SD and mevap ∼ 3.18
GeV for SI coupling cases, and can be lowered further in future searches in neutrinoChapter 8

196

detectors such as the Hyper-Kamiokande, with possible improvement of the sensitivity
to cross sections by an order of magnitude.

Figure 8.8: Evaporation rate versus WIMP mass for SD coupling (left) and SI coupling(right) cases. Figure is taken from (229).

Below the evaporation mass, the annihilation rate decreases and the evaporation
rate increases. Therefore the capture rate of WIMPs with lighter mass may be set
in equilibrium with evaporation rate, instead of annihilation rate. As a result, theannihilation rate, therefore the number of WIMP neutrinos detected in a neutrino
detector, may become independent of the scattering cross section.
Recent theoretical work by Busoni et al. (229) points out that ven below evaporation mass, a neutrino detector can study the WIMP scattering property by solving theEq. 2.14 with evaporation term. With the canonical value for the thermally-averaged
annihilation rate < σv > = 3 ×10−26 cm3 /s, one can find that for the WIMP with SD(SI) coupling cross-section of 10−41 cm2 (10−42 cm2 ), the minimum mass testable ina neutrino detector is as small as ∼2 GeV. Figure 8.9 shows the test-minimum mass
versus cross-section (229).

8.4.5

Summary

In conclusion, the related uncertainties to the capture process are discussed. Theseerrors are added in quadrature and the resultant numbers are shown in Tab. 8.17. The
effects of them on the result from the SK analysis are indicated by the shadowed regionsin Fig. 8.10 and Fig. 8.11. The combined effects of all errors loosen the upper limit of
scattering cross-section maximally 42% for SD and 46% for SI coupling WIMPs.

Chapter 8

197

Figure 8.9: The test-minimum WIMP mass is shown on the parameter space of the
WIMP mass versus the SD (left) and SI (right) scattering cross sections. The shaded
regions are the parameter space which can not be tested by a neutrino detector. Figure
is taken from (229).

mχ
(GeV)
4
6
10
20
50
100
200

SD
SI
SD
SI
SD
SI
SD
SI
SD
SI
SD
SI
SD
SI

upper

lower

19%
22%
19%
23%
22%
25%
30%
30%
40%
35%
41%
40%
42%
46%

−12%
−11%
−12%
−12%
−15%
−13%
−20%
−21%
−25%
−36%
−20%
−44%
−17%
−51%

Table 8.17: Relative magnitudes of total changes in the capture rate due to uncertainties of form factor, solar model and astrophysics together are shown for several WIMP
masses, respectively for SD and SI coupling cases.

198

WIMP-proton SD Cross section[pb]Chapter 8

10DAMA/LIBRA (channeling)

1
10
10

-1

DAMA/LIBRASIMPLE

IceCube (b b )PICASSOBAKSAN (bb )-2

IceCube (W + W - )

10
10
10

-3

+

-

BAKSAN (W W )BAKSAN ( τ + τ - )

-4

τ+τ-

bb

W +W -

-5

10

10

2

WIMP mass[GeV/c 2]

Figure 8.10: 90% C. L. upper limits on SD WIMP-proton cross section calculated
for the DarkSUSY (83) default are shown for the τ + τ − channel in red solid; bb in
red dashed; W + W − in red dot-dashed with uncertainty bands to take account form
factor, solar model and velocity distribution uncertainties. Also shown are limits fromother experiments: IceCube (54) bb (brown dashed) / W + W − (brown dot-dashed);
BAKSAN (59) τ + τ − (pink solid) / bb (pink dashed) / W + W − (pink dot-dashed);
PICASSO (202) (blue solid); SIMPLE (203) (green solid). The black shaded regions
are the 3σ C.L. signal claimed by DAMA/LIBRA (62; 204).

199

WIMP-nucleon SI Cross section[pb]Chapter 8

10

10

-3

XENON10 S2-only-4

DAMA/LIBRA (channeling)
CoGeNT

10

10

10CDMSlite
DAMA/LIBRA-5CRESSTII-6

CDMS II Si-7XENON100
LUX
+ -

τ τ

10

SuperCDMS

bb

W +W 102

WIMP mass[GeV/c 2]

Figure 8.11: 90% C. L. upper limits on the SI WIMP-nucleon cross section calculatedusing the DarkSUSY (83) default are shown for the τ + τ − channel in red solid; bb
in red dashed; W + W − in red dot-dashed with uncertainty bands to take accountform factor, solar model, form factor and velocity distribution uncertainties. Also
shown are signal claims from other experiments: DAMA/LIBRA (black shaded regions,
3σ C.L.); CoGENT (48) (magenta shaded region, 90% C.L.); CRESSTII (45) (violet
shaded regions, 2σ C.L.); CDMS II Si (46) (blue shaded region, 90% C.L.); and limits:
SuperCDMS (51) (cyan solid); CDMSlite (205) (blue dashed); XENON10 S2-only (52)(dark green dashed); XENON100 (49) (green solid); LUX (50) (brown solid).

Chapter 8

8.5

Uncertainties in particle physics

8.5.1

Overview

200

Throughout the majority of the WIMP searches including this work, we constrain
a WIMP candidate to be a single type of WIMP which explains the entire signal
detected solely, and to fulfill the observed local density 0.3 GeV/cm3 of the WIMPs
in the Milky way. Typically, no assumption is made on the WIMP mass and analyses
test several WIMP masses. Annihilation cross-section is also free from particle-physics
models under the condition that the equilibrium state of the WIMP is achieved; theuncertainty in the equilibrium condition itself will be discussed shortly. As mentioned
before, typically solar WIMP searches make an assumption of the single-annihilation
channel and test two or three channels, which are picked up to represent relatively“soft” and “hard” expected neutrino spectra.
The scattering properties of WIMPs have several uncertainties highly depending
one the particle physics model. A common way to interpret the signal usually makes
following assumptions for the scattering interaction:A) contact interaction,B) single type of interaction: SD couping only or SI couping only,C) for SD coupling case, ‘odd group assumption’ for amplitudes of Sp and Sn ,D) for SI coupling case, isospin invariance for amplitudes of σp and σn .
In the literature, there have been discussions on long-ranged interactions of dark
matter with nuclei, in elastic and inelastic scatterings. To some of them a neutrino
detector is expected to have a strong sensitivity but in this dissertation, we restraint
our discussion in contact interaction.
The assumption B) is left as a pure assumption, that we interpret the signal as if
it is from pure SD or pure SI coupling.
The SD capture in the Sun is effectively governed by hydrogen, therefore we neglect
the contribution from heavier elements. This allows the situation to be extremely
simple compared to direct detection experiments using heavy elements in two ways;
first, for heavier nuclei than hydrogen, in many cases the “odd-group” model is used,which assumes that total nuclear spin is carried by either protons or neutrons, whichever
are more unpaired. So only one of either ap < Sp(N ) > or ap < Sn(N ) > in the WIMPnucleon scattering cross-section at zero momentum transfer (1.5.2) is assumed to be
non zero. However, for the SD scattering of WIMP to hydrogen, an < Sn(N ) > isnaturally not needed to be considered. Secondly we don’t have to take the decoherence
into account by introducing a nuclear form factor suppression, as mentioned in theprevious chapter. As a result, for the solar WIMP search, we can set upper limit on
SD WIMP-proton cross-section without assumption on C).

Chapter 8

8.5.2

201

Isospin violation

WIMP-nucleon SI Cross section[pb]When the SI interaction is assumed, the further assumption D) that the fp /fn ratio
to be 1, or “isospin conservation”, is typically made. Although fp /fn is predicted to bevery close to 1 in most of the WIMP models found in MSSM, it is not guaranteed in
more general scenarios such as sneutrinos found in nMSSM. In these cases, an additional
assumption on the ratio between fp and fn is needed to be made. As mentioned in Chap.
1, the phenomenologically motivated value fp /fn = −0.7 has interesting sensitivity fora neutrino detector. Thus, in addition to the conventional interpretation to isospin
conserving model, we show our result on isospin-violating model as shown in Fig. 8.12.

10

10

10

10

-1

XENON100
CDMS_Ge
CoGeNT 8.4kg-days 2008, WIMP-nucleon
DAMA/LIBRA 2008, no ion channeling
CRESST II 730kg-days 2011
CDMS_Si_c58 2013, 90%

-2

-3

-4

+ −

SKI-IV, bb-bar / τ τ channel
10

-5

5

6

7

8

9 10

20

WIMP mass(GeV)

Figure 8.12: 90% upper limits on the SI IVDM-proton cross-section are shown forthe τ + τ − channel in red solid; bb in red dashed. Also shown are limits from otherexperiments: XENON100 (green solid); CDMS-Si (dark blue solid); CDMS-Ge (darkblue shaded region); CoGENT (pink shaded region); CRESSTII (yellow shaded region);
DAMA/LIBRA (light blue shaded region).

Chapter 8

8.5.3

202

Equilibrium condition

Given the age of the Sun, it is frequently assumed that the capture rate and theannihilation rate of WIMPs have been equilibrated. As discussed in Chap. 2, the instant thermalization assumption can be more easily achieved for low mass WIMPs withhigh scattering cross-section, and the relation of annihilation cross section to scatteringcross section is well-defined but model dependent. With the typically assumed value
3 ×10−26 cm3 s−1 velocity-independent s-wave annihilation cross section, the capture
rate of 1025 s−1 requires <σ > 10−30 cm3 s−1 (65) and in this case equilibrium could
safely be reached.
However, as several assumptions are involved in this scenario, for reliable usage ofresults from solar WIMP searches by neutrino experiments, the equilibrium conditionneeds to be evaluated for specific WIMP scenarios in individual studies (219; 66; 76).
This has been done for example for cMSSM models (219; 66) and pMSSM models (76),and other scenarios in general can be scanned in future.

8.6

Summary“Indirect” detection methods are in general regarded to suffer from uncertainties inbackgrounds, astrophysics (effects are generally larger than direct detection) and propagation of annihilation products to detector. However, the detection of solar WIMP
neutrino is not the case.
The backgrounds for WIMP search in neutrino detectors are in general atmospheric
neutrinos, which are well understood enough to precisely constraint neutrino oscillation
parameters as shown in Chap. 5. Also the propagation of the neutrinos produced inside
the Sun to detector is straightforward (88). The detector response to both signal andthe back-ground neutrinos are well studied as well.
A neutrino detector for solar WIMP search especially has a good sensitivity tospin-dependent coupling WIMP search. In this case, the nuclear physics- and particle
physics-related uncertainties in the scattering process are negligible due to the simple
structure of the hydrogen. Also the solar model is not a source for the uncertainty of
WIMP-proton SD coupling as its uncertainty is more relevant to the distribution of theheavy nuclei inside the Sun. In a scenario of isospin-violating dark matter, the relative
limit from a neutrino detector gets stronger as it gets weaker suppression, compared
to direct detection experiments.
The dark matter phase distribution is regarded to be a large source of uncertainty
for WIMP searches. However, the uncertainty in the local density will affect the indirect
detection in the same size of the direct detection, and the result of the study of theuncertainties from the velocity distribution shows a mild change as the capture isgiven by an integral over wide range of the velocity distribution. Also because thecapture process is more efficient from low-velocity region, it is insensitive to the criticaluncertainty in high-velocity tail of the dark matter halo. Possibly existing co-rotating
structure with the Sun can largely boost the signal and hence make the interpretation

Chapter 8of solar WIMP search more conservative.

203

Chapter 9
Conclusion
Light WIMPs annihilating in the Sun were searched using the Super-Kamiokande
I-IV 3903-days data. To increase signal acceptance for a few-GeV to a few tens of GeV
WIMPs, the analysis used vertex-contained neutrino events in addition to neutrinoinduced up-going muon events. Also νe , ν e , ντ and ν τ fluxes were used as well asνµ , ν µ fluxes. As a result, the signal acceptance has increased 47 times comparedto the previous analysis which used up-going muons only for example for 10-GeV
WIMP annihilating to bb. To discriminate the overwhelming atmospheric neutrino
back-ground events which increase as ∝ E −2.7 at low energy, we performed a leastsquares fit which makes full use of the energy, angle and flavor information. Under
the assumption that SK data is composed of events from atmospheric neutrinos and
WIMP neutrinos, we fit data to find the best contribution of WIMP-induced neutrino
events.
As a result, we found no significant excess for 4–200-GeV WIMP hypotheses andthe result is interpreted as upper limits on the WIMP-nucleon elastic scattering crosssections. The derived upper limit on the SD WIMP-proton cross section shows themost stringent constraint to date for WIMP masses below 200 GeV even with theconservative assumption of the bb̄ annihilation channel. In the case that the branch of
WIMP annihilation into τ + τ − is 100%, we exclude the entire DAMA/CoGeNT-claimed
signal regions and most parts of the CDMS signal region. This result is consistent
with the recent null results from XENON100, LUX and SuperCDMS. We also exclude
new region for WIMP masses below 6 GeV with τ + τ − channel. If it is the case that
WIMP violates isospin invariance and experiences destructive interference, our mostconservative channel bb̄ also becomes inconsistent with the CDMS Si signal.
We investigated how robust the limits were against astrophysical uncertainties using
DarkSUSY simulation code. We have performed a detailed study of the effect of thevelocity distribution of dark matter on capture rate in the Sun, therefore effect onindirect WIMP search using the neutrino flux from the Sun. We examined four sourcesof uncertainties: orbital speed of the Sun, escape velocity of dark matter from the halo,dark matter velocity distribution functions and the existence of a dark disc.
We found that even extreme cases currently discussed do not decrease the sensitivity
204

Chapter 9

205of indirect detection significantly, because the capture is achieved over a broad range ofthe velocity distribution. The effect of the uncertainty in the high-velocity tail of dark
matter halo is very marginal as the capture process is rather inefficient in this region.
For a WIMP mass of 20 GeV, we found the uncertainties for SD (SI) coupling case areabout 18 (14)% from local circular speed, 1 (1)% from high-velocity cut for the tested
range, and 16 (10)% from the dark matter halos taken from cosmological simulations.
The total sum of the uncertainties was found to be less then 24% (17%) for 20 GeV
SD (SI) coupling WIMPs in the case absence of the dark disc. For heavier WIMPs,
the uncertainty was larger but still below 50%; for SD (SI) coupling, they are 45 (32)%from local circular speed, 1 (1)% from high-velocity cut, 11 (16)% from deviation of the
VDF from Maxwellian, in the case of 500 GeV WIMP. We concluded that the overall
uncertainties from velocity distribution in solar WIMP searches were moderate. With
the dark matter disc structure expected from ΛCDM cosmology, the impact could besignificant as the low-velocity population, which is easy to be captured, is enriched.
The signal could be enhanced as much as 6 (15) times for 100 GeV SI (SD) coupling
WIMPs. However, the impact of a dark disc on direct detections, which had beenpreviously shown, is expected to be milder. Hence the possible existence of a dark discwould only change the interpretation of the limits from the solar WIMP searches on
the conservative side.
We also studied how the changes made by various source of uncertainties on thelimits of solar WIMP search differ from that of direct detection. We found that thesolar WIMP search and direct detection would have uncorrelated sensitivities to thesources of uncertainties in the tail of the VDF, and anti-correlated responses toward
the uncertainties which shift the overall WIMP population. We confirmed that the solar neutrino search for light WIMP masses could be a good complementary method tobreak degeneracies among scattering cross section, WIMP mass and various astrophysical uncertainties that could significantly alter the interpretation of direct detection
experiments. For high mass WIMP scenarios, sensitivity of solar WIMP searches tothe low-velocity region can be exploited to understand properties such as the dark disccomponent to which direct detection experiments are rather insensitive.
For the first time the astrophysical uncertainties including the conclusion of thestudy of the velocity distribution was reflected on the derivation of upper limit in solar
WIMP search. The upper limit on SD and SI WIMP-nucleon scattering cross sections
obtained in this work carry an uncertainty of about 30% for 4 GeV WIMP, which
increases for higher WIMP masses and resulting in up to 100% for 200 GeV WIMP.
The new limits on cross sections by the Super-Kamiokande is shown together withthe astrophysical uncertainty bands, which is relevant for precise comparison to direct
detection experiment limits.
By quantitative study of uncertainties, it is possible to compare solar WIMP neutrino searches with direct detection. In terms of not only good sensitivity for signal but
also unique sensitivities and insensitivities to astrophysical properties, the usefulness of
this result and searches from future neutrino detectors such as the Hyper-Kamiokande
and the PINGU upgrade are highlighted.

Chapter 9

206

The uncertainties which are expected to decrease the sensitivity, for example theuncertainty of the Q2 spectrum in deep inelastic scattering, could be studied more in
future to improve the sensitivity. Also generating signal MC with larger statistics, orin different strategy, will improve the sensitivity. However, the sensitivity is governedby the statistical uncertainty as shown in Chap. 7.5. The future improvement ofsensitivity to detection of WIMPs can be obtained by increasing statistics of data.
With five-years data from the next generation neutrino detector Hyper-Kamiokande,
which has about 25 times larger fiducial volume compared to the Super-Kamiokande,
the sensitivity to ∼ 50 GeV WIMPs annihilating to bb̄ channel (20 GeV WIMPs forτ + τ − channel) will be improved by a factor 3.5 compared to this work. For heavier
WIMPs, the 18 times enlarged effective area compared to the SK will give about 3
times improvement compared to this work.
In direct detection experiments, the sensitivity to WIMP mass is degenerated withthat to the cross section. However, with the new SK WIMP analysis method, it ispossible to distinguish the signals coming from WIMPs with different masses by theirenergy spectra. The SK, as well as future neutrino detectors sensitive to differential
energy spectrum of WIMP neutrinos, will have power to determine the WIMP massby performing a two-dimensional χ2 test instead of making an assumption on WIMP
mass. It will be so in particular for light WIMP scenario of WIMP mass below about
100 GeV, where the signal events are well accepted by contained event categories andspread over many energy bins.
The mass determination by solar WIMP search for light WIMP scenarios is not only
expected to be powerful, but also important in terms of astrophysical uncertainties. In
the light WIMP mass region, the uncertainty in the high-velocity tail of the velocity
distribution of WIMPs will significantly affect the event rate of direct detection experiments, therefore entangling together with the mass and cross sections of WIMPs.
However, solar WIMP search is affected less than 1% by the high-velocity tail. In caseof multiple observations by direct detection experiments and solar WIMP searches, thesolar WIMP neutrino detection will take important role in revealing the properties of
WIMPs.

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