An Analysis of the Oscillation of
Atmospheric Neutrinos
byShimpei Tobayama

B.Sc., Keio University, 2010

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
in
The Faculty of Graduate and Postdoctoral Studies(Physics)

THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)December 2016c Shimpei Tobayama 2016⃝

Abstract
This thesis presents an analysis of the oscillation of atmospheric neutrinos
observed in Super-Kamiokande, a large underground water Cherenkov detector in Japan. The observed atmospheric neutrino events are reconstructed
and selected using a newly developed maximum likelihood event reconstruction algorithm, and a Markov chain Monte Carlo technique is employed
to present the results on neutrino oscillation parameters as marginalized
Bayesian posterior probabilities.
The result of analyzing the SK-IV data of 2520 days exposure shows apreference for normal mass hierarchy with the posterior probability of 85.9%,
and the mode and the 68% credible interval of each oscillation parameter’s
marginalized 1D posterior probability distribution for normal hierarchy are+0.17−3 eV2 .
2
sin2 θ23 = 0.606+0.044−0.118 and ∆m32 = 2.13−0.38 × 10iiPreface
This thesis is ultimately based on the work by the past and present membersof the Super-Kamiokande(SK) and T2K collaborations, which are both large
international collaborations comprising hundreds of members. My specificcontributions are below.
For the Photosensor Test Facility described in Section 3.7, I performed
the early-stage designing of the magnetic field compensation system using
finite element analysis, and also designed, constructed and tested the waterproof optical head unit in collaboration with Philip Lu and Chapman Lim
at TRIUMF.
For the atmospheric neutrino simulation in Chapter 4, using the latest simulation softwares developed by the SK and T2K collaborations, I
produced the atmospheric neutrino MC dataset for the SK collaborationand performed extensive validation, making detailed comparisons to the oldsimulation to understand its behaviour. This MC dataset was used for theatmospheric neutrino analysis presented in Chapter 8.
I developed the SK event reconstruction algorithm described in Chapter 6 in collaboration with the T2K collaborators in Canada and US. Thebase software framework was developed by myself and Michael Wilking (formerly at TRIUMF, now at Stony Brook U.), who also developed and optimized the π 0 reconstruction in Section 6.9. The time window algorithm
described in Section 6.6.2 was developed by Patrick de Perio (formerly at
U. Toronto, now at Columbia U.) and Andrew Missert at CU Boulder.
The T2K νe appearance analysis described in Chapter 7 has been published by the T2K collaboration as: K. Abe et al., “Observation of Electron Neutrino Appearance in a Muon Neutrino Beam”, Phys. Rev. Lett.,
112:061802, 2014. I performed the validation studies for the far detector(SK) data in addition to my involvement in the development of the new π 0
rejection method using the event reconstruction mentioned above.
The SK atmospheric neutrino oscillation analysis presented in Chapter 8
was performed largely by myself. I studied event selection methods using
the new event reconstruction mentioned above. Systematic uncertainties onflux and cross section described in Section 8.2 are mostly inherited from theiii

Preface
existing atmospheric neutrino analysis framework at SK, while I introduced
new methods to propagate FSI uncertainties as described in Section 8.2.3.
Uncertainties on detector response and reconstruction was evaluated by thecollaborators in the SK energy scale working group which I was part of,and main contributions were made by Miao Jiang at Kyoto U. and Yusuke
Suda at U. Tokyo. Based on the basic Markov chain Monte Carlo (MCMC)
analysis software developed in T2K, I developed an analysis framework to
perform Bayesian oscillation analysis of atmospheric neutrinos using MCMC
and performed sensitivity studies and the final data analysis. This is thefirst atmospheric neutrino analysis at SK which uses MCMC.

iv

Table of Contents
Abstract. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ii

Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iii

Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

List of Tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 The Standard Model and Neutrinos . . . . . . . . . . . . . .
1.2 Neutrino Oscillations . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Neutrino Mass and Mixing . . . . . . . . . . . . . . .
1.2.2 Neutrino Oscillations in Vacuum . . . . . . . . . . . .
1.2.3 CP Violation in Neutrino Oscillations . . . . . . . . .
1.2.4 Neutrino Oscillations in Matter . . . . . . . . . . . .
1.3 Experimental Status . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Solar and Reactor Neutrinos . . . . . . . . . . . . . .
1.3.2 Atmospheric and Accelerator Neutrinos . . . . . . . .
1.3.3 Measurement of θ13 . . . . . . . . . . . . . . . . . . .
1.4 Unresolved Issues . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Probing the Unknowns with Atmospheric Neutrinos . . . . .
1.6 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
3
3
4
6
6
9
9
11
14
15
16
19

2 Super-Kamiokande . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Detector Overview . . . . . . . . . . . . . . . . . . . . . . . .
2.2 A Water Cherenkov Detector . . . . . . . . . . . . . . . . . .
2.3 Detector Phases and SK-IV . . . . . . . . . . . . . . . . . . .
2.4 Inner Detector . . . . . . . . . . . . . . . . . . . . . . . . . .

20
20
22
23
24

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Table of Contents
2.5
2.6
2.7

Outer Detector . . . . . . . . . . . . . . . . . . . . . . . . . .
Water and Air Purification . . . . . . . . . . . . . . . . . . .
Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . .

25
26
26

3 Detector Calibration . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Relative PMT Gain Calibration . . . . . . . . . . . . . . . .
3.2 Absolute PMT Gain Calibration . . . . . . . . . . . . . . . .
3.3 PMT Quantum Eﬃciency . . . . . . . . . . . . . . . . . . . .
3.4 Relative Timing Calibration . . . . . . . . . . . . . . . . . .
3.5 Water Property Measurement . . . . . . . . . . . . . . . . .
3.6 PMT and Black Sheet Reflectivity . . . . . . . . . . . . . . .
3.7 Photosensor Test Facility . . . . . . . . . . . . . . . . . . . .

28
28
29
30
31
33
37
38

4 Atmospheric Neutrino Simulation . . . . . . . . . . . . . . .
4.1 Atmospheric Neutrino Flux . . . . . . . . . . . . . . . . . . .
4.2 Neutrino Interaction . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Elastic and Quasi-Elastic Scattering . . . . . . . . . .
4.2.2 Meson Exchange Current . . . . . . . . . . . . . . . .
4.2.3 Single Meson Production . . . . . . . . . . . . . . . .
4.2.4 Deep Inelastic Scattering . . . . . . . . . . . . . . . .
4.2.5 Final State Interaction . . . . . . . . . . . . . . . . .
4.3 Detector Simulation . . . . . . . . . . . . . . . . . . . . . . .

43
43
45
46
47
48
48
49
50

5 Data Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 First and Second Reduction . . . . . . . . . . . . . . . . . .
5.2 Third Reduction . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Fourth Reduction . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Fifth Reduction . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Final FC Selection . . . . . . . . . . . . . . . . . . . . . . . .

52
52
53
54
54
55

6 Event Reconstruction . . . . . . . . . . . . . . . . . . . . . . .
6.1 Likelihood Function . . . . . . . . . . . . . . . . . . . . . . .
6.2 Predicted Charge . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Predicted Charge from Direct Light . . . . . . . . . .
6.2.2 Cherenkov Emission Profile . . . . . . . . . . . . . . .
6.2.3 Solid Angle Factor . . . . . . . . . . . . . . . . . . . .
6.2.4 Light Transmission Factor . . . . . . . . . . . . . . .
6.2.5 PMT Angular Acceptance . . . . . . . . . . . . . . .
6.2.6 Predicted Charge from Indirect Light . . . . . . . . .
6.2.7 Parabolic Approximation . . . . . . . . . . . . . . . .

56
56
57
57
58
60
61
61
61
64

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Table of Contents
6.3
6.4

Unhit Probability and Charge Likelihood . . . . . . . . . . . 66
Time Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.4.1 Direct Light Time Likelihood . . . . . . . . . . . . . . 67
6.4.2 Indirect Light Time Likelihood . . . . . . . . . . . . . 69
6.4.3 Merging Direct and Indirect Light Time Likelihoods . 69
6.5 Vertex Pre-fit . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.6 Subevent Algorithm . . . . . . . . . . . . . . . . . . . . . . . 75
6.6.1 Peak Finder . . . . . . . . . . . . . . . . . . . . . . . 75
6.6.2 Defining Time Windows and Final Subevents . . . . . 77
6.6.3 Performance of the Subevent Algorithm . . . . . . . . 78
6.7 Single-Ring Fit . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.7.1 Single-Ring Electron & Muon Fit . . . . . . . . . . . 79
6.7.2 Performance of the Single-Ring e & µ Fitter . . . . . 79
6.7.3 e/µ Particle Identification . . . . . . . . . . . . . . . 80
6.7.4 In-Gate Decay Electron Fit . . . . . . . . . . . . . . . 85
6.8 Upstream-Track π + Fit . . . . . . . . . . . . . . . . . . . . . 87
6.8.1 µ/π + Identification . . . . . . . . . . . . . . . . . . . 88
6.9 π 0 Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.9.1 e/π 0 Identification . . . . . . . . . . . . . . . . . . . . 91
6.10 Multi-Ring Fit . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.10.1 Initial Multi-Ring Fit . . . . . . . . . . . . . . . . . . 92
6.10.2 Sequential Multi-Ring Fit . . . . . . . . . . . . . . . . 95
6.10.3 Performance of the Multi-Ring Fitter . . . . . . . . . 96
6.11 fiTQun in SK-I to III . . . . . . . . . . . . . . . . . . . . . . 102
7 First Application of fiTQun: T2K νe Appearance . . . . . 105
7.1 T2K Experiment . . . . . . . . . . . . . . . . . . . . . . . . . 105
7.2 νe Appearance Analysis . . . . . . . . . . . . . . . . . . . . . 108
7.3 νe Event Selection . . . . . . . . . . . . . . . . . . . . . . . . 110
7.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
8 Atmospheric Neutrino Oscillation Analysis . . . . . . . . . 116
8.1 Event Selection and Binning . . . . . . . . . . . . . . . . . . 116
8.1.1 Selection Criteria . . . . . . . . . . . . . . . . . . . . 117
8.1.2 ν/ν̄ Separation for Single-Ring Events . . . . . . . . . 118
8.1.3 Multi-Ring νe /ν̄e Selection . . . . . . . . . . . . . . . 119
8.1.4 Sample Statistics and Purities . . . . . . . . . . . . . 123
8.1.5 Binning . . . . . . . . . . . . . . . . . . . . . . . . . . 123
8.1.6 Observing the Oscillation Eﬀects . . . . . . . . . . . . 127
8.2 Systematic Uncertainties . . . . . . . . . . . . . . . . . . . . 132
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Table of Contents

8.3

8.4

8.5
8.6

8.2.1 Atmospheric Neutrino Flux . . . . . . . . . . . . . . . 132
8.2.2 Neutrino Interaction . . . . . . . . . . . . . . . . . . . 134
8.2.3 Final State Interaction . . . . . . . . . . . . . . . . . 135
8.2.4 Detector Response and Reconstruction . . . . . . . . 137
Event Rate Prediction . . . . . . . . . . . . . . . . . . . . . . 139
8.3.1 Oscillation Weighting . . . . . . . . . . . . . . . . . . 139
8.3.2 Propagating Variations of Systematics . . . . . . . . . 140
Bayesian Analysis using Markov Chain Monte Carlo . . . . . 142
8.4.1 Likelihood Function . . . . . . . . . . . . . . . . . . . 143
8.4.2 Bayesian Posterior Probability . . . . . . . . . . . . . 143
8.4.3 Marginalization and Parameter Estimation . . . . . . 144
8.4.4 Markov Chain Monte Carlo . . . . . . . . . . . . . . . 145
Sensitivity Studies . . . . . . . . . . . . . . . . . . . . . . . . 147
Data Analysis Results . . . . . . . . . . . . . . . . . . . . . . 154
8.6.1 Posterior Distributions . . . . . . . . . . . . . . . . . 155
8.6.2 Zenith Angle Distributions . . . . . . . . . . . . . . . 160
8.6.3 Goodness of Fit . . . . . . . . . . . . . . . . . . . . . 162
8.6.4 Result Comparison to Other Experiments . . . . . . . 162

9 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . 168
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Appendices
A List of Systematic Parameters . . . . . . . . . . . . . . . . . . 180
B Atmospheric Neutrino Selection Improvement by fiTQun

182

viii

List of Tables
1.1

Fermions in the Standard Model . . . . . . . . . . . . . . . .

1

2.1

Definition of the event triggers . . . . . . . . . . . . . . . . .

27

6.1

Decay electron detection eﬃciencies for stopping cosmic-ray
muons for data and MC . . . . . . . . . . . . . . . . . . . . .

78

7.1
7.2

Assumed values for the oscillation parameters . . . . . . . . . 111
Expected numbers of signal and background events passing
each selection stage . . . . . . . . . . . . . . . . . . . . . . . . 112

8.1

Interaction mode breakdown and the total event rates for each
event sample . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Default values for the oscillation parameters which are used
for event rate calculation . . . . . . . . . . . . . . . . . . . . . 128
True oscillation parameters for the Asimov dataset production 147
Priors for the oscillation parameters . . . . . . . . . . . . . . 148
The 1D posterior mode and the 68% HPD credible intervals . 155
Posterior probabilities for the mass hierarchy and the θ23 octant160
Oscillation parameters which are used to calculate the event
rate predictions shown in Figures 8.26 to 8.28. . . . . . . . . 162

8.2
8.3
8.4
8.5
8.6
8.7

9.1

Expected posterior probabilities for for favouring the correct
mass hierarchy for diﬀerent SK exposure . . . . . . . . . . . . 169

A.1 List of systematic parameters. Continues to Table A.2. . . . . 180
A.2 List of systematic parameters. Continues from Table A.1. . . 181
B.1 Interaction mode breakdown and the total MC event rates foreach event sample categorized by the APFIT-based selection

183

ix

List of Figures
1.1
1.2
1.3
1.4
1.5
1.6

2.1
2.2
2.3
2.4
2.5
2.6

3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8

Examples of weak interaction processes involving neutrinos .
Coherent scattering of neutrinos in matter . . . . . . . . . . .
8 B solar neutrino flux measured by SNO through diﬀerent
channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Zenith angle distributions observed at Super-Kamiokande . .
The ordering of the neutrino mass . . . . . . . . . . . . . . .
The oscillation probability P (νµ → νe ) and the ratio of oscillated to unoscillated νe flux for atmospheric neutrinos reaching the Super-Kamiokande detector . . . . . . . . . . . . . . .
A sketch of the Super-Kamiokande detector . . . . . . . . . .
The supporting structure and the PMTs . . . . . . . . . . . .
A schematic drawing of Cherenkov radiation . . . . . . . . . .
An event display of a single electron event . . . . . . . . . . .
A schematic drawing of the Hamamatsu 20-inch PMT . . . .
Quantum eﬃciency of the photocathode as a function of wavelength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The observed charge distribution for single p.e. hits obtained
from the nickel source calibration data . . . . . . . . . . . . .
A schematic view of the timing calibration system . . . . . .
The TQ distribution for an ID PMT . . . . . . . . . . . . . .
The timing distribution of ID PMTs at a charge bin ∼1 p.e. .
A schematic view of the laser injector system for measuring
the water property and the reflectivity of the PMTs . . . . .
The TOF-subtracted hit time distributions for the laser injector data taken at 405 nm . . . . . . . . . . . . . . . . . . .
Light absorption and scattering coeﬃcients as a function of
wavelength . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Time variation of the light absorption and scattering coeﬃcients measured at various wavelengths . . . . . . . . . . . . .

3
6
10
13
16

17
21
21
22
23
24
25

30
31
32
33
34
35
36
37

x

List of Figures
3.9

A schematic view of the apparatus for black sheet reflectivity
measurement . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.10 A schematic drawing of the PTF and the external view of the
PTF magnetic field compensation system . . . . . . . . . . .
3.11 A SK PMT submerged in the tank filled with water . . . . .
3.12 The interior of the water-tight optical head box . . . . . . . .
3.13 Local variations of the gain of a SK PMT . . . . . . . . . . .
4.1
4.2
4.3
4.4
4.5

Zenith angle distributions of the atmospheric neutrino flux at
Super-K averaged over the azimuth . . . . . . . . . . . . . . .
The direction-averaged atmospheric neutrino flux at Super-K
as a function of energy . . . . . . . . . . . . . . . . . . . . . .
νµ and ν̄µ CC cross sections per nucleon divided by neutrinoenergy, plotted as a function of energy . . . . . . . . . . . . .
π + -12 C scattering cross sections . . . . . . . . . . . . . . . . .
Low momentum π + absorption cross sections compared between experimental data and the variations of the NEUT pioncascade model . . . . . . . . . . . . . . . . . . . . . . . . . . .

38
39
40
41
42

44
45
47
50

51

6.1

Schematic diagram describing the variables relevant to predicted charge calculation . . . . . . . . . . . . . . . . . . . . . 58
6.2 Cherenkov emission profile g(p, s, cos θ) for electrons . . . . . 59
6.3 Cherenkov emission profile g(p, s, cos θ) for muons . . . . . . . 60
6.4 Angular acceptance of the PMT . . . . . . . . . . . . . . . . 62
6.5 A schematic diagram describing the variables relevant to thescattering table . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.6 The photon acceptance factor J(s) overlaid with an approximating parabola . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.7 The unhit probability P (unhit|µ) with and without the correction for the PMT threshold eﬀect . . . . . . . . . . . . . . 66
6.8 The normalized charge likelihood fq (q|µ) . . . . . . . . . . . . 68
6.9 The direct light residual time likelihood ftdir (tres ) for 300 MeV/celectrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.10 The direct light residual time likelihood ftdir (tres ) for 2000 MeV/celectrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.11 The direct light residual time likelihood ftdir (tres ) for 450 MeV/cmuons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6.12 The direct light residual time likelihood ftdir (tres ) for 2000 MeV/cmuons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

xi

List of Figures
6.13 Distribution of the scanned goodness G(x, t) as a function oftime t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.14 Single-ring electron and muon vertex resolution for FC truefiducial CCQE events in atmospheric neutrino MC, compared
between APFIT and fiTQun . . . . . . . . . . . . . . . . . . .
6.15 Single-ring electron and muon vertex resolution plotted asa function of true momentum, for FC true-fiducial CCQE
events in atmospheric neutrino MC, compared between APFIT and fiTQun . . . . . . . . . . . . . . . . . . . . . . . . .
6.16 Single-ring electron and muon direction resolution for FC
true-fiducial CCQE events in atmospheric neutrino MC, compared between APFIT and fiTQun . . . . . . . . . . . . . . .
6.17 Single-ring electron and muon direction resolution plotted asa function of true momentum, for FC true-fiducial CCQE
events in atmospheric neutrino MC, compared between APFIT and fiTQun . . . . . . . . . . . . . . . . . . . . . . . . .
6.18 Single-ring electron and muon momentum resolution plotted
as a function of true momentum, for FC true-fiducial CCQE
events in atmospheric neutrino MC, compared between APFIT and fiTQun . . . . . . . . . . . . . . . . . . . . . . . . .
6.19 Event displays of simulated single electron and muon events .
6.20 Likelihood separation of single-ring electron and muon eventsin the FC true-fiducial CCQE event sample in the atmospheric neutrino MC . . . . . . . . . . . . . . . . . . . . . . .
6.21 Misidentification rate of single-ring electron and muon eventsin the FC true-fiducial CCQE event sample in the atmospheric neutrino MC, plotted as a function of true momentum
6.22 Event displays of an MC event which has an in-gate decay
electron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.23 Charge distribution for hits that are associated with each ingate subevent, for the same event which was shown in Figure 6.22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.24 A schematic diagram of a charged pion which undergoes ahadronic scattering on a nucleus and the rings it produces,
and an event display of a simulated single π + event . . . . . .
6.25 A schematic diagram describing how an upstream-track hypothesis is constructed in fiTQun . . . . . . . . . . . . . . . .
6.26 Likelihood separation of νµ CCQE and NCπ + events . . . . .
6.27 A schematic diagram showing how the π 0 hypothesis is constructed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

81

81

82

82

83
83

84

84
86

86

87
88
89
90
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List of Figures
6.28 CC single electron and NC single π 0 events in the FC truefiducial atmospheric neutrino MC sample, separated using theπ 0 fit variables . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.29 Misidentification rate for the π 0 rejection cut Equation 6.28,
for the CC single electron and NC single π 0 events in the FCtrue-fiducial atmospheric neutrino MC sample, plotted as afunction of the true particle momentum . . . . . . . . . . . . 92
6.30 A tree diagram showing how the tree of the multi-ring hypotheses evolve as the number of rings is increased . . . . . . 93
6.31 Distribution of the square root of the log likelihood ratio between the best-fit single-ring electron hypothesis and the 2R
hypothesis assuming electron as the first ring, for FC sub-GeV
atmospheric neutrino MC events with no decay electrons detected . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.32 Reconstructed event categories of νe CC events with no visible
pions in final state in the FC atmospheric neutrino MC forfiTQun and APFIT . . . . . . . . . . . . . . . . . . . . . . . . 97
6.33 Reconstructed event categories of NC events with a single
π 0 and no visible π ± in final state in the FC atmospheric
neutrino MC for fiTQun and APFIT . . . . . . . . . . . . . . 97
6.34 Reconstructed event categories of νe CC events with a single
π 0 and no visible π ± in final state in the FC atmospheric
neutrino MC for fiTQun and APFIT . . . . . . . . . . . . . . 98
6.35 Reconstructed event categories of νµ CC events with no visible
pions in final state in the FC atmospheric neutrino MC forfiTQun and APFIT . . . . . . . . . . . . . . . . . . . . . . . . 99
6.36 Reconstructed event categories of νµ CC events with a single
π 0 and no visible π ± in final state in the FC atmospheric
neutrino MC for fiTQun and APFIT . . . . . . . . . . . . . . 99
6.37 Reconstructed invariant mass calculated using the 2nd andthe 3rd rings for events reconstructed as 3e ring by fiTQun
and APFIT in the FC atmospheric neutrino MC . . . . . . . 100
6.38 Reconstructed invariant mass calculated using the 2nd andthe 3rd rings for events reconstructed as 1µ + 2e ring byfiTQun and APFIT in the FC atmospheric neutrino MC . . . 101
6.39 Distributions of the log likelihood ratio between the fiTQun
single-ring electron and muon fits for the FC sub-GeV singlering atmospheric neutrino events in each SK phase . . . . . . 103
6.40 Time variation of the light attenuation length in water . . . . 104

xiii

List of Figures
7.1
7.2
7.3
7.4
7.5
7.6
7.7

Schematic drawing of the T2K beam production site . . . . . 105
T2K νµ beam spectrum at diﬀerent oﬀ-axis angles and the νµsurvival probability at SK . . . . . . . . . . . . . . . . . . . . 106
A schematic drawing of ND280 . . . . . . . . . . . . . . . . . 107
Predicted T2K neutrino flux at SK . . . . . . . . . . . . . . . 108
π 0 rejection cut which was used in the previous νe appearance
analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
2D distribution of the fiTQun π 0 cut variables after the Eνrec <1250MeV cut, for the dataset shown in Table 7.2 . . . . . . . 114
The best-fit (for normal hierarchy) pe −θe distribution plottedwith the observed data . . . . . . . . . . . . . . . . . . . . . . 115

8.1

Distribution of the variables which are used for the Multi-GeV
Multi-ring E-like(MME) likelihood selection . . . . . . . . . . 121
8.2 Distribution of the sum of ln(Lsig /Lbkg ) for all four MME
selection variables . . . . . . . . . . . . . . . . . . . . . . . . 122
8.3 Distribution of the variables which are used for the multi-ringνe /ν̄e likelihood separation . . . . . . . . . . . . . . . . . . . . 124
8.4 Distribution of the sum of ln(Lνe /Lν̄e ) for all three νe /ν̄e separation variables . . . . . . . . . . . . . . . . . . . . . . . . . 125
8.5 Binning for each event sample . . . . . . . . . . . . . . . . . . 127
8.6 The eﬀect of sin2 θ23 variations on the distributions of thecosine of the zenith angle for sub-GeV samples . . . . . . . . 129
8.7 The eﬀect of sin2 θ23 variations on the distributions of thecosine of the zenith angle for Multi-GeV samples . . . . . . . 129
8.8 The eﬀect of δCP variations on the distributions of the cosineof the zenith angle for Sub-GeV samples . . . . . . . . . . . . 130
8.9 The eﬀect of the mass hierarchy on the distributions of thecosine of the zenith angle for Multi-GeV e-like samples . . . . 131
8.10 Uncertainty on the flux normalization√
as a function of energy 132
8.11 The fractional error matrix sgn(Vij ) × |Vij | for the sub-GeV
samples . . . . . . . . . . . . . . . . . √. . . . . . . . . . . . . 136
8.12 The fractional error matrix sgn(Vij ) × |Vij | for the multiGeV samples . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
8.13 Reconstructed vertex Z position of stopping cosmic-ray muons
entering from the top of the detector . . . . . . . . . . . . . . 138
8.14 The response coeﬃcient fij for the MAQE systematic for theνe CC events in each bin in the sub-GeV one-ring e-like 0decay
sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

xiv

List of Figures
8.15 ∆m232 and − ln P (⃗o, f⃗|D), the negative-log of the posterior
probability, plotted against the number of steps taken in eachchain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
8.16 The marginalized 2D posterior distribution for ∆m232 andsin2 θ23 given the Asimov dataset . . . . . . . . . . . . . . . . 150
8.17 The 2D marginalized posterior distributions for sin2 θ23 andδCP given the Asimov dataset . . . . . . . . . . . . . . . . . . 152
8.18 The marginalized 1D posterior distributions for sin2 θ23 , |∆m232 |and δCP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
8.19 The mean and the σ of the marginalized 1D posterior foreach systematic parameter in the case of assuming normal
hierarchy, for the Asimov dataset . . . . . . . . . . . . . . . . 154
8.20 The marginalized 2D posterior distribution for ∆m232 andsin2 θ23 given the SK-IV FC data . . . . . . . . . . . . . . . . 156
8.21 The marginalized 2D posterior distributions for |∆m232 | andsin2 θ23 for the cases of assuming normal and inverted hierarchy157
8.22 The marginalized 2D posterior distributions for sin2 θ23 andδCP for the cases of assuming normal and inverted hierarchy . 158
8.23 The marginalized 1D posterior distributions for sin2 θ23 , |∆m232 |and δCP for the cases of assuming normal and inverted hierarchy159
8.24 The marginalized 1D posterior distributions for sin2 θ23 andδCP after marginalizing over the mass hierarchy . . . . . . . . 160
8.25 The mean and the σ of the marginalized 1D posterior foreach systematic parameter in the cases of assuming normal
and inverted hierarchy . . . . . . . . . . . . . . . . . . . . . . 161
8.26 Distributions of the cosine of the zenith angle for the sub-GeV
event samples . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
8.27 Distributions of the cosine of the zenith angle for the multiGeV e-like event samples . . . . . . . . . . . . . . . . . . . . . 164
8.28 Distributions of the cosine of the zenith angle for the multiGeV µ-like and other event samples . . . . . . . . . . . . . . 165
8.29 Distribution of the log likelihood ratio between the observed
data and fake data which is used for the goodness of fit test . 166
8.30 90% credible and confidence regions for ∆m232 and sin2 θ23 forthe results from various experiments . . . . . . . . . . . . . . 167
B.1 90% HPD credible regions for ∆m232 and sin2 θ23 compared
between the cases of using APFIT or fiTQun for event selection, on Asimov datasets . . . . . . . . . . . . . . . . . . . . . 184

xv

Acknowledgements
First and foremost, I thank my supervisor, Hiro Tanaka, for all of his support. His generosity and patience for letting me pursue my interests and theguidance throughout were essential for completing the long PhD journey.
I also thank Akira Konaka, for stimulating my interests in Super-K andatmospheric neutrinos through a number of inspiring discussions. Many
thanks to Mike Wilking, without whom the most important part of mywork, fiTQun, would have never happened. Working with Patrick de Perio
was always fun, and I thank him for showing me what it is to be a successful
graduate student and also for broadening my horizons in many aspects.
I am grateful to the members of the SKanada group: Sophie Berkman,
John Martin, Corina Nantais and Roman Tacik, and the US collaborators
including Andy Missert, Eric Zimmerman and Cris Vilela, for all the progress
made over the years in fiTQun and various other new eﬀorts at Super-K.
I also thank Scott Oser, whose deep insights in statistics helped me form
the basis of my knowledge in statistics. I thank all the past and present
members of the T2K-Canada group: not only was it a great pleasure working
with this extraordinary group of talent, the collaboration “meatings” we hadin Tokai were the highlights of my life as a graduate student.
It was a great privilege working with the collaborators on the Super-K
experiment. Special thanks to the SK E-scale team: Shunichi Mine, Yusuke
Suda and Miao Jiang, without whose help I would have not been able tofinish my analysis in such a timely manner. I also thank Roger Wendell for
helping me figure out the various details in atmospheric neutrino analysis,
and Richard Calland for the great help on MCMC.
I’d like to oﬀer my special thanks to Tomonobu Tomura and my friends
in Kamioka for making the life in Kamioka such a wonderful experience.
I am grateful to Haruka Haruno, Minami Kaido, Kirara Amanogawa
and Towa Akagi who, during the most diﬃcult times, gave me hope andenlightened the path to complete my PhD: a long dream since childhood.
Finally, I’d like to sincerely thank my parents, grandparents, sister andmy entire family. I am where I am only because of their continuous supportand encouragement. I say to them all: hontou-ni arigato-gozaimasu!
xviChapter 1

Introduction
Neutrino physics has achieved remarkable progress in the past two decades.
Once thought to be massless, neutrinos are now confirmed to have nonzero
mass by the discovery of neutrino oscillations which necessitates the revision of the established theory of the Standard Model of particle physics.
Despite the recent progress, many of the properties of neutrinos still remain
unknown, and improving our knowledge of neutrinos leads the path toward
the fundamental theories governing the universe.

1.1

The Standard Model and Neutrinos

The Standard Model of particle physics includes six quarks and six leptons
as shown in Table 1.1, which are all spin 1/2 Dirac fermions and each fermion
has a corresponding antiparticle which has opposite charge.

quarksleptons

Charge
+2/3
−1/3
−1
0

1st generation
up
down
electronelectron neutrino

2nd generation
charm
strangemuon
muon neutrino

3rd generation
topbottomtau
tau neutrino

Table 1.1: Fermions in the Standard Model.
There are three generations of quarks and each generation has two typesof quarks, one with the electric charge of +2/3 and the other with −1/3.
Similar to quarks there are three generations of leptons, and each generation
has a charged lepton with −1 electric charge and a neutrino which is electrically neutral. The six quarks and the three charged leptons are massive,
and the mass gets larger as the generation increases, for instance, a charm
quark is heavier than an up quark, and a muon is heavier than an electron.
Neutrinos are assumed to be massless in the Standard Model, and until the
discovery of neutrino oscillations which is explained in the following section,
experimental results were consistent with zero neutrino mass.

11.1. The Standard Model and Neutrinos
There are three fundamental interactions in the Standard Model which
are the strong, the electromagnetic and the weak interactions, and they are
mediated by gauge bosons which are spin 1 particles. The strong interaction
is mediated by gluons which are massless, and they only interact with quarks
and other gluons but not with leptons. The electromagnetic interaction ismediated by photons which are also massless, and they interact with allelectrically charged particles; i.e. all fermions except for neutrinos interact
electromagnetically. Finally, weak interactions occur to all fermions andare mediated by the weak bosons: W ± which have ±1 electric charge andZ 0 which is neutral. Unlike the other gauge bosons the weak bosons are
massive.
The above interactions in the Standard Model are fundamentally definedby local gauge symmetries which also require the gauge bosons to be massless, and the strong interaction is described by a SU (3) symmetry in theinteractions between the quark and the gluon fields. The electromagnetic
and the weak interactions arise from the fundamental electroweak interaction which is described by a SU (2) × U (1) symmetry. The corresponding
gauge fields interact with the Higgs field which spontaneously breaks thesymmetry and results in the massive weak bosons mediating the weak interaction and the massless photons mediating the electromagnetic interaction.
The symmetry breaking caused by the Higgs field also gives rise to the massof the quarks and the charged leptons, and introduces a massive spin 0 Higgs
boson.
As stated earlier, neutrinos interact by the weak interaction only and areinsusceptible to the electromagnetic or the strong interactions, and thus theyrarely interact with matter. The three generations, which are also called theflavours of neutrinos are labeled as νe , νµ and ντ , and each of them respectively couples via the charged current(CC) weak interaction mediated by
the W ± boson to the charged lepton of the corresponding flavour: electron, muon and tau. A CC interaction involves a neutrino and a charged
lepton of the same flavour only, or rather, neutrino flavours are defined by
the flavour of the charged lepton with which each neutrino partners via the
CC interaction. The CC interaction is responsible for various weak decay
processes involving neutrinos such as the muon decay illustrated on the leftof Figure 1.1. The interaction mediated by the Z 0 boson as on the right of
Figure 1.1 is called the neutral current(NC) weak interaction, and neutrinos
do not change their type by NC interactions.

21.2. Neutrino Oscillations
⌫¯e

Time

⌫µ

⌫e

e

W

e

Z0
⌫e

µ

e

Figure 1.1: Examples of weak interaction processes involving neutrinos. The
left diagram is the decay of a muon into an electron and neutrinos via
the charged current interaction, and the right diagram is a neutral current
electron-neutrino scattering.

1.2

Neutrino Oscillations

The neutrino mass is experimentally known to be small. The observationthat neutrinos are produced with definite helicity[1] suggests that neutrinos
are massless, and the results of any direct mass measurements to date byprecision kinematic measurement of various weak decay processes are consistent with zero mass[2]. However, the discovery of neutrino oscillations
has proven that neutrino mass is, in fact, not zero, which contradicts theassumption made in the Standard Model. This section describes the basic
theories behind neutrino oscillations.

1.2.1

Neutrino Mass and Mixing

Suppose now that neutrinos do have nonzero mass, and consider free neutrinos propagating in vacuum. Since there are three neutrino flavours there
are three orthogonal states |να ⟩ (α = e, µ, τ ) which are the eigenstates of the
weak interaction. One can also define the mass eigenstates |νi ⟩ (i = 1, 2, 3)
which are the eigenstates of the free particle Hamiltonian with the possibly
diﬀerent mass eigenvalues mi (i = 1, 2, 3), and since each of them form acomplete basis the two bases are related by a unitary matrix U :|να ⟩ =3∑∗Uαi
|νi ⟩.

(1.1)

i

A general N × N unitary matrix has N 2 real free parameters, N (N −1)/2 of which can be considered as rotation angles as in the case for areal orthogonal matrix of the same size, and the remaining N (N + 1)/23

1.2. Neutrino Oscillations
real parameters can be attributed to complex phases. Therefore, the 3 × 3
unitary matrix U can be characterized by 3 rotation(mixing) angles and6 complex phases. However, assuming that neutrinos are standard Diracfermions, one can freely define the phase of each of the flavour and themass eigenstates without changing any physical consequences, and 5 of thephase parameters can be therefore factored out from the mixing matrix asthey are physically not meaningful. Thus, the fundamental matrix U which
describes the mixing of the flavour eigenstates and the mass eigenstates has
three mixing angles and a single complex phase. The matrix is analogous
to the Cabibbo-Kobayashi-Maskawa(CKM) matrix[3] which describes thequark mixing in weak interactions, and it is known as the Pontecorvo-MakiNakagawa-Sakata(PMNS) matrix[4][5] which is usually parameterized as:




1
0
0
cos θ13
0U =  0 cos θ23 sin θ23  
0 − sin θ23 cos θ23− sin θ13 eiδCP

0 sin θ13 e−iδCP

1
0

0
cos θ13cos θ12 sin θ12 0×  − sin θ12 cos θ12 0  , (1.2)
001where θ12 , θ13 and θ23 are the mixing angles and δCP is the complex phase.
Two extra complex phases α1 and α2 , known as the Majorana phases, need
to be introduced in the mixing matrix as U → U × diag(1, eiα1 , eiα2 ) if
neutrinos are Majorana fermions as opposed to Dirac fermions, i.e. if aneutrino is its own antiparticle. In such case, one can no longer freely choose
the phase of the mass eigenstates and the corresponding degrees of freedom
must remain in the unitary matrix.

1.2.2

Neutrino Oscillations in Vacuum

Consider now the time evolution of a flavour eigenstate in Equation 1.1
as it propagates in vacuum. The transition amplitude between the flavour
eigenstates να and νβ after time t is (c, h̄ = 1 in the following discussions):⟨νβ |e−iĤt |να ⟩ =3∑∗e−iEi t Uβi Uαi
,

(1.3)

i

√

where Ĥ is the free particle Hamiltonian and Ei ≡ p2 + m2i is the energy
eigenvalue of the free particle with momentum p. It can be seen from theequation that the transition amplitude between diﬀerent flavour eigenstates41.2. Neutrino Oscillations
can be nonzero as the time elapses, and the transition probability from thestate να to νβ is:3∑

P (να → νβ )(t) = |⟨νβ |e−iĤt |να ⟩|2 =∗∗ −i(Ei −Ej )tUαi
Uβi Uαj Uβj
e
.

(1.4)

i,j

Note that the Majorana phases α1 and α2 mentioned in the previous section
do not aﬀect the transition probability at all, as the phases cancel when thematrix elements are multiplied by their complex conjugates. Thus, θ12 , θ13 ,θ23 and δCP are the only parameters in the mixing matrix which aﬀect thetransition probability, regardless of whether neutrinos are Dirac or Majorana
fermions.
In the ultra-relativistic limit of p2 ≫ m2i ,Ei − Ej =√p2 + m2j −√p2 + m2i ≈

∆m2ij
,
2E

(1.5)

where E ≈ |p| is the characteristic energy and∆m2ij ≡ m2i − m2j .

(1.6)

Using Equation 1.5 and also replacing the time t with the distance L theneutrino traveled, Equation 1.4 now reads:
(

3
∑

)

∆m2ij L∗∗Uαi
Uβi Uαj Uβj
exp −iP (να → νβ ) =
,
2E
i,j

(1.7)

and the unitarity condition of U further leads the equation to:
P (να → νβ ) = δαβ − 4

∑

[

Re

∗
∗
Uαi
Uβi Uαj Uβj

]

sin

i>j

+ 2

∑
i>j

[

Im

∗
∗
Uαi
Uβi Uαj Uβj

(

2

]

(

∆m2ij L4E))

∆m2ij Lsin
. (1.8)2E

From this equation it can be seen that neutrinos can change its flavour
as it travels in space, and since the transition probability varies periodicallyaccording to the phase ∝ L/E the phenomenon is referred to as the neutrino
oscillation. In order for the oscillation to happen, the following two conditions are required, one of which is that the mixing matrix U ̸= I, i.e. theflavour and the mass eigenstates are not identical. The other condition isthat ∆m2ij ̸= 0, i.e., at least one of three mass eigenvalues has to be diﬀerent
from the others. Thus, an observation of neutrino oscillations is an evidenceof nonzero neutrino mass.
5

1.2. Neutrino Oscillations

Time

⌫↵

p, n, e

W±

Z0
⌫↵

⌫e

e

p, n, e⌫eeFigure 1.2: Coherent scattering of neutrinos in matter. The left diagram isa NC scattering which involves all neutrinos, and the right diagram is a CC
scattering which involves electron neutrinos only. Note that such scattering
also occurs for antineutrinos, while in such case the CC scattering is described by an “annihilation-creation-type” diagram, i.e., the right diagram
rotated by 90 degrees.

1.2.3

CP Violation in Neutrino Oscillations

For antineutrinos, mixing as in Equation 1.1 is described by the complex
conjugate of the mixing matrix:|ν̄α ⟩ =3∑

Uαi |ν̄i ⟩,

(1.9)

i

and since the mixing matrix is diﬀerent between neutrinos and antineutrinos
if δCP is neither 0 nor π, δCP is referred to as the CP violating phase. In suchcase, it immediately follows that the sign of the third term in Equation 1.8
flips for antineutrinos, which suggests that the oscillation probability will
be diﬀerent between neutrinos and antineutrinos. Thus, CP violation in the
lepton sector can be probed through neutrino oscillation measurements.

1.2.4

Neutrino Oscillations in Matter

When neutrinos propagate in matter, the interactions between the neutrinos and the matter change the oscillation behaviour. The phenomenon,
which was first introduced by Wolfenstein[6] and subsequently elaboratedby Mikheev and Smirnov[7], is known as the matter eﬀect or the MSW
eﬀect of neutrino oscillations.
When neutrinos propagate in matter, all three flavours of neutrinos undergo neutral current coherent scattering with protons, neutrons and electrons as shown on the left of Figure 1.2. Such interaction introduces an6

1.2. Neutrino Oscillationseﬀective potential VNC which simply shifts the energy levels of all types ofneutrinos by a same amount. Since ordinary matter contains electrons but
not muons or taus, for electron neutrinos only the coherent charged current
scattering as on the right of Figure 1.2 also occurs in addition to the NC
scattering, and the process introduces an additional potential VCC which
only aﬀects electron neutrinos:√
VCC = ± 2GF Ne .
(1.10)In the equation, GF is the Fermi coupling constant, Ne is the number densityof electrons in the matter and the positive and the negative signs are forneutrinos and antineutrinos respectively.
In order to see the eﬀect on the oscillations, for simplicity, consider now
a case of two flavour oscillations between two flavour eigenstates νe and νx ,where νx is a non-electron state which is insusceptible to VCC and can beconsidered as some combination of νµ and ντ . The flavor states are related
to the mass eigenstates νi with energies Ei by a mixing matrix U :
(

νe
νx

)

(

=U

ν1
ν2

)

(

≡

cos θ sin θ− sin θ cos θ

)(

ν1
ν2

)

(1.11).

Let H ≡ diag(E1 , E2 ) ≡ H0 + H1 be the free particle Hamiltonian in themass basis where H0 ≡ (E1 + E2 )/2 × diag(1, 1) and
H1 ≡

∆m2
E2 − E1
× diag(−1, 1) =× diag(−1, 1),2
4E(1.12)

where Equation 1.5 was used and ∆m2 ≡ m22 − m21 . Since adding a term
proportional to the unit matrix to the Hamiltonian merely adds a common
phase to all neutrino types and does not aﬀect oscillations, H0 can be ignored, and the vacuum Schrödinger equation in the flavour basis becomes:
d
i
dt

(

νe
νx

)

(

= U H1 U
=

∆m2
4E

T

(

νe
νx

)

− cos 2θ sin 2θ
sin 2θ cos 2θ

)(

νe
νx

)

,

(1.13)

which describes the oscillation of flavour states in vacuum. Solving theequation, the transition probability is:
P (νe → νx ) = sin2 2θ sin2

∆m2 L
,
4E

(1.14)
7

1.2. Neutrino Oscillations
where the substitution t → L was done as before.
Consider now the oscillations in matter, in which case the Hamiltonian
has an additional potential term due to matter. The potential is diagonal in
the flavour basis in the form diag(VCC + VNC , VNC ), and shifting the matrix
by a multiple of unit matrix the Schrödinger equation in matter becomes:
d
i
dt

(

νe
νx

)

∆m2
4E

=

(

− cos 2θ + x
sin 2θ
sin 2θ
cos 2θ − x

)(

νe
νx

)

, (1.15)

where

2EVCC
.
(1.16)∆m2
One can then define the eﬀective mixing angle θM and the mass splitting∆m2M such that:x≡sin 2θ
,sin 2θM ≡ √sin2 2θ + (cos 2θ − x)2∆m2M ≡ ∆m2√(1.17)

sin2 2θ + (cos 2θ − x)2 ,(1.18)and Equation 1.15 now becomes:
d
i
dt

(

νe
νx

)

=

∆m2M
4E

(

− cos 2θM sin 2θM
sin 2θM cos 2θM

)(

νe
νx

)

.

(1.19)

This is of the same form as Equation 1.13 but the original mixing angle andthe mass splitting θ and ∆m2 are now replaced by the eﬀective values inmatter θM and ∆m2M . This is the matter eﬀect, and the oscillation amplitude
and the frequency are changed from the case in vacuum due to the CC
interaction with matter shifting the energy levels.
A dramatic consequence follows when:x = cos 2θ.

(1.20)

In such case, sin 2θM = 1, and the mixing becomes maximal regardless ofthe original value of the mixing angle θ. Such eﬀect is known as the MSW
resonance, and it occurs when the energy level splitting in matter ∆m2M
becomes minimal. Note that the resonance is sensitive to the sign of ∆m2
as well as whether it is a neutrino(VCC > 0) or an antineutrino(VCC < 0): if∆m2 > 0 the resonance only happens for neutrinos, whereas if ∆m2 < 0 it
only happens for antineutrinos.
8

1.3. Experimental Status

1.3

Experimental Status

Through the observation of neutrinos from various sources, neutrino oscillation is now an established fact, and the majority of the oscillation parameters have been measured. This section summarizes the diﬀerent neutrino
oscillation experiments which have been performed to date.

1.3.1

Solar and Reactor Neutrinos

The sun produces electron neutrinos through various nuclear fusion and βdecay processes at its interior, with their energy generally ranging in a few
to ten MeV. The first measurement of solar neutrinos was made by theHomestake experiment in 1968[8], which detected the neutrinos by counting
the decays of the radioisotope of argon produced in a chlorine tank via thereaction:νe + 37 Cl → 37 Ar + e− .
(1.21)The observed neutrino flux was ∼ 1/3 of the prediction from the standard
solar model(SSM)[9], and this was the first observed indication of neutrino
flavour transitions in history. Several experiments which followed also observed a significant deficit of the solar neutrino flux compared to the SSM
prediction. However, the eﬀect which was eventually known as the solar neutrino problem remained unresolved for more than 30 years since its initialobservation.
The first conclusive measurement to resolve the problem was made by the
SNO experiment, which was capable of measuring both the electron neutrino
flux and the total neutrino flux from the sun using heavy water[10][11]. In
SNO, neutrinos can be detected by the CC and NC interactions on deuterium
and also by elastic scattering (ES) on electron as the following:
CC : νe + d → e− + p + p,

(1.22)

NC : νx + d → νx + p + n,

(1.23)−−

ES : νx + e → νx + e .

(1.24)

νx above represents any of the three neutrino flavours, and while the NC
interaction on deuterium occurs equally for all neutrino flavours, only νeundergoes the CC interaction, and the experiment is thus capable of measuring the total flux and the νe flux separately through the two channels.
For the elastic scattering on electron, the channel is sensitive to all neutrino
flavours via NC interactions, however, since νe can also undergo CC interactions with electron the cross section of this process for νe is larger by a9qµo (× 10 6 cm -2 s-1)

1.3. Experimental Status

BS05

qSSM 68% C.L.

6

NCqµ o 68%, 95%, 99% C.L.

5
4
3
SNO

2

qCC 68% C.L.
SNO

qNC 68% C.L.
1

SNO

qES 68% C.L.
SK

qES 68% C.L.
0
0

0.5

1

1.5

2

2.5

3

3.5

qe (× 106 cm-2 s-1)

Figure 1.3: 8 B solar neutrino flux measured by SNO through diﬀerent channels. The horizontal axis is the νe flux and the vertical axis is the sum ofthe νµ and ντ flux. 68% allowed regions from the CC, NC and ES measurements are shown as the red, blue and green bands respectively, and the ES
measurement result from Super-Kamiokande is shown as a gray band. The
dashed lines represents the SSM prediction of the total neutrino flux. The
point and the contours represent the best estimate and the allowed regions
at 68, 95 and 99% confidence levels by combining the CC and NC results.
Figure taken from [12].
factor of ∼6 compared to νµ,τ (“ES” refers to all elastic scattering on electron via CC and NC interactions). As shown in Figure 1.3, the result of theexperiment showed a clear deficit of the νe flux by a factor of ∼ 1/3 relative
to the total flux which agreed with the SSM prediction, which is a decisive
evidence for the flavour transition of solar neutrinos. Combining the SNO
results with another solar neutrino measurement by Super-Kamiokande[13]as well as the reactor neutrino measurement by KamLAND[14], it is now
confirmed that the flavour transition of the solar neutrinos occurs via themechanism known as the large mixing angle(LMA) MSW solution[6][7].
According to the LMA-MSW solution, due to the high matter density at
the core of the sun, the matter eﬀect described in Section 1.2.4 significantly
changes the energy levels of the neutrinos such that at energies ∼ 10 MeV
the second mass eigenstate ν2 becomes nearly equal to the νe state, i.e.,
101.3. Experimental Status
neutrinos at that energy are produced in a nearly pure ν2 state at the core.
As the neutrinos propagate through the sun and exit to space, the matter
density gradually decreases such that the neutrino state changes adiabatically, keeping the neutrinos produced at the core in a nearly-pure ν2 state
until they exit the sun while the flavour composition of the neutrino statechanges according to the change in the matter density. The neutrinos thenpropagate until they get detected on earth spending most of the time invacuum, during which the neutrino flavours do not oscillate as it is alreadyin a mass eigenstate. An observer on earth then projects the mass-state neutrinos onto flavour states, and the flavour change νe → νx of solar neutrinos
is thus observed. The observation of the solar neutrino flavour transition
measures the parameters θ12 and ∆m221 .
Another method to measure θ12 and ∆m221 is to study the “disappearance” of ν̄e ’s coming from nuclear reactors, by measuring the deficit in theν̄e flux due to the transition ν̄e → ν̄x in a O(100 km) path in the air,which is eﬀectively a vacuum. The KamLAND experiment[14] measured asignificant deficit of the ν̄e flux from the various nuclear reactors in Japan
compared to the expectations assuming no oscillations, measuring the oscillation parameters precisely and thus contributing to resolving the solar
neutrino problem.
From the global analysis of various solar and reactor neutrino data, thecurrent known values for θ12 and ∆m221 are[15]:1.3.2+0.014,sin2 θ12 = 0.304−0.013(1.25)∆m221 = 7.53 ± 0.18 × 10−5 eV2 .

(1.26)

Atmospheric and Accelerator Neutrinos

Atmospheric neutrinos are produced by the cosmic ray particles impinging
on earth’s atmosphere. Primary cosmic ray particles are mostly protons andhelium nuclei, and when they interact with the nuclei in the atmosphere,
they produce hadronic showers which consist mostly of pions as well assome kaons. Neutrinos are produced when the produced hadrons decay, forexample, through the decays of charged pions and the subsequent decays ofmuons:
π + → µ+ + νµ → νµ + e+ + ν̄µ + νe ,−−−π → µ + ν̄µ → ν̄µ + e + νµ + ν̄e .

(1.27)(1.28)

Thus, the atmospheric neutrino flux contains neutrinos and antineutrinos ofboth electron and muon type with the flavour ratio (νµ + ν̄µ )/(νe + ν̄e ) ∼ 2,11

1.3. Experimental Status
while the ratio increases at higher energy since µ± have a higher chance
to reach the ground without decay. Since the cosmic ray flux is close toisotropic the neutrinos come from all directions, and while the ones producedin the atmosphere right above a detector and coming downward only travelfor ∼ 10 km before reaching the detector, the ones produced on the otherside of the earth coming upward travel > 10, 000 km across the earth. The
path length of atmospheric neutrinos therefore ranges between 10 − 104 km
depending on the zenith angle of neutrino’s direction, and together withenergies ranging from 10−1 to 103 GeV, atmospheric neutrinos cover a wide
range of L/E to which the oscillation phase depends on, making them aperfect source for studying neutrino oscillations.
In the late 1980’s and early 1990’s, experiments using large undergroundwater Cherenkov detectors such as Kamiokande[16][17] and IMB-3[18][19]observed a large deficit in the flavour ratio (νµ + ν̄µ )/(νe + ν̄e ): the measured
ratio was nearly half of what was expected. The results suggested neutrino
oscillations of a diﬀerent kind from that involving solar neutrinos.
In 1998, Super-Kamiokande[20] reported a large deficit in the upwardgoing νµ events compared to the expectations assuming no oscillations while
the downward-going νµ ’s as well as νe ’s from all directions were rather consistent with expectations, as shown in Figure 1.4. This is a strong evidencefor νµ oscillating to ντ or to unknown sterile states, and the data was, infact, consistent with a two flavour νµ → ντ mixing described by:
P (νµ → ντ ) = sin2 2θ sin2

∆m2 L
,
4E

(1.29)

with a maximal mixing of sin2 2θ = 1(θ = 45◦ ) and |∆m2 | ≈ O(10−3 eV2 ).
The oscillation eﬀect observed in atmospheric neutrinos were then confirmed by experiments using artificial neutrinos produced from accelerators.
In such experiments, a high energy proton beam is collided with a target producing hadronic showers which subsequently produce an intense νµbeam, and the neutrinos are then observed at a large far detector locatedat O(100 km) distance. Unlike atmospheric neutrinos, accelerator neutrino
beams are very pure in νµ ’s with small contaminations from νe ’s or antineutrinos, and the neutrino energy is tuned ∼ 1 GeV such that the νµdisappearance eﬀect νµ → νx becomes maximal at the far detector based onthe ∆m2 observed at Super-Kamiokande. K2K[21] and MINOS[22] are thefirst of such long-baseline accelerator neutrino experiments, and they bothobserved significant deficit in the observed νµ events as well as distortions inthe νµ energy spectrum, producing results consistent with the atmospheric
neutrino observation at Super-Kamiokande.
121.3. Experimental Status

Figure 1.4: Zenith angle distributions for electron-like(top) and muonlike(bottom) events in sub-GeV(left) and multi-GeV(right), observed at
Super-Kamiokande. cos θ < 0 and cos θ > 0 represent upward- anddownward-going events respectively. The hatched bands show the expectations assuming no oscillations, while the black lines are the best-fit todata assuming νµ → ντ oscillations. Figure taken from [20].
There are many ongoing experiments which are currently measuring theparameters θ23 and ∆m232 , which govern νµ → ντ oscillations, with increasing precision, and their results are in general still consistent with the maximal mixing of θ23 = 45◦ , and |∆m232 | is measured to be ∼ 2.5×10−3 eV2 with
∼ 5% uncertainty[15]. The latest results from the long-baseline experimentT2K[23] is (assuming ∆m232 > 0):+0.046sin2 θ23 = 0.532−0.068,(1.30)−3|∆m232 | = 2.545+0.081
eV2 .
−0.084 × 10

(1.31)

Latest results on θ23 and ∆m232 from the atmospheric and accelerator neutrino experiments are discussed again in Section 8.6.4.
Note that oscillation experiments are often insensitive to the sign of∆m232 as can be seen from the expressions for oscillation probabilities Equa13

1.3. Experimental Status
tions 1.8 and 1.29, and although the absolute value of ∆m232 has been measured rather precisely, its sign is still not determined. This is not the case for∆m221 , since the matter eﬀect observed in solar neutrinos are only possible
if ∆m221 > 0 as discussed in Section 1.2.4. Similarly, one way to determine
the sign of ∆m232 is to observe the oscillations driven by ∆m232 for neutrinos
which travel through significant amount of matter, which will be discussed
more in Section 1.5.

1.3.3

Measurement of θ13

The mixing angle θ13 was considered to be small, as the solar and atmospheric neutrino results were able to be explained assuming it is zero. The
first attempt to directly measure θ13 was made by CHOOZ, by observing theν̄e disappearance ν̄e → ν̄x from a nuclear reactor at ∼ 1 km, which is much
shorter than KamLAND. At such distance and the ν̄e energy of ∼ 3 MeV,
rather than the oscillation driven by the “solar” oscillation parameters θ12and ∆m221 , the oscillation driven by θ13 and ∆m231 :P (ν̄e → ν̄e ) ≈ 1 − sin2 2θ13 sin2

∆m231 L,
4E

(1.32)

becomes dominant, and θ13 can be measured by such short-baseline reactor
antineutrino experiments. The result showed no evidence for nonzero θ13 ,giving un upper limit of sin2 2θ13 < 0.15 at 90% confidence level[24].
In 2012, new generations of short-baseline reactor experiments Double
Chooz[25], Daya Bay[26] and RENO[27] separately reported that sin2 2θ13 ≈0.1 by observing clear deficit in the reactor ν̄e flux, where Daya Bay and
RENO excluded θ13 = 0 at 5.2σ and 4.9σ respectively.
θ13 can also be measured via the appearance of νe in a pure accelerator
νµ beam, whose probability is expressed to the leading order as:
P (νµ → νe ) ≈ sin2 2θ13 sin2 θ23 sin2

∆m231 L.
4E

(1.33)As in the equation, the leading term in the appearance probability is proportional to sin2 2θ13 and νe appearance is thus greatly suppressed if θ13 = 0. In
2011, T2K reported the first indication of νe appearance excluding θ13 = 0
at 2.5σ significance[28], and the updated result in 2013 showed the firstdecisive evidence for νe appearance at 7.3σ[29].
The current best measurements for θ13 come from the short-baseline
reactor experiments, and the averaged values from them are[15]:sin2 2θ13 = 8.5 ± 0.5 × 10−2 .

(1.34)
14

1.4. Unresolved Issues

1.4

Unresolved Issues

Although many of the parameters involved in neutrino oscillations have beenmeasured, there are still important open questions which need to be answered in order to fully understand the nature of neutrinos.
First of all, it is not known whether θ23 mixing is exactly maximal (45◦ ),and if not, whether the angle is larger or smaller than 45◦ . This is known as
the θ23 octant, and the cases θ23 <45◦ and θ23 >45◦ are referred to as the first
and the second octant respectively. νµ disappearance is the primary oscillation mode which is sensitive to θ23 , however, since the oscillation probability
has a parameter dependence of the form sin2 2θ23 , this mode alone cannot
determine the octant. Therefore, one needs to look at other subdominant
oscillation eﬀects such as the ones involving νe in order to resolve the octant,for instance, the νµ → νe oscillation whose leading term depends on sin2 θ23
as in Equation 1.33.
While the squared mass diﬀerences between the neutrino mass states
have been measured rather precisely, the sign of ∆m232 (or ∆m231 ) is still
yet to be determined. As illustrated in Figure 1.5, the case ∆m232 > 0 isreferred to as normal hierarchy(NH) in which case the mass eigenstate m3
has the largest mass, whereas ∆m232 < 0 is called inverted hierarchy(IH) in
which case m3 has the smallest mass. The absolute mass is also unknown
since oscillation experiments can only measure the mass diﬀerences. Fromcosmological measurements, the sum of the mass of the three neutrinos aremeasured to be less than 0.3 eV[2], and combining the knowledge of themass hierarchy from oscillation experiments we may be able to fully reveal
the mass structure of the neutrinos in near future. Furthermore, the originof the neutrino mass and the reason for its smallness compared to otherparticles is still not known. Some of the appealing theories which can explain
this assume that neutrinos are Majorana fermions, i.e. a neutrino is its
own antiparticle, which is a fundamentally new type of particles which havenever been observed. The next generation of neutrino-less double beta decay
experiments can determine whether or not neutrinos are Majorana fermionsin case the mass hierarchy is inverted, but not necessarily if it is normal
hierarchy[30]. Therefore, information regarding the mass hierarchy provides
valuable input in order to unveil the fundamental nature of the massive
neutrinos. Knowledge of the mass hierarchy also resolves the degeneracies in
the oscillation probabilities such that the sensitivity of various ongoing andfuture neutrino oscillation experiments to δCP can be significantly improved.
Finally, and probably most importantly, it is still an open question
whether CP is violated for leptons, i.e., whether δCP is nonzero. Some lead15

1.5. Probing the Unknowns with Atmospheric Neutrinos

Normalm2

Invertedνe νμ ντ

m23

m22
m21

m232
m22
m21

m221

m23

?
0
Figure 1.5: The ordering of the neutrino mass. For normal hierarchy m3 is
the heaviest state and ∆m232 > 0, whereas for inverted hierarchy m3 is thelightest and ∆m232 < 0. The absolute mass is also still unknown and only
the relative mass diﬀerences have been measured.
ing theories suggest that CP violation in the lepton sector can produce thematter-antimatter asymmetry which is essential to the formation of the current matter-dominated universe[31]. Although the observable CP violationin the currently known leptons itself may not necessarily be directly related
to such process[32][33], it is nonetheless of fundamental interest whether CP
violation can be observed in neutrino oscillations.

1.5

Probing the Unknowns with Atmospheric
Neutrinos

The primary oscillation eﬀect which is seen in atmospheric neutrinos is theνµ disappearance, and from the probability P (νµ → νµ ) one can measuresin2 2θ23 and |∆m232 |. However, due to the wide range of energies and thelong path length in matter, atmospheric neutrinos provide other informationon the oscillation parameters through various sub-leading oscillation eﬀects,
allowing us to probe the remaining unknown properties of neutrinos.
As mentioned in Section 1.3.2, the path length of atmospheric neutrinos
varies as a function of the zenith angle of neutrinos’ direction. Since neutrino
oscillation is characterized by the oscillation phase proportional to L/E,
oscillation eﬀects for atmospheric neutrinos manifest in the two-dimensional
distributions of zenith angle versus energy. Figure 1.6 shows the oscillation
probability P (νµ → νe ) and the ratio of oscillated to unoscillated νe flux
16

1.5. Probing the Unknowns with Atmospheric Neutrinos

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

10

20

0.7

-0.8
-1 -1
10

Eν (GeV)

1

10

10

2 0.6

Eν (GeV)Figure 1.6: The oscillation probability P (νµ → νe )(left) and the ratio of oscillated to unoscillated νe flux(right) for atmospheric neutrinos reaching theSuper-Kamiokande detector. The horizontal axes are the neutrino energy
and the vertical axes are the cosine of the zenith angle of neutrino direction where cos θ = 1 is downward-going and cos θ = −1 is upward-going.
Oscillation probabilities are calculated by assuming the following oscillation
parameters: (∆m221 , ∆m232 , sin2 θ12 , sin2 θ23 , sin2 θ13 , δCP ) = (7.7×10−5 eV2 ,+2.1×10−3 eV2 , 0.3, 0.5, 0.04, 0◦ ). Taken from [36].
for atmospheric neutrinos reaching the Super-Kamiokande detector. The
oscillation probabilities are calculated by fully considering the three-flavour
oscillation eﬀects in matter, using the methods by Barger et al.[34] andassuming the radial density structure of the earth described in [35]. In
the figure, discontinuities can be seen at cos θ ≈ −0.45 and cos θ ≈ −0.84
which correspond to the upper and the lower mantle boundary and thelower mantle and the outer core boundary respectively, which is due to thediscrete change in the matter density causing the matter induced oscillation
probabilities to change discretely. Some of the prominent oscillation eﬀects
are described in the following.
The enhancement of the νµ → νe oscillation probability seen in 2 −10 GeV upward going region is due to the MSW resonance eﬀect whichhappens while the neutrinos traverse the earth’s core. Following a discussion
similar to Section 1.2.4, when ∆m232 ≫ ∆m221 , the transition probability inconstant density matter is approximated as[37]:P (νµ → νe ) ≈ sin2 2θ13,M sin2 θ23 sin2∆m232,M L,
4E(1.35)
17

1.5. Probing the Unknowns with Atmospheric Neutrinos
where the eﬀective mixing angle and mass splitting in matter are:sin 2θ13,M = √sin 2θ13
sin 2θ13 + (cos 2θ13 − x32 )22,(1.36)√∆m232,M = ∆m232 sin2 2θ13 + (cos 2θ13 − x32 )2 ,
while
x32 ≡

2EVCC
,
∆m232

(1.37)(1.38)√and VCC = ± 2GF Ne as before. MSW resonance happens when the resonance conditionx32 = cos 2θ13
(1.39)
is satisfied, and νe appearance due to θ13 becomes maximal despite thesmall value of the actual θ13 parameter. Since the sign of VCC flips between
neutrinos and antineutrinos, the enhancement happens for neutrinos only if∆m232 >0 and for antineutrinos only if ∆m232 <0. Thus, mass hierarchy canbe probed by observing the excess in multi-GeV upward-going νe and ν̄e flux.
Furthermore, since the appearance probability Equation 1.35 is proportional
to sin2 θ23 and not sin2 2θ23 , this oscillation eﬀect also provides sensitivity
to the θ23 octant.
The oscillation patterns seen below 1 GeV on the left plot are the oscillations driven by θ12 and ∆m221 which are relevant for solar neutrino oscillations. Assuming θ13 = 0 and constant matter density, the oscillation
probabilities in such region are given as[38]:P (νe → νe ) = 1 − Pex ,(1.40)P (νe → νµ ) = P (νµ → νe ) = Pex cos2 θ23 ,(1.41)

where
Pex = sin2 2θ12,M sin2

∆m221,M L.
4E(1.42)

sin 2θ12,M and ∆m221,M are expressed simply by replacing the mixing angle
and mass splitting in Equation 1.36 and Equation 1.37 by θ12 and ∆m221 .
Consider now the eﬀect of this oscillation on the net observed νe flux. Lettingr ≡ Φ0µ /Φ0e

(1.43)

18

1.6. Thesis Overview
be the ratio of νµ and νe flux before oscillation, the fractional diﬀerence inthe observed νe flux due to oscillation, considering the contributions fromboth νe and νµ produced in the atmosphere, is expressed as:
(
)
Φe
2
−
1
=
P
r
cos
θ
−
1
.
ex
23
Φ0e

(1.44)As mentioned previously, for low energy atmospheric neutrinos the flux
flavour ratio r is roughly 2, and the terms in the parenthesis cancel whencos2 θ23 = 0.5, i.e. when θ23 mixing is maximal. When θ23 <45◦ an excess of
sub-GeV νe flux is observed, while if θ23 >45◦ a deficit will be seen instead,and this oscillation eﬀect therefore provides sensitivity to the θ23 octant.
When the ∆m221 driven oscillations depending on θ13 is also considered,i.e. when not taking the approximation θ13 = 0 in the discussion above,
the interference between the θ13 -dependent oscillations and the oscillations
described by Equation 1.41 introduces a dependence of oscillation probabilities on δCP . The interference gives an overall ∼3% eﬀect on the observed
total νe flux in sub-GeV depending on the value of δCP [39], and although
this is not particularly a large eﬀect this provides sensitivity to δCP .
The oscillation eﬀects discussed here will be revisited in Section 8.1.6.

1.6

Thesis Overview

In this thesis, atmospheric neutrino data observed at the Super-Kamiokande
detector is analyzed in order to the measure the neutrino oscillation parameters, namely, sin2 θ23 , ∆m232 including its sign and δCP , to which atmospheric
neutrino data has sensitivity. Chapter 2 describes of the Super-Kamiokande
detector, and its calibration methods are summarized in Chapter 3. Chapter 4 describes the atmospheric neutrino simulation which is used in conjunction with the observed data in the oscillation analysis. Chapter 5 gives
an overview of the data reduction process to select atmospheric neutrinos
from the data acquired at Super-Kamiokande. Chapter 6 details the maximum likelihood event reconstruction algorithm “fiTQun” which was recently
developed for Super-Kamiokande, which provides input information to theoscillation analysis. As an example of the application of fiTQun, Chapter 7
summarizes the T2K νe appearance analysis which was the first physics
analysis in which fiTQun was used. Chapter 8 details the methods for theatmospheric neutrino oscillation analysis and presents the results of analyzing Super-Kamiokande data. The conclusions and the future outlook of theanalysis are summarized in Chapter 9.
19

Chapter 2

Super-Kamiokande
Super-Kamiokande(Super-K, SK) is a large underground water Cherenkov
detector located in the Kamioka mine in Mt. Ikenoyama, Gifu Prefecture,
Japan. The detector consists of a water tank holding 50 kt of ultra-purewater, and the photomultiplier tubes (PMTs) lining the tank interior detect
the Cherenkov radiation which is produced by charged particles which propagate in the water. The primary purposes of the detector are nucleon decay
searches and the detection of neutrinos from various sources: solar neutrinos, astrophysical neutrinos such as the ones from supernovae, atmospheric
neutrinos and accelerator neutrinos in long-baseline neutrino oscillation experiments such as K2K and T2K.

2.1

Detector Overview

Figure 2.1 shows the sketch of the detector and the surrounding experimental
area. The water tank is a vertical cylinder of 41.4 m in hight and 39.3
m in diameter holding 50 kt of ultra-pure water in total, and the volume
is optically separated into two concentric cylinders: the inner detector(ID)which is the main detector region holding 32 kt of water with 36.2 m in heightand 33.8 m in diameter, and the outer detector(OD) which is a cylindricalshell volume surrounding the ID with its thickness being 2.05 m on the top
and the bottom and 2.2 m on the side. The OD serves as a veto detector
for cosmic ray backgrounds entering from outside of the detecter as well asa passive shield against other entering backgrounds such as neutrons andgamma rays produced in the surrounding rock. The two detector regions
are separated by a 55 cm wide stainless steel supporting structure covered
with opaque materials as shown in Figure 2.2, and the inner side of the
supporting structure is mounted with 11,129 20-inch PMTs facing inward
viewing the ID volume, while the outer side is mounted with 1,885 8-inch
PMTs facing outward viewing the OD. Since the PMTs are sensitive tomagnetic fields, 26 Helmholtz coils line the wall of the tank and reduce thegeomagnetic field to ∼50 mG.
The detector has a ∼1000 m overburden of rock which has a shielding20

2.1. Detector Overview

Figure 2.1: A sketch of the Super-Kamiokande detector. Taken from [40].

Figure 2.2: The supporting structure and the PMTs. Taken from [40].

21

2.2. A Water Cherenkov Detector

ct/n
θC
βct

Figure 2.3: A schematic drawing of Cherenkov radiation.
eﬀect against cosmic rays equivalent to 2700 m of water. The rate of cosmic
rays reaching the detector is reduced to ∼3 Hz which is low enough that various nucleon decay and neutrino studies can be performed without concerns
about cosmic ray muon backgrounds.

2.2

A Water Cherenkov Detector

Cherenkov radiation is an electrodynamic phenomenon which is analogous
to the sonic boom produced by an object travelling faster than the speedof sound. When a charged particle travels in a dielectric medium withrefractive index n at velocity v=βc which exceeds the phase velocity c/n oflight in the medium, the local disturbances to the dielectric caused by themoving charge radiate electromagnetic waves which form an electromagnetic
shock wave as schematically shown in Figure 2.3. This is the Cherenkov
light, and the light is emitted as a cone with its opening angle θC relative
to the particle’s direction satisfying:cos θC =

1
.
nβ

(2.1)

Since n≈1.34 in water at the typical visible light wavelength the detector issensitive to, for a particle traveling at β=1 Cherenkov light is therefore emitted at the angle θC ≈42◦ . The number of emitted photons per wavelength
per unit travel distance of the particle is given as:
(

)

d2 N
1
2πα
= 2 1− 2 2 ,
dxdλ
λ
n β

(2.2)

where λ is the wavelength and α is the fine-structure constant.
22

2.3. Detector Phases and SK-IV
Super-Kamiokande IV
Run 73002 Sub 376 Event 78389592
14-09-22:19:45:35
Inner: 3647 hits, 9221 pe
Outer: 3 hits, 2 pe
Trigger: 0x10000007
D_wall: 575.8 cm
Evis: 979.9 MeV
e-like, p = 979.9 MeV/c

Charge(pe)>26.7
23.3-26.7
20.2-23.3
17.3-20.2
14.7-17.3
12.2-14.7
10.0-12.2
8.0-10.0
6.2- 8.0
4.7- 6.2
3.3- 4.7
2.2- 3.3
1.3- 2.2
0.7- 1.3
0.2- 0.7< 0.2
840
0 mu-edecays

672
504
336
168
0
0

500

1000

1500

2000

Times (ns)Figure 2.4: An event display of a single electron event in the atmospheric
neutrino data, where the unrolled view of the ID is shown in the figure. Eachdot represents a PMT which detected Cherenkov photons, and the colour
scale indicates the integrated charge which corresponds to the number ofphotons detected at each PMT.
Since β>1/n is required in order for a particle to emit Cherenkov radiation, the momentum threshold at which this occurs for various particle
types assuming n=1.34 are roughly: 0.57 MeV/c for an electron, 118 MeV/c
for a muon, 156 MeV/c for a π + and 1052 MeV/c for a proton.
In Super-Kamiokande, neutrinos are detected when they interact withthe water and produce charged particles above the Cherenkov threshold,and the cone of light emitted by a charged particle is observed as a ring by
the PMTs on the detector wall as shown in Figure 2.4. For each event thetime of the first photon arrival as well as the integrated charge at each PMT
is recorded, and from that information the type and the kinematics of theparticles are inferred as described in Chapter 6.

2.3

Detector Phases and SK-IV

Since Super-K started its operation in 1996, there have been four distinct
detector phases. The initial phase, referred to as SK-I, started in April 1996
and continued until the detector was shut down for a scheduled maintenancein July 2001. On November 12, 2001 when the detector was being refilledwith water after the completion of the maintenance, one of the PMTs atthe bottom imploded, causing a shock wave which destroyed more than half23

2.4. Inner Detector

Figure 2.5: A schematic drawing of the Hamamatsu 20-inch PMT used forthe ID. Taken from [40].
of the PMTs in the detector. After the accident the detector was operated temporarily by redistributing the surviving PMTs evenly, while each
ID PMT was enclosed in a cover made of acrylic and fibre-reinforced plastic(FRP) in order to prevent another implosion accident. The second phaseof the detector, SK-II, ran in such configuration with less than half of theoriginal photo-coverage between October 2002 and October 2005. The detector was then brought back to its original full capacity after installing
additional PMTs, and the SK-III phase started in July 2006. Finally, in
September 2008 the detector underwent a brief shut down in order to upgrade the electronics, and the SK-IV phase continues since then until thepresent day.
This thesis focuses on analyzing the data from SK-IV only, and theconfigurations of the detector in this phase are described in the following.

2.4

Inner Detector

Figure 2.5 shows a schematic drawing of the Hamamatsu 20-inch PMT which
is used in the ID. 11,129 of the PMTs are evenly placed on the wall of the
ID on a 70 cm square grid realizing 40% coverage of the ID surface area by
the photocathodes, and Cherenkov photons produced by the particles in the24

2.5. Outer Detector

Figure 2.6: Quantum eﬃciency of the photocathode as a function of wavelength. Taken from [40].
ID are detected with high eﬃciency and resolution. The photocathodes arecomposed of a bialkali (Sb-K-Cs) material which has quantum eﬃciency as
shown in Figure 2.6 with a peak eﬃciency being around 22% at 360 nm wavelength. The 11-stage dynode in the PMT has a gain of 107 when suppliedwith high voltage ranging in 1700-2000V, and the collection eﬃciency of thefirst dynode is over 70%. The transit time spread for a single photoelectron
signal is 2.2 ns.
In order to protect the detector against PMT implosions as happened in
2001, each ID PMT is enclosed in a FRP housing with an acrylic window
covering the photocathode area so that the shock wave does not occur even
if one of the PMTs imploded. The spaces on the ID wall between the PMTs
are covered with opaque black sheets made of polyethylene terephthalate
ensuring the ID and the OD are optically well separated.

2.5

Outer Detector

The outer detector consists of evenly-spaced 1,885 8-inch PMTs facing outward to the OD water. Unlike the ID, the spaces between the PMTs arecovered with a reflective material Tyvek which has ∼90% reflectivity at 400
nm in order to increase the photon collection eﬃciency in the OD. To further enhance the light collection, each PMT is also surrounded by a 60 cm
square wavelength-shifting plates which absorbs the ultra-violet Cherenkov
photons and reemits blue-green visible light to which the PMTs are sensitive.
25

2.6. Water and Air Purification
Although the timing resolution of the OD PMTs slightly degrade from 13
ns to 15 ns due to the reemission process, the wavelength shifter increases
the light collection eﬃciency in the OD by 60% which is more important
than the timing resolution considering the role of the OD as a veto detector
for entering backgrounds. In order to better distinguish between entering,exiting and through-going particle events, the barrel region of the OD is
optically separated from the top and the bottom regions by Tyvek.

2.6

Water and Air Purification

As Cherenkov light needs to travel tens of metres in the detector water before reaching the PMTs to be detected, maintaining high transparency of
the water is crucial for this experiment. The water which is originally takenfrom the spring water in the mine undergoes multiple filtering, sterilizing
and degassing processes in order to remove particle, bacterial and radioactive contaminants from the water. The water is continuously purified andcirculated at a rate of ∼50 t/hr, and its temperature is kept at ∼13 ◦ Cwith temporal variations within 0.01 ◦ C using a heat exchanger in order tosuppress bacterial growth as well as stabilizing the detector response.
As the rock in the mine has high radon content, the air in the mine isnaturally rather high in radioactivity. In order to reduce the radon level in
the air especially so that it does not contaminate the detector water, filteredRn-free air is continuously supplied from outside of the mine into the experimental area, keeping the area at positive pressure. Furthermore, the rocksurrounding the experimental hall is entirely coated with a polyurethanematerial in order to prevent the radon in the rock from being released into
the air.

2.7

Data Acquisition

When a photon hits the photocathode of a PMT and produces a photoelectron(p.e.), the signal is amplified by the dynodes in the PMT and produces
a pulse current signal. In SK-IV, such analog signal from a PMT is first
fed into a charge-to-time convertor(QTC) which was specially designed forSuper-K[41]. When the input signal from a PMT exceeds a discriminator
threshold(we will call this a “hit” for brevity), the QTC starts integrating
the charge from the signal over the following 400 ns, and then outputs a
rectangular pulse signal whose rising edge representing the time the input
analog signal crossed the threshold and the width of the pulse representing
26

2.7. Data Acquisition
the integrated charge. The output rectangular signal from the QTC is thendigitized by a time-to-digital convertor(TDC), and the digitized charge andtime, which represent the number of the detected p.e.’s and the the production time of the first p.e. at the PMT respectively, are then sent for
further processing and are eventually used in physics analyses. The QTC
demonstrates an overall dynamic range of 0.2-2500 pC which corresponds toa maximum signal of ∼1000 p.e., and the non-linearity of charge is measured
to be below 1% across the entire dynamic range. Such high dynamic range
and linearity enables precise analyses of neutrinos events across a wide range
of energy.
Since it is not feasible to store the information of every single hit in
the detector, an event trigger is issued when the number of PMTs thatare simultaneously hit exceeds a certain threshold, and only the hits thatare accompanied by a trigger are saved. Event triggering in SK-IV is done
entirely by software using the digitized hit time information from the detector, and depending on the purpose there are five diﬀerent types of event
triggers which are summarized in Table 2.1. A triggering computer continuously monitors the observed data and calculates the variable N200 which
represents the number of hits in a sliding 200 ns time window at a given
time. When N200 exceeds the defined threshold of a trigger, the trigger isissued and the charge and time information for the hits which are containedin the entire event time window around the trigger is recorded. For the
OD trigger N200 is calculated using the hits in the OD PMTs only, while
for the other triggers hits in the ID are used. The thresholds for the SLE
and SHE triggers were lowered in May 2015 and September 2011 respectively. The events accompanying the high-energy(HE) trigger are used inthis atmospheric neutrino analysis.
Trigger Type
SLE
LE
HE
SHE
OD

N200 Threshold
34→31
47
50
70→58
22 (in OD)

Event Time Window (µs)[−0.5, +1.0][−5, +35][−5, +35][−5, +35][−5, +35]

Table 2.1: Definition of the event triggers. The triggers based on ID hits are:Super Low Energy(SLE), Low Energy(LE), High Energy(HE) and Special
High Energy(SHE) triggers. The OD trigger is based on OD hits.

27

Chapter 3

Detector Calibration
This chapter describes the procedures for calibrating the Super-Kamiokande
detector. The calibration results are used for producing the detector simulation which is described in Section 4.3 as well as for analyzing the observed
data. Further details on the calibration methods can be found in [42].
As mentioned in Chapter 2, what Super-K records as data is ultimately
the charge and the time information from the PMTs through the electronics,
and their charge and time response therefore needs to be calibrated precisely.
The dominant factors which characterize the charge response are the quantum eﬃciency and the gain of the PMTs, and as the characteristics of each
PMT have non-negligible variations, such properties need to be calibrated
on an individual PMT basis. The time response also has significant variations between diﬀerent PMT channels. In addition to the response of the
PMTs and the electronics, the optical properties of water and other components in the detector also need to measured in order to precisely model the
Cherenkov photon propagation in the detector simulation.

3.1

Relative PMT Gain Calibration

The supply high voltage for the PMTs are set individually for each PMT
so that the charge response for a given light intensity is approximately thesame across all PMTs. In order to measure and correct for the remaining
individual variations in the PMT gain, the following measurement is done.
The measurement uses the same apparatus as the timing calibration
which is described in detail in Section 3.4. As schematically shown in Figure 3.2, pulsed laser light is injected isotropically at a fixed position near
the centre of the ID, and the hits and the charge at the ID PMTs are thenmeasured repeatedly by flashing the laser at two diﬀerent intensities.
One set of measurements is performed by flashing the laser at high intensity IH such that every PMT detects several photons at a time. In suchcase, the average measured charge Q(i) at the i-th PMT is proportional tothe intensity IH of the injected light as well as the individual gain of the28

3.2. Absolute PMT Gain Calibration
PMT G(i):Q(i) ∝ IH × a(i) × ϵ(i) × G(i).

(3.1)

In this equation, the measured charge is also proportional to the light acceptance a(i), which depends on the PMT’s geometrical configuration within
the detector, and the quantum eﬃciency ϵ(i) of the PMT.
Another set of measurements is made by flashing the laser at very low
intensity IL so that only a few PMTs get hit at a time. At such low intensity
a PMT detects at most one photon at a time, and the average number of hits
N (i) at each PMT, which is equivalent to the hit probability, is proportional
to the intensity of the injected light but is almost independent of the PMT
gain due to the low hit discriminator threshold:N (i) ∝ IL × a(i) × ϵ(i).

(3.2)

Since the position of the light source is unchanged and the only diﬀerence
between the two sets of measurements is the intensity of the injected light,
by taking the ratio Q(i)/N (i) the factors which vary between individual
PMTs cancel except for the gain G(i), and the relative individual variationof the PMT gain is thus obtained from the measured ratio:
G(i) ∝ Q(i)/N (i).

(3.3)

According to the measurement, the variation of the gain across all PMTs has
a standard deviation of 5.9%, and the individual relative gain factor obtained
above is used to correct the observed charge at each PMT to eﬀectively makethe charge response of the PMTs uniform across the entire detector when
the data is analyzed.

3.2

Absolute PMT Gain Calibration

The absolute gain, which relates the measured charge to the number ofdetected p.e.’s at a PMT, is obtained by a calibration measurement using a “nickel source”: a low energy gamma ray source made of a sphereof nickel which emits 9 MeV gamma rays isotropically upon capturing theneutrons emitted from a 252 Cf neutron source which is placed at the centre. The source is placed at the centre of the ID and produces on average
0.004 p.e./event at each PMT, and more than 99% of the produced hits will
therefore be single p.e. hits.
After correcting for the relative gain variation for each PMT as describedin Section 3.1, the observed charge distribution for the single p.e. hits produced by the nickel source is obtained by accumulating the hits from all29

3.3. PMT Quantum Eﬃciency
106
105
104
103
102
10
1
0

5

10 15 20 25 30 35 40 45 50
pC

Figure 3.1: The observed charge distribution for single p.e. hits obtained
from the nickel source calibration data. Figure taken from [42].
PMTs as is shown in Figure 3.1. From the average of this distribution, theconversion factor between the observed charge and the number of p.e.’s isdetermined to be 2.658 pC/p.e. The obtained single p.e. charge distributionis also used in the detector simulation which is discussed in Section 4.3.

3.3

PMT Quantum Eﬃciency

In addition to the gain, the quantum eﬃciency(Q.E.) also varies between
diﬀerent PMTs and the eﬃciency for each PMT needs to be measured. The
measurement is done using the same nickel source as described in Section 3.2.
When the intensity of light reaching a PMT is small enough that the
expected number of produced p.e. is significantly smaller than one as in
the nickel source data, the average number of hits at a PMT, i.e. the hit
probability, is proportional to the quantum eﬃciency as well as the intensityof the light as described by Equation 3.2. Therefore, the hit probabilities
measured by the nickel source data can be used to calibrate the Q.E. foreach PMT. Since it is not possible to express the light acceptance a(i) in
Equation 3.2 in closed form and accurately correct for it to extract the Q.E.
ϵ(i) directly from the measured hit probability, the nickel source events are
simulated assuming no individual variations for the Q.E. so that it can becompared to the observed data. The hit probability obtained from the simulated events, which accounts for the factor a(i) via simulation but does notinclude the variations in ϵ(i), is compared to the hit probability calculated30

3.4. Relative Timing Calibration
0.CUGT

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

5-+&

8CTKCDNGHKNVGT&KHHWUGTDCNN
Figure 3.2: A schematic view of the timing calibration system. Figure takenfrom [42].
for each PMT using the real nickel source data and the Q.E. for each PMTϵ(i) is obtained. The resulting Q.E. for each individual PMT is used in thedetector simulation.

3.4

Relative Timing Calibration

The time interval between the time a photoelectron is initially produced
at a PMT and the time the signal from the PMT is registered as a hit by
the electronics varies between the PMTs due to many factors including thediﬀerence in the length of the cables connecting the PMTs to the electronics.
Also, since the time it takes for an analog signal from a PMT to cross thehit discriminator threshold depends on the pulse height, the time response
depends on the observed charge.
In order to calibrate such variations in the time response, a fast pulsed
laser light is injected isotropically at the centre of the ID as schematicallyshown in Figure 3.2. The time of the laser injection is monitored directlyfrom the light source by a fast 2-inch monitoring PMT, and using that31

3.4. Relative Timing Calibration

T (ns)

tqmap of cable # 00010
3102

0 1 2 3 4 5 6 7 8 9 10
1230

10

Q (p.C.)
1.0
0.9

1220

0.8
0.7

1210
0.6
1200

0.5
0.4

1190
0.3

Entries 561679
1180

0.2
0.1

1170
0

20

40

60

80

100

120

140

160

180

0.0

QBin

Figure 3.3: The TQ distribution for an ID PMT. The horizontal axis represents the charge where the scale in pC is shown at the top: the scale islinear up to 10pC and then goes to log scale. The vertical axis represents
the TOF-corrected hit time where a larger value corresponds to earlier hits.
Figure taken from [42].
injection time as a reference the relative oﬀsets between the hit time of
the ID PMTs are obtained from the time-of-flight(TOF) corrected hit time
which is defined as the hit time subtracted by the time required for a photon
to travel from the light source to the PMT. Since the timing also depends onthe charge as stated before, calibration data is taken at various laser intensityand for each PMT the 2D distribution of the observed TOF-corrected timeand charge is produced which is referred to as the “TQ distribution”.
Figure 3.3 shows an example of the TQ distribution for a PMT. In the
figure, the horizontal axis represents the charge and the vertical axis represents the TOF-corrected hit time. Once the TQ distribution is obtainedfor each PMT, the distribution is fitted by an asymmetric Gaussian in eachcharge bin, and the peak position of the Gaussians is then fitted by a polynomial as a function of charge. The fitted peak position of the TQ distribution,
named the TQ map, is saved into a database for each PMT and is then used
to correct the hit timing as a function of charge on an individual PMT basis.
This ensures that the time response of the PMTs in the entire detector issynchronized.
32

3.5. Water Property Measurement

Entries per 0.52 ns

3

10
700
600
500
400
300
200
100
0
-50

-40

-30

-20

-10

0

10

20

30

40

50

T (ns)
Figure 3.4: The timing distribution of ID PMTs at a charge bin ∼1 p.e.,
produced using hits from all PMTs. The horizontal axis represents thetiming after TQ map correction, where a larger value corresponds to earlier
hits. The red data points represent the data obtained from calibration, andthe blue curve is the fitted asymmetric Gaussian. Figure taken from [42].
The same dataset is also used to obtain the PMT timing resolution
which is put in to the detector simulation. At each charge bin, a timing
distribution is produced by accumulating the hits from all PMTs after TQ
map correction, and the distribution is fitted by an asymmetric Gaussian
as shown in Figure 3.4. The detector simulation then simulates the PMTtime response by an asymmetric Gaussian using the widths as a function of
charge which is obtained from the calibration.

3.5

Water Property Measurement

In order to accurately model the Cherenkov photon propagation in water, anempirical model for light scattering and absorption in the detector simulation
is tuned based on a calibration measurement using a laser injector.
In the detector simulation, the attenuation of light in water is characterized as exp (−l/L(λ)), where l is the distance the light traveled and L(λ) is

33

3.5. Water Property Measurement

Figure 3.5: A schematic view of the laser injector system for measuring thewater property and the reflectivity of the PMTs. Figure taken from [42].
a wavelength-dependent attenuation length defined as:
L(λ) = (αsym (λ) + αasy (λ) + αabs (λ))−1 .

(3.4)

The coeﬃcient αsym (λ) represents the amplitude of the “symmetric scattering” which takes into account the eﬀects from Rayleigh scattering andsymmetric Mie scattering, and it has the angular distribution of the form
1 + cos2 θ where θ is the photon scattering angle. αasy (λ) represents the“asymmetric scattering” which accounts for forward Mie scattering, and it
has the angular dependence of cos θ in the forward direction while there isno backward scattering. Finally, αabs (λ) is for absorption. These coeﬃcientsare empirical functions of the wavelength λ which are determined based on
the calibration data from a laser injector system as shown in Figure 3.5.
In the setup, a collimated laser beam is injected at the top of SK vertically down toward the bottom, and the light scattered in water and reflectedfrom the bottom is detected by the PMTs on the barrel and the top wallof the detector. The TOF-subtracted hit time distribution, i.e. the PMT
hit timing distribution after subtracting the time required for photons totravel from the beam spot at the bottom to each PMT, for the PMTs ineach detector region as indicated in Figure 3.5 is shown in Figure 3.6. The
sharp peaks on the right between 1830-1900 ns are due to the photons reflected at the beam spot from the bottom of the detector, whereas the hits
at earlier times are caused by photons which scattered in water before reaching the bottom and arrived at the barrel or the top wall. The scattering34

3.5. Water Property Measurement

0.004
0.003
0.002
0.001

0.004
0.003
0.002
0.001
0.004
0.003
0.002
0.001
0.004
0.003
0.002
0.001
0.004
0.003
0.002
0.001
0.004
0.003
0.002
0.001

1500

1600

1700

1800

1900 ns

Figure 3.6: The TOF-subtracted hit time distributions for the laser injector
data taken at 405 nm. The top plot is for the PMTs on the top wall, andthe following five plots correspond to the the five regions on the barrel as
indicated in Figure 3.5. The black circles represent the measured data andthe red histograms are the result from the laser injector simulation after
tuning the water properties and PMT reflectivity to data. The data before
1830 ns and between the two vertical blue lines on the left is used for thewater measurement, and the data in the range 1830-1900 ns is used fortuning the PMT reflectivity. Figure taken from [42].

35

3.5. Water Property Measurement

Inverse water transparency (1/m)

10-1
Total

Absorption

Symmetric

Asymmetric

10-2

10-3

10-4
300

325

350

375

400

425

450

475

500

Wavelength (nm)Figure 3.7: Light absorption and scattering coeﬃcients as a function of
wavelength. The data points are the data taken in April 2009, and the red,
blue and magenta lines represent the tuned αabs (λ), αsym (λ) and αasy (λ)
respectively. The black line represents the sum of the three, which is theinverse of the attenuation length L(λ). Figure taken from [42].
and absorption coeﬃcients αsym (λ), αasy (λ) and αabs (λ) are tuned using thelatter, the data before 1830 ns, such that the tuned simulation best agrees
within the model parameter space with the measured data taken at various
wavelengths. Figure 3.7 shows the scattering and the absorption coeﬃcientsas functions of wavelength which are tuned using the data taken in April
2009, and the red histograms in Figure 3.6 are the timing distributions forthe best-tuned laser injector simulation. The measurement leads to an overall attenuation length of ∼120 m at 400 nm wavelength. The apparatus issituated permanently in the detector and constantly takes data during the
SK operation so that the water parameters can be constantly monitored.
Figure 3.8 shows the time variation of the measured coeﬃcients at various
wavelengths.
36

3.6. PMT and Black Sheet Reflectivity

Figure 3.8: Time variation of the light absorption and scattering coeﬃcientsmeasured at various wavelengths. The coeﬃcients αsym , αasy and αabs areshown in purple, black and blue respectively. Some of the laser units were
replaced in 2009 with some changes in the wavelength, and the vertical blacklines indicate when the replacement took place. Figure taken from [42].

3.6

PMT and Black Sheet Reflectivity

The laser injector data used in Section 3.5 for water measurement is also
used to tune the reflectivity of the PMTs in the detector simulation. Ignoring the reflections at the PMT internal structure, the detector simulation
models light reflections at the PMT surface by considering the interface
of four diﬀerent layers: the water, the PMT glass, the photocathode andthe vacuum inside the PMT. Each layer is characterized by a wavelengthdependent refractive index, and since the photocathode layer absorbs light acomplex refractive index is assigned to the photocathode which is treated as
an eﬀective tuning parameter for PMT reflectivity. Using the laser injector
data in the time range 1830-1900 ns which contains the peaks from the lightreflected at the tank bottom as shown in Figure 3.6, the complex refractive
index of the photocathode is tuned such that the laser injector simulation
37

3.7. Photosensor Test Facility

Figure 3.9: A schematic view of the apparatus for black sheet reflectivity
measurement. Figure taken from [42].
best agrees with the measured data in this region. The tuning is performedat the wavelengths 337, 365, 400 and 420 nm, and the result is used whensimulating physics events such as neutrinos.
The reflectivity of the black sheet is measured in the SK detector using a
dedicated apparatus. As shown in Figure 3.9, a laser injector unit to whicha specimen of the black sheet is attached is placed at the centre of SK,
and the reflected laser from the black sheet is projected horizontally to thebarrel of the ID. The intensity and the profile of the reflected light is thenmeasured for each of the the incident angles 30◦ , 45◦ and 60◦ and for the
wavelengths 377, 400 and 420 nm. The intensity of the injected light directly
projected onto the wall without reflection is also measured for normalizing
the reflected light data, and the reflectivity calculated from the measured
result is used in the detector simulation.

3.7

Photosensor Test Facility

Properties of the SK PMTs have been measured over the years throughvarious in-situ calibration measurements in SK as described above. While
such methods have served well in calibrating the overall characteristics of
the PMTs so far, the increasing physics data statistics and the growing
complexities of physics analyses require more detailed understanding of the38

3.7. Photosensor Test Facility

Figure 3.10: Left: A schematic drawing of the PTF, showing the watertank, mechanical arms, optical heads and the PMT which is being measured.
Right: External view of the PTF magnetic field compensation system.
detector, so that we can improve the detector simulation for better data-MC
agreement and also reduce the systematic uncertainties in the detector.
The Photosensor Test Facility (PTF) at TRIUMF (Vancouver, Canada)
is a facility which was constructed to perform detailed ex-situ measurementsof the responses and the passive optical properties of large photosensors inwater. The left of Figure 3.10 shows the schematic drawing of the facility.
The facility consists of a large water tank in which a 20-inch SK PMT canbe placed upright and fully submerged in filtered water provided by a water
purification system. An optical head unit equipped with a collimated lightsource is mounted on each of the two automated mechanical arms whichmove the optical heads with three translation and two rotation degrees of
freedom with a position and direction accuracy of 1 mm and 1◦ . This allows
the detailed measurement of the PMT response for light incident at various
positions on the PMT. Each optical head is also equipped with a small
receiver PMT which is used to detect reflected light, and by using bothmechanical arms at the same time and receiving the light emitted by theother optical head, PMT reflectivity measurements can also be performed.
Figure 3.11 shows the interior of the tank with a SK PMT submerged in39

3.7. Photosensor Test Facility

Figure 3.11: A SK PMT submerged in the tank filled with water. The two
black boxes are the water-tight optical head units mounted on the mechanical arms.
water, along with the optical heads.
The optical head unit is a water-tight box. Light from a pulsed laser ora Xe lamp is delivered via an optical fibre which is connected in the box to alight source unit, which collimates and linearly polarizes the light as shownin Figure 3.12. The collimated beam is then split in two by a beam splitter,one of which is projected onto the target PMT outside of the box while theother is detected by a small monitoring PMT in the box which monitors
the intensity variation of the incident light. Another 18 mm square receiver
PMT is placed next to the light source unit, which views out of the box andis used for reflectivity measurements.
The response of large photosensors are greatly aﬀected by magnetic
fields. In order to compensate the geomagnetic field and the field from thecyclotron at TRIUMF, active field cancellation using three-axis Helmholtz
coils and passive magnetic shielding are employed as shown on the right of
Figure 3.10. During a measurement, local magnetic fields are constantly
monitored by a magnetometer placed in the optical head unit. Studies are
ongoing to automatically tune the coil current based on real time field measurements so that the magnetic field can be reduced to ∼10 mG level, which
is lower than the ∼50 mG field in SK.

40

3.7. Photosensor Test Facility

Figure 3.12: Top: The interior of the water-tight optical head box. Light isdelivered via an optical fibre which is connected to the light source unit. A
receiver PMT is placed next to the light source and receives the light fromoutside. Bottom: The light source unit, which consists of a collimator, apolarizer, a beam splitter and a monitoring PMT.

41

3.7. Photosensor Test Facility

Figure 3.13: Local variations of the gain of a SK PMT. The plot shows
the map of the gain variation viewing the PMT vertically down. This isa relative gain variation measurement and the colour scale is in arbitrary
units. Note that the horizontal and vertical band structure crossing near
the centre is produced by bands which are placed on the surface of the PMT
for calibrating the position and directions of incident beam and is not theresult of the characteristics of the PMT itself.
Finally, the entire apparatus is surrounded by two layers of dark curtains
which produce a dark environment adequate for measuring PMT responses.
The facility has been making various measurements of diﬀerent photosensors including the SK PMT, such as the measurement of local variationsin the photon detection eﬃciencies and the gain. Figure 3.13 shows an example result of a relative gain variation measurement for a SK PMT performed
in air, in which the peak position of the 1 p.e. distributions are measured
by shining laser at diﬀerent positions on the SK PMT. Although the results
have not been incorporated into SK analyses yet, it is expected that the results from the PTF measurements will be used to improve the PMT models
in the SK detector simulation and also to better understand the systematics
related to detector response in near future.

42

Chapter 4

Atmospheric Neutrino
Simulation
Physics analyses at Super-K are done by comparing the observed data to thepredictions obtained from the Monte Carlo(MC) simulations. This chapter
describes the atmospheric neutrino MC simulation which is used in the oscillation analysis. Producing the MC involves simulating the following threemajor components: the atmospheric neutrino flux reaching the detector, theinteraction of the neutrinos with water and the response of the detector to
the particles produced by the neutrino interaction.

4.1

Atmospheric Neutrino Flux

As the analysis of atmospheric neutrino oscillations is essentially done byinvestigating the change in the observed atmospheric neutrino flux at thedetector due to oscillations, it is essential to have an accurate model for the
production of neutrinos in the atmosphere. Super-K adopts the flux modelby Honda et al.[43] for various analyses concerning atmospheric neutrinos,
and the model is briefly described below.
The primary cosmic ray flux, which consists mostly of hydrogen nuclei,
is modeled based on the measurements by AMS[44] and BESS[45][46]. The
primary flux is aﬀected by the solar wind and depending on whether the solar
activity is at maximum or minimum the primary flux at ∼1 GeV varies bynearly a factor of two. The eﬀect of the geomagnetic field manifests in the
up-down and east-west asymmetry of the neutrino flux at Super-K as well as
the low-energy cutoﬀ in the flux due to the deflection of low energy primarycosmic rays out to space.
The US-standard atmosphere ’76[47] is adopted as the model for thedensity structure of the atmosphere, and for the production of secondary
cosmic ray particles in the atmosphere, DPMJET-III[48] is used for hadronicinteractions above 32 GeV while for energy below 32 GeV JAM[49] is used
for better agreement with low energy cosmic ray muon data. The hadron434.1. Atmospheric Neutrino Flux

Figure 4.1: Zenith angle distributions of the atmospheric neutrino flux at
Super-K averaged over the azimuth at 0.32 GeV(left), 1.0 GeV(middle) and
3.2 GeV(right). cos θ=1 represents vertically downward going and cos θ=-1
represents vertically upward going neutrinos. Figure taken from [43].
production models are further tuned based on the cosmic ray muon measurements at various altitudes such as [46], [50] and [51]. During the flux
calculation the trajectories of the cosmic ray particles in the atmosphere are
fully simulated in three-dimensions.
Figure 4.1 shows the zenith angle distributions of the atmospheric neutrino flux at Super-K calculated by the Honda model. The flux is peaked at
the horizon largely since the cosmic ray particles can travel longer distancesin the atmosphere horizontally rather than vertically, allowing more time fordecay. The large up-down asymmetry as seen in the left plot is due to thegeomagnetic field, and the eﬀect gets smaller at higher energies since higher
energy cosmic rays are aﬀected less by the geomagnetic field.
Figure 4.2 shows the direction-averaged neutrino flux at Super-K as afunction of energy. The plots compare the predictions from the Honda modelwith other flux models such as the Fluka model[52] and the Bartol model[53],which have diﬀerences in hadronic interaction simulations and particle trajectory calculations as well as the cosmic ray dataset the models are tuned
to. The diﬀerences in the flux predictions between the models are used toestimate the systematic uncertainties related to the atmospheric neutrino
flux.
The MC events are generated using the predicted flux at Super-K without considering any neutrino oscillations, i.e., assuming that all neutrinos

44

4.2. Neutrino Interaction

Figure 4.2: The direction-averaged atmospheric neutrino flux at Super-K as
a function of energy. The left plot is the absolute flux and the right plotis the various flux ratios. Red represents the Honda flux model which isused in the Super-K MC, and predictions from other models are used for
estimating the systematic uncertainties for the flux. Figure taken from [43].
reach the detector in their original flavour at the production in the atmosphere. Oscillations are applied at the time of performing an analysis byreweighting each MC event according to its oscillation probability based on
the neutrino energy and direction, and the procedure is discussed in moredetail in Section 8.3.1.

4.2

Neutrino Interaction

As stated in Section 2.2, neutrinos are detected at Super-K when they interact with the water in the detector. Neutrino interactions at Super-K are
simulated by the neutrino event generator NEUT[54]. The software simulates the interaction of neutrinos with oxygen and hydrogen nuclei in water,while the interactions with electrons are neglected in atmospheric neutrino
simulations due to the three orders of magnitude smaller cross sections relative to that for nuclei.
Neutrino interactions can be categorized into Charged Current (CC)and Neutral Current (NC) interactions. A CC interaction happens whena neutrino interacts with a target by exchanging a W ± boson producing acharged lepton whose flavour corresponds to that of the neutrino. Since the45

4.2. Neutrino Interaction
flavour of the charged lepton, i.e. electron or muon, can be identified inSuper-K as it will be discussed in Chapter 6, one can identify the flavourof the incoming neutrino in a CC event, which is important in observing
the eﬀects from neutrino flavour change in an oscillation analysis. On theother hand, a NC interaction is mediated by a Z 0 boson in which case theoutgoing lepton is also a neutrino and does not leave any clear signal in
the detector concerning the flavour. The NC events are therefore in generalconsidered as backgrounds in atmospheric neutrino oscillation analyses.
Although the original neutrino flux produced in the atmosphere only
contains electron and muon neutrinos, tau neutrinos are also detected at
Super-K because of oscillations. Due to the large 1.78 GeV/c2 rest mass ofthe tau lepton ντ CC interactions only occur at energies above several GeV,
resulting in a significantly smaller number of produced events compared toνe and νµ . Nevertheless, ντ interactions are simulated in the same manneras νe and νµ CC interactions by NEUT. The produced tau leptons undergo
complex decays into multiple mesons and leptons which are simulated by
TAUOLA[55].
The neutrino interaction modes which are simulated by NEUT are summarized in the following, and the cross sections as functions of energy forthe dominant interaction modes are shown in Figure 4.3. Except for mesonexchange currents, each interaction mode is modeled for both CC and NC.
In addition to the initial interaction of the incident neutrino with the target
nucleus, NEUT also simulates the subsequent interaction of the produced
particles within the nucleus, which is described in Section 4.2.5.

4.2.1

Elastic and Quasi-Elastic Scattering

NC elastic scattering is a process in which a neutrino simply scatters oﬀ
a nucleon target by transferring some momentum without producing any
new particles. Quasi-elastic (QE) scattering is an equivalent process fora CC interaction in which case the incident neutrino is converted into a
corresponding charged lepton while the target nucleon also converts in orderto preserve the total electric charge. For example, a νµ CCQE interactionis the following process:νµ + n → µ− + p.
(4.1)
In NEUT, such interactions on a free nucleon target are simulated by the
Llewellyn-Smith model[58], and for a nucleon bound in an oxygen nucleus
the relativistic Fermi gas model by Smith and Moniz[59] is used in order totake into account the Fermi motion and the Pauli blocking of the nucleon.
The model contains a form factor which is characterized by the axial vector
46

4.2. Neutrino Interaction

Figure 4.3: νµ (left) and ν̄µ (right) CC cross sections per nucleon divided
by neutrino energy, plotted as a function of energy. Experimental datasummarized in [56] is compared to the predictions from NEUT, and thecontributions from quasi-elastic scattering(QE), resonance production(RES)
and deep inelastic scattering(DIS) are shown separately in diﬀerent colours.
Figure taken from [57].
mass parameter MAQE , and the shape of the diﬀerential cross section as afunction of Q2 (square of the four-momentum transfer) as well as the total
cross section depend on the parameter. MAQE is set to 1.21 GeV/c2 basedon the results from K2K[60] and MiniBooNE[61].

4.2.2

Meson Exchange Current

In the CCQE interaction on a bound nucleon described above, an approximation is made such that a neutrino interacts with a single nucleon only
while neglecting any direct correlations between the nucleons in the nucleus.
However, in reality the interactions among the nucleons must introduce somedirect correlations such that an incident neutrino interacts simultaneouslywith multiple nucleons. The tension observed between the existing CCQE
model and the data from experiments such as MiniBooNE[61] also suggests
the existence of such process, and multi-nucleon correlations have been agrowing interest in the field of neutrino research in the past several years.
The Meson Exchange Current(MEC) model by Nieves et al.[62] is one ofthe prominent models which attempt to go beyond the preexisting simplistic
model of CCQE, and their model has been recently introduced to NEUT.
In the simulated process, the incident neutrino interacts with a pair of two
nucleons as opposed to a single nucleon target as:να + N N ′ → lα + N ′′ N ′′′ ,

(4.2)
47

4.2. Neutrino Interaction
where N N ′ and N ′′ N ′′′ are the nucleon pair before and after the interaction.
Although multi-nucleon correlations should in principle be present inany neutrino interactions on a bound nucleon, the CC MEC interaction
mentioned above is the only such process which is considered in NEUT at
the present moment.

4.2.3

Single Meson Production

Production of a single meson in a neutrino interaction is modeled in NEUT
primarily as a resonance production, in which a baryon resonance excitation
produces a single meson in the final state as in:να + N → lα + N ∗ → lα + N ′ + m,

(4.3)

where N and N ′ are the nucleon in the initial and the final state, N ∗ is theintermediate baryon resonance and m is a meson such as π, K and η. lα isthe outgoing lepton which is a charged lepton for CC interaction while it is aneutrino for NC. Such interaction is considered in NEUT for W < 2 GeV/c2 ,where W is the invariant mass of the hadronic final state. The simulation
is done adopting the Rein-Sehgal model[63] with the revised form factor by
Graczyk and Sobczyk[64], and the parameters characterizing the form factor
were determined using bubble chamber data[65][66].
Coherent pion production is also considered, in which the incident neutrino interacts with the entire oxygen nucleus producing a pion:να + 16 O → lα + 16 O + π.

(4.4)

Due to the low momentum transfer in this process the outgoing lepton andpion produced through this mechanism are peaked in the forward direction.
The interaction is simulated according to the model by Rein and Sehgal[67].

4.2.4

Deep Inelastic Scattering

In deep inelastic scattering (DIS), the incident neutrino interacts with aconstituent quark in the target nucleon and often produce multiple hadrons.
In NEUT the process is considered for the cases with hadronic invariant
mass W > 1.3 GeV/c2 and becomes increasingly dominant in multi-GeV
energies. The nucleon structure functions used in the model are based on
the GRV98 parton distribution functions[68], and the corrections by Bodek
and Yang[69] are applied in order to use the model in the low Q2 region.

48

4.2. Neutrino Interaction
For W < 2 GeV/c2 , only pions are considered as outgoing hadrons, andpion multiplicity in this region is estimated from bubble chamber experiments [70][71]. Since there is an overlap with the resonance pion productionin this region, only the cases of producing two or more pions are consideredin the DIS channel for W < 2 GeV/c2 . For W > 2 GeV/c2 , production ofheavier mesons such as K and η are also considered and the hadronic final
states are determined by PYTHIA/JETSET[72].

4.2.5

Final State Interaction

When a neutrino interaction happens in a nucleus, the hadrons which areproduced at the initial interaction must propagate through the nuclear
medium before getting detected outside of the nucleus, and during suchtime the hadrons often interact hadronically with the nuclear medium. Suchinteractions are referred to as final state interactions (FSI) and it is an important eﬀect for neutrino analyses at Super-K since it directly alters theobservable particle final state from what is initially produced at the neutrino interaction. NEUT considers the FSI for mesons and nucleons, andfor nucleons[73] and heavy mesons such as K[74][75][76] and η[77] the FSI
is simulated based on the cascade models and external data.
The hadrons which are most frequently seen at Super-K are, by far,the pions, and it is therefore important to accurately model the FSI for pions. Pion FSI is simulated by the NEUT pion cascade model[78] in which
pions are propagated classically in finite steps through a nuclear medium
described by the Woods-Saxon nuclear density profile[79]. At each step, apion may undergo the following processes according to the interaction probabilities given in the model: quasi-elastic scattering in which a single pion of
the same charge remains after the interaction, charge exchange in which acharged pion is converted into a π 0 and vice versa, pion absorption in which
no pions remain after the interaction and pion production in which multiple
pions are produced. The model has been tuned based on various pionnucleon and pion-nucleus scattering data, and Figure 4.4 shows the tuned
π + -12 C scattering cross section simulated in NEUT compared with experimental data. The uncertainty in the model is evaluated by comparing themodel variations against pion scattering data. As shown in Figure 4.5, by simultaneously varying the FSI model parameters and comparing the resulting
cross sections against various pion scattering data, representative parameter
sets are chosen which span a “1σ error surface” on the multi-dimensional
FSI model parameter space. There are in total six model parameters whichcharacterize the interaction probabilities for quasi-elastic scattering, charge
494.3. Detector Simulation

Figure 4.4: π + -12 C scattering cross sections, where the cross section foreach interaction process is shown separately in the colour described in thelegend. The data points are the measured results from π + -12 C scatteringexperiments and the solid lines are the cross sections calculated by the tuned
NEUT pion cascade model. The dashed lines are for the previous NEUT
pion FSI model. Figure taken from [78].
exchange, absorption and pion production in two energy ranges(pion momentum of below or above 500 MeV/c), and 16 representative parameter
sets are obtained as the 1σ variations[80].

4.3

Detector Simulation

The particles which are produced by NEUT are then propagated in thedetector by a GEANT3[81]-based detector simulation called SKDETSIM.
GEANT simulates the interaction of the particles with water, particle decays
as well as the production of Cherenkov photons. Hadronic interactions are
simulated by GCALOR[82] except for the pions below 500 MeV/c for which
the NEUT pion cascade model described in Section 4.2.5 is used.
The Cherenkov photons produced by the particles are then propagated in
the detector using a custom code which models the scattering and absorption
50

4.3. Detector Simulation

Figure 4.5: Low momentum π + absorption cross sections compared between
experimental data and the variations of the NEUT pion cascade model. In
the figure, the interaction probabilities for quasi-elastic(qe), absorption(ab)and charge exchange(cx) for pions < 500 MeV/c are simultaneously varied
in NEUT, and the resulting cross sections for eight representative variationsof the parameter set are shown in diﬀerent colours. Figure taken from [78].
of light in water based on the calibration results in Section 3.5. Reflectivityof the PMT and the black sheet are also modeled using the results obtainedin Section 3.6.
The charge and time response of the PMTs and the electronics are also
simulated based on the calibration measurements, and the detector simulation at the end provides the same data structure as the observed data sothat the two can be analyzed in the same manner.

51

Chapter 5

Data Reduction
While SK detects roughly 10 atmospheric neutrino events each day, the highenergy event trigger is fired O(106 ) times per day, most of which are due
to background events caused by cosmic ray muons, radioactivity and noisefrom the electronics. In order to remove such backgrounds and select theneutrino events for physics analyses, a series of reduction cuts are applied
to the observed data.
The events which are targeted in the neutrino oscillation analysis presented in Chapter 8 are the atmospheric neutrinos which interact with thewater in the ID, whose daughter particles are then detected as Cherenkov
rings by the ID PMTs with high resolution. Such neutrino events can beclassified into two categories based on the activity in the OD: an event iscategorized as Fully-Contained(FC) if all produced particles stop within the
ID and leave no signals in the OD, whereas it is categorized as PartiallyContained(PC) if any of the daughter particles exit to the OD and leave
signals there. The FC event sample has large statistics and since all the
Cherenkov photons are deposited in the ID the events can be precisely reconstructed and used in physics analyses. The oscillation analysis in Chapter 8 therefore uses the FC atmospheric neutrino data, and the proceduresto select the FC events are described in the following. The selection process
consists of a series of five data reduction stages which gradually eliminate
the backgrounds in data.

5.1

First and Second Reduction

The first two stages of the reduction process are designed to quickly filter
out the obvious backgrounds from cosmic ray muons, electrical noise andlow energy backgrounds such as those caused by radioactivity.
In the first reduction, events whose total charge from the ID PMTs isless than 200 p.e. are rejected as low energy backgrounds. This thresholdroughly corresponds to the charge deposited by a 22 MeV/c electron. As
will be mentioned later, only the events with energy greater than that of a
30 MeV/c electron are used in the oscillation analysis, and this cut removes52

5.2. Third Reduction
the majority of low energy backgrounds while keeping the targeted neutrino
events. In addition, in order to reject cosmic ray muons which usually leave
signals in the OD, events with more than 50 OD hits within a 800 ns time
window around the event trigger are rejected.
In the second reduction, the threshold for the above OD activity cut islowered to 30 OD hits for events with the total ID charge less than 100,000
p.e. so that lower energy muons can be further reduced. Furthermore, an
event is rejected if more than half of the total ID charge originates froma single PMT. This cut is intended to reject low energy events which arecaused by electrical noise such as a discharge in a PMT.
After the first and the second reduction the number events reduces to∼200 events/day, which is already more than three orders of magnitude
reduction from the initial dataset defined by the high-energy triggers.

5.2

Third Reduction

The third reduction targets several specific background sources which cannot
be removed by the first two reduction stages.
One of the major remaining backgrounds are the cosmic ray muons which
enter from outside of the detector but leave low activity in the OD. In
order to remove through-going muons which enter and exit the detector, theremaining events are processed by the through-going muon fitter which isa fast algorithm to reconstruct the entering and the exiting point on the
ID wall assuming a muon. Once the events are reconstructed, a goodnessof fit is evaluated based on the deviation of the observed PMT hit timings
from the predictions assuming a through-going muon, and events which
agree well with the muon hypothesis get rejected. In addition to the cut
based the goodness of fit, events which have more than 9 OD hits within
8m from the estimated entrance or exit points are also rejected. Similar
rejection methods are also applied assuming a stopping muon which is amuon entering the detector but stopping within the ID.
Another type of a muon background is the cable hole muons. At the
top of the SK detector there are twelve holes where the bundles of signal
and high voltage cables for the PMTs go through, and comic ray muons canenter the detector through the holes without leaving any signals in the OD.
Such muons are rejected based on the signals from the plastic scintillation
counters installed on top of the holes and whether the estimated entering
point is within 4 m from a cable hole.
In order to reduce the remaining low energy backgrounds, events are53

5.3. Fourth Reduction
rejected if they have less than 50 hits within a 50 ns time window in the thetime-of-flight corrected hit time distribution. The time-of-flight correction isdone using a vertex roughly estimated using the hit time information. The
cut threshold corresponds to the energy equivalent to a 9 MeV/c electron.
A flasher PMT event is an event caused by an electrical discharge ina PMT. To reduce such background, an event is rejected if the minimum
number of ID hits in a 100 ns time window sliding from 300 ns to 800 ns
after the event trigger is greater than 19. Since flasher PMT events typicallyhave broader hit timing distributions compared to real particle events, events
having a long tail in the hit timing distribution are rejected by this cut.
After the third reduction the number events reduces to ∼45 events/day.

5.3

Fourth Reduction

The fourth reduction employs a pattern matching algorithm to remove theremaining flasher PMT events. As flasher PMT events tend to repeat withsimilar PMT hit patterns, correlations of the charge pattern is calculated
between diﬀerent events. For a given event the charge correlation is calculated between the 10,000 events neighboring in time, and events with highcorrelations are rejected.
The event rate after the fourth reduction is ∼18 events/day.

5.4

Fifth Reduction

The fifth reduction is designed to reject the remaining cosmic ray muon andflasher PMT events.
A cosmic ray muon whose momentum is near the Cherenkov thresholdmay not issue an event trigger by itself, but instead its decay electron canbe detected as an event. Such events are rejected if there are more than 9
OD hits per 200 ns during any time between 8800 ns and 100 ns before theevent trigger.
Further reduction of flasher PMT events are done by requiring tighter
criteria for the tails in the hit timing distribution compared to the cut inthe third reduction. This is done by using the time-of-flight corrected hit
time based on the reconstructed vertex rather than the raw hit times.
Approximately 16 events/day remain after the fifth reduction.

54

5.5. Final FC Selection

5.5

Final FC Selection

After the five reduction stages are applied, the final FC event sample forphysics analyses are selected by requiring the following:• Number of hit PMTs in the largest OD hit cluster is less than 16.
• Visible energy is greater than 30 MeV.
• Reconstructed vertex position is more than 2 m away from the ID wall.
The first cut ensures that the OD activity is low enough that it can beclassified as an FC event rather than PC. Atmospheric neutrino analyses
only use the events with energy greater than that of a 30 MeV/c electron as
the atmospheric neutrino flux falls oﬀ at such low energy, and the second cut
is applied to reject any potential low energy events which are not relevant.
The third cut is the fiducial volume(FV) cut which defines a 22.5 kt fiducial
volume of the SK detector. The visible energy and the vertex is obtainedusing the event reconstruction algorithm described in Chapter 6 and moredetails can be found there as well as in Section 8.1.1.
The remaining backgrounds after the five reduction stages are mostly
cosmic ray muons and flasher events, both of which are then largely rejectedby the FV cut since they both originate on the ID wall. The backgroundcontamination in the final selected FC sample is estimated to be ∼0.1%
based on the eye-scan of the selected events.
The overall eﬃciency of the FC sample selections for the true neutrino
events which happen in the FV is estimated to be greater than 98% basedon the simulation, and the average observation rate of the final FC sample
is ∼8 events/day.

55

Chapter 6

Event Reconstruction
In order to use the observed events in physics analyses, one needs to reconstruct the type of the particles within an event as well as their kinematics.
This chapter describes the event reconstruction algorithm which is used as
an input for the oscillation analysis. The algorithm, named fiTQun, employsa maximum likelihood method using the charge and time information thatis observed by the PMTs to reconstruct particle types and kinematics in thedetector. The algorithm is based on the methods which are developed for
the MiniBooNE experiment[83], but is developed from scratch for Super-K
with additional features such as the advanced multi-ring reconstruction.
In the following, Sections 6.1 to 6.4 describe the core algorithm of thefiTQun maximum likelihood event reconstruction, and the subsequent sections detail the practical procedures of reconstructing Super-K events.

6.1

Likelihood Function

The basis of the fiTQun reconstruction algorithm is the likelihood function.
Let x denote a particle event hypothesis, which specifies the type of theparticles in the event as well as their vertices and kinematics. An event inSuper-K contains the information on which PMTs registered a hit, and for
PMTs that were hit the integrated charge as well as the time of the first
photon arrival are also provided. Using such observed information, one canconstruct the following likelihood function given the hypothesis x:
L(x) =unhit∏j

Pj (unhit|x)hit∏{1 − Pi (unhit|x)}fq (qi |x)ft (ti |x).

(6.1)

i

In this equation, the index j runs over all PMTs which did not register ahit, and for each such PMT the unhit probability Pj (unhit|x) which is theprobability the PMT does not register a hit given hypothesis x is multiplied.
For the PMTs which did register a hit, the hit probability as well as the
charge likelihood and the time likelihood are multiplied, where qi and ti
represent the observed charge and time respectively. The charge likelihood
56

6.2. Predicted Chargefq (qi |x) is a conditional probability density function of observing charge qi
at the i-th PMT given hypothesis x provided that the PMT registered ahit. Similarly, the time likelihood ft (ti |x) is the probability density for a hit
being produced at time ti , given x. Once the likelihood function is defined,
one can look for the hypothesis x which maximizes L(x) and use that inphysics analyses as the estimate for the particle configuration in the event.

6.2

Predicted Charge

In practice, when calculating the likelihood, the process of particle and optical photon propagation are decoupled from the response of the PMTs andthe electronics by introducing the predicted charge µi , which is the mean
number of photoelectrons that are produced at the i-th PMT given a hypothesis. µi explicitly depends on the hypothesis x, and Equation 6.1 isthen rewritten using µi as
L(x) =unhit∏

Pj (unhit|µj )hit∏{1 − Pi (unhit|µi )}fq (qi |µi )ft (ti |x),

(6.2)

i

j

where the unhit probability and the charge likelihood now depend on x
only through the predicted charge. Pi (unhit|µi ) and fq (qi |µi ) are purely theproperties of the PMTs and the electronics and do not explicitly depend on
the processes of Cherenkov photon emission and propagation.
Likelihood evaluation is then performed in two steps: for a given hypothesis x the predicted charge µi for each PMT is first calculated, and using
the calculated µi and the observed information the likelihood function Equation 6.2 is evaluated. For a given single particle hypothesis the predictedcharge contribution from direct and indirect light are calculated separately,
and the sum of the two is used in likelihood calculation. For a multi-particle
hypothesis, predicted charge is first calculated separately for each particle
in the hypothesis, and then the sum of the predicted charge from all theparticles are used in evaluating Equation 6.2. The method for predictedcharge calculation for a single particle is described in the following.

6.2.1

Predicted Charge from Direct Light

The predicted charge from direct light produced by a single particle is expressed as an integral along the particle track:
∫

µ

dir

= Φ(p)

ds g(p, s, cos θ)Ω(R)T (R)ϵ(η),

(6.3)
57

6.2. Predicted Charge
s

R

PMT

Figure 6.1: Schematic diagram describing the variables relevant to predictedcharge calculation. The white dot is the initial position of the particle.
where s is the distance the particle traveled along the track from its initial
position. p is the initial momentum of the particle, and variables R, θ, ηcharacterize the relative orientation of the particle and the PMT in concern
which are all functions of s, as described in Figure 6.1. The following sections
describe each of the following factors which appear in the above equation:
Φ which represents the photon yield, the Cherenkov emission profile g, thePMT solid angle factor Ω, the light transmission factor T and the PMTangular acceptance ϵ.

6.2.2

Cherenkov Emission Profile

The Cherenkov emission profile g(p, s, cos θ) describes the number of photons emitted per unit track length per unit solid angle at angle θ with respect
to the particle direction, at the instance when the particle with initial momentum p has traveled a distance s along the particle track. g(p, s, cos θ) is
normalized such that∫g(p, s, cos θ)dsdΩ = 1.

(6.4)

For each particle type, the profiles are generated at a range of discrete
initial momentum values using particle gun MC, which is produced by injecting a single particle in the GEANT3-based detector simulation(SKDETSIM)
mentioned in Section 4.3. Representative distributions of the Cherenkov
profile for electrons and muons are shown in Figures 6.2 and 6.3. When
producing the profile, event-by-event randomness due to multiple scattering
58

400

0.8

300

s (cm)s (cm)

6.2. Predicted Charge
400

0.6

300

0.4

200

0.2

100

0

0

0.5
0.4
0.3

100
0

0.6

0.8

s (cm)

e 50MeV/c

1
cosθ

400

0.2
0.15

200

0.1
0.6

0.8

e 200MeV/c
0.25

300

0.2

s (cm)

200

1
cosθ

400
300

0.1

200

0.1
100
0

0.05
0.6

e 500MeV/c

0.8

1
cosθ

0

0

0.05
100
0

0.6

e 1000MeV/c

0.8

1
cosθ

0

Figure 6.2: Cherenkov emission profile g(p, s, cos θ) for electrons at diﬀerent
initial momentum. Horizontal axes represent the cosine of the angle fromthe particle direction, and vertical axes are the distance traveled from theinitial position of the particle.
etc. is averaged over. It can be seen in the figure that photon emission forelectrons are peaked at cos θ ≈ 0.75 which corresponds to the opening angleof the Cherenkov cone from a particle travelling at the velocity β = 1 inwater, regardless of the momentum. Also, electromagnetic showers caused
by electrons produce rather broad angular distributions. For muons on theother hand, the angular distribution of photon emission is much sharperand depends much stronger on momentum compared to electrons, and we
see the collapse of the Cherenkov cone as the particle travels and loses its
momentum. Such diﬀerences in the emission profile between diﬀerent particle types provide particle identification capability, which will be discussed
in more detail in later sections.
The factor Φ(p) is a normalization factor which is proportional to the
59

400
4

s (cm)s (cm)

6.2. Predicted Charge

300

400
0.8
300
0.6

3
200

200

0.4

2

0

0.6

s (cm)µ 160MeV/c

0.8

1
cosθ

1

100

0

0

400
0.3
0.25

300

0.2
0.6

µ 300MeV/cs (cm)

100

0.8

1
cosθ

400
0.15
300

0.2
200

0.15
0.1

100

0

0.1
200
0.05

100

0.05
0

0.6

µ 600MeV/c

0.8

1
cosθ

0

0

0.6

µ 1000MeV/c

0.8

1
cosθ

0

Figure 6.3: Cherenkov emission profile g(p, s, cos θ) for muons at diﬀerent
initial momentum. Horizontal axes represent the cosine of the angle fromthe particle direction, and vertical axes are the distance traveled from theinitial position of the particle.
average total number of photons that are emitted by a particle with initialmomentum p. Φ(p) also absorbs the constant factors that are not accounted
for by other factors in Equation 6.3 such as the quantum eﬃciencies of thePMTs, and enforces proper normalization to the predicted charge.

6.2.3

Solid Angle FactorΩ(R) represents the solid angle subtended by a PMT viewed from its normal
direction at distance R. In order to reduce computation time, the factor issimply approximated as the area of a circle normalized by the distance:Ω(R) =

πa2,
R2 + a2(6.5)
60

6.2. Predicted Charge
where a = 25.4 cm is the radius of the PMT. The approximation holds
suﬃciently well at distance R > 1 m.

6.2.4

Light Transmission Factor

T (R) is the attenuation factor of direct light due to absorption and scatteringin water, and it is written as
T (R) = exp(−R/Latt ),(6.6)

where Latt is the wavelength-averaged attenuation length for Cherenkov
radiation. The attenuation length is obtained from the detector simulationand its value for SK-IV is 7496.46 cm.

6.2.5

PMT Angular Acceptanceϵ(η) is the PMT angular acceptance, which is a function of the angle η
between the PMT normal and the direction of the particle position viewed
from the PMT. Figure 6.4 is the acceptance curve obtained from the detector
simulation, which also includes the eﬀect of the shadowing by neighboring
PMTs at high incident angle. The distribution is fitted with a joint polynomial as indicated by the solid line in the figure, and the fit function isused as ϵ(η) in event reconstruction. We adopt a normalization condition
ϵ(η = 0) = 1.

6.2.6

Predicted Charge from Indirect LightIn addition to direct light which was just discussed, indirect light must also
be taken into account in order to properly predict the amount of charge
deposited at each PMT. This includes the light scattered in water as wellas the reflected light coming from detector components such as the black
sheet and PMTs themselves. Similar to Equation 6.3 for direct light, thepredicted charge from indirect light is written as∫µsct = Φ(p)ds1
ρ(p, s)Ω(R)T (R)ϵ(η)A(s),
4π

where
ρ(p, s) ≡

(6.7)

∫

g(p, s, cos θ) dΩ(6.8)

is the fraction of photons emitted per unit track length, at position s along
the particle trajectory. When the factor A(s) is removed from Equation 6.7,
61

6.2. Predicted Charge

2500
2000
1500
1000
500
0
0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
cosη

Figure 6.4: Angular acceptance of the PMT plotted as a function of cos ηwhich is obtained from detector simulation, with a fitted curve overlaid as asolid line. Note that the vertical axis in the figure is in arbitrary unit; after
the shape of the curve is extracted, the fitted function is renormalized sothat its value is 1 at η = 0.
the integral represents the amount of direct light deposited from an imaginary isotropic light source whose trajectory and the total light intensity are
the same as the charged particle described by Equation 6.3. In other words,
photon emission described in Equation 6.3 is averaged over all directions at
each point on the particle track. The object
A(s) = A(xPMT , zvtx , Rvtx , φ, θ, ϕ) ≡dµsct,
dµiso,dir(6.9)

which is referred to as the scattering table, is the ratio of the diﬀerential
predicted charges deposited from point s along the track, by the indirect
Cherenkov light from the concerned charged particle and the direct lightfrom the imaginary isotropic source which was discussed above. Assumingthat the Cherenkov opening angle does not change as a function of momentum, the momentum dependence is factored out from the scattering table bytaking the ratio. The scattering table then depends only on the positionaland directional alignment of the particle and the PMT, and a common table62

6.2. Predicted Charge

Rvtx
φ
zvtx
(θ,ϕ)

zPMT

Figure 6.5: A schematic diagram describing the variables relevant to thescattering table. For PMTs on the top or the bottom wall, the distance
RPMT of the PMT position from the vertical axis at the tank center is used,instead of zPMT .
can be used for particles at any momentum. Note, again, that this is an approximation which assumes the Cherenkov opening angle to be independentof momentum and is fixed at the opening angle for particles at β = 1, whencalculating the distribution of the indirect light.
Assuming the azimuthal symmetry of the detector, the scattering table
is suﬃciently expressed as a function of six variables which describe theorientations of the PMT and the particle in the detector, as schematicallyshown in Figure 6.5. Note that the figure is for the PMTs on the side wall,in which case the variable xPMT in Equation 6.9 corresponds to the PMTz position zPMT . For the PMTs on the top or the bottom wall, the PMT’s
distance RPMT from the vertical axis at the tank center is used, instead ofzPMT .
The scattering table is produced by generating a particle gun MC where
3 MeV/c electrons are placed randomly within the detector while multiple
scattering is turned oﬀ. Such electrons can be considered as point Cherenkov
sources with β = 1. During the simulation, the number of reflection andscattering of each photon which arrives at a PMT is recorded, and direct
and indirect photons arriving at the PMTs are counted and filled separately
into six-dimensional histograms, each axis corresponding to one of the six
636.2. Predicted Charge
variables in Equation 6.9. Then, at each bin in the six-dimensional parameter space, the numerator dµsct in Equation 6.9 is obtained from thehistogram for indirect photons, while the denominator dµiso,dir is calculated
by averaging the histogram for direct photons over the particle direction.
The scattering table is thus obtained as a six-dimensional histogram, and it
is linearly interpolated in terms of the six variables at the time of evaluating
the predicted charge.

6.2.7

Parabolic Approximation

As in Equations 6.3 and 6.7, evaluation of the predicted charge involves
integrals along the particle trajectory, and it is not computationally feasible
to perform the integrals at the time of reconstructing an event. The following
approximation is therefore made in order to avoid the need of evaluating the
integral at the time event reconstruction.
Let J(s) denote the last three factors in Equation 6.3, which describes
how the photons are received by the PMT:
J(s) ≡ Ω(R)T (R)ϵ(η) ≈ j0 + j1 s + j2 s2 .

(6.10)

This factor varies slowly as a function of s, and it is therefore reasonably
approximated by a parabola, an example of which is shown in Figure 6.6.
The coeﬃcients jn are obtained by evaluating J(s) at three points along theparticle trajectory, namely, the particle’s initial position, the point at which
90% of Cherenkov light is emitted, and their midpoint.
Once J(s) is approximated as a parabola, Equation 6.3 reduces to
∫

µ

dir

= Φ(p)

ds g(s)J(s) ≈ Φ(p)(I0 j0 + I1 j1 + I2 j2 ),

where
In ≡

(6.11)∫

ds g(s)sn .

(6.12)The integral in Equation 6.12 is performed in advance, and In is tabulated
as a function of three parameters which specify the initial condition of theparticle and its relative orientation to the PMT; namely, the initial valuesof the momentum p, the distance R from the particle to the PMT, and theangle θ between the particle direction and the line connecting the particle
and the PMT positions. Once the integral Equation 6.12 is performed at
discrete bins in R, θ and p, In is fitted as a function of p at each R, θ bin inorder to smooth the likelihood surface. When reconstructing an event, In is64

6.2. Predicted ChargeJ(s)0.0012Full

0.00118

Parabolic0.001160.001140.001120.00110.001080.00106

0

20

40

60

80

100

120

140

160
s (cm)

Figure 6.6: The photon acceptance factor J(s) (black) overlaid with anapproximating parabola (red). The coeﬃcients for the parabola are obtainedby evaluating J(s) at the three points indicated by the hollow circles. The
plot is for the initial conditions R = 500 cm, θ = 90◦ , η = 0◦ .
linearly interpolated in terms of R and θ, after the fit function is evaluatedat momentum p at the relevant neighboring R, θ bins.
Since the scattering table A(s) also tends to vary slowly as a function ofs, for indirect light the approximation
J(s)A(s) ≈ k0 + k1 s + k2 s2(6.13)

is made, which reduces Equation 6.7 to
∫

µ

sct

= Φ(p)ds11
ρ(s)J(s)A(s) = Φ(p) (k0 + K1 k1 + K2 k2 ),
4π
4π

where
Kn ≡

(6.14)∫ds ρ(s)sn .

(6.15)

Kn is a function of p only, and it is therefore fitted as a function of p. The
coeﬃcients kn are obtained in the same way as jn in the direct light case.

65

6.3. Unhit Probability and Charge Likelihood

6.3

Unhit Probability and Charge Likelihood

Once the direct and indirect predicted charge is evaluated for all the particles involved in the event hypothesis, the predicted charge is summed up
and the unhit probability Pi (unhit|µi ) and the charge likelihood fq (qi |µi ) in
Equation 6.2 are evaluated for each PMT.
Since the predicted charge µ is the mean number of photoelectrons produced at a PMT, the actual number of the produced photoelectrons shouldobey a Poisson distribution with mean µ. Therefore, the probability of no
photoelectron being produced given the predicted charge µ is e−µ . However,there are cases in which the signal from a photoelectron does not cross the
PMT hit threshold such that it does not produce a hit, and in order to take
into account such PMT threshold eﬀects, a correction up to the third order
is applied in the expression of the unhit probability:
P (unhit|µ) ≈ (1 + a1 µ + a2 µ2 + a3 µ3 )e−µ .

(6.16)P(unhit| µ)

The coeﬃcients an are obtained from the detector simulation, and Figure 6.7
shows the unhit probability with and without the correction, compared withthe actual values obtained from the simulation.
1
10-1

w/o corr.
w/ 3rd order corr.
MC true

10-2
10-3
10-4
10-5
0

2

4

6

8

10

12
µ(p.e.)

Figure 6.7: The unhit probability P (unhit|µ) with(red) and without(blue)
the correction for the PMT threshold eﬀect. The data points show the values
obtained from detector simulation.

66

6.4. Time Likelihood
The charge likelihood fq (q|µ) is obtained by directly generating photoelectrons following Poisson statistics with mean µ at the PMTs in thedetector simulation, and taking the distribution of the measured charge forthe hit PMTs. Figure 6.8 shows the obtained normalized charge likelihood
distribution at a range of predicted charge values. In order to smooth thelikelihood surface, the normalized charge distributions are then fitted by apolynomial as a function of µ at each fixed value of q. When evaluating
the likelihood, the fit parameters are linearly interpolated in terms of theobserved charge q, and the resulting polynomial is evaluated at µ.

6.4Time Likelihood

The time likelihood ft (ti |x) in Equation 6.2 depends on the event hypothesis x as well as the position of the PMT in a complex way. Since it isimpractical to accurately account for all such dependencies, the following
approximations are made for the time likelihood calculation.
The approximation starts by expressing the time likelihood in terms of
the residual hit time tres
i which is defined based on the raw hit time ti asi
tresi := ti − t − smid /c − |RPMT − x − smid d|/cn ,(6.17)

where x, t are the particle vertex position and time, d is the particle direcition, RPMT
is the position of the i-th PMT and smid represents half of theparticle track length.
cn := c/n, n = 1.38
(6.18)
is the group velocity of Cherenkov light in water. tres
i is therefore the residual
hit time after subtracting the expected direct photon arrival time from theraw hit time, assuming all the photons are emitted when the particle is at
the track midpoint.
As described below, time likelihoods for direct photon hits and indirect
photon hits are calculated separately using tres
i , and the two likelihoods are
merged based on the relative intensities of the direct and indirect light toobtain the final time likelihood that is used in Equation 6.2. The direct
and indirect light time likelihoods for a single particle is calculated as thefollowing.

6.4.1

Direct Light Time Likelihood

In order to calculate the direct light time likelihood, an approximation ismade so that the time likelihood depends only on tres
i , the predicted charge
676.4. Time Likelihood

0.7

0.7

0.6

µ= 0.30 p.e.

0.5

0.6

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0
0

5

10

15

20
25
q (p.e.)0.7

00µ= 1.50 p.e.

0.5

10

0.6

15

20
25
q (p.e.)

0.4

0.3

0.3

0.2

0.2

0.1

0.1
5

10

15

20
25
q (p.e.)µ= 3.00 p.e.

0.5

0.4

0
0

5

10

15

20
25
q (p.e.)

0.7

0.7
0.6

µ= 6.00 p.e.

0.5

0.6

0.4

0.3

0.3

0.2

0.2

0.1

0.1
5

10

15

20
25
q (p.e.)µ=12.00 p.e.

0.5

0.4

0
0

5

0.7

0.6

0
0

µ= 0.70 p.e.

0.5

0
0

5

10

15

20
25
q (p.e.)Figure 6.8: The normalized charge likelihood fq (q|µ) at a range of predictedcharge µ. The data points are obtained from the detector simulation, andthe solid lines indicate the fitted function, which is used when evaluating
the likelihood.

68

6.4. Time Likelihoodfrom direct light µdir and the particle’s momentum p. Since a hit is produced by the first photon arriving at a PMT, the width of the residual time
distribution decreases as photon statistics increases, and such eﬀect is characterized well by the predicted charge. The momentum also translates tothe width of the residual time distribution since the photon time-of-flightcorrection in Equation 6.17 becomes less accurate as the particle track becomes longer, resulting in a wider distribution. All other dependencies on
the configurations of the particle and the PMT are averaged out in thisapproximation, and part of such information is embedded in the predictedcharge.
The residual time likelihood distribution for direct light ftdir (tres ) is assumed to be a Gaussian, whose peak position and σ depending on µdir andmomentum p. In order to produce the residual time distribution, particle
gun MC samples are generated at various fixed momentum, and at each
momentum a 2D histogram of tres -log µdir is filled for every hit caused bya direct photon. The tres distribution is then fitted by a Gaussian at eachlog µdir bin, and its mean and σ are fitted by a sixth-order polynomial inlog µdir . Finally, each of the polynomial fit parameters is fitted as a functionof momentum. The direct light residual time likelihood ftdir (tres ) is thus
parameterized as a smooth function of µdir and p. The resulting likelihood
as well as the original distribution obtained from the MC are shown forelectrons and muons in Figures 6.9 to 6.12.

6.4.2

Indirect Light Time Likelihood

The residual time PDF for indirect light is currently modeled in fiTQun as:(√

ftsct (tres ) = 1/)π
σ + 2γ ×2{exp(−τ 2 /2σ 2 )(τ < 0), (6.19)(τ /γ + 1) exp(−τ /γ) (τ > 0)

where τ = tres − 5 ns, σ = 8 ns and γ = 25 ns. This is an empirical function
which was determined so that it reproduces a typical shape of the indirect
light residual time distribution, and it has a characteristic long right side
tail due to reflected light. No µsct or momentum dependence is consideredat the present moment.

6.4.3

Merging Direct and Indirect Light Time Likelihoods

The direct and indirect light time likelihoods obtained above are then merged
to produce the final overall time likelihood function which appears in Equation 6.2. A further assumption is made here that all indirect photons arrive
696.4. Time Likelihood

0.15

ep=300MeV/cµ= 0.26 p.e.

0.1

0.05

0

ep=300MeV/cµ= 1.05 p.e.

0.1

0.05

-20

0

20

40
t [ns]

0

res

0.2

e-

0.15

p=300MeV/cµ= 3.80 p.e.

-20

0

20

40
tres [ns]ep=300MeV/cµ=15.14 p.e.

0.3
0.2

0.1
0.1

0.05
0

-20

0

20

40
tres [ns]

0

-20

0

20

40
tres [ns]Figure 6.9: The direct light residual time likelihood ftdir (tres ) for 300 MeV/celectrons, at a range of predicted charge µdir . The data points indicate theoriginal distributions obtained from the MC simulation, and the red curves
are the Gaussian likelihood function which is parameterized as a function ofµdir and p.

70

6.4. Time Likelihood

0.1

e-

e-

p=2000MeV/cµ= 0.26 p.e.

p=2000MeV/cµ= 1.05 p.e.

0.1

0.05
0.05

0

-20

0

0.2

20

40
t [ns]

ep=2000MeV/cµ= 3.80 p.e.

0.15

-20

0

20

40
tres [ns]e-0.3p=2000MeV/cµ=15.14 p.e.

0.2

0.1

0.1

0.05
0

0

res

-20

0

20

40
tres [ns]

0

-20

0

20

40
tres [ns]Figure 6.10: The direct light residual time likelihood ftdir (tres ) for2000 MeV/c electrons, at a range of predicted charge µdir . The data points
indicate the original distributions obtained from the MC simulation, andthe red curves are the Gaussian likelihood function which is parameterized
as a function of µdir and p.

71

6.4. Time Likelihoodµp=450MeV/cµ= 0.26 p.e.

0.1

0.05

0

µ-

0.15

p=450MeV/cµ= 1.05 p.e.

0.1

0.05

-20

0

20

40
t [ns]

0

res

0.2

µ-

0.15

p=450MeV/cµ= 3.80 p.e.

-20

0

20

40
tres [ns]µp=450MeV/cµ=15.14 p.e.

0.3
0.2

0.1
0.1

0.05
0

-20

0

20

40
tres [ns]

0

-20

0

20

40
tres [ns]Figure 6.11: The direct light residual time likelihood ftdir (tres ) for 450 MeV/cmuons, at a range of predicted charge µdir . The data points indicate theoriginal distributions obtained from the MC simulation, and the red curves
are the Gaussian likelihood function which is parameterized as a function ofµdir and p.

72

6.4. Time Likelihood

0.08

µ-

0.06

p=2000MeV/cµ= 0.26 p.e.

0.04

µ-

0.1

p=2000MeV/cµ= 1.05 p.e.

0.05

0.02
0

-20

0

20

40
t [ns]

µ-

-20

0

20

40
tres [ns]µ-0.3p=2000MeV/cµ= 3.80 p.e.

0.15

p=2000MeV/cµ=15.14 p.e.
0.2

0.1

0.1

0.05
0

0

res

-20

0

20

40
tres [ns]

0

-20

0

20

40
tres [ns]Figure 6.12: The direct light residual time likelihood ftdir (tres ) for2000 MeV/c muons, at a range of predicted charge µdir . The data points
indicate the original distributions obtained from the MC simulation, andthe red curves are the Gaussian likelihood function which is parameterized
as a function of µdir and p.

73

6.5. Vertex Pre-fit
after any of the direct photons; i.e., an indirect photon can produce a hit only
if there were no direct photons observed at the PMT. The time likelihood
can then be written as:
dir res
sct resft (tresi ) = wft (ti ) + (1 − w)ft (ti ),where(6.20)

1 − e−µ,(6.21)
1 − e−µdir e−µsct
sct resand ftdir (tresi ), ft (ti ) are the normalized residual time likelihoods for direct light and indirect light respectively, at the given predicted charge andmomentum values. The time likelihood is thus constructed on-the-fly by
combining the time likelihoods for direct and indirect light based on theirpredicted charge values.
This method is extended to a multi-particle hypothesis by ordering the
involving particles by the calculated residual time at each PMT, and thenassuming that all the direct photons from a particle with smaller tres always
arrive earlier than any of the direct photons from a particle with larger tres .
dir

w≡

6.5

Vertex Pre-fit

As it was mentioned at the beginning of this chapter, fiTQun is a maximum
likelihood event reconstruction which searches for a particle event hypothesis x which maximizes the likelihood function defined in Equation 6.1. Inpractice, this is done by minimizing the negative log likelihood − ln L(x)with respect to x, using the minimization package MINUIT[84]. In order toensure the minimizer properly locates and converges at the global minimum
rather than being stuck at a local minimum, it is important to seed theminimizer with reasonably accurate parameter values before execution.
The vertex pre-fitter is a fast algorithm which uses only the hit time
information from the PMTs to roughly estimate the vertex position andtime which can then be used as the parameter seed for − ln L minimization,and it is run at the beginning of processing an event before the minimization
is run. The estimation is done by searching for the vertex position x andtime t which maximizes the vertex goodness which is defined as:hit∑iexp(−(Tres/σ)2 /2),(6.22)ii
Tres:= ti − t − |RPMT− x|/cn .

(6.23)G(x, t) :=

i

where

74

6.6. Subevent Algorithm
i is the residual hit time which is similar to Equation 6.17, however, aTres
point light source is assumed to be located at the initial particle vertex inthis case. When the vertex position and time get close to their true values,
i distribute near zero, which results in a large value of the goodness. The
Tres
pre-fitter first maximizes the goodness by performing an iterative coarse
grid-search in space and time, gradually shrinking the grid size and σ. Afterthe grid search is done, a MINUIT minimization of −G(x, t) is performed
with σ = 4 ns, which provides the final seed values for vertex and time.

6.6

Subevent Algorithm

The term “subevent” refers to particle activities in the detector which happen closely in time. An event in Super-K, which is defined as detector
activities in a O(10 µs) time window around an event trigger, may contain multiple subevents; for example, in case there is a neutrino interaction
which produces a decay electron, the initial particles produced at the neutrino interaction and the subsequent decay electron are counted as separate
subevents since they are well separated in time due to the relatively longmuon lifetime of ∼ 2 µs. A subevent algorithm is a mechanism to searchfor such subevent activities around the event trigger and associate the PMT
hits produced by each of them so that precise maximum likelihood reconstruction can be applied to each subevent. The procedures are applied atthe very beginning of processing an event by fiTQun, prior to any of thelikelihood reconstruction.

6.6.1

Peak Finder

The subevent algorithm starts by searching and locating in time the subevent
activities around the event trigger, which is done by the peak finder.
As stated in Section 6.5, the vertex goodness Equation 6.22 takes a large
value when it is evaluated at the vertex position x and time t which are close
to where the actual particle vertex is located, since in such case the residual
hit time distributes near zero for hits that are produced by direct light emitted by the particle. Using this fact, the peak finder searches for subevents
by fixing the vertex position x at the value provided by the vertex pre-fitter,and scanning the goodness while varying the time t. Assuming that all thevertex positions of the primary particles and the decay electrons lie close to
the pre-fit vertex, subevents appear as large peaks in the distribution of thegoodness scanned as a function of t.

75

6.6. Subevent Algorithm

800
700
600
500
400
300
200
100
0

700

800

900 1000 1100 1200 1300 1400
t (ns)

Figure 6.13: Distribution of the scanned goodness G(x, t) as a function oftime t, for an example event with a muon primary and a decay electron.
The vertical dashed lines indicate true particle time, the black dots are thegoodness scan points, and the blue and green curves represent the thresholdcurves which are used for subevent peak selection. The red vertical lines
indicate the time of the subevent peaks which is found by the algorithm.
First, a rough interaction vertex search is done by running the vertex pre-fitter described in Section 6.5 using the PMT hits that are in the[−100 ns, +400 ns] time window around the event trigger. Then, the peakfinder scans and evaluates the goodness every 8 ns step in t while x is fixed at
the pre-fit vertex, starting at ∼300 ns before the event trigger. For this scan,
all the hits within the time window of [−200 ns, 15000 ns] around the event
trigger is used. The parameter σ in Equation 6.22 is set to 6.3 ns, which was
optimized for the step size. Figure 6.13 is an example of the distribution of
the goodness for an event with a single muon and a decay electron, wherethe vertical dashed lines indicate true simulated particle time and black dotsshow the scan points.
In order to select subevent peaks in this distribution, a goodness threshold curve F (t) is defined as:
F (t) := 0.25 max{G(x, ti )f (t − ti )} + η,
i∈M

where
1
f (τ ) :=, γ=
1 + (τ /γ)2{

25 ns (τ < 0)
70 ns (τ > 0).

(6.24)(6.25)
76

6.6. Subevent Algorithm
M represents all local maxima of goodness scan points, i.e. a scan point
which is higher than the neighboring two points, which has a goodness value
larger than the constant threshold η = 9. The blue curve in the figure
represents F (t), and the green curve is 0.6F (t). The first subevent peakis defined as the first local maximum scan point which lies above the blue
curve. The scan then continues onto later time searching for another peak,
and a local maximum is ignored even if it is above the blue curve if there
is no scan point after the previous found peak lying below the green curve.
This double-threshold criterion is employed in order not to double count a
subevent which produces multiple peaks, which often occurs since the photon
time-of-flight subtraction assuming a point light source in Equation 6.23 does
not work perfectly and a single subevent may produce multiple peaks.

6.6.2

Defining Time Windows and Final Subevents

Once the subevent candidates are found by the peak finder, a time window
is defined around each of the peaks which contains the hits associated withthe peak, in which likelihood reconstruction can be performed. Within atime window a PMT can produce a hit at most once, and only the hits thatlie within the time window will be used for likelihood reconstruction; i.e.,
the PMT hits that lie outside of the time window are considered as unhit in
Equation 6.2.
In order to define the time window for a given peak, hit times are first
converted to residual time with respect to the peak using Equation 6.23, andthe earliest and the latest hits in raw hit time ti which satisfy −180 ns <i < 800 ns define the start and the end time of the time window respecTres
tively. The procedure is repeated for all the peaks identified by the peakfinder, and time windows that overlap are merged into a single time window.
After the time windows are defined, the vertex pre-fitter and the peakfinder are run once again in each time window, this time, by only using thehits that are contained within the respective time window. This procedureis necessary especially for the cases where the distance between the primary
and the decay electron vertices is large, since in such case the original pre-fit
vertex is far from the actual decay electron vertex and the photon time-offlight correction in Equation 6.23 works poorly, which results in the peakfinder finding fake subevent peaks around the true decay electron subevent.
The remaining peaks after re-running the peak finder are considered as
the final subevents found in fiTQun, and precise likelihood reconstruction isapplied to each of them as discussed in the following sections. The numberof decay electrons, which is a variable that is often used in various physics
776.7. Single-Ring Fit
analyses, is calculated as the number of the found subevents minus one.

6.6.3

Performance of the Subevent Algorithm

The performance of the subevent algorithm has been studied using the data
and MC samples of stopping cosmic-ray muons, i.e., cosmic-ray muons thatenter the detector and stop in the ID. Stopping muon events are clean single
muon events which almost always leaves a single decay electron, and they
are therefore an ideal source for studying the performance of the algorithm.
Table 6.1 shows the detection eﬃciencies of decay electrons from stoppingmuons for data and MC in sub-GeV and multi-GeV energy range, where theenergy range is defined by whether the muon momentum is below or above
1.33 GeV/c. The eﬃciency here is defined as the number of stopping muon
events in which multiple subevents are found divided by the total numberof events. Note that the detection eﬃciency gets lower at higher energy,
since the more PMTs are hit by the primary muon, the harder it becomesto detect decay electrons that are produced shortly after the parent muon.
The diﬀerence in eﬃciencies between data and MC is taken as the systematic
uncertainty on the decay electron detection eﬃciency.
Energy range
Sub-GeV
Multi-GeV

Data
88.41 ± 0.08%
67.77 ± 0.05%

MC87.81 ± 0.15%
68.02 ± 0.09%

Table 6.1: Decay electron detection eﬃciencies for stopping cosmic-ray
muons for data and MC. The energy range is defined by whether the muon
momentum is below or above 1.33 GeV/c.

6.7

Single-Ring Fit

After the subevents are defined by the subevent algorithm, precise maximum
likelihood event reconstruction as discussed in Sections 6.1 to 6.4 is applied
to each of the subevents. The most basic reconstruction which is applied
to every subevent is the single-ring fitter which is a fit to obtain the singleparticle hypothesis that maximizes the likelihood, which is described in thissection.
fiTQun considers three types of single-ring hypotheses: electron, muonand π + , and the three fits are applied to every subevent that was foundby the subevent algorithm. The electron and muon hypotheses contain78

6.7. Single-Ring Fit
seven parameters, which are the vertex position x, time t, zenith angle andazimuth of the direction θ, ϕ, and momentum p. The π + hypothesis has
one additional parameter and it is therefore an eight parameter hypothesis,
which is described separately in detail in Section 6.8.

6.7.1

Single-Ring Electron & Muon Fit

The single-ring electron fit starts by setting the vertex initially at the position determined by the vertex pre-fit which was run in the time window
to which the subevent belongs, and the time to the peak time. In order todetermine the initial value for the direction, a likelihood scan of the direction is performed by sampling the likelihood at 400 points that are equally
spaced on the unit sphere. During the direction scan, momentum is fixed atthe value which is roughly estimated using the total observed charge. After the direction is determined by the grid scan, the momentum seed value
is set by performing another likelihood scan, this time, by only varying the
momentum. Once all the seven track parameters are seeded, the parameters
are then simultaneously fit by minimizing the − ln L in Equation 6.2 using
the SIMPLEX algorithm in the MINUIT package, which provides the final
maximum likelihood estimation of the particle track parameters assuming asingle electron.
Following the single-ring electron fit is the single-ring muon fit, whichtakes the vertex, time and direction from the electron fit result as their seedvalues. The momentum is then estimated by a likelihood scan as it wasdone in the electron fit, after which the full simultaneous minimization ofthe − ln L is performed.

6.7.2

Performance of the Single-Ring e & µ FitterIn this section, the performance of the single-ring electron and muon fits
described above are compared to APFIT, an event reconstruction algorithm
which preexisted at Super-K before fiTQun. The algorithm of APFIT isfundamentally diﬀerent from fiTQun which is detailed in [85].
Figure 6.14 shows the distribution of the distance between the reconstructed and true vertices for single electron and muon events in the FC
single-ring CCQE event sample in the atmospheric neutrino MC whose true
interaction vertex lies within the fiducial volume(as opposed to the reconstructed vertex as in Section 5.5), compared between APFIT and fiTQun.
The resolution is defined as the 68 percentile of the respective distributions.
Figure 6.15 shows the resolution plotted as a function of true momentum.
796.7. Single-Ring Fit
The angle between the reconstructed and true directions as well as direction
resolution are shown similarly in Figures 6.16 and 6.17.
Figure 6.18 shows the momentum resolution for the same set of events.
For these plots, the distribution of the fractional diﬀerence between thereconstructed and true momenta are fitted by a Gaussian at each true momentum bin, and the width of the fitted Gaussian is plotted.
It can be seen in the figures that fiTQun has higher resolution comparedto APFIT, especially for vertex and momentum reconstruction.

6.7.3

e/µ Particle Identification

As mentioned in Section 6.2.2, Cherenkov light emission happens diﬀerently
between particle types, and the ring pattern they produce will therefore
be diﬀerent. As shown in Figure 6.19, electrons produce fuzzy rings due
to electromagnetic shower while muons which do not shower produce clear
rings with sharp edges, and the diﬀerence arises from the diﬀerences in the
Cherenkov emission as seen in Figures 6.2 and 6.3.
In fiTQun, particle identification (PID) is done by comparing the bestfit likelihood values between diﬀerent particle hypotheses. For example,
electrons and muons are distinguished by making a cut on ln (Le /Lµ ), thelog likelihood ratio between the best-fit electron and muon hypotheses.
Figure 6.20 shows the likelihood separation of the single-ring electron andmuon events in the FC true-fiducial CCQE event sample in the atmospheric
neutrino MC. In order to properly evaluate the PID performance for low
momentum muons, for a muon event it is also required that a decay electron
is found by the subevent algorithm, in order to remove the low momentum
muon events which was triggered on the decay electron. The vertical axesin the plots are ln (Le /Lµ ), and the horizontal axes are the reconstructed
single-ring electron fit momentum. As it can be seen in the figure, electron
events are clearly identified and separated from muon events by making acut:ln (Le /Lµ ) > −10,
(6.26)
as indicated by the magenta line. The cut threshold was determined basedon the MC considering the misidentification rate for both electrons andmuons.
The misidentification rate of the same electron and muon events as above
are shown as a function of true momentum in Figure 6.21. As shown in the
figure, for both particle types the misidentification rate is well below 1% andis smaller for fiTQun than APFIT.

80

6.7. Single-Ring Fit

300
200

Resolution:
APfit : 29.4cm
fiTQun: 20.1cm

100
0
0

300
200

Resolution:
APfit : 22.7cm
fiTQun: 19.3cm

100
0
0

50
100
Dist. btw. tru/rec vtx. [cm]50
100
Dist. btw. tru/rec vtx. [cm]100fiTQun
APFIT

50

0
0

500
1000
1500
True momentum [MeV/c]

Vertex Resolution [cm]Vertex Resolution [cm]Figure 6.14: Single-ring electron(left) and muon(right) vertex resolution for
FC true-fiducial CCQE events in atmospheric neutrino MC, compared between APFIT(black) and fiTQun(red). The resolution is defined as the 68
percentile of the respective distributions.

100

fiTQun
APFIT

50

0

500
1000
1500
True momentum [MeV/c]

Figure 6.15: Single-ring electron(left) and muon(right) vertex resolution
plotted as a function of true momentum, for FC true-fiducial CCQE
events in atmospheric neutrino MC, compared between APFIT(black) andfiTQun(red).

816.7. Single-Ring Fit

200
150

Resolution:
APfit : 2.91°
fiTQun: 2.81°

300
200

Resolution:
APfit : 1.75°
fiTQun: 1.73°

100
100

50
0
0

0
0

5
10
Ang. btw. tru/rec dir. [deg]510Ang. btw. tru/rec dir. [deg]15

10

fiTQun
APFIT

5

0
0

500
1000
1500
True momentum [MeV/c]

Direction Resolution [deg]Direction Resolution [deg]Figure 6.16: Single-ring electron(left) and muon(right) direction resolution
for FC true-fiducial CCQE events in atmospheric neutrino MC, compared
between APFIT(black) and fiTQun(red). The resolution is defined as the68 percentile of the respective distributions.

15

10

fiTQun
APFIT

5

0

500
1000
1500
True momentum [MeV/c]

Figure 6.17: Single-ring electron(left) and muon(right) direction resolution plotted as a function of true momentum, for FC true-fiducial CCQE
events in atmospheric neutrino MC, compared between APFIT(black) andfiTQun(red).

82

10
8
6

fiTQun
APFIT

4
2
0
0

500
1000
1500
True momentum [MeV/c]

Momentum Resolution [%]Momentum Resolution [%]6.7. Single-Ring Fit

1086

fiTQun
APFIT

4
2
0

500
1000
1500
True momentum [MeV/c]

Figure 6.18: Single-ring electron(left) and muon(right) momentum resolution plotted as a function of true momentum, for FC true-fiducial CCQE
events in atmospheric neutrino MC, compared between APFIT(black) andfiTQun(red).

Super-Kamiokande IV

Super-Kamiokande IV

Run 999999 Sub 0 Event 40

Run 999999 Sub 0 Event 21

11-10-30:15:38:38

11-10-30:15:41:52

Inner: 3974 hits, 8035 pe

Inner: 2526 hits, 7078 pe

Outer: 2 hits, 1 pe

Outer: 5 hits, 6 pe

Trigger: 0x03

Trigger: 0x03

D_wall: 678.5 cm

D_wall: 560.0 cm

Charge(pe)Charge(pe)>26.7
23.3-26.7
20.2-23.3
17.3-20.2
14.7-17.3
12.2-14.7
10.0-12.2
8.0-10.0
6.2- 8.0
4.7- 6.2
3.3- 4.7
2.2- 3.3
1.3- 2.2
0.7- 1.3
0.2- 0.7
< 0.2

>26.7
23.3-26.7
20.2-23.3
17.3-20.2
14.7-17.3
12.2-14.7
10.0-12.2
8.0-10.0
6.2- 8.0
4.7- 6.2
3.3- 4.7
2.2- 3.3
1.3- 2.2
0.7- 1.3
0.2- 0.7
< 0.2
1300

500
0 mu-edecays1040

1 mu-e
decay

400

780

300

520

200

260

100

0

0
0

500

1000

Times (ns)

1500

2000

0

500

1000

1500

2000

Times (ns)Figure 6.19: Event displays of simulated single electron(left) andmuon(right) events. Electrons produce fuzzy rings due to electromagnetic
shower, while muons, which do not shower, produce clear rings with sharpedges.

83

6

2000

ln(Le/Lµ)ln(Le/Lµ)6.7. Single-Ring Fit

4

0

30

2000

20

0

10

2
-2000

-2000
0

200 400 600 800 1000
Rec. e momentum (MeV/c)00

200 400 600 800 1000
Rec. e momentum (MeV/c)

0

5
4

e-

3
2
1
0
0

Miss-ID rate [%]Miss-ID rate [%]Figure 6.20: Likelihood separation of single-ring electron(left) andmuon(right) events in the FC true-fiducial CCQE event sample in the atmospheric neutrino MC. The vertical axes are ln (Le /Lµ ), and the horizontal
axes are the reconstructed single-ring electron fit momentum. The magentalines indicate the cut criteria for electron-muon separation.

5
4

µ-

3
2
1

500
1000
1500
True momentum [MeV/c]

0
0

500
1000
1500
True momentum [MeV/c]

Figure 6.21: Misidentification rate of single-ring electron(left) andmuon(right) events in the FC true-fiducial CCQE event sample in the atmospheric neutrino MC, plotted as a function of true momentum. The redmarkers indicate the performance of fiTQun, and the black markers are for
APfit.

84

6.7. Single-Ring Fit

6.7.4

In-Gate Decay Electron Fit

As stated in Section 6.6.2, the initial time window which was defined aroundeach subevent peak found by the peak finder can be merged with anothertime window in cases where two subevents happen closely in time, andin such case, the time window will contain multiple subevents. The decay
electrons which are contained in the same time window as the parent primaryparticles are referred to as “in-gate” decay electrons, and fitting such events
requires a special treatment since the hits that are contained in the time
window have to be distributed among the subevents. By employing the hit
separation scheme described below, the single-ring electron, muon and π +fits are applied to all subevents including the in-gate ones.
For a given in-gate subevent, a PMT hit is considered to be associatedwith the subevent if the hit satisfies:
i−30 ns < Tres
< 60 ns,(6.27)

i is the residual time defined in Equation 6.23 calculated using thewhere Tres
pre-fit vertex and the peak time of the subevent. In order to reconstruct
the in-gate subevent, single-ring fits are performed by using the hits thatsatisfy Equation 6.27 only, while all the other hit PMTs are ignored. This
procedure eﬀectively masks the PMT hits that are caused by the particle
activities that are not contained in the subevent in concern. This masking
and fitting procedure is applied to each of the in-gate subevents in the time
window except for the first, primary subevent.
The primary subevent is reconstructed by masking all the hits that areassociated with the other in-gate subevents, i.e., a hit PMT is considered to
be unhit if the hit satisfies Equation 6.27 for any of the in-gate subevents.
In order to see how the procedures described above work, event displaysof an MC event which has an in-gate decay electron are shown in Figure 6.22.
In this event, in addition to the rings from the primary particles, there isa ring from an in-gate decay electron in the upper half of the figures which
was observed considerably later compared to the other rings, as it is clear
from the hit time distribution on the right. The peak finder finds two in-gate
subevents in this event, and after the hit allocation among the subevents is
performed, the charge distributions for hits that are associated with eachof the two subevents become as shown in Figure 6.23. As shown in the
figures, hits from the primary particles and the in-gate decay electron are
clearly separated. Single-ring fits are performed on each of the subeventsusing these allocated hits.

85

30
2000

25
20

0

4000

2000

0

15
-2000

-2000

10
5

-4000

-4000

-2000

0

2000

4000

-4000

0

-4000

-2000

0

2000

4000

1400
1350
1300
1250
1200
1150
1100
1050
1000
950
900

Time (ns)

35

4000

Charge (p.e.)

6.7. Single-Ring Fit

30
2000

25
20

0

14
4000

12
2000

10
8

0

6

15
-2000

-2000

4

10
5

-4000

-4000

-2000

0

2000

4000

0

Charge (p.e.)

35

4000

Charge (p.e.)

Figure 6.22: Event displays of an MC event which has an in-gate decay
electron. The left plot shows the distribution of the charge of each hit in thetime window, and the right plot shows the hit time. In addition to the rings
due to the primary particles, a ring from an in-gate decay electron is visible
in the upper half of the figures, and they are clearly separated in time as it
is seen in the right figure.

2

-4000

-4000

-2000

0

2000

4000

0

Figure 6.23: Charge distribution for hits that are associated with each ingate subevent, for the same event which was shown in Figure 6.22. The left
figure shows the hits allocated to the first subevent, and the right is for the
in-gate second subevent. As it can be seen from the figures, rings from theprimary particles and the in-gate decay electron are clearly separated.

866.8. Upstream-Track π + Fit

6.8

Upstream-Track π + Fit

Another single-ring hypothesis which has not been discussed yet is the π + hypothesis. As schematically shown on the left of Figure 6.24, when a charged
pion propagates in water it often interacts hadronically with the nuclei inwater, which results in an abrupt change in its direction due to hard scattering or even absorption. The “upstream” portion of the π + trajectory, i.e.,
the part before the hadronic interaction happened, thus produces a thin andsharp ring pattern which is quite distinct from that of electrons or muonsas seen in the event display on the right of Figure 6.24.
Super-Kamiokande IV

Upstream ringRun 999999 Sub 0 Event 5
11-10-31:20:15:13
Inner: 1180 hits, 2736 pe
Outer: 3 hits, 2 pe
Trigger: 0x03
D_wall: 270.6 cm

Charge(pe)>26.7
23.3-26.7
20.2-23.3
17.3-20.2
14.7-17.3
12.2-14.7
10.0-12.2
8.0-10.0
6.2- 8.0
4.7- 6.2
3.3- 4.7
2.2- 3.3
1.3- 2.2
0.7- 1.3
0.2- 0.7< 0.2
240
2 mu-edecays

192
144

Downstream ring

96
48
0
0

500

1000

1500

2000

Times (ns)Figure 6.24: On the left is a schematic diagram of a charged pion which
undergoes a hadronic scattering on a nucleus and the rings it produces. The
thin cyan ring represents the “upstream” ring which is produced by theparticle track before the scattering. On the right is an event display of asimulated single π + event. The pion underwent multiple hadronic scatteringand several upstream rings are seen as a result.
The “upstream-track” π + hypothesis is designed to reconstruct such athin upstream ring which is produced from a truncated charged pion track.
Figure 6.25 schematically shows how an upstream-track hypothesis is constructed in fiTQun. The “anti-track” starts at the assumed position of thehadronic interaction where the π + track gets truncated, and the predictedcharge deposited by the anti-track is subtracted from the predicted charge
from the full track which originates at the initial π + vertex. The remaining charge will then correspond to the predicted charge from the upstream
track which is represented by the outer cyan ring in the figure. Predictedcharges for both the full track and the anti-track are calculated using thesame framework as in the regular single-ring hypothesis.
876.8. Upstream-Track π + Fit

Anti-track

Full track

Figure 6.25: A schematic diagram describing how an upstream-track hypothesis is constructed in fiTQun. The predicted charge from the anti-track(dashed red arrow) is subtracted from the predicted charge from the fulltrack (solid blue arrow), in order to produce the predicted charge distribution for the upstream track which is represented by the outer cyan ring.
In practice, the upstream-track hypothesis has an additional parameter
Eloss which represents the kinetic energy that was lost in the upstream-track,in addition to the seven parameters which the regular single-ring hypothesis
has. The momentum of the anti-track is determined by the energy left after
subtracting Eloss from the initial total energy. By assuming constant kinetic
energy loss per unit track length, the vertex of the anti-track is calculated byconverting Eloss to the distance using a constant conversion factor. Finally,
for time likelihood, residual time as in Equation 6.17 is calculated using the
midpoint of the upstream track, not the full track.
The upstream-track single-ring π + fit is performed after the single-ringmuon fit is done, taking the muon fit results as parameter seeds. This is aneight-parameter fit and all parameters are simultaneously fitted by MINUIT.

6.8.1µ/π + Identification

As muon and π + have similar mass, in absence of any hadronic interaction
the two produce similar ring pattern. π + ’s are therefore often hard to distinguish from muons, and they often become backgrounds to muon signals in
Super-K. The upstream-track π + fitter, however, provides means to separate
the two particle types.
Figure 6.26 shows the likelihood separation of νµ CCQE and NCπ + eventsin the T2K accelerator neutrino MC simulation, which has a narrow-band
neutrino flux spectrum peaked at ∼ 600 MeV. The vertical axes are the88

6.8. Upstream-Track π + Fit

Figure 6.26: Likelihood separation of νµ CCQE(left) and NCπ + (right)
events, in the T2K accelerator neutrino MC simulation. The vertical axes
are ln(Lπ+ /Lµ ) and the horizontal axes are Eloss .
log likelihood ratio ln(Lπ+ /Lµ ) between the best-fit upstream-track π + fit
and the muon fit, and the horizontal axes are the Eloss parameter fromthe π + fit. It can be seen in the figures that, although there are someindistinguishable events, a significant fraction of π + events can be clearly
separated from muons. A study shows that introducing a cut as indicated by
the black lines can reduce the NCπ + background contamination in the T2K
νµ signal sample by ∼ 60%. Although the µ/π + identification is not usedin the atmospheric neutrino analysis at the present moment, it is expectedthat introducing such selection in future analyses will reduce the NCπ +backgrounds in the signal single muon event samples.
It is interesting to note that the log likelihood ratio for muon events distribute near zero as in the left of Figure 6.26, which means that muons are
fit almost equally well by the muon and the π + hypotheses (although thereconstructed kinematics may be biased). As muons and π + ’s are both nonshowering particles and have similar mass, the Cherenkov emission profileof the two are similar, and the π + hypothesis can therefore reproduce muonrings well by increasing the Eloss parameter to the point where the momentum at the assumed upstream-track endpoint drops below the Cherenkov
threshold. The fact that muon rings can be fit by the upstream-track π +hypothesis is used in the multi-ring fitter, which will be described later inthis chapter.

89

6.9. π 0 Fit

0Conversion point

Figure 6.27: A schematic diagram showing how the π 0 hypothesis is constructed. Rings from two electron tracks, as indicated by the green arrows,
are combined in this hypothesis, and the two tracks point back to a single
vertex while the actual origin of each track has some displacement along the
track which corresponds to the photon conversion length.

6.9

π 0 FitA π 0 in most cases decays into two gamma rays, and as a gamma ray produces a ring pattern which is almost indistinguishable to an electron in awater Cherenkov detector, a decayed π 0 produces two electron-like rings
which point back to a common single vertex. The π 0 fitter in fiTQun is a
dedicated two-ring fitter which is designed to fit such π 0 events.
Figure 6.27 schematically shows how a π 0 hypothesis is constructed in
fiTQun. Two electron rings are combined in this hypothesis, and the rings
are placed so that the two tracks point back to a single vertex which corresponds to the assumed π 0 vertex. Since it takes some finite conversion
length for a γ to convert into a e+ + e− pair and produce electromagnetic
shower, the vertex of each electron ring is shifted along its track by a free parameter. The π 0 hypothesis therefore has twelve parameters: the direction
θi , ϕi , momentum pi and the conversion length dconv
for each ring (i = 1, 2),iand the common vertex x, y, z, t.
The π 0 fit starts by making a rough estimate for the twelve fit parameters
using a dedicated seeding procedure. Taking the result of the single electron
fit as the seed and fixing the parameters there for the first ring, the secondring is searched by placing the second ring and looking for the direction of thesecond ring which minimizes the − ln L at 100 points uniformly distributedon the unit sphere surface. During this search, the momentum of the secondring is set arbitrarily to 50 MeV/c, and the conversion lengths of the tworings are set to 50 cm. After the second ring search is done, the momentaof the rings p1 , p2 are fit simultaneously while the other ten parameters are
fixed at the seed values to improve the momentum estimation. Finally, all
90

300

60

200

40

100

20

0
0

1.5

400

π

80

ln(L 0/Le)

400πln(L 0/Le)

6.9. π 0 Fit

300

1

200
0.5

100
0

100

200
300
400
Rec. mπ0 (MeV/c2)

0

0

100

200
300
400
Rec. mπ0 (MeV/c2)

0

Figure 6.28: CC single electron(left) and NC single π 0 (right) events in the
FC true-fiducial atmospheric neutrino MC sample, separated using the π 0
fit variables. The vertical axes are the log likelihood ratio ln(Lπ0 /Le ) andthe horizontal axes are the reconstructed invariant mass of the π 0 fit.
the twelve parameters are fit simultaneously to get the final best-fit values forthe parameters. As in the single-ring fit case, the fit is done by minimizingthe − ln L using the SIMPLEX algorithm in MINUIT.

6.9.1

e/π 0 Identification

π 0 s were known as one of the main backgrounds to electron signals in water
Cherenkov detectors as a π 0 becomes indistinguishable to electrons if oneof the two gamma rings are missed by event reconstruction, which happens
often for high momentum π 0 s which have significant Lorentz boost. Oneof the main applications of the π 0 fitter is to eﬃciently separate π 0 s fromelectrons and reduce the background contamination in electron signal event
samples.
Figure 6.28 shows the CC single electron and NC single π 0 events in the
FC true-fiducial atmospheric neutrino MC sample, separated using the π 0 fit
variables. In the plots, the vertical axes are ln(Lπ0 /Le ), the log likelihood
ratio between the best-fit π 0 and electron hypotheses, and the horizontal
axes mπ0 are the reconstructed invariant mass calculated from the π 0 fit
result. As shown in the figures, the two particle types are separated well inthe 2D distributions. By introducing a π 0 rejection line cut:ln(Lπ0 /Le ) < 175 − 0.875 × mπ0 (MeV/c2 ),(6.28)

as indicated by the magenta lines in the plots, the two event categories canbe separated by the misidentification rate shown in Figure 6.29.
91

50
40

Miss-ID rate [%]Miss-ID rate [%]6.10. Multi-Ring Fit

e-

30
20
10
0

25
20

π0

15
10
5

500
1000
1500
True momentum [MeV/c]

0

500
1000
1500
True momentum [MeV/c]

Figure 6.29: Misidentification rate for the π 0 rejection cut Equation 6.28, for
the CC single electron(left) and NC single π 0 (right) events in the FC truefiducial atmospheric neutrino MC sample, plotted as a function of the true
particle momentum. For electrons the misidentification rate is calculated as
the fraction of events above the cut line in Figure 6.28, while for π 0 ’s thefraction below the cut line is plotted.

6.10

Multi-Ring Fit

In atmospheric neutrino analyses, it is essential to reconstruct and use theevents with multiple particles, since a large fraction of events in multi-GeV
have multi-particle final states. This section describes the multi-ring fitter,
which is designed to identify and reconstruct up to six rings in an event.
The multi-ring fitter is applied only to the first subevent in an event; i.e.,the detector activities caused by the particles which immediately follow theinitial neutrino interaction, and not the subsequent decay electrons.

6.10.1

Initial Multi-Ring Fit

The process of reconstructing a multi-ring event starts by doing an exhaustive search for any possible ring candidates. This is done by doing an iterative search for an additional ring by increasing the number of rings oneby-one, considering electron and upstream-track π + ring hypotheses. The
electron rings are intended to fit the electron or gamma-ray rings, while theupstream-track π + rings are expected to fit muon or π + rings since they canboth be fit by π + rings as mentioned in Section 6.8.1.
The tree diagram in Figure 6.30 shows how the multi-ring hypotheses
evolve as the number of rings is increased in this ring search procedure.
Starting from the single-ring electron fit result, an additional electron ring92

6.10. Multi-Ring Fit

1R

e

2R

ee

3R
4R

eπ

eee
eeee

eeπ
eeeπ

eeπe

eπe
eeππ

eπee

eππ
eπeπ

eππe

eπππ

Figure 6.30: A tree diagram showing how the tree of the multi-ring hypotheses evolve as the number of rings is increased. The diagram is for the caseassuming the first ring as electron, and equivalent procedure is done for thecase of assuming π + as the first ring.
is attached at the same vertex as the first electron ring, and the likelihood is
evaluated at diﬀerent directions for the second ring at 400 points which are
uniformly distributed on the unit sphere surface. For this scan the secondring momentum is arbitrarily assumed to be 50 MeV/c. Then, the scanpoint with the minimum − ln L is chosen and the momentum of the secondring is fitted while all the other parameters are fixed. After the momentumof the second ring is roughly estimated, the direction and the momentum of
the second ring are simultaneously fitted, while the common vertex and the
direction and the momentum of the first ring are fixed. The momenta ofthe two rings are then fit simultaneously while the vertex and the directionsof the two rings are fixed in order to more properly distribute the observed
visible energy among the rings, as the single-ring fit momentum tends tooverestimate the momentum of the first ring due to the influence of thecharge from the additional rings. Finally, the directions and momenta ofthe two rings as well as the rings’ common vertex are all simultaneously fit
to get the final best-fit two-ring result. After the two-ring fit assuming anelectron second ring is done, the procedure above is repeated using a π + ringas the second ring, this time, also fitting the Eloss parameter simultaneouslywith the other fitted parameters.
Once the two-ring fits are done, for each of the two particle hypotheses
for the new ring, whether the fitted second ring is an actual true ring ischecked by comparing the likelihoods between the hypotheses before andafter adding the new ring. Figure 6.31 shows the square root of the log
likelihood ratio between the best-fit single-ring electron hypothesis and the93

6.10. Multi-Ring Fit

600

SK4 Data
νeCC 1e
νeCC oth.
νµCC 1µνµCC oth.
NC

400
200
0

0

10
20
30
40
50
p
fQ(L
RC
likelihood
sgn ⇥ | ln
/L1Re ) |2Rex

Figure 6.31: Distribution of the square root of the log likelihood ratio between the best-fit single-ring electron hypothesis and the 2R hypothesis assuming electron as the first ring, for FC sub-GeV atmospheric neutrino MC
events with no decay electrons detected. MC distribution is subdivided by final state categories and shown in diﬀerent colours. The data points overlaid
are the SK-IV FC data.
better of the 2R hypotheses assuming electron as the first ring(for brevity,
such variable will be called the “ring-counting likelihood” henceforth), forFC sub-GeV atmospheric neutrino MC events with no decay electrons detected. In the plot, MC events are categorized by the final states, basedon whether there are any π 0 ’s or visible π ± ’s or protons in the final state.
As shown in the figure, true single-ring events such as the CC single electron events(blue) have relatively small value for the ring-counting likelihood,
while the value is large for multi-ring events, for example for NC events(grey)
which is dominated by single π 0 events. A cut on the ring-counting likelihood is made at a constant threshold(9.35 in this case), and if the likelihood
is below the threshold the new ring is assumed to be fake and no further
action is done. If the likelihood exceeds the threshold, on the other hand,
it is assumed that the new ring is a true ring, and another ring is searchedbased on the two-ring fit result and examined using the procedures describedabove.
As schematically shown in Figure 6.30, the above procedures of adding,946.10. Multi-Ring Fit
fitting and examining the new ring is iterated until either the newly added
ring fails the true ring criterion or reach six rings which is the maximum
number of rings fiTQun can handle. The simultaneous fit of all parameters
including the direction of the rings and the common vertex is only done
for the two-ring fit case, and for anything higher, only the simultaneous
momentum fit of all rings is done. This is done mainly to save computation
time, and it is assumed that once the simultaneous two-ring fit is done thefit vertex gives a reasonable estimate for the true interaction vertex. Also,
for the fifth and the sixth ring, only electron hypothesis is considered to savecomputation time.
The entire procedure above is also repeated assuming π + as the first ring,and once all the branches of the multi-ring hypothesis tree are terminated,the hypothesis with the smallest − ln L is chosen as the “seeding multiring hypothesis” among the hypotheses that terminates a branch, i.e., thehypotheses which do not contain any fake rings. The chosen hypothesis willbe the seed hypothesis for the next step of the multi-ring fit.

6.10.2

Sequential Multi-Ring Fit

The final multi-ring fit hypothesis obtained in Section 6.10.1 often contains
fake rings as well misidentified rings due to the limitations of the seeding
procedure. In order to improve the fit result, each ring is refitted sequentiallyand its particle type is reevaluated following the procedures below.
The rings in the final seeding multi-ring hypothesis obtained in Section 6.10.1 are first reordered in terms of the visible energy of the ring, i.e.,
the reconstructed kinetic energy above Cherenkov threshold. Then, for themost energetic ring, the angle between the directions of the ring and eachlower energy ring is calculated, and if the angle for a given lower energy ringis smaller than 20◦ , the lower energy ring is merged to the highest energy
ring by adding the visible energy. After that, the highest energy ring is
re-fitted assuming three particle types: electron, muon and π + , while all theother remaining rings which were not merged in the ring-merge procedure
are fixed at their original configurations. This is essentially a single-ring fit
in the presence of other fixed rings, and all the ring parameters are fit simultaneously including the vertex, which will in the end provide a multi-ring
hypothesis with the vertex position of the rings not necessarily being at asingle location. The type of the most energetic ring, whether it is a showering(such as electron and gamma) or a non-showering(such as muon andπ + ) ring, is then examined by comparing the best-fit likelihoods between
the hypotheses assuming the most energetic ring as electron or π + . The
95

6.10. Multi-Ring Fit
criterion for selecting it as a showering(electron) ring is:ln (Le /Lπ+ ) > −10,

(6.29)

which is analogous to Equation 6.26, but instead of the muon the π + ring
hypothesis is used here since this way both muons and π + ’s can be distinguished from electrons. The ring type selected by this cut is accepted
as the final type of the ring(showering or non-showering), and the ring in
the original multi-ring hypothesis is replaced by the selected re-fitted hypothesis (the cases for muon and π + is described below in more detail).
After the most energetic ring is re-fitted, the above procedure is repeated
for the subsequent lower energy rings which remained after ring merging,in the descending order of the ring energy until all the remaining rings are
re-fitted.
At the present moment, the above e/π + cut is the only particle identification cut that is applied in the sequential fit procedure, and the µ/π +
identification as described in Section 6.8.1 is not applied. If the mostenergetic ring is determined to be non-showering(π + -like) as opposed toshowering(electron-like), the ring is always assumed to be a muon, sincesuch case usually happens only for νµ CC events with a true muon ring andrarely happens for NC events with a true π + ring being most energetic. For
other rings, a ring which is identified as π + -like rather than electron-like isalways assumed as a π + .
The fit result obtained after the entire sequential fit procedure above isthe final result which will be used as fiTQun’s best estimate for the event’s
final state and is used in the atmospheric neutrino analysis.

6.10.3

Performance of the Multi-Ring FitterIn this section, the ring-counting and particle identification performance ofthe fiTQun multi-ring fitter is compared to APFIT, the preexisting Super-K
reconstruction which was mentioned in Section 6.7. The dataset used hereis the FC atmospheric neutrino MC normalized to 2519.89 days exposure
which is the same as the data exposure in SK-IV.
Figures 6.32 and 6.33 show the reconstructed event categories for theνe CC single electron events and the NC single π 0 events respectively, compared between fiTQun and APFIT. As shown in the figures, fiTQun has
a higher fraction of NC1π 0 events which are properly reconstructed as twoelectron ring events(shown in green), while the fraction of reconstructing CC
single electron events properly as single electron ring event(shown in grey)
is at a similar level between fiTQun and APFIT. Furthermore, Figure 6.34
96

6.10. Multi-Ring Fit

νeCC0π00π±: fiTQun Ctg.

νeCC0π00π±: APfit Ctg.

2000

2000

1000

1000

0

-1

-0.5

0 0.5 1
Log (E [GeV])
10

0

-1

-0.5

vis

0 0.5 1
Log (E [GeV])
10

vis

Figure 6.32: Reconstructed event categories of νe CC events with no visible
pions in final state in the FC atmospheric neutrino MC, for fiTQun(left)
and APFIT(right). The horizontal axes are the log of the visible energy inGeV. Reconstructed single electron ring event category, which corresponds
to the true category, is shown in grey.

NC1π00π±: fiTQun Ctg.

NC1π00π±: APfit Ctg.

800

800

600

600

400

400

200

200

0

-1

-0.5

0 0.5 1
Log (E [GeV])
10

vis

0

-1

-0.5

0 0.5 1
Log (E [GeV])
10

vis

Figure 6.33: Reconstructed event categories of NC events with a single π 0and no visible π ± in final state in the FC atmospheric neutrino MC, forfiTQun(left) and APFIT(right). The horizontal axes are the log of thevisible energy in GeV. Reconstructed two electron ring event category, whichcorresponds to the true category, is shown in green.

976.10. Multi-Ring Fit

νeCC1π00π±: fiTQun Ctg.

νeCC1π00π±: APfit Ctg.

200

200

150

150

100

100

50

50

0

-1

-0.5

0 0.5 1
Log (E [GeV])
10

vis

0

-1

-0.5

0 0.5 1
Log (E [GeV])
10

vis

Figure 6.34: Reconstructed event categories of νe CC events with a single
π 0 and no visible π ± in final state in the FC atmospheric neutrino MC,
for fiTQun(left) and APFIT(right). The horizontal axes are the log of thevisible energy in GeV. Reconstructed three electron ring event category,
which corresponds to the true category, is shown in blue.
shows that fiTQun’s eﬃciency for properly reconstructing νe CC 1π 0 eventsas three electron ring events(shown in blue) is significantly higher comparedto APFIT. Similarly, reconstructed event categories for νµ CC single muonand νµ CC 1π 0 events are shown in Figures 6.35 and 6.36, which lead to asimilar conclusion.
In order to see how well the multi-ring kinematics are reconstructed,
Figure 6.37 shows the reconstructed invariant mass calculated from the 2nd
and the 3rd rings in the events categorized as 3e ring events by fiTQun and
APFIT. The plots show that the true νe CC 1π 0 events(shown in blue) which
are targeted by the ring selection are selected with much higher eﬃciency andpurity by fiTQun than APFIT, and a clear π 0 mass peak is seen. Figure 6.38
shows the equivalent plots for νµ CC 1π 0 events.
From these distributions, it is clearly seen that fiTQun’s multi-ring fitter has a significantly better performance in properly identifying and reconstructing the rings in multi-ring events compared to the preexisting APFIT
reconstruction.

986.10. Multi-Ring Fit

νµCC0π00π±: fiTQun Ctg.

νµCC0π00π±: APfit Ctg.

1500

1500

1000

1000

500

500

0

-1

-0.5

0 0.5 1
Log (E [GeV])
10

0

-1

-0.5

vis

0 0.5 1
Log (E [GeV])
10

vis

Figure 6.35: Reconstructed event categories of νµ CC events with no visible
pions in final state in the FC atmospheric neutrino MC, for fiTQun(left)
and APFIT(right). The horizontal axes are the log of the visible energy inGeV. Reconstructed single muon ring event category, which corresponds tothe true category, is shown in brown.

νµCC1π00π±: fiTQun Ctg.

νµCC1π00π±: APfit Ctg.

200

200

150

150

100

100

50

50

0

-1

-0.5

0 0.5 1
Log (E [GeV])
10

vis

0

-1

-0.5

0 0.5 1
Log (E [GeV])
10

vis

Figure 6.36: Reconstructed event categories of νµ CC events with a single
π 0 and no visible π ± in final state in the FC atmospheric neutrino MC,
for fiTQun(left) and APFIT(right). The horizontal axes are the log of thevisible energy in GeV. Reconstructed 1µ + 2e ring event category, whichcorresponds to the true category, is shown in cyan.

996.10. Multi-Ring Fit

0.32 < Evis < 1.00GeV : fiTQun sel.

0.32 < Evis < 1.00GeV : APfit sel.

νeCC1π0:100.77 evts.
Purity: 61.2 %νeCC1π0:58.65 evts.
Purity: 44.1 %

20
10
0
0

10

100
200
300
Rec. Invar. Mass[MeV/c2]

1.00 < Evis < 3.16GeV : fiTQun sel.

20
15

20

νeCC1π0:71.81 evts.
Purity: 62.9 %00

1.00 < Evis < 3.16GeV : APfit sel.

20
15

10

10

5

5

0
0

100
200
300
Rec. Invar. Mass[MeV/c2]

100
200
300
Rec. Invar. Mass[MeV/c2]

00νeCC1π0:23.73 evts.
Purity: 45.6 %

100
200
300
Rec. Invar. Mass[MeV/c2]

Figure 6.37: Reconstructed invariant mass calculated using the 2nd and the
3rd rings for events reconstructed as 3e ring by fiTQun(left column) and
APFIT(right column), in the FC atmospheric neutrino MC. The upper row
is for visible energy in the range 0.32 − 1.00 GeV, and the lower row is for1.00 − 3.16 GeV. Each true final state is shown in diﬀerent colour in theplots, and the true νe CC 1π 0 events which are targeted by the ring selection
is shown in blue. As shown in the plots, a clearer and larger peak is seenin the fiTQun plots near the π 0 mass ∼ 135MeV/c2 , which indicates thatthe π 0 was reconstructed properly with higher eﬃciency. The event rate
and the purity of the νe CC 1π 0 events which pass the invariant mass cut as
indicated by the magenta arrows are also shown in the plots.

100

6.10. Multi-Ring Fit

0.32 < Evis < 1.00GeV : fiTQun sel.

4030

νµCC1π0:124.43 evts.
Purity: 57.4 %0.32 < Evis < 1.00GeV : APfit sel.

40
30

20

20

10

10

0
0

100
200
300
Rec. Invar. Mass[MeV/c2]

1.00 < Evis < 3.16GeV : fiTQun sel.

3020

νµCC1π0:114.15 evts.
Purity: 72.4 %

10
0
0

0
0

νµCC1π0:57.81 evts.
Purity: 44.8 %

100
200
300
Rec. Invar. Mass[MeV/c2]

1.00 < Evis < 3.16GeV : APfit sel.

3020

νµCC1π0:27.27 evts.
Purity: 49.3 %

10

100
200
300
Rec. Invar. Mass[MeV/c2]

0
0

100
200
300
Rec. Invar. Mass[MeV/c2]

Figure 6.38: Reconstructed invariant mass calculated using the 2nd and the
3rd rings for events reconstructed as 1µ + 2e ring by fiTQun(left column)
and APFIT(right column), in the FC atmospheric neutrino MC. The upper
row is for visible energy in the range 0.32 − 1.00 GeV, and the lower row
is for 1.00 − 3.16 GeV. Each true final state is shown in diﬀerent colourin the plots, and the true νµ CC 1π 0 events which are targeted by the ring
selection is shown in cyan. As shown in the plots, a clearer and larger peakis seen in the fiTQun plots near the π 0 mass ∼ 135MeV/c2 , which indicatesthat the π 0 was reconstructed properly with higher eﬃciency. The event
rate and the purity of the νµ CC 1π 0 events which pass the invariant mass
cut as indicated by the magenta arrows are also shown in the plots.

1016.11. fiTQun in SK-I to III

6.11

fiTQun in SK-I to III

As mentioned in Section 2.3, there are four distinct detector phases in SuperK, and only the data from the latest SK-IV phase is used in the atmospheric
neutrino oscillation analysis which is presented in Chapter 8. The data fromthe old SK-I to III phases is not used in the analysis at the present moment
due to the large data-MC discrepancies observed in the reconstructed variable distributions from fiTQun, and usage of such data requires improvements in the detector simulation in those phases.
As an example of such data-MC discrepancy, Figure 6.39 shows the dataMC comparison of the distribution of the log likelihood ratio between the
single-ring electron and muon fits ln (Le /Lµ ) discussed in Section 6.7.3, for
the FC sub-GeV (i.e. visible energy below 1330 MeV) single-ring atmospheric neutrino events. It can be seen from the plots that while the distribution agrees well in SK-IV, there are rather large discrepancies between
data and MC for SK-I and III, namely, the single electron events in data
are shifted to the left such that the electron-muon separation appears to beworse in data compared to MC.
In general, SK-IV is the most stable of the four detector phases due tothe various hardware improvements such as the electronics and the water
system, and because of the sophisticated calibration methods described in
Chapter 3 which have significantly improved over the years the detector
simulation is also more detailed and better tuned in SK-IV. The data-MC
agreement is therefore overall better in SK-IV compared to the other phases.
One of the potential improvements one can make in the detector simulation for SK-I to III is the time response of the PMTs. As discussed in
Section 3.4, the time resolution of the ID PMTs is obtained from the thetiming calibration data using a laser system, and the time response is asymmetric having a longer tail in later time as Figure 3.4 shows. For SK-IV, suchasymmetric time response is simulated by an asymmetric Gaussian based on
the measured time resolution, however, for SK-I to III the response is simulated by a simple symmetric Gaussian. As the fiTQun likelihood uses thedetailed time information, such diﬀerences in the time response between thereality and the simulation can directly manifest as the data-MC discrepancies in the fitted likelihoods. By reprocessing the old TQ calibration data,
it is possible to improve the simulation for SK-I to III to incorporate theasymmetric time response.
Diﬀerences in the light attenuation length in water also cause diﬀerencesin the fiTQun outputs. Figure 6.40 shows the variation of the measured
light attenuation length over time, and as the figure shows the water quality1026.11. fiTQun in SK-I to III

800
600
400
200
0

200

100

0

SK1 Data
νeCC 1e
νeCC oth.
νµCC 1µνµCC oth.
NC

-2000

400

200

0

2000
ln(Le/Lµ)

SK3 Data
νeCC 1e
νeCC oth.
νµCC 1µνµCC oth.
NC

-2000

0

1000

500

0

2000
ln(Le/Lµ)0

SK2 Data
νeCC 1e
νeCC oth.
νµCC 1µνµCC oth.
NC

-1000

0

1000
ln(Le/Lµ)02000
ln(Le/Lµ)

SK4 Data
νeCC 1e
νeCC oth.
νµCC 1µνµCC oth.
NC

-2000

Figure 6.39: Distributions of the log likelihood ratio between the fiTQun
single-ring electron and muon fits ln (Le /Lµ ) for the FC sub-GeV single-ringatmospheric neutrino events in each SK phase. The plot on the top-left, topright, bottom-left and bottom-right is for SK I, II, III and IV respectively.
Points with error bars represent the data, and MC expectations are broken
down by final state categories where 1e and 1µ represent the events in which
the only visible particle being an electron or a muon.

103

Attenuation length (m)6.11. fiTQun in SK-I to III

120

100

80

60
2000

2005

2010

2015
Year

Figure 6.40: Time variation of the light attenuation length in water. The
four detector phases are shown in diﬀerent colours, where SK I, II, III and
IV are shown in black, green, blue and red respectively.
in SK-IV is much more stable compared to the other phases. The variations in water quality such as the changes in absorption or scattering aﬀects
the charge pattern of the Cherenkov rings and hence aﬀects the likelihood.
Currently, atmospheric neutrino MC is produced without considering suchvariations in the detector properties over time, and the simulation is doneusing the calibration constants measured at a single point in time in eachdetector phase. For instance, the water in SK-IV is simulated based on
the water calibration data taken in April 2009. When the property of thewater significantly deviates in data compared to the simulation, the difference manifests as data-MC discrepancies in the fiTQun likelihoods. One
potential solution to improve the data-MC agreement is to simulate the time
variations in the MC based on the available calibration data.
It is thus expected that the detector simulation in SK-I to III can be
improved in near future by using the available calibration data so that one
can benefit from the advanced reconstruction performance of fiTQun whenanalyzing the data from the old SK phases.

104

Chapter 7First Application of fiTQun:
T2K νe Appearance
As an example application of fiTQun, this chapter briefly describes the T2K
νe appearance analysis which was the first application of fiTQun in a physics
analysis. In the analysis, fiTQun is used to reduce the NC π 0 background in
the signal νe event sample observed at Super-K, resulting in ∼62% reductionof the NC background compared to the previous π 0 rejection method while
the loss in the signal eﬃciency is only ∼2%. The analysis result has been
published by the T2K collaboration in [29], which reports the discovery ofthe νµ → νe oscillation.

7.1

T2K ExperimentT2K is a long-baseline accelerator neutrino oscillation experiment located inJapan. A nearly pure νµ beam produced at the J-PARC accelerator facilityin Tokai is detected at Super-K at the distance of 295 km, and the flavourchange of the neutrinos are measured precisely.
As schematically shown in Figure 7.1, at the beam production site, protons accelerated to 30 GeV at the J-PARC main ring are collided with agraphite target producing mainly charged pions with some kaons, and thepions are subsequently focused by three magnetic horns. The focused pions

Figure 7.1: Schematic drawing of the T2K beam production site. Figure
taken from [23].
1057.1. T2K Experiment

Figure 7.2: T2K νµ beam spectrum at diﬀerent oﬀ-axis angles. The νµsurvival probability at SK is also shown on the top. Figure taken from [86].
then undergo the decay π + → µ+ + νµ in a decay volume, producing an
intense νµ beam. The produced µ+ ’s are mostly stopped at the beam dump
so that their decay ν̄µ and νe , which become backgrounds to the oscillation
measurements, are at very low energies and are not focused towards thebeam direction. By reversing the current to the magnetic horns, one caneither focus π + and defocus π − or vice versa and thus choose whether toproduce a νµ or ν̄µ beam. SK is located 2.5◦ oﬀ-axis from the beam direction, which results in a narrow neutrino energy spectrum peaked at 0.6 GeV
at SK as shown in Figure 7.2. The peak energy is where the ∆m232 -induced
oscillation eﬀects become maximal given the known value for ∆m232 and the
295 km baseline. The neutrino beam is pulsed, and a precise GPS timing
system allows one to only select the T2K beam neutrino events at SK withbeam-unrelated backgrounds being ∼0.002%. The neutrinos before oscillation are measured at a near detector complex located at 280 m from thetarget, which consists of an on-axis detector(INGRID) and an oﬀ-axis detector(ND280) at 2.5◦ . Figure 7.3 shows a schematic drawing of ND280 which
is a suite of several sub-detectors enclosed in a magnet producing magnetic
field of 0.2 T. The primary components of ND280 used in the oscillation
analysis are the fine-grained detectors (FGDs) which are plastic scintillator
1067.1. T2K Experiment

Figure 7.3: A schematic drawing of ND280. Figure taken from [87].
trackers and also serve as the target for neutrino interaction, and the time
projection chambers (TPCs) which are placed next to the FGDs and areused to determine the type and the momentum of the particles produced in
the FGDs. The near detectors are used to precisely measure the propertiesof the neutrino beam and neutrino cross sections in order to constrain thesystematic uncertainties in oscillation analyses.
The initial goals of T2K are the discovery of νµ → νe oscillation induced
by nonzero θ13 and the precision measurement of θ23 and ∆m232 through νµdisappearance, i.e. the measurement of the survival probability νµ → νµ .
Given the value of θ13 which is recently precisely measured by the reactor
experiments, CP violation can also be probed by T2K through a CP-odd
sub-leading term in the νµ → νe oscillation probability which depends onsin δCP :∆m231 L4E
∆m221 Lsin 2θ12 sin 2θ23∆m231 Lsin−
sin2 2θ13 sin2
sin δCP2 sin θ13
4E
4E
+ (CP even, solar and matter eﬀect terms).
(7.1)

P (νµ → νe ) ≈ sin2 θ23 sin2 2θ13 sin2107

7.2. νe Appearance Analysis

Figure 7.4: Predicted T2K neutrino flux at SK. Figure provided by the T2K
beam working group.

7.2

νe Appearance Analysis

The initial goal of the T2K νe appearance analysis is to discover the νµ → νeoscillation and precisely measure its oscillation probability, whose leading
term depends on sin2 2θ13 as shown in Equation 7.1. This is done by observing beam-induced νe events appearing at SK in a beam which is originallyproduced consisting almost entirely of νµ . For a precise measurement, it isessential to accurately predict the neutrino flux at SK and also constrain theuncertainties related to neutrino interactions at SK, so that neutrino events
can be simulated accurately and properly compared to the observed datafor oscillation analyses.
For the neutrino flux prediction, hadron production at the graphite target is simulated using FLUKA[88] and the data from a dedicated hadronproduction experiment NA61/SHINE[89][90], and the propagation of thesecondary particles are simulated by GEANT3[81] with GCALOR[82]. Figure 7.4 shows the predicted neutrino flux at SK based on the simulation.
The flux consists mostly of νµ , however, there are some contaminations fromν̄µ and also νe and ν̄e which become irreducible backgrounds to νe appearance analysis. The overall uncertainty on the absolute flux is estimated to be
10−15%, and it is dominated by the uncertainties in the hadron production.
1087.2. νe Appearance Analysis
Other than the neutrino flux, T2K neutrino event simulation at SK is done
following the same procedures as the atmospheric neutrino MC described in
Chapter 4, using NEUT neutrino event generator and the GEANT3-based
detector simulation SKDETSIM.
In order to constrain the systematic uncertainties related to the neutrino
flux and cross sections, the data from ND280 is analyzed, namely, by performing a fit to the νµ CC interaction data in the FGDs. Taking the prioruncertainties on the flux simulation and the prior constraints on cross section model parameters estimated from external datasets such as [91][92][93],the neutrino energy spectrum and the NEUT interaction models are further
constrained based on a simultaneous fit of the flux and cross section model
parameters to the ND280 data. The resulting uncertainties on the model
parameters after the fit are then propagated to the event predictions at SK
and used in oscillation analysis. After the ND280 fit, the fractional total
uncertainty on the expected number of νe candidate events observed at SK
reduces from 27.2% to 8.8% when assuming sin2 2θ13 = 0.1.
The signal for νe appearance at SK is observed as νe CC events which
are coincident with the beam. As CCQE is the dominant neutrino interaction mode at ∼ 600 MeV where the energy of the expected νe appearance
signal peaks at, single electron events are selected as candidate events forνe appearance following the event selection criteria detailed in Section 7.3.
According to the selection and assuming sin2 2θ13 = 0.1, 21.6 νe candidate
events are expected to be observed at the given beam exposure, out of which
4.3 events are backgrounds from beam-intrinsic νe CC and NC events. Thus,
the observation of a significant number of the selected νe candidate events
will be a clear evidence of νe appearance.
Once νe candidate events at SK are selected, an extended maximum
likelihood fit is performed to obtain the neutrino oscillation parameters fromthe observed SK data. The likelihood is defined as:
L(Nobs , ⃗x|⃗o, f⃗) = Lnorm (Nobs |⃗o, f⃗) × Lshape (⃗x|⃗o, f⃗) × Lsyst (f⃗),(7.2)

where Nobs is the number of observed νe candidate events, ⃗x represents thereconstructed kinematic variables for the candidate events, ⃗o is the oscillation parameters and f⃗ represents the parameters for systematic uncertainties which include the parameters for the flux, neutrino interaction and the
SK selection eﬃciencies. Lnorm is the likelihood of observing Nobs events
given the model parameters ⃗o and f⃗, and it is expressed as a Poisson distribution at the predicted event rate calculated with ⃗o and f⃗. For eachobserved event, its reconstructed momentum pe and the angle θe between
1097.3. νe Event Selection
the directions of the neutrino beam and the observed electron are used as“shape” information in the likelihood term Lshape which is a 2D probability
distribution for pe − θe given the model parameters. This is to improve thesensitivity to νe appearance by taking advantage of the fact that the shape of
the pe − θe distribution diﬀers between the appearance signal and the background events. The predicted event rate for Lnorm and pe − θe distribution
for Lshape are both calculated using the T2K MC simulation. Lsyst is theconstraint on the parameters for systematic uncertainties, and the resultsfrom the ND280 fit mentioned above are used for the applicable parameters.
During an oscillation fit, the parameters for systematic uncertainties are numerically integrated over at each point in the oscillation parameter space,
and the resulting marginalized likelihood is then maximized with respect tothe oscillation parameters of interest to obtain the best fit values.

7.3

νe Event Selection

Below summarizes the full event selection criteria for the νe appearance analysis. In order to achieve high sensitivity to νe appearance via θ13 -induced
oscillation, it is essential to keep the backgrounds unrelated to the oscillation signal as low as possible. The targeted event category for the signal
is νe CCQE which is a single electron event with no decay electrons. As
discussed in Section 6.9.1, in SK, π 0 ’s are one of the major backgrounds tosingle electron events, and fiTQun is used in order to reduce the π 0 background which contaminate the νe candidate event sample. Other than theπ 0 rejection cut, the νe event selection described below is done based on
the information from APFIT[85], the preexisting reconstruction algorithm
at SK before fiTQun was introduced as mentioned in Chapter 6.
T2K neutrino events at SK are first selected by a reduction process using
a precise GPS timing system mentioned earlier, and then the FC selection
as described in Section 5.5 using APFIT information is applied to selectneutrino events whose interaction occur in the fiducial volume(FV) and arecontained in the ID (T2K neutrino events passing such selection is referred
to as FCFV). Then, single-ring events whose ring type is determined to beelectron-like as opposed to muon-like are selected. In addition, in order toreject the events in which only a decay electron is seen, the reconstructed
momentum of the electron ring is required to be > 100 MeV/c. Since thetargeted νe CCQE events do not produce any decay electrons, it is also
required that the number of detected decay electrons following the primaryinteraction is zero. Furthermore, only the events with the reconstructed1107.3. νe Event Selection
Parametersin2 θ12
sin2 θ23∆m221|∆m232 |δCP

Fixed value
0.306
0.50
7.6 × 10−5 eV2
2.4 × 10−3 eV20Table 7.1: Assumed values for the oscillation parameters.
neutrino energy Eνrec being less than 1250 MeV are selected, since the eventswith larger Eνrec are mostly beam-intrinsic νe background events. Eνrec isdefined as:Eνrec =

m2p − (mn − Eb )2 − m2e + 2(mn − Eb )Ee,2(mn − Eb − Ee + pe cos θe )(7.3)

where mp and mn are the proton and the neutron mass respectively, Eb =27 MeV is the neutron binding energy in oxygen nucleus, me is the electronmass and Ee is the reconstructed total energy of the electron. Ignoring the
Fermi momentum of the target neutron, the above equation gives the energyof the incident νe based on the two body kinematics of νe + n → e + p, using
the observed pe and θe .
After the above selection cuts based on APFIT are applied, the final
cut to reduce the remaining NC π 0 backgrounds are applied using fiTQun,
namely, the cut described by Equation 6.28. Table 7.2 shows the expected
number of signal and background events after each stage of the selection cuts,
assuming sin2 2θ13 = 0.1, normal hierarchy and other oscillation parameters
set to the values in Table 7.1. As in the table, the NC background remaining
after the single-ring electron-like selections, which is dominated by NC π 0
events, is drastically reduced after applying the fiTQun π 0 cut, while the loss
in the νµ →νe CC appearance signal events by the cut is small. The final
sample selected after all the cuts above has a νµ →νe CC signal purity of
80.2 %, while the background contamination from NC events is only 4.4 %.
The remaining backgrounds are predominantly the irreducible backgrounds
from CC interactions from the beam-intrinsic νe and ν̄e , i.e. the νe and ν̄ecomponents at neutrino flux production as shown in Figure 7.4.
In the previous T2K νe appearance analysis[28], a diﬀerent method was
employed for π 0 rejection. Using a dedicated π 0 reconstruction algorithm
which assumes two electron-like rings and force-reconstructs an event asa π 0 based on the light patterns[94], an invariant mass is calculated fromthe reconstructed momenta and the directions of the two rings, and π 0111

7.4. Results

Selection
FCFVSingle-ringElectron-like PID
pe > 100MeV/c
No decay-e
Eνrec < 1250MeVPrevious π 0 cut
fiTQun π 0 cut
Fraction [%]Eﬃciency [%]νµ +ν̄µCC
247.75142.44
5.63
3.66
0.69
0.21
0.13
0.07
0.3
0.0

Expectedνe +ν̄e
NC
BG
CC
total
15.36 83.02 346.13
9.82 23.46 175.72
9.74 16.35 31.72
9.68 13.99 27.32
7.87 11.84 20.40
3.73
8.99
12.94
3.41
2.55
6.08
3.24
0.96
4.27
15.0
4.4
19.8
21.1
1.2
1.2

Data
νµ →νe
CC
26.22
22.72
22.45
22.04
19.63
18.82
17.69
17.32
80.2
66.0

377
193
60
57
44
39
31
28
-

Table 7.2: Expected numbers of signal and background events passing eachselection stage with the assumption sin2 2θ13 = 0.1, normal hierarchy andother oscillation parameters set to the values in Table 7.1. The fractionof each event category in the final selected sample is also shown, and theeﬃciency of the overall selection is calculated taking the FCFV event rates
as the denominator. The shaded row also shows the event rates in the caseof applying the previous π 0 rejection method[28] as opposed to the fiTQun
π 0 cut. Data taken from January 2010 to May 2013 which has the neutrino
beam exposure of 6.57 × 1020 POT(protons on target) is also shown on therightmost column, and the MC expectations are normalized to the same
exposure.
rejection is done by selecting the events with the invariant mass being lessthan 105 MeV/c2 as shown in Figure 7.5. The predicted event rates for thesignal and backgrounds in the case of employing the previous π 0 rejection
method, instead of the fiTQun π 0 cut, are shown in the shaded row inTable 7.2. As in the table, the remaining NC background in the final selected
sample is reduced by ∼62% by replacing the previous method with thefiTQun cut, while the eﬃciency loss for the appearance signal is only ∼2%.

7.4

Results

The results of analyzing the data taken from January 2010 to May 2013
which has the neutrino beam exposure of 6.57 × 1020 POT(protons on target) is the following. As shown in Table 7.2, 28 events are observed as νe112

7.4. Results

RUN1-4 data

(6.570× 10 POT)

Number of events

20

Osc. νe CC
νµ+νµ CCνe+νe CCNC
(MC w/ sin22θ13=0.1)

30

20

10

0
0

100

200

300

Invariant mass (MeV/c2)
Figure 7.5: π 0 rejection cut which was used in the previous νe appearance
analysis[28]. The reconstructed invariant mass calculated by a dedicated
π 0 reconstruction algorithm is required to be less than 105 MeV/c2 for an
event to be selected as a νe candidate. As in the plot, the NC backgrounds
which are dominated by NC π 0 events produce a peak near the π 0 massof ∼ 135 MeV/c2 and are largely rejected by the cut. The data and the
expectations shown in the plot are for the dataset shown in Table 7.2.
candidates, and Figure 7.6 shows the 2D distribution of the fiTQun π 0 cut
variables for the observed data after the Eνrec < 1250MeV cut.
Using the observed data, a maximum likelihood fit as described in Section 7.2 is performed by fitting for sin2 2θ13 only, while other oscillation
parameters are fixed at the values in Table 7.1. The fit results in the best-fit
value and the 68% confidence interval of sin2 2θ13 = 0.140+0.038−0.032 assuming+0.0452
normal hierarchy and sin 2θ13 = 0.170−0.037 assuming inverted hierarchy,
and the best-fit prediction of the pe − θe distribution overlaid with the observed data are shown in Figure 7.7.
The significance for nonzero θ13 was calculated by generating a large
number of toy experiments assuming θ13 = 0 and calculating a p-value.
While generating the toy experiments, other oscillation parameters are fixedat the values on Table 7.1, and systematic uncertainties are randomly thrown113

7.4. Results

MC w/
sin22θ13=0.1

π

fiTQun ln(L 0/Le)

400

200

0
0

100

200

300
2

fiTQun mγ γ (MeV/c )Figure 7.6: 2D distribution of the fiTQun π 0 cut variables after the Eνrec <1250MeV cut, for the dataset shown in Table 7.2. The horizontal axis is
the reconstructed invariant mass from the π 0 fit, and the vertical axis is thelog likelihood ratio between the π 0 and the single-ring electron fits. The
black markers indicate the observed data and the gray box histograms are
MC expectations. The events below the blue line are selected as the final νecandidate events.
according to their prior uncertainties. Then, a test statistic defined as:best∆χ2 := −2 × (ln L′ (sin2 2θ13 = 0) − ln L′ (sin2 2θ13
)),

(7.4)

is calculated for each toy experiment, where L′ is the likelihood marginalbest is the best-fit value obtained from dataized over systematics and sin2 2θ13
mentioned above. The p-value is then defined as the fraction of toy experiments which have ∆χ2 > 53.64, where 53.64 is the ∆χ2 for the observed
data. After producing 1015 fake experiments the p-value is calculated to be
1.0 × 10−13 , which is equivalent to the exclusion of θ13 = 0 at 7.3σ significance and shows a clear discovery νe of appearance.

1147.4. Results

Data
Best fit
Background component

10

Angle (degrees)

5

180

1

150

Data
Best fit

120

0.8

90

0.6

60

0.4

30

0.2

0

5

0

0
500
1000
1500
Momentum (MeV/c)

Figure 7.7: The best-fit (for normal hierarchy) pe − θe distribution plottedwith the observed data. The 1D plots on the top and the left are the 1D peand θe distributions respectively. Figure taken from [29].

115Chapter 8

Atmospheric Neutrino
Oscillation Analysis
In this chapter, a neutrino oscillation analysis using the SK atmospheric
neutrino data is described in detail. As mentioned in Chapter 1, in atmospheric neutrino data, oscillation eﬀects manifest in the energy and thezenith angle distributions of the observed neutrino events. In this analysis,
the atmospheric neutrino data is first selected into several subsamples inorder to separate the events in terms of neutrino type, and for each sample
the events are binned in the 2D distribution of the observed momentum
and the zenith angle of the direction. The oscillation parameters are thenextracted by comparing the binned data to the expectation calculated from
MC simulation at diﬀerent oscillation parameters.
The dataset used in this analysis is the FC data in SK-IV, and reconstructed particle information from fiTQun described in Chapter 6 is used
for selecting and binning the data. For oscillation parameter estimation, aMarkov Chain Monte Carlo technique is employed to make Bayesian predictions on the oscillation parameters based on the observed data. The
following sections describe each of the analysis elements in detail.

8.1

Event Selection and BinningThis section describes the event selection and the binning which is used
as an input for the oscillation analysis. FC data is categorized into subsamples based on the reconstructed particle information from fiTQun, andthen binned in reconstructed momentum and zenith angle. Since the oscillation eﬀects manifest diﬀerently between the flavours of atmospheric neutrinos as described in Chapter 1, it is important to implement an event
selection scheme which is capable of categorizing the observed neutrino
events by neutrino flavour. In addition to separating the events by neutrino
flavour(electron or muon) based on the reconstructed ring type, statisticalseparations of neutrinos and antineutrinos are done in order to improve the116

8.1. Event Selection and Binningsensitivity to the oscillation eﬀects such as the ones from the mass hierarchy,
which happen diﬀerently for neutrinos and antineutrinos.

8.1.1

Selection Criteria

The events which are used in the analysis are the FC events as described in
Chapter 5 which have the visible energy greater than 30 MeV. The visibleenergy, Evis , is defined as the sum of the reconstructed kinetic energy above
Cherenkov threshold for all rings, taking into account the assumed particle
type (electron, muon or π + ) of each ring. The fiducial volume cut is also
applied, which requires that the distance from the reconstructed vertex of
the most energetic ring to the nearest point on the wall to be greater than
200 cm. The events which pass these basic selections are further divided into
13 event categories based on the reconstructed information from fiTQun as
follows:
1. Sub-GeV one-ring e-like 0 decay
Evis <1330MeV, one e-like ring with momentum >100MeV/c, ndcy =0.
2. Sub-GeV one-ring e-like ≥1 decay
Evis <1330MeV, one e-like ring with momentum >100MeV/c, ndcy ≥1.
3. Sub-GeV one-ring µ-like 0 decay
Evis <1330MeV, one µ-like ring with momentum >200MeV/c, ndcy =0.
4. Sub-GeV one-ring µ-like 1 decay
Evis <1330MeV, one µ-like ring with momentum >200MeV/c, ndcy =1.
5. Sub-GeV one-ring µ-like ≥2 decay
Evis <1330MeV, one µ-like ring with momentum >200MeV/c, ndcy ≥2.
6. Sub-GeV two-ring π 0
Evis <1330MeV, two e-like rings, ndcy =0, reconstructed invariant masscalculated from the two rings in the range 85-215MeV/c2 .
7. Multi-GeV one-ring e-like νeEvis >1330MeV, one e-like ring, ndcy ≥1
8. Multi-GeV one-ring e-like ν̄eEvis >1330MeV, one e-like ring, ndcy =0
9. Multi-GeV one-ring µ-like
Evis >1330MeV, one µ-like ring117

8.1. Event Selection and Binning
10. Multi-GeV multi-ring e-like νeEvis >1330MeV, two or more rings, highest momentum ring is e-like,passes the MME selection cut, passes the multi-ring νe -like cut.
11. Multi-GeV multi-ring e-like ν̄eEvis >1330MeV, two or more rings, highest momentum ring is e-like,passes the MME selection cut, fails the multi-ring νe -like cut.
12. Multi-GeV multi-ring µ-like
Evis >600MeV, two or more rings, highest momentum ring is µ-like,
momentum of the µ-like ring >600MeV/c.
13. Multi-GeV multi-ring e-like other
Evis >1330MeV, two or more rings, highest momentum ring is e-like,
fails the MME selection cut.
In the selection criteria above, ndcy is the number of decay electrons,
which is defined as the number of subevents minus 1 as described in Section 6.6.2. As discussed in Section 8.1.2, neutrinos and antineutrinos can bestatistically separated using the decay electron information. In addition to
the e-like and µ-like samples which are the enriched samples of νe CC andνµ CC events respectively, the two-ring π 0 sample is included in the analysis in order to constrain the NC background in the e-like samples. The
definitions of the MME selection cut and the multi-ring νe -like cut which
are applied to the multi-GeV multi-ring e-like samples are detailed in Section 8.1.3. In multi-GeV, while the e-like samples are statistically separated
into νe and ν̄e enriched samples, no neutrino-antineutrino separation is doneto the µ-like samples, mainly due to the lack of statistics to give any noticiable improvement in the mass hierarchy sensitivity.

8.1.2

ν/ν̄ Separation for Single-Ring Events

A large fraction of the events selected as single-ring are CCQE events. In a
CCQE event, the particles which are left after the neutrino interaction are
the outgoing charged lepton and a nucleon, a proton or a neutron dependingon whether it is a neutrino or antineutrino interaction. As the lepton charge
is indistinguishable in a water Cherenkov detector and the emitted proton isusually below Cherenkov threshold, separating neutrinos and antineutrinosin the CCQE channel is therefore diﬃcult.
The situation is, however, diﬀerent for CC single charged pion events
which are categorized as single-ring due to the pion momentum being below
118

8.1. Event Selection and Binning
Cherenkov threshold. Consider the CC1π ± interaction for a neutrino andan antineutrino:να + N → lα− + N + π + ,(8.1)

ν̄α + N → lα+ + N + π − .

(8.2)

For the case of neutrino, the produced π + often decays into a µ+ , which
subsequently produces a decay positron which will be detected. For antineutrino, on the other hand, the emitted π − in most cases gets absorbedby the nuclei in water, and the interaction therefore leaves no decay electron. Thus, a single-ring event with an extra decay electron being detected
has a higher chance of being a neutrino event, rather than antineutrino.
The event selection criteria in Section 8.1.1 thus subdivides the single-ring
events further into neutrino-enriched and antineutrino-enriched samples by
the number of decay electrons. For instance, the “Sub-GeV one-ring e-like≥1 decay” sample has high concentration of νe with significantly less ν̄e , andthe “Sub-GeV one-ring µ-like ≥2 decay” sample, in which in most cases the
decay electrons from the primary muon and a π + are both detected, consists
mostly of νµ with small ν̄µ content.

8.1.3

Multi-Ring νe /ν̄e Selection

As the fraction of multi-ring events increases at higher energy, multi-ring
event samples are also included in the analysis in multi-GeV. Due to thecomplexity of multi-ring events, it is in general more diﬃcult to achieve
high purity of the targeted event types in a multi-ring event category compared to the single-ring case. The multi-ring e-like event sample, as selectedby requiring the highest momentum ring to be e-like, has rather high contamination of νµ CC and NC events.
In order to reduce such backgrounds, a likelihood selection is applied
to subdivide the Multi-GeV Multi-ring E-like(MME) sample into a νe +
ν̄e CC enriched sample and a background enriched sample. Below are thedescriptions of the four variables which are input to the likelihood selection,
whose distributions are shown in Figure 8.1:
1. Energy fraction of the Most Energetic Ring
The fraction of the visible energy of the Most Energetic Ring(MER) to
the total visible energy. The definition of the visible energy is statedin Section 8.1.1. The variable tends to be large for νe + ν̄e CC events
since the outgoing electron often carries a large fraction of the event’s
energy, whereas the value is small for νµ + ν̄µ CC and NC events in
119

8.1. Event Selection and Binning
the same visible energy range since often in such events the MER is a
gamma ring from a π 0 and the hadronic system carrying large energy
produces large number of rings and deposit larger energy outside of
the MER.
2. Energy fraction of the Most Energetic π + -like Ring
The fraction of the visible energy of the Most Energetic π + -like(MEπ + )ring to the total visible energy. The value is set to zero if there is no
π + -like ring found. The νµ + ν̄µ CC events which are categorized as
multi-ring e-like usually have a rather low momentum muon ring whose
visible energy is lower than another ring such as the γ ring from a π 0
decay. For such events the muon ring is often reconstructed as a π + like ring, and the ring tends to have higher energy than a true π +ring which is produced in νe + ν̄e and NC events. The variable thus
eﬀectively discriminates νµ + ν̄µ CC events from the rest.
3. Number of decay electrons
Number of decay electrons as defined in Section 8.1.1. The value tends
to be larger for νµ + ν̄µ CC and NC events since the hadronic system
carrying large energy produces large number of decay electrons forsuch events.
4. Distance to the farthest decay electron
The distance between the vertices of the most energetic ring and themost distant decay electron, normalized by the visible energy of themost energetic ring. The variable is set to zero if there is no decay
electron detected. The variable tends to be larger for νµ + ν̄µ CC events
since the muon in such events travel longer distance than charged pionsin νe + ν̄e CC events.
As shown in Figure 8.1, the variables provide power to discriminate between
νe + ν̄e CC events and the backgrounds. For event selection, the normalized
distributions for signal(νe + ν̄e CC) and background(νµ + ν̄µ CC and NC) for
each variable are produced and used as likelihood functions. For a given
event, ln(Lsig /Lbkg ), the log likelihood ratio of assuming the event as asignal and a background, is evaluated for each of the four variables using
the likelihood functions, and the sum of the four log likelihood ratios isthen used to make a selection cut. Figure 8.2 is the distribution of the cut
variable for inclusive multi-GeV multi-ring e-like events where the signal andthe background are shown separately, and the signal(νe + ν̄e CC) enrichedsample is selected by applying a cut as >-0.25 which was determined based120

8.1. Event Selection and Binning

MRe : 2.51GeV < Evis < 5.01GeV

MRe : 2.51GeV < Evis < 5.01GeV

102
30
10
20

1

10

10-1

0
0

10-2
0.2
0.4
0.6
0.8
1
0
Energy fraction of MER

MRe : 2.51GeV < Evis < 5.01GeV

0.2

0.4
0.6
0.8
1
Energy fraction of MEπ+

MRe : 2.51GeV < Evis < 5.01GeV

102
100

10
1

50

0

10-1

0

2

4

10-2
6
8
10
0
# of decay-e

0.2
0.4
0.6
0.8
1
Distance to decay-e / EMER

Figure 8.1: Distribution of the variables which are used for the MultiGeV Multi-ring E-like(MME) likelihood selection, for the multi-ring e-likeevents with visible energy in the range 2.51-5.01 GeV. Distributions areshown separately for νe CC(green), ν̄e CC(blue), νµ CC(red), ν̄µ CC(yellow)
and NC(gray).

121

8.1. Event Selection and Binning

100

50

0
-10

Total pass: 1217.8 events
Mode: Fraction (Eff.)
ν e CC: 56.9% (77.8%)
ν e CC: 21.3% (93.3%)
ν µ CC: 3.8% (16.4%)
ν µ CC: 0.3% (14.5%)
NC: 17.6% (49.1%)

-5

0
5
10
Lklhd. sep. by all variables: ln(L /LBkg )Sig

Figure 8.2: Distribution of the sum of ln(Lsig /Lbkg ) for all four MME selection variables, for inclusive multi-GeV multi-ring e-like events. The distribution is shown separately for signal(νe +ν̄e CC) and background(νµ +ν̄µ CC andNC) in cyan and magenta respectively. The MME selection cut is applied
at -0.25 as indicated by the red arrow, and the interaction mode breakdownof the selected sample is shown in the figure. The eﬃciency for passing the
cut is shown in the parenthesis for each interaction mode. For comparison,
the fraction of νe CC and ν̄e CC events before the selection cut is applied are
46.4% and 14.5% respectively.
on sensitivity. The events which are rejected by this cut are categorized as
the “Multi-GeV multi-ring e-like other” sample.
Once the νe + ν̄e CC enriched sample is selected by the MME selection
cut, another likelihood selection is applied, this time, to separate the νe CC
events from ν̄e CC events. The selection is done in a similar manner as above
using the following three variables:
1. Number of rings
2. Number of decay electrons
3. Transverse momentum fraction1228.1. Event Selection and Binning
Defined as:
ptran :=∑

iEvis
sin θi /Evis ,

(8.3)

i

where θi is the angle between each ring and the most energetic ring,i is the visible energy of the ring and E
Evis
vis is the sum of the visibleenergies of all rings.
The variables are motivated by the fact that νe scattering tends to be lessforward peaked and have larger energy transferred to the hadronic systemcompared to ν̄e , and as shown in Figure 8.3, the variables tend to be largerfor νe than ν̄e . Similar to what was done for the MME selection, likelihood
functions for νe CC and ν̄e CC are produced using the distributions, andthen for a given event ln(Lνe /Lν̄e ), the likelihood ratio between the νe andν̄e hypotheses, is evaluated for each variable. The distribution of the sum ofln(Lνe /Lν̄e ) for all variables are shown in Figure 8.4, and the events passing
the cut at >0 are selected as the “Multi-GeV multi-ring e-like νe ” sample,
while the ones rejected are categorized as “Multi-GeV multi-ring e-like ν̄e ”.

8.1.4

Sample Statistics and Purities

Table 8.1 shows the statistics and the interaction mode breakdown of each
event sample selected by the selections in Section 8.1.1. Event rates for data
and MC expectation are both shown in the table, and the expected event
rates are calculated assuming two-flavour νµ → ντ oscillations at sin2 2θ =1.0 and ∆m2 = 2.5 × 10−3 eV2 and are normalized to 2519.89 days exposure
which is the same as data. Note that the expectations for ντ + ν̄τ CC events
are not shown on the table, however, they are properly considered in theactual oscillation analysis. It can be seen from the table that the event
selection achieves the intended separation of observed events by neutrino
type, namely, the νe -enriched samples as selected in Sections 8.1.2 and 8.1.3
have high purity of νe with low ν̄e contamination.
As demonstrated in Section 6.10.3, fiTQun achieves a significantly better
performance in reconstructing multi-ring events compared to the preexisting
APFIT algorithm. In order to see the improvement in the atmospheric
neutrino selection made by fiTQun, the event sample breakdown for anequivalent event selection using APFIT is shown in Appendix B.

8.1.5

Binning

Figure 8.5 shows the binning scheme for each event sample. The events ineach sample are binned in the observed momentum and zenith angle in order123

8.1. Event Selection and Binning

MRe : 2.51GeV < Evis < 5.01GeV

MRe : 2.51GeV < Evis < 5.01GeV
1

0.4
0.5
0.2

0

0

2

4

6
# of rings

0

0

2

4

6

8
10
# of decay-e

MRe : 2.51GeV < Evis < 5.01GeV
0.3

0.2

0.1

0
0

0.2
0.4
0.6
0.8
1
Transverse momentum fraction

Figure 8.3: Distribution of the variables which are used for the multi-ringνe /ν̄e likelihood separation, for the multi-ring e-like events passing the MME
selection cut with visible energy in the range 2.51-5.01 GeV. Distributions
are shown separately for νe CC(green) and ν̄e CC(blue). Each distribution isarea normalized for easier comparison of the shape.

124

8.1. Event Selection and Binning

150

100

Total pass: 602.9 events
Mode: Fraction (Eff.)
ν e CC: 60.1% (52.3%)
ν e CC: 12.4% (28.7%)
ν µ CC: 5.4% (69.6%)
ν µ CC: 0.4% (59.1%)
NC: 21.7% (61.0%)

50

0

-4

-2

0
2
4
Lklhd. sep. by all variables: ln(Lν /Lν )ee

Figure 8.4: Distribution of the sum of ln(Lνe /Lν̄e ) for all three νe /ν̄eseparation variables, for multi-GeV multi-ring e-like events passing the
MME selection. The distribution is shown separately for νe CC, ν̄e CC andbackground(νµ + ν̄µ CC and NC) in green, blue and black respectively. Theνe -like selection cut is applied at 0 as indicated by the red arrow, and theinteraction mode breakdown of the selected sample is shown in the figure.
The eﬃciency for passing the cut is shown in the parenthesis for each interaction mode. For comparison, the fraction of νe CC and ν̄e CC events before
the selection cut is applied are 56.9% and 21.3% respectively.

1258.1. Event Selection and Binning%
1Re 0dcy 1Re ≥1dcy 1Rµ 0dcy 1Rµ 1dcy 1Rµ ≥2dcy
νe CC
73.22
91.30
0.92
0.02
0.00
ν̄e CC
23.96
2.14
0.35
0.00
0.00
νµ CC
0.10
2.46
69.82
68.15
94.86
ν̄µ CC
0.02
0.60
10.08
29.38
3.57
NC
2.70
3.50
18.83
2.45
1.57
Total event rate
MC4876.2450.7
851.7
4378.1
356.5
Data
4677
450
814
4412
378
%
1Re νe 1Re ν̄eνe CC 75.39
56.11
ν̄e CC
8.26
38.35
νµ CC
4.55
0.22
ν̄µ CC
0.93
0.05
NC
10.87
5.27
Total event rate
MC
161.2
839.1
Data
174
773

2Rπ 0
5.00
1.49
0.81
0.08
92.61
1130.5
1095

1Rµ
0.02
0.02
59.71
40.20
0.05

MRe νe
60.15
12.38
5.36
0.38
21.73

MRe ν̄e
53.70
30.14
2.30
0.26
13.61

MRµ
0.92
0.11
72.56
24.07
2.34

MRe oth.
28.21
2.68
33.95
3.27
31.89

763.0
721

602.9
585

614.9
536

1627.0
1562

699.0
807

Table 8.1: Interaction mode breakdown and the total event rates for each
event sample, where the upper and the lower table is for sub-GeV and multiGeV samples respectively. Event rates for MC and data are both shown in
the tables. MC expectations are calculated with two-flavour oscillations at
sin2 2θ=1.0 and ∆m2 =2.5 × 10−3 eV2 and is normalized to the same 2519.89
days exposure as data.

126

Sub-GeV

Multi-GeV

8.1. Event Selection and Binning
Log10P[GeV/c]
0.7
0.4
0.0
1Re
νe

1Re
ν̄e

1Rμ

1Re
0dcy

1Re
≧1dcy

1Rμ
0dcy

MRe
νe

MRe
ν̄e

MRμ

MRe
Oth.

Log10P[GeV/c)]
0.0
-0.2
-0.4
-0.6
-0.8
1Rμ
1dcy

1Rμ
≧2dcy

2Rπ0

Figure 8.5: Binning for each event sample. In the figure, the momentum binedges are shown in the log10 of the momentum in GeV/c, and the highest
and the lowest momentum bins in each sample are overflow and underflow
bins respectively. The unshaded samples are further subdivided into 10 binsin the cosine of the zenith angle equally spaced in [−1, +1], while there isno zenith angle subdivision for the shaded samples.
to extract the oscillation pattern as mentioned before, and for single-ring
events the momentum and the zenith angle of the direction of the observed
ring are used. For multi-ring events, the total visible energy Evis is used as
the “generalized momentum”, and the direction of the reconstructed total
3-momentum vector is used for the zenith angle. The zenith angle is binned
in 10 bins equally spaced in the cosine of the zenith angle, except for somesamples whose statistics are too low such that the statistical fluctuation of
MC expectation becomes problematic, or have no directional sensitivity.

8.1.6

Observing the Oscillation Eﬀects

Once the events are selected and binned, the binned data is compared against
the expectations and the oscillation parameters are estimated. In order tosee how the oscillation eﬀects are observed in each sample, the eﬀects ofvarying each oscillation parameter on the predicted zenith distributions are127

8.1. Event Selection and Binning
Parametersin2 θ12
sin2 θ13
sin2 θ23
∆m221
∆m232
δCP

Default Value
0.304
0.02190.50
7.53 × 10−5 eV2+2.39 × 10−3 eV2 (NH)0Table 8.2: Default values for the oscillation parameters which are used forevent rate calculation.
shown below. Full three-flavour oscillations including the matter eﬀect areconsidered, with the base values for the oscillation parameters set to what are
shown in Table 8.2. The event rates are normalized to the data-equivalent2519.89 days exposure. In the zenith distributions, cos θZ <0 is for upwardgoing events and cos θZ >0 is for downward-going. The discussions below
follow the discussions of atmospheric neutrino oscillations in Section 1.5.
Figure 8.6 shows the zenith distributions for the sub-GeV e-like and µlike samples when sin2 θ23 is varied as 0.4, 0.5 and 0.6. Since the µ-likesample mainly observes νµ disappearance which depends on sin2 2θ23 , theobserved event rate is minimized when θ23 =45◦ , i.e. sin2 θ23 =0.5, and any
deviation from that will increase the event rates. For the e-like sample on
the other hand, the ∆m221 -induced oscillation which depends on cos2 θ23 isimportant in sub-GeV, and thus an excess of event rate is seen when θ23 <45◦
while a deficit is seen when θ23 >45◦ . Thus, the sub-GeV e-like sample issensitive to the θ23 octant.
Figure 8.7 shows the eﬀect of sin2 θ23 on the multi-GeV e-like samples.
These samples observe the enhanced νe appearance by the MSW resonanceeﬀect which is proportional to sin2 θ23 . Therefore, we see a corresponding
excess or deficit of the upward-going e-like events depending on whether
sin2 θ23 is greater or smaller than 0.5.
The most noticeable eﬀect from δCP is the enhancement of sub-GeV νeevents when −π<δCP <0, as seen in Figure 8.8. Since this eﬀect somewhat
competes with the eﬀect from sin2 θ23 shown in Figure 8.6, the correlations
between the two parameters will be seen in the oscillation analysis discussed
in Section 8.6.
Finally, Figure 8.9 compares the zenith distributions for the multi-GeV
e-like samples for normal and inverted hierarchy. The MSW resonance enhancement of electron type neutrino appearance only happens for neutrinos
when it is normal hierarchy, while for inverted hierarchy it only happens for128

8.1. Event Selection and Binning

Sub-GeV 1Rµ 1dcy

Sub-GeV 1Re 0dcy
600

600

400

400

200

200

0
-1

-0.5

0

0.5

1

cosθZ

0
-1

-0.5

0

0.5

1

cosθZ

Figure 8.6: The eﬀect of sin2 θ23 variations on the distributions of the cosineof the zenith angle for sub-GeV samples. The left is for one-ring e-like 0decay
and the right is for one-ring µ-like 1decay. The plots show the expectations
at sin2 θ23 set to 0.4(solid green), 0.5(solid black) and 0.6(dashed red).

Multi-GeV 1Re νe

Multi-GeV 1Re νe

20

100

10

50

0
-1

-0.5

0

0.5

1

cosθZ

0
-1

-0.5

0

0.5

1

cosθZ

Figure 8.7: The eﬀect of sin2 θ23 variations on the distributions of the cosineof the zenith angle for Multi-GeV samples. The left is for one-ring e-like νeand the right is for one-ring e-like ν̄e . The plots show the expectations at
sin2 θ23 set to 0.4(solid green), 0.5(solid black) and 0.6(dashed red).

1298.1. Event Selection and Binning

Sub-GeV 1Re ≥1dcy

Sub-GeV 1Re 0dcy
600
400

400

200

200

0
-1

-0.5

0

0.5

1

cosθZ

0
-1

-0.5

0

0.5

1

cosθZ

Figure 8.8: The eﬀect of δCP variations on the distributions of the cosine of
the zenith angle for Sub-GeV samples. The left is for one-ring e-like 0decay
and the right is for one-ring e-like ≥1decay. The plots show the expectations
at δCP set to −π/2(solid green), 0(solid black) and +π/2(dashed red).
antineutrinos. As in the left plots, the multi-GeV νe -enriched event samples, selected by the νe /ν̄e separation scheme as described in Sections 8.1.2
and 8.1.3, therefore observe an excess of upward-going events in the caseof normal hierarchy compared to inverted hierarchy. On the other hand,the ν̄e -enriched samples on the right, which have much closer to equal mixture of νe and ν̄e as in Table 8.1, show almost no sensitivity to the mass
hierarchy, since the eﬀect of flipping the hierarchy almost cancels between
neutrinos and antineutrinos. Employing the νe /ν̄e separation and having
the νe -enriched samples thus makes the eﬀect from the mass hierarchy more
prominent and improves the sensitivity.

1308.1. Event Selection and Binning

Multi-GeV 1Re νe

Multi-GeV 1Re νe

20

100

10

50

0
-1

-0.5

0

0.5

1

cosθZ

0
-1

Multi-GeV MRe νe

0

0.5

1

cosθZ

Multi-GeV MRe νe
100

100

50

50

0
-1

-0.5

-0.5

0

0.5

1

cosθZ

0
-1

-0.5

0

0.5

1

cosθZ

Figure 8.9: The eﬀect of the mass hierarchy on the distributions of the cosineof the zenith angle for Multi-GeV e-like samples. The left side is for the νe enriched samples and the right side is for the ν̄e -enriched samples. The solid
green lines are the expectations for normal hierarchy while the dashed redlines are for inverted hierarchy.

1318.2. Systematic Uncertainties

8.2

Systematic Uncertainties

This section describes the sources of systematic uncertainties which are considered in the oscillation analysis. While comparing the observed data to
prediction in order to extract the oscillation parameters, the parameter foreach systematic is also varied and its eﬀect is propagated to the predicted
event rate for each bin. In total there are 82 model parameters considered in
the analysis, six of which are the oscillation parameters. Out of the remaining 76 parameters for systematics, 18 are for the atmospheric neutrino flux,
19 are for neutrino interaction, 27 are for FSI and 12 are for the detector
response and reconstruction, and the full list of the systematic parameters
can be found in Appendix A. The prior uncertainties on the systematic parameters are all treated as Gaussians where the 1σ widths are given below,
and apart from the FSI parameters they are assumed to be uncorrelated.

8.2.1

Atmospheric Neutrino Flux

Normalization Error (%)

Absolute Normalization
The uncertainty on the absolute flux normalization is given by Honda etal.[95] as an energy-dependent error as in Figure 8.10. Scaling parameters
for below and above 1 GeV independently scale the flux according to thefunction shown. In order to account for the additional diﬀerences between
the base Honda flux model[95] and other flux models such as FLUKA[52]and Bartol[53] above 10 GeV, an additional 5% normalization error on multiGeV events is considered as an independent parameter.
25
20
15
10
5
0
10-1

1

10

102

10

3

Eν (GeV)Figure 8.10: Uncertainty on the flux normalization as a function of energy.
132

8.2. Systematic Uncertainties

Flux Ratios
The uncertainties on the various flux ratios are estimated by the comparisonof the Honda flux[95] to the FLUKA[52] and the Bartol[53] flux models. For
the flavour ratio (νµ + ν̄µ )/(νe + ν̄e ), the uncertainty is 2% for Eν < 1 GeV,
3% for 1 GeV < Eν < 10 GeV, 5% for 10 GeV < Eν < 30 GeV and then linearly increases in log10 Eν to 30% at 1 TeV. For ν̄e /νe ratio the uncertainty
is 5% for Eν < 10 GeV, 8% for 10 GeV < Eν < 100 GeV and then linearly
increases in log10 Eν to 30% at 1 TeV. Finally, the uncertainty on ν̄µ /νµratio is 2% for Eν < 1 GeV, 6% for 1 GeV < Eν < 50 GeV and then linearly
increases in log10 Eν to 60% at 1 TeV. For these ratio uncertainties, the fluxin the following three energy regions are varied independently according toa scaling parameter in each region: Eν < 1 GeV, 1 GeV < Eν < 10 GeV
and Eν > 10 GeV. The uncertainty on the up/down ratio and the horizontal/vertical ratio of the flux is also estimated by comparing the threeflux models, and the diﬀerences in the zenith distributions between the flux
models are taken as the uncertainty.
K/π Ratio
While the atmospheric neutrino flux below 10 GeV is mainly from pion decays, contributions from kaons become increasingly dominant as the energy
increases. Based on the K/π production ratio measurement by the SPY
experiment[96], the uncertainty on the ratio of the flux induced by kaonsand pions is estimated to be 5% for Eν < 100 GeV and then linearly increases with energy to 20% at 1 TeV.
Solar Activity
The activity of the sun cycles with 11 year period, and the resulting variations in the magnetic field aﬀects the cosmic-ray flux. The variation in theneutrino flux caused by a ±1 year variation in the solar cycle is taken as theuncertainty on the flux.
Neutrino Path Length
The production height of atmospheric neutrinos depends on the density
structure of the atmosphere, and the uncertainty on that needs to be propagated to the oscillation probability which depends on the path length ofneutrinos. The uncertainty in the density of the atmosphere is estimatedto be 10% based on the comparison between the US-standard 76 and theMSISE-90 models for the atmosphere[97], and the resulting variations in the
production height of atmospheric neutrinos are propagated to the oscillation
1338.2. Systematic Uncertainties
probability calculation.
Matter Eﬀect
Although the matter density structure of the earth is rather well known[35],
there is an uncertainty on the electron density which depends on the exact
chemical composition of the core, and such uncertainty needs to be taken
into account when calculating the matter-induced oscillation probabilities
through the Earth. As the core is assumed to consist of heavy elements
such as iron whose electron density is about 6.8% less than that of lightelements, the uncertainty on the electron density in the core is estimated tobe 6.8% and is considered during oscillation probability calculation.

8.2.2

Neutrino Interaction

CCQE
For CCQE interaction the uncertainties on the absolute total cross section
as well as the cross section ratios ν̄/ν and (νµ + ν̄µ )/(νe + ν̄e ) are assigned
according to the diﬀerences between the base model of relativistic Fermi gas
and the model by Nieves et al.[98]. In addition, 20% uncertainty is assignedon the axial vector mass MAQE .
MECAs there are still large uncertainties on the MEC model and the cross section
has not been explicitly measured at the present moment, 100% uncertainty
is assigned to the total cross section of the MEC process.
Single Meson Production
The uncertainties on the parameters which characterize the Graczyk-Sobczyk
form factor[64] were obtained from the data from bubble chamber experiments [65][66]. Additional uncertainties in the total cross sections are assigned based on the diﬀerences between the base model in NEUT to themodel by Hernandez et al.[99].
DIS
For DIS interaction, 10% uncertainty is assigned to the total cross section,and in order to account for the larger uncertainties in the energy below 10
GeV the diﬀerences between the base model of GRV98 with the Bodek-Yang
correction and the CKMT model[100] are considered as an additional uncertainty. Also, the uncertainty on the Q2 distributions of the DIS interaction
is assigned by the diﬀerences between the GRV98 model with and without1348.2. Systematic Uncertainties
the Bodek-Yang correction. Separate parameters for W < 1.3 GeV/c2 andW > 1.3 GeV/c2 vary the Q2 distributions in the two regions independently.
Coherent Pion Production
As there was no evidence for νµ CC coherent pion production observed by
the SciBooNE experiment[101], 100% uncertainty is assigned to the total
cross section of coherent pion production.
NC/CC Ratio
20% uncertainty is assigned to the ratio of the inclusive NC and CC crosssections.
ντ CC Cross Section
25% uncertainty is assigned to the inclusive ντ CC total cross section.

8.2.3

Final State Interaction

As final state interactions(FSI) change the observable final state of particles
which exit the nucleus, information such as the number of rings and decay
electrons which the event selection is based on is directly aﬀected by the
FSI. It is therefore important to consider the uncertainties in the FSI modelin the oscillation analysis.
As described in Section 4.2.5, based on the external pion scattering data
the uncertainties on the NEUT pion cascade model parameters are given
as representative sets of the FSI model parameters which span the 1σ error
surface in the multi-dimensional FSI parameter space. In order to incorporate this uncertainty in the analysis, a covariance matrix is formed among
the event rates of the selected event samples based on the allowed FSI variations. Figure 8.11 shows the error matrix for the fractional event ratevariations for the sub-GeV samples by the simultaneous variations of the
FSI parameters on the 1σ surface. In addition to the FSI which happens
within the initial target nucleus, secondary interactions(SI) of pions outside
the target nucleus are also varied simultaneously since they are simulated
in the detector simulation using the same NEUT pion cascade model. In
the oscillation analysis, instead of directly manipulating the underlying FSI
model parameters, a systematic parameter to scale the event rate is introduced for each sample, and the prior uncertainty on the scaling parameters
are treated as a multivariate Gaussian with the errors and the correlations
given in Figure 8.11. For the sub-GeV event samples, each selected sample
is further subdivided into two momentum bins and there are thus 12 scaling135

8.2. Systematic Uncertainties
0.15

2Rπ0 #2
2Rπ0 #1

0.1

1Rµ >1dcy #2
1Rµ >1dcy #1

0.05

1Rµ 1dcy #2
1Rµ 1dcy #10

1Rµ 0dcy #2
1Rµ 0dcy #1-0.05

1Re >0dcy #2
1Re >0dcy #1-0.1

1Re 0dcy #22Rπ0 #2

2Rπ0 #1

1Rµ >1dcy #2

1Rµ >1dcy #1

1Rµ 1dcy #2

1Rµ 1dcy #1

1Rµ 0dcy #1

1Rµ 0dcy #2

1Re >0dcy #2

1Re >0dcy #1

1Re 0dcy #2

1Re 0dcy #1

1Re 0dcy #1

-0.15

√

Figure 8.11: The fractional error matrix sgn(Vij ) × |Vij | for the sub-GeV
samples, where Vij is the covariance for the fractional variations in the event
rate of each sample by FSI and SI parameter variations. Each sample isfurther subdivided into two momentum bins, for below(#1) and above(#2)
400 MeV/c momentum. The diagonal elements represent the fractional error
on the event rate of each sample which can be up to 13% depending on thesample.
parameters in total.
For multi-GeV, while there is no subdivision of the samples by momentum, the e-like samples are subdivided into νe CC, ν̄e CC and background(νµ +ν̄µ CC and NC) based on the true interaction mode. Since the multiGeV samples are not as pure as the sub-GeV samples, the composition ofeach sample is also allowed to change this way as well as the total event
rate during the oscillation analysis. When forming the covariance matrix
for multi-GeV samples, in addition to the parameters for FSI and SI, pion
multiplicity is also varied based on the diﬀerences between NEUT and theresult from the CHORUS experiment[102] in order to account for the uncertainties which may not be covered by the FSI/SI variations. Figure 8.12
is the resulting error matrix for the multi-GeV samples. There are thus 15
scaling parameters for multi-GeV samples which are introduced to the anal136

8.2. Systematic Uncertainties

MR oth.
MRµMRνe bkg.
MRνe νeCC
MRνe νeCC
MRνe bkg.
MRνe νeCC
MRνe νeCC
1Rµ1Rνe bkg.
1Rνe νeCC
1Rνe νeCC
1Rνe bkg.
1Rνe νeCC
1Rνe νeCC

0.1

0.05

0

-0.05

MR oth.

1Rνe νeCC
1Rνe νeCC
1Rνe bkg.
1Rνe νeCC
1Rνe νeCC
1Rνe bkg.
1RµMRνe νeCC
MRνe νeCC
MRνe bkg.
MRνe νeCC
MRνe νeCC
MRνe bkg.
MRµ

-0.1

√

Figure 8.12: The fractional error matrix sgn(Vij ) × |Vij | for the multiGeV samples, where Vij is the covariance for the fractional variations in theevent rate of each sample by FSI, SI and pion multiplicity variations. The
diagonal elements represent the fractional error on the event rate of eachsample which can be up to 11% depending on the sample.
ysis whose prior correlations are given by the figure. No prior correlation is
introduced between the sub-GeV and the multi-GeV parameters.

8.2.4

Detector Response and Reconstruction

FC Reduction
The uncertainty on the FC reduction eﬃciency is estimated to be 1.3% basedon the comparison of the relevant cut variables between data and MC. The
uncertainty on the eﬃciency of the FC/PC separation cut is estimated to be
0.2%. Contaminations from non-neutrino backgrounds such as the flasher
PMT events and cosmic-ray muon events are estimated by eye-scanning to
be within 0.1%.
Fiducial Volume
The fiducial volume uncertainty is estimated to be 2%, which corresponds
1378.2. Systematic Uncertainties

Number of Events

SK4: Muon vertex (after dwall cut, top-entering)
70000
60000

Mean: Data: 1806.634± 0.066
MC: 1808.400± 0.110
Data: 1806.63±0.07cm
Error: 0.10± 0.01%
MC:1808.40±0.11 cm

50000
40000
30000
20000
10000
0
1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400
Z (cm)

Figure 8.13: Reconstructed vertex Z position of stopping cosmic-ray muons
entering from the top of the detector. The red histogram is for MC and theblack data points are for data, and the mean of each distribution is shown on
the top-right of the plot. The top wall of the ID is located at Z = 1810 cm.
to a systematic inward/outward shift of the vertex position by ∼10 cm near
the fiducial boundary. The reconstructed vertex distributions of stopping
cosmic-ray muons for data and MC agree well within this uncertainty as
shown in Figure 8.13.
Ring Counting
The uncertainty on the ring-finding eﬃciency is evaluated by comparing
the distributions of ring-counting likelihood as described in Section 6.10.1
between the FC data and MC. The uncertainty ranges from 0.3% to 2.8%
depending on the event category, and it is in general larger in multi-GeV.
Particle Identification
Particle identification uncertainties are estimated by comparing the distributions of e/π + likelihood ratio as discussed in Section 6.10.2 between the
FC data and MC. The errors are estimated separately for single-ring andmulti-ring events and vary from 0.1% to 0.4% depending on the event category.
Energy Calibration
1388.3. Event Rate Prediction
The uncertainty on the absolute energy scale is estimated by the data-MC
comparison of the energy scales in three control samples: stopping cosmicray muons, decay electrons and the mass peak of NCπ 0 . The largest diﬀerence between data and MC across all samples is taken as the uncertainty on
the absolute energy scale and is estimated to be 2.1%. The up/down asymmetry of the energy scale is also estimated from the momentum of decay
electrons with diﬀerent directions and is estimated to be 0.3%.
Decay Electron Tagging
As in Section 6.6.3, the uncertainty on decay electron detection eﬃciency isestimated by the data-MC comparison of the detection eﬃciency in stopping
cosmic-ray muon samples. The uncertainty is estimated to be 0.8% acrossall energies.

8.3

Event Rate Prediction

This section describes the procedures to calculate the predicted event rates
in the analysis bins for a given set of oscillation and systematic parameters.
The calculation is done by manipulating the base atmospheric neutrino MC
which was described in Chapter 4, and the method to reflect the variationsof each parameter depends on the type of the parameter.

8.3.1

Oscillation Weighting

The predicted the event rates at a given oscillation parameter set and nominal systematic parameters are calculated as the following.
The atmospheric neutrino MC is produced assuming no oscillations. The
oscillation is reflected to the event rate predictions by reweighting the MCfrom unoscillated flux to oscillated flux on event-by-event basis. For eachMC event, the ratio of the calculated oscillated flux over the original unocsillated flux is taken as a weight factor, and the event is filled into areconstructed momentum and zenith bin as is done for the observed data
but with the calculated weight. When reweighting a MC event of neutrino
type α(=e, µ, τ ), the following weight factor is applied:ω=

TData 1
{Φe P (νe → να ) + Φµ P (νµ → να )} .
TMC Φ0α(8.4)

The first factor in the equation adjusts the exposure of the original MC
to the data exposure where TData and TMC are the exposure for the data
and the MC respectively. TMC is 500 years for the MC which is used in139

8.3. Event Rate Prediction
this analysis. Φ0α is the atmospheric neutrino flux from which the MC was
generated, evaluated at the energy and the direction of the neutrino in theevent, and the Honda flux at middle solar activity before oscillations wasused for each neutrino type. For ντ and ν̄τ event generation the flux for νµand ν̄µ respectively was used, since tau neutrinos are not produced in theatmosphere via pion and kaon decays and only appear after oscillations. The
terms in the middle parentheses represent the total flux of neutrino type α
after oscillations where Φe and Φµ are the electron and muon neutrino flux
respectively before oscillations, and as the original atmospheric neutrino
flux consists of electron and muon neutrinos there are contributions fromboth once the oscillation is considered. The flux used here is the Honda
flux as described in Section 4.1 weighted according to the solar activity
corresponding to the data period as:Φβ = (1 − η)Φmin+ ηΦmax
β
β ,

(8.5)

where Φminand Φmax
are the flux at the minimum and maximum solarββ
activity respectively. Φβ is thus linearly interpolated between the two witha weight factor η, and for this analysis the weight is set to 0.35 which
represents the average solar activity during the data period. Finally, theobject P (νβ → να ) is the transition probability from a neutrino of type β
to type α calculated at the energy and the direction of the neutrino in theevent.
The oscillation probability is calculated using the Prob3++ software
package developed by the Super-Kamiokande collaboration[103] which is
publicly available and has been widely used in various neutrino oscillation
analyses. Following the methods given by Barger et al.[34] the software
algebraically calculates the full three-flavour oscillation probabilities for atmospheric neutrinos including the matter eﬀect in the earth based on theradial density structure of the earth given by the PREM model[35].

8.3.2

Propagating Variations of Systematics

Variations in the atmospheric neutrino flux parameters are applied on eventby-event basis at the time of filling the MC events into analysis bins, by
manipulating the weight for each MC event according to the variations in
the flux systematics. For example, if the absolute flux at the energy fora given event is supposed to be increased by 5%, the weight for the event
is increased by 5%. The weights from all flux parameters are multiplied
together to get the overall weight for the flux systematic variations for each
event. The solar activity systematic is propagated by accordingly changing
140

8.3. Event Rate Prediction
the mixture of the fluxes at minimum and maximum solar activity when
linearly combining them as described in Section 8.3.1. For the neutrino
path length and matter eﬀect systematics, the variations are reflected to theoscillation probabilities by directly manipulating the path length and thematter density which are used in Prob3++ for probability calculation.
Unlike the flux uncertainties which can be parameterized rather simply
based on the neutrino type, energy and direction, the cross section uncertainties have complex dependencies on the event configurations and it istherefore not computationally feasible to do an event-by-event reweighting.
The cross section systematic variations are propagated instead by directly
manipulating the predicted event rate at each analysis bin, after the MC
events are reweighted by flux and oscillation parameters and binned. Assuming a linear response of the bin contents to each cross section systematicparameter, the cross section parameter variations are applied to each bin as:
λi → λi∏()

1 + fij ϵj ,

(8.6)

j

where λi is the predicted event rate in the i-th bin(before the systematic
variations are applied), ϵj is the variation of the j-th systematic from its
nominal value in the unit of σ: the magnitude of the uncertainty as describedin Section 8.2. The object fij is a linear response coeﬃcient which gives thefractional change in the content of the i-th bin by the variation of the jth systematic by 1σ, and it is calculated for each systematic and analysis
bin by reweighting the MC events event-by-event using the detailed MC
truth information, prior to executing the software for oscillation parameter
estimation. Figure 8.14 shows an example of such pre-calculated response
coeﬃcients. In order to allow the interaction mode composition of each bin
to change, the MC events are first binned separately for νe CC, ν̄e CC, νµ CC,
ν̄µ CC, ντ +ν̄τ CC and NC, and after the response is applied as in Equation 8.6
separately for the event rates from each interaction mode, the total predicted
event rate in each bin is obtained by combining the contributions from allinteraction modes.
Most of the detector response and reconstruction systematic parameters
are treated in the same way as the cross section systematics, by applying
the response linearly according to Equation 8.6. The exceptions are theenergy scale parameters, which are applied event-by-event by manipulating
the reconstructed momentum before the MC events are binned, in order tomore properly treat the migration of events between the momentum bins
due to energy scale change.
1418.4. Bayesian Analysis using Markov Chain Monte Carlo

cosθZ

SGeV 1Re 0dcy , νe CC : neut_axial_mass

0.1

1

0.08

0.5

0.06
0
0.04
-0.5
-1
-1

0.02
-0.8

-0.6

-0.4

-0.2
0
0.2
Log (P[GeV/c])

010

Figure 8.14: The response coeﬃcient fij for the MAQE systematic for theνe CC events in each bin in the sub-GeV one-ring e-like 0decay sample. The
entries represent the fractional variations in the expected event rate at eachbin caused by a +1σ variation in the systematic parameter.
Finally, for the FSI parameters, the predicted rate at each bin is simply
scaled according to the scaling parameter for the event sample in concern.
The correlations between the scaling parameters for diﬀerent samples arethen enforced using the covariance which was disucussed in Section 8.2.3,
and the details will be discussed again later in Section 8.4.2.

8.4

Bayesian Analysis using Markov Chain
Monte Carlo

Once the events in the observed data are binned and the predicted event
rate at each bin is ready to be calculated, the oscillation parameters can beobtained by searching for the set of the parameters for which the prediction best matches with the observed data. This analysis employs a Markov
Chain Monte Carlo method to make Bayesian predictions on the oscillation
parameters based on the observed data, and the procedures are detailed in
the following.

142

8.4. Bayesian Analysis using Markov Chain Monte Carlo

8.4.1

Likelihood Function

In order to compare the observed binned data to MC expectations, this analysis defines the following likelihood function which is based on the observed
data and the model parameters in the MC simulation:
L(D|⃗o, f⃗) :=

n
∏
λNi
i

Ni !
i=1

exp(−λi ),(8.7)

where D represents the observed data, ⃗o is the vector of model parameterswhich we are interested in such as the oscillation parameters and f⃗ is thevector of nuisance parameters: the model parameters which we are notinterested in such as the parameters for systematic uncertainties. Ni is theobserved number of events in the i-th bin where there are n bins in total,and λi = λi (⃗o, f⃗) is the expected event rate in the i-th bin given the model
parameters ⃗o, f⃗ which is calculated as described in Section 8.3. A likelihood
function is the probability for obtaining the observed data given an assumed
model, and as the data is binned in this analysis, the likelihood is definedas the product of the probability at each bin of observing Ni events given
the mean event rate λi based on the assumed model, which is given by the
Poisson distribution with mean λi .

8.4.2

Bayesian Posterior Probability

The Bayes’ theorem states that the probability of a hypothesis being true
given data is given by:
P (H|D, I) =P (D|H, I)P (H|I),
P (D|I)(8.8)

where H represents the hypothesis, I is the prior knowledge on the hypothesis and D represents the observed data. P (H|I) is the prior probability forthe hypothesis H, and the object P (D|H, I) is the likelihood, which is theprobability for observing the data given the hypothesis and the prior knowledge. The denominator P (D|I) can simply be considered as a normalizationterm:∑
P (D|I) =P (D|H, I)P (H|I),
(8.9)
{H}

where the sum runs over all possible hypotheses in the given model space,
as is clear from the normalization condition of applying the model sum toeach side of Equation 8.8.
143

8.4. Bayesian Analysis using Markov Chain Monte Carlo
The discussions above is directly applicable to the problem of estimating
the oscillation parameters from the atmospheric neutrino data in this analysis. The probability distribution for the model parameters ⃗o, f⃗ given theobserved data D, i.e. the posterior probability given data, is written as:
P (⃗o, f⃗|D) = ∫L(D|⃗o, f⃗)p(⃗o, f⃗),
L(D|⃗o, f⃗)p(⃗o, f⃗) d⃗o df⃗(8.10)

where L(D|⃗o, f⃗) is the likelihood function defined in Equation 8.7. p(⃗o, f⃗)is the prior probability distribution for the model parameters ⃗o, f⃗, and in
this analysis, except for some oscillation parameters for which flat priors areused, it is assumed that the prior is described as a multivariate Gaussian:p(⃗o, f⃗) = p(⃗o)p(f⃗),
1 Tp(⃗o) ∝ exp(− d⃗o Vo−1 d⃗o ),2
1 T
p(f⃗) ∝ exp(− d⃗f Vf−1 d⃗f ),2(8.11)(8.12)(8.13)

where d⃗o and d⃗f are the deviations from the nominal values for ⃗o and f⃗
respectively. Vo and Vf are the covariance matrices for the prior constraints
on the parameters ⃗o and f⃗, and apart from the systematic parameters related
to the FSI uncertainties which are correlated as described in Section 8.2.3,
the parameter priors are assumed to be uncorrelated, i.e., only the diagonal
elements in the matrices are nonzero for the majority of the parameters.
Once the Bayesian posterior probability Equation 8.10 is defined, one
can obtain the joint probability distribution for the model parameter sets⃗o, f⃗ given the observed data in two steps: by calculating the expected event
rate λi at each bin for the assumed values of ⃗o and f⃗, and then evaluating
Equation 8.10.

8.4.3

Marginalization and Parameter Estimation

In the oscillation analysis, we are only interested in certain oscillation parameters while not caring about the rest of the model parameters. Therefore,
rather than the joint probability P (⃗o, f⃗|D), we want the probability for theinteresting parameters ⃗o only, disregarding the values of the nuisance parameters f⃗. This is done by marginalization, i.e., by integrating out thenuisance parameters from the posterior probability P (⃗o, f⃗|D):P (⃗o|D) =∫P (⃗o, f⃗|D) df⃗.

(8.14)
144

8.4. Bayesian Analysis using Markov Chain Monte Carlo
The probability P (⃗o|D) is now simply the probability for the interesting
parameters ⃗o given the observed data, taking into consideration all the possibilities for the values for the nuisance parameters based on their prioruncertainties and data. Obtaining such marginalized posterior probabilities for the oscillation parameters based on the SK data is the goal of thisoscillation analysis.

8.4.4

Markov Chain Monte CarloIn this analysis there are 82 model parameters in total, and performing the
integration in Equation 8.14 in such high dimensionality numerically can bechallenging. Markov Chain Monte Carlo provides means to perform suchhigh dimensional integrals eﬃciently.
Markov Chain Monte Carlo(MCMC) is essentially a directed random
walk in a multi-dimensional parameter space which is used to sample from amulti-dimensional probability distribution. After randomly sampling large
number of points from the multi-dimensional parameter space following thisprocedure, the density distribution of the sampled points will eventually
approximate the multi-dimensional probability distribution in concern.
Metropolis-Hastings algorithm[104] is the method which is used in thisanalysis to produce a Markov chain, and the procedures to produce sampled
points {Xt } from a probability distribution P (X) for a multi-dimensionalparameter set X follow as below:
1. Randomly chose the initial state X0 for the parameters.
2. Define a proposal function Q(X → Y ) which gives the probability
distribution for proposing the next state Y given the current state X.
3. After t steps when the chain is currently at the state Xt , propose a
new state Y using Q(Xt → Y ).
4. Accept the proposed state Y with the acceptance probability:(α(Xt → Y ) := min 1,)

P (Y ) Q(Y → Xt ),P (Xt ) Q(Xt → Y )(8.15)

i.e., move to the proposed new state(Xt+1 = Y ) with the probabilityα(Xt → Y ), discard the proposed state and remain in the current
state(Xt+1 = Xt ) with the probability 1 − α(Xt → Y ).
5. Repeat the procedures from 3.
145

8.4. Bayesian Analysis using Markov Chain Monte CarloIn the case when the proposal function is symmetric, i.e.,
Q(X → Y ) = Q(Y → X),(8.16)

Equation 8.15 reduces to:()

P (Y )α(Xt → Y ) = min 1,.
P (Xt )

(8.17)

It is interesting to note that although the number of steps required forthe chain to reasonably converge to the sampled distribution depends onthe choice for the proposal function Q(X → Y ), given suﬃcient numberof sampled points the chain will eventually approximate the distribution
regardless of the choice of the functional form of Q(X → Y ), as long as thetransition probability to any point where P (Y ) ̸= 0 is nonzero.
For the oscillation analysis, the distribution which is sampled using
the above algorithm is the posterior probability distribution P (⃗o, f⃗|D) in
Equation 8.10, and a Gaussian proposal function is chosen which draws
a Gaussian-distributed random number centered at the current parametervalue. Since this is a symmetric proposal function, proposed steps are accepted with the probability:()

P (o⃗′ , f⃗′ |D)α(⃗o, f⃗ → o⃗′ , f⃗′ ) = min 1,.
P (⃗o, f⃗|D)(8.18)

Note that since the denominator in Equation 8.10 is a constant which does
not depend on ⃗o and f⃗, the above equation further reduces to:()

L(D|o⃗′ , f⃗′ )p(o⃗′ , f⃗′ )α(⃗o, f⃗ → o⃗′ , f⃗′ ) = min 1,,
L(D|⃗o, f⃗)p(⃗o, f⃗)(8.19)

which eliminates the need of evaluating the integral in Equation 8.10.
Once a large number of points are sampled from the probability distribution, marginalization as in Equation 8.14 is readily done: one can simply
project the sampled points onto the axes of interesting parameters and binthem in a histogram. In this oscillation analysis, the posterior probability
density distribution for the model parameters Equation 8.10 is sampled using
MCMC, and the sampled points are then binned into histograms of oscillation parameters which, after normalizing, gives the marginalized posterior
probability distributions for the oscillation parameters which is presented as
the final result.
146

8.5. Sensitivity Studies

8.5

Sensitivity Studies

In order to test the oscillation analysis framework and understand its behaviour before analyzing the real data, a sensitivity study was performed byrunning the analysis on an Asimov dataset, i.e., taking the expected event
rates without any statistical fluctuations as a dataset. The expectations arecalculated at the oscillation parameters in Table 8.3, and all the systematic parameters are set to the nominal values. The exposure is taken to be
2519.89 days which is the same as the real data.
Parametersin2 θ12
sin2 θ13
sin2 θ23
∆m221
∆m232
δCP

True value
0.304
0.02190.50
7.53 × 10−5 eV2+2.39 × 10−3 eV2 (NH)−π/2

Table 8.3: True oscillation parameters for the Asimov dataset production.
The MCMC oscillation analysis as described in Section 8.4 is performedon the above dataset, which in the end provides Bayesian predictions on theoscillation parameters estimated from the Asimov data. During the analysis
the priors for the oscillation parameters are taken as shown in Table 8.4,
while the priors for the systematic parameters are all Gaussian with theuncertainties assigned as in Section 8.2. Prior constraints on sin2 θ12 , ∆m221
and sin2 θ13 are based on the PDG2015 average values[15]. Flat priors are
assumed for sin2 θ23 , ∆m232 and δCP to which the atmospheric neutrino data
is sensitive, and the results on the three parameter will therefore be purely
driven by sensitivity of the Super-K data. Since we expect a large gap
between the normal and inverted hierarchy peaks in the ∆m232 posterior, aspecial treatment is made for the proposal function for ∆m232 so that the signof the proposed ∆m232 flips by 50% probability at each step, in addition to
the Gaussian proposal for its absolute value. This ensures that the Markov
Chain properly samples the posterior probability in both hierarchies within
the limited number of MCMC steps. The resulting posterior can be used to
predict the mass hierarchy as well as the value for ∆m232 .
For this sensitivity study, 50 MCMC chains were produced, each having
120,000 steps. Figure 8.15 shows the evolution of ∆m232 and − ln P (⃗o, f⃗|D),
the negative-log of the posterior probability, as the MCMC processes progress,
where the horizontal axes are the number of steps taken within each chain.
147

8.5. Sensitivity Studies
Parametersin2 θ12
sin2 θ13
sin2 θ23
∆m221
∆m232
δCP

Prior
Gaussian, 0.304 ± 0.014
Gaussian, 0.0219 ± 0.0012
Flat in [0, 1]
Gaussian, 7.53 ± 0.18 × 10−5 eV2Flat, P(NH)=P(IH)=0.5
Flat in [−π, +π]Table 8.4: Priors for the oscillation parameters which are used in the analysis. Constraints on sin2 θ12 , ∆m221 and sin2 θ13 are based on the global fit
and averaged results listed in [15].
It can be seen especially from the − ln P (⃗o, f⃗|D) plot that for the first ∼5000
steps the distribution of the variable depends on the number of steps taken,
which indicates that the random walk has not reached an equilibrium during this period. Such period is referred to as the “burn-in” period, andsince the steps taken during this time do not properly represent the posterior distribution, certain number of steps at the beginning of the chainsneed to be discarded. In this analysis, the initial 20,000 steps in each chain
are discarded as burn-in, and as it can be seen from the plots the MCMC
process reaches an equilibrium after 20,000 steps and is evenly sampling theposterior distribution. Thus, there are 5×106 post-burn-in steps used forthis sensitivity study.
Once the posterior probability is sampled by MCMC, the marginalized
posterior distributions for the oscillation parameters are readily obtained
simply by projecting the sampled points into histograms for the oscillation
parameters. Figure 8.16 shows the marginalized 2D posterior distribution for∆m232 and sin2 θ23 where all the other oscillation and systematic parameters
have been marginalized over. The blue cross indicates the mode of this2D posterior, i.e. the point with the highest posterior probability density
which is analogous to the “best fit” point in a more-common frequentist
analysis, and the cyan star is the true point where the Asimov dataset was
produced. The dashed and solid contours indicate the 68% and 90% Highest
Posterior Density(HPD) credible regions respectively. A α% HPD credible
region is defined such that the integrated probability within the region is
α% while the posterior density in any point within the region is higher
than any point outside. Thus, the point estimate and the allowed regions
for the oscillation parameters can be obtained from the Bayesian posterior
probability distributions produced using MCMC.

148

delm2_23

8.5. Sensitivity Studies

0.005
0.004
0.003

3000

0.002
0.001
2000

0
-0.001-0.0021000-0.003-0.004-0.005
0

20

40

60

80

100

120

0

LogL

×103 steps-57030-570403000-570502500-570602000-570701500-570801000-57090500-57100-57110
0

20

40

60

80

100

120

0

×103 steps
Figure 8.15: ∆m232 (upper plot) and − ln P (⃗o, f⃗|D) (lower plot), thenegative-log of the posterior probability, plotted against the number of steps
taken in each chain. Steps from all the produced MCMC chains are combined in the plots.

149

8.5. Sensitivity Studies∆m232 (10-3 eV2)

P(NH)=P(IH)=0.5

5
3
4
3

2.5

2
2

1
0

1.5

-1
1

-2
-3

0.5
-4
-5

0
0.3

0.4

0.5

0.6

0.7
sin2θ23

Figure 8.16: The marginalized 2D posterior distribution for ∆m232 andsin2 θ23 given the Asimov dataset, where the blue cross indicates the modeof this 2D posterior and the cyan star is the true point where the Asimov
dataset was produced. The dashed and solid contours indicate the 68% and
90% HPD credible regions respectively.

150

8.5. Sensitivity Studies
As in Figure 8.16, the posterior is distributed for both positive and negative signs for ∆m232 , and one can obtain the posterior probabilities for themass hierarchy by marginalizing over all other information, i.e., by simply
integrating the ∆m232 posterior for above and below zero which is equivalent
to counting the number of steps taken in the normal and inverted hierarchies. For this Asimov dataset the ratio of the number of steps taken in eachhierarchy is NH:IH=2901738:2098262, which yields the posterior probability
for normal hierarchy to be 58.0%.
By selecting the steps in either one of the hierarchies only, one can also
obtain the posteriors in the case of assuming one of the hierarchy to be true.
For example, Figure 8.17 shows the 2D posteriors for sin2 θ23 and δCP in thecase of fixed mass hierarchy. Figure 8.18 is the 1D posteriors for the threeparameters for each hierarchy where the 68% and 90% HPD credible regions
are indicated by the dark gray and light gray regions respectively.
It is interesting to note that, although the mode of the 2D posterior
is located near the true point as in the upper plot of Figure 8.17, once the
distribution is further marginalized over sin2 θ23 the mode of the 1D posteriorfor δCP drifts away from the true value. This is due to the correlations
between the two variables as seen in Figure 8.17, namely, although the local
density is higher near the true point at δCP = −0.5π and sin2 θ23 = 0.5,
the distribution is wider in sin2 θ23 at δCP ≈ −0.7π. Being able to fully
marginalize over the nuisance parameters with complex correlations in suchway without having to assume the Gaussian-ness of the posterior is one ofthe distinct features of a Bayesian analysis such as this analysis, as opposedto the more common frequentist analysis by χ2 minimization.
Figure 8.19 shows the mean and the standard deviation(σ) of the 1D
marginalized posterior for each systematic parameter in the case of assuming normal hierarchy. The enumeration of the systematic parameters can befound in Appendix A. In the plot the markers and the error bars represent
the mean and the σ of the posterior respectively, where the nominal value ofeach systematic parameter is taken to be zero and the deviation from nominal is normalized by the prior uncertainty. Although it is a small eﬀect,
the posterior mean for some of the parameters deviate from zero due to thecorrelations with other parameters. This plot essentially shows how eachsystematic parameter can be constrained by the atmospheric neutrino data
itself, and for the parameters whose posterior is significantly narrower thanthe prior, i.e. the parameters whose width of the error bar is significantly
smaller than one, the constraint from the data is strong. For instance, theflux normalization below 1 GeV (parameter #1) which has ∼10% prior uncertainty is strongly constrained by the large statistics of sub-GeV samples.
151

8.5. Sensitivity Studies

sin2θ23

NH
0.7
3

0.6

2

0.5
0.4

1

0.3
-1

-0.5

0

0.5

1
δCP (π)0sin2θ23

IH
0.7
3
0.6
2

0.5
0.4

1

0.3
-1

-0.5

0

0.5

1
δCP (π)0Figure 8.17: The 2D marginalized posterior distributions for sin2 θ23 andδCP given the Asimov dataset, where the upper plot is for assuming normal
hierarchy to be true while the lower plot is for inverted hierarchy. The
blue cross indicates the mode of this 2D posterior and the cyan star is thetrue point where the Asimov dataset was produced. The dashed and solid
contours indicate the 68% and 90% HPD credible regions respectively.

152

8.5. Sensitivity Studies

IH

NH

6

6

4

4

2

2

0
0.2

0.4

0.6

0.8
sin2θ23

0
0.2

0.4

0.8
sin2θ23

IH

NH

1

0.6

0.8
0.6
0.5

0.4
0.2

0
1

2

3

4

5

0
1

2

3

|∆m232| (10-3 eV2)
NH

0.8

0.6

0.4

0.4

0.2

0.2

0
-1

-0.5

0

IH

0.8

0.6

0.5

1
δCP (π)0-145|∆m232| (10-3 eV2)

-0.5

0

0.5

1
δCP (π)Figure 8.18: The marginalized 1D posterior distributions for sin2 θ23 (top),|∆m232 |(middle) and δCP (bottom) for the cases of assuming normal (left)
and inverted (right) hierarchy, given the Asimov dataset. The vertical bluelines indicate the mode of the 1D posteriors while the cyan lines indicate thetrue point where the Asimov dataset was produced. The dark gray and lightgray regions indicate the 68% and 90% HPD credible regions respectively.

1538.6. Data Analysis Results

In units of prior σPosterior Mean & RMS of Systematic Parameters : NH
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2

10

20

30

40

50

60
70
Systematic parameter index

Figure 8.19: The mean and the σ of the marginalized 1D posterior foreach systematic parameter in the case of assuming normal hierarchy, for
the Asimov dataset. The enumeration of the systematic parameters can befound in Appendix A. The markers and the error bars represent the mean
and the σ of the posterior respectively, where the nominal value of eachsystematic parameter is taken to be zero and the deviation from nominal isnormalized by the prior uncertainty.
A similar sensitivity study was performed using APFIT for event selection instead of fiTQun, and the comparison of the sensitivity between thetwo cases can be found in Appendix B.

8.6

Data Analysis Results

The result of analyzing the Super-K data is presented in this section. The
dataset which is used in this analysis is the FC data from SK-IV with 2519.89
days exposure. The observed number of events in each selected event category has been summarized in Table 8.1.
The analysis is performed in the same manner as in the sensitivity study
in Section 8.5, taking the priors for the oscillation parameters as describedin Table 8.4. 200 MCMC chains of 120,000 steps were produced where theinitial 20,000 steps in each chain are discarded as burn-in, which leaves intotal 2×107 post-burn-in steps for this analysis.

154

8.6. Data Analysis Results

8.6.1

Posterior Distributions

Figure 8.20 shows the marginalized 2D posterior distribution for ∆m232 andsin2 θ23 . It can be seen in the figure that the posterior is distributed morein the upper half which indicates that there is a larger preference for normal
hierarchy, which is due to the observed overall excess in the upward-going
multi-GeV νe events as shown in Figure 8.27. The posteriors for the caseof assuming one of the hierarchy to be true are also shown in Figure 8.21.
The mode of the posteriors are located at sin2 θ23 = 0.606 and ∆m232 =+2.15 × 10−3 eV2 for normal hierarchy, and at sin2 θ23 = 0.406 and ∆m232 =−1.85 × 10−3 eV2 for inverted hierarchy.
Figure 8.22 shows the 2D posteriors for sin2 θ23 and δCP for each of thehierarchy assumption. As was discussed in Section 8.1.6, since decreasing
sin2 θ23 from 0.5 and decreasing δCP from 0 both have the eﬀect of increasing
the sub-GeV e-like events, the competing eﬀects introduce high correlationsin the posterior for the two parameters and the preference for δCP is diﬀerent
depending on the θ23 octant.
Figure 8.23 shows the marginalized 1D posteriors for the three oscillation parameters plotted separately for normal and inverted hierarchy, andthe mode and the 68% HPD credible region for each 1D posterior are summarized in Table 8.5. The 1D posteriors in the case mass hierarchy is marginalized are also shown in Figure 8.24.
Parametersin2 θ23|∆m232 |(10−3 eV2 )δCP (π)Normal Hierarchy
0.606[0.488,0.650]
2.13[1.75,2.30]-0.92[-1.00,-0.17]∪[0.72,1.00]

Inverted Hierarchy
0.406[0.375,0.475]∪[0.538,0.625]
1.88[1.7,2.3]
0.92[-1.00,-0.17]∪[0.61,1.00]

Table 8.5: The 1D posterior mode and the 68% HPD credible intervals.
From the posterior in Figure 8.20, the posterior probabilities for normal(∆m232 >0) and inverted (∆m232 <0) hierarchy as well as the first (sin2 θ23 <0.5)
and the second (sin2 θ23 >0.5) θ23 octant are calculated by integrating the
2D posterior in each of the four quadrants. The results are summarized inTable 8.6, and the data shows mild preferences for the normal hierarchy andthe second θ23 octant.

155

8.6. Data Analysis Results∆m232 (10-3 eV2)

P(NH)=P(IH)=0.5

5
4

6

3
5
2
1

4

0
3
-1
-2

2

-3
1
-4
-5

0
0.3

0.4

0.5

0.6

0.7
sin2θ23

Figure 8.20: The marginalized 2D posterior distribution for ∆m232 andsin2 θ23 given the SK-IV FC data, where the blue cross indicates the modeof this 2D posterior. The dashed and solid contours indicate the 68% and
90% HPD credible regions respectively.

156

|∆m232| (10-3 eV2)

8.6. Data Analysis Results

4

6

3

4

2

2

1

|∆m232| (10-3 eV2)

NH

5

0.3

0.4

0.5

0.6

0.7
sin2θ23

0

IH

5

6
4
4

3

2

2
1

0.3

0.4

0.5

0.6

0.7
sin2θ23

0

Figure 8.21: The marginalized 2D posterior distributions for |∆m232 | andsin2 θ23 for the cases of assuming normal (top) and inverted (bottom) hierarchy, given the SK-IV FC data. The blue cross indicates the mode of eachposterior. The dashed and solid contours indicate the 68% and 90% HPD
credible regions respectively.

157

8.6. Data Analysis Results

sin2θ23

NH
0.7

4

0.6

3

0.5

2

0.4
1
0.3
-1

-0.5

0

0.5

1
δCP (π)0sin2θ23

IH
2.5

0.7

2

0.6

1.5

0.5

1
0.4
0.5
0.3
-1

-0.5

0

0.5

1
δCP (π)0Figure 8.22: The marginalized 2D posterior distributions for sin2 θ23 andδCP for the cases of assuming normal (top) and inverted (bottom) hierarchy,
given the SK-IV FC data. The blue cross indicates the mode of each 2D
posterior. The dashed and solid contours indicate the 68% and 90% HPD
credible regions respectively.

158

8.6. Data Analysis Results

NH

IH

6
4
4
2

2
0
0.2

0.4

0.6

0.8
sin θ23

0
0.2

0.4

0.6

0.8
sin2θ23

2

NH

IH

1.5

1.5

1

1

0.5
0
1

0.5
2

3

4

5

0
1

2

3

|∆m232| (10-3 eV2)
IH

NH

0.6

0.6

0.4

0.4

0.2

0.2

0
-1

-0.5

0

4
5
|∆m232| (10-3 eV2)

0.51δCP (π)

0
-1

-0.5

0

0.5

1
δCP (π)Figure 8.23: The marginalized 1D posterior distributions for sin2 θ23 (top),|∆m232 | (middle) and δCP (bottom) for the cases of assuming normal (left)
and inverted (right) hierarchy, given the SK-IV FC data. The vertical bluelines indicate the mode of the 1D posteriors. The dark gray and light grayregions indicate the 68% and 90% HPD credible regions respectively.

159

8.6. Data Analysis Results
MH Marginalized

MH Marginalized

6
0.6
4
0.4
2
0
0.2

0.2
0.4

0.6

0.8
sin θ23

0
-1

-0.5

0

2

0.5

1
δCP (π)Figure 8.24: The marginalized 1D posterior distributions for sin2 θ23 andδCP after marginalizing over the mass hierarchy, given the SK-IV FC data.
The vertical blue lines indicate the mode of the 1D posteriors. The darkgray and light gray regions indicate the 68% and 90% HPD credible regions
respectively.

NormalInvertedSum

1st Oct.
0.275
0.076
0.351

2nd Oct.
0.584
0.065
0.649

Sum
0.859
0.141
1.000

Table 8.6: Posterior probabilities for the mass hierarchy and the θ23 octant.
The 1D posterior distributions for the other oscillation parameters havethe mean and the σ of the following: sin2 θ12 = 0.304 ± 0.014, sin2 θ13 =0.0219 ± 0.0012 and ∆m221 = 7.53 ± 0.18 × 10−5 eV2 . These values arenearly identical to the prior constraints listed in Table 8.4 due to the lackof sensitivity of atmospheric neutrino data to these parameters.
Figure 8.25 shows the mean and the σ of the marginalized 1D posterior
for each systematic parameter where the posteriors for normal and inverted
hierarchy are shown separately. The enumeration of the systematic parameters can be found in Appendix A.

8.6.2

Zenith Angle Distributions

Figures 8.26 to 8.28 show the zenith angle distributions for each event sample comparing the data with the expectations calculated with normal andinverted hierarchy. For each hierarchy, the expectations are calculated by
setting each oscillation and systematic parameter to the value at the modeof the parameter’s 1D posterior for the given choice of mass hierarchy. For
160

8.6. Data Analysis Results

In units of prior σPosterior Mean & RMS of Systematic Parameters : NH
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2

10

20

30

40

50

60
70
Systematic parameter index

In units of prior σPosterior Mean & RMS of Systematic Parameters : IH
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2

10

20

30

40

50

60
70
Systematic parameter index

Figure 8.25: The mean and the σ of the marginalized 1D posterior foreach systematic parameter in the cases of assuming normal(top) and inverted(bottom) hierarchy, for the SK-IV FC data. The enumeration of thesystematic parameters can be found in Appendix A. The markers and theerror bars represent the mean and the σ of the posterior respectively, where
the nominal value of each systematic parameter is taken to be zero and thedeviation from nominal is normalized by the prior uncertainty.

161

8.6. Data Analysis Results
the oscillation parameters sin2 θ23 , ∆m232 and δCP only, since there are large
complex correlations between the parameters, their values are set to what
are shown in Table 8.7 which are close to the mode of the 2D posteriors. The
observed overall excess in the upward-going multi-GeV νe events contributes
to the preference for the normal hierarchy.
Parametersin2 θ23∆m232 (10−3 eV2 )δCP (π)Normal Hierarchy
0.61
+2.15
-0.9

Inverted Hierarchy
0.41
-1.85
+0.7

Table 8.7: Oscillation parameters which are used to calculate the event rate
predictions shown in Figures 8.26 to 8.28.

8.6.3

Goodness of Fit

In order to test whether or not the models assumed in this analysis disagreewith the observed data, a goodness of fit test is performed according to theprescriptions described in [105]. First, 2000 points are randomly sampledfrom the MCMC chains which were used in the data analysis, which produces
a reduced set of sample points which represents the posterior distribution.
For each sampled point, a fake dataset is thrown using the predictions at thatpoint, and the likelihood Equation 8.7 is compared between the fake data andthe real data. Figure 8.29 shows the distribution of the log likelihood ratio
between the real and the fake data for all points. A p-value is then calculated
as the fraction of points at which the real data having a larger likelihood
value than the fake data. The calculated p-value, which the authors of[105] refer to as the posterior predictive p-value, is essentially the classicalp-value as used in the standard goodness of fit test in frequentist analyses
averaged over the entire posterior. This test is therefore assessing the fitnessof the entire Bayesian posterior model to the observed data, rather than ata single best-fit point. The posterior predictive p-value for this analysis iscalculated to be 10.3%, which is large enough that the assumed models are
not excluded by the data according to this test.

8.6.4

Result Comparison to Other Experiments

Figure 8.30 compares the results of this analysis for ∆m232 and sin2 θ23
with the latest results from other experiments. In the plots, 90% HPD
credible regions are shown for Bayesian analyses, whereas for frequentist
162

8.6. Data Analysis Results

Sub-GeV 1Re ≥1dcy

Sub-GeV 1Re 0dcy
600
400
400
200

200

0
-1

-0.5

0

0.5

1

cosθZ

0
-1

Sub-GeV 1Rµ 0dcy

-0.5

0

0.5

1

cosθZ

Sub-GeV 1Rµ 1dcy
600

100
400
50
200

0
-1

-0.5

0

0.5

1

cosθZ

0
-1

Sub-GeV 1Rµ ≥2dcy

-0.5

0

0.5

1

cosθZ

Sub-GeV 2Rπ0

400
1000
300
200

500

100
0
-1

-0.5

0

0.5

1

cosθZ

0
-1

-0.5

0

0.5

1

cosθZ

Figure 8.26: Distributions of the cosine of the zenith angle for the sub-GeV
event samples. The black markers indicate the data, and the solid green
and the dashed red lines represent the expectations calculated with normal
hierarchy and inverted hierarchy respectively. For each hierarchy, the expectation is calculated by setting the oscillation and systematic parametersat the mode of the posteriors for the respective hierarchy.

163

8.6. Data Analysis Results

Multi-GeV 1Re νe

Multi-GeV 1Re νe

30
100

20

50

10

0
-1

-0.5

0

0.5

1

cosθZ

0
-1

Multi-GeV MRe νe
100

-0.5

0

0.5

1

cosθZ

Multi-GeV MRe νe
80
60

50

40
20

0
-1

-0.5

0

0.5

1

cosθZ

0
-1

-0.5

0

0.5

1

cosθZ

Figure 8.27: Distributions of the cosine of the zenith angle for the multiGeV e-like event samples. The black markers indicate the data, and thesolid green and the dashed red lines represent the expectations calculatedwith normal hierarchy and inverted hierarchy respectively. For each hierarchy, the expectation is calculated by setting the oscillation and systematicparameters at the mode of the posteriors for the respective hierarchy.

164

8.6. Data Analysis Results

Multi-GeV 1R
1RµµMulti-GeV MR
MRµ
µ

100

200
150

50

100
50

0
-1

-0.5

0

0.5

1

cosθZ

0
-1

-0.5

0

0.5

1

cosθZ

Multi-GeV MRe oth.
100

50

0
-1

-0.5

0

0.5

1

cosθZ

Figure 8.28: Distributions of the cosine of the zenith angle for the multi-GeVµ-like and other event samples. The black markers indicate the data, and thesolid green and the dashed red lines represent the expectations calculatedwith normal hierarchy and inverted hierarchy respectively. For each hierarchy, the expectation is calculated by setting the oscillation and systematicparameters at the mode of the posteriors for the respective hierarchy.

165

8.6. Data Analysis Results60

p = 206/2000= 10.3%

40

20

0

-60

-40

-20

0

20
40
lnLData-lnLThrow

Figure 8.29: Distribution of the log likelihood ratio between the observed
data and fake data which is used for the goodness of fit test. The posterior
predictive p-value is calculated to be 10.3%.
analyses 90% confidence regions are shown instead. The plots show thelatest preliminary results from the conventional SK atmospheric neutrino
analysis[106][107](labeled as “SK1-4 Freq.” in the plots) which employs afrequentist analysis technique by χ2 minimization and event selection basedon APFIT, the preexisting event reconstruction which was mentioned in
Chapter 6. The analysis uses the full SK-I to IV data of 5326 days exposure
which is more than double the data statistics compared to the SK-IV-only
data which is used in the Bayesian analysis with fiTQun event selection
presented in this chapter, and the diﬀerence in sensitivity between the twoanalyses is dominantly due to the additional data in the APFIT-based analysis. The data from the old SK-I to III phases is not included in the fiTQunbased analysis due to the observed large data-MC discrepancies as discussed
in Section 6.11, and inclusion of such data requires improvements in the detector simulation which is expected to be achievable in near future. The
T2K result[108] is from a Bayesian analysis which uses a similar MCMC
technique described in Section 8.4 and produces Bayesian credible regions.
The results from MINOS[109] and NOνA[110] are both from frequentist
analyses which report their results as confidence regions. One should notethat the Bayesian credible regions and frequentist confidence regions are different and are not comparable in a strict sense, although such comparison
is interesting as a rough measure.
1668.6. Data Analysis Results∆m232 (10-3 eV2)

Normal Hierarchy
4SK4 Bayes.
SK1-4 Freq.
T2K

MINOS
NOνA

3

2

1

0.3

0.4

0.5

0.6

0.7
sin2θ23∆m232 (10-3 eV2)

Inverted Hierarchy
-1

-2

-3
SK4 Bayes.
SK1-4 Freq.
T2K

-4

0.3

0.4

MINOS
NOνA

0.5

0.6

0.7
sin2θ23

Figure 8.30: 90% credible and confidence regions for ∆m232 and sin2 θ23 forthe results from various experiments. The red contours labeled as “SK4
Bayes.” are the results from the analysis presented in this chapter which
is a Bayesian analysis using SK-IV atmospheric neutrino data selected byfiTQun event reconstruction. The upper and the lower plots are the casesof assuming normal and inverted hierarchy respectively.
167

Chapter 9

Conclusions and Outlook
An oscillation analysis of the Super-Kamiokande atmospheric neutrino data
was presented. The analysis employed a new maximum likelihood event
reconstruction algorithm to reconstruct and select atmospheric neutrino
events, and oscillation parameters were estimated using a Markov chain
Monte Carlo technique which produces marginalized Bayesian posterior probability distributions for neutrino oscillation parameters based on the observed data.
Analyzing the data of 2520 days exposure from SK-IV, the result prefers
normal hierarchy with the posterior probability of 85.9%, and the modeand the 68% highest posterior density credible interval of each oscillation
parameter’s 1D posterior probability distribution for normal hierarchy are+0.17
2−3 eV2 . The analysis
sin2 θ23 = 0.606+0.044−0.118 and ∆m32 = 2.13−0.38 × 10
also has small sensitivity to δCP , however, nothing conclusive can be said
regarding whether CP violation is observed.
The Bayesian analysis method using MCMC allows one to fully marginalize over nuisance parameters and report the result on the parameters of interest as marginalized Bayesian posterior probabilities. When parameters
have highly non-Gaussian behaviours, which is the case for the oscillation
parameters in this analysis, being able to actually marginalize over nuisanceparameters is advantageous compared to the more common frequentist technique of profiling the likelihood, i.e. to minimize the negative-log-likelihoodor the χ2 over nuisance parameters, which becomes equal to marginalization
only when the parameters have Gaussian behaviours.
The new event reconstruction algorithm “fiTQun” significantly improves
particle identification and vertex and kinematic reconstruction performances
compared to the preexisting event reconstruction at SK. The new algorithm
was first used in the T2K νe appearance analysis for π 0 rejection, which
resulted in more than 60% reduction of the NC background in the signal
νe event sample compared to the previous π 0 rejection method. However,
the application of fiTQun in T2K has been limited to the π 0 rejection only,
and the SK atmospheric neutrino analysis presented in this thesis is thefirst physics analysis which makes an extensive use of the new algorithm
168

Chapter 9. Conclusions and Outlookby employing an entirely fiTQun-based event selection. There are currently
ongoing eﬀorts to extend the usage of fiTQun in various other studies suchas the nucleon decay analyses at SK[111], T2K oscillation analyses[112] andstudies for future water Cherenkov detectors[113].
Only the data from SK-IV was used in the analysis presented in thisthesis due to the large data-MC discrepancy observed in SK-I to III as mentioned in Section 6.11. However, by improving the detector simulation, it isexpected that the data from SK-I to III can also be included in the oscillation analysis using fiTQun in near future. Table 9.1 shows the expected
sensitivity to the mass hierarchy at each of the following SK exposure: 2520
days which is the the current SK-IV exposure used in the presented data
analysis, 5300 days which is roughly the current total exposure from SK-I to
IV, and 8500 days which is the expected accumulated exposure in 10 years
when the next generation neutrino oscillation experiments such as HyperKamiokande[114] and DUNE[115] are expected to start. As in the table,sensitivity to the mass hierarchy continues to improve significantly as thedata statistics increase.
Exposure
True NH
True IH

2520 days
67.2%
61.8%

5300 days
81.6%
71.4%

8500 days
91.0%
80.9%

Table 9.1: Expected posterior probabilities for for favouring the correct mass
hierarchy for diﬀerent SK exposure, based on the Asimov dataset produced
at either normal or inverted hierarchy, sin2 θ23 = 0.6 and other oscillation parameters set to the values in Table 8.3. The middle row shows the posterior
probabilities for favouring NH for the dataset produced with NH, and thebottom row are the probabilities for favouring IH for the datasets producedwith IH.
Long baseline accelerator neutrino experiments such as T2K and NOνA
are rapidly accumulating data, and the experiments have sensitivity to δCP
and mass hierarchy. However, the competing eﬀects from the unknowns
produce degeneracies in the oscillation probabilities which limit the experiments’ sensitivities. By combining their results and the results from SK
atmospheric neutrino analysis which strongly constrains the mass hierarchy,
one can disentangle the diﬀerent oscillation eﬀects and improve the sensitivity to δCP and the mass hierarchy compared to the analyses of individual
experiments. It is expected that decisive measurements of the neutrinomass hierarchy and CP violation would require the next generation neu169

Chapter 9. Conclusions and Outlook
trino oscillation experiments such as Hyper-Kamiokande and DUNE which
are planned to start their operation in the late 2020’s. However, by combining the current global eﬀorts we can further explore the remaining mysteriesof neutrino oscillations in the interim.

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maximum-likelihood event reconstruction for Super-Kamiokande”.
Poster presentation at NEUTRINO 2016.
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and future experiments”. Poster presentation at NEUTRINO 2016.
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[115] R. Acciarri et al.. “Long-Baseline Neutrino Facility (LBNF) and Deep
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2016.

179

Appendix A

List of Systematic
Parameters
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25

Absolute flux E<1GeV
Absolute flux E>1GeV
Flux νµ /νe ratio E<1GeV
Flux νµ /νe ratio 1<E<10GeV
Flux νµ /νe ratio E>10GeV
Flux ν̄e /νe ratio E<1GeV
Flux ν̄e /νe ratio 1<E<10GeV
Flux ν̄e /νe ratio E>10GeV
Flux ν̄µ /νµ ratio E<1GeV
Flux ν̄µ /νµ ratio 1<E<10GeV
Flux ν̄µ /νµ ratio E>10GeV
ν path length
Matter eﬀect
Solar activity
CCQE xsec.
CCQE ν/ν̄ ratio
CCQE νµ /νe ratio
MAQE
MEC xsec.
Single meson xsec.
Single meson MA
Single meson C5A (0)Single meson bkg.
Single meson π 0 /π ± ratio
Single meson ν̄/ν ratio

26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50

DIS model diﬀerence
DIS xsec.
DIS Q2 shape low-W
DIS Q2 shape high-W
Coherent π xsec.
NC/CC ratio
K/π ratio
Up/down ratio
Horizontal/vertical ratio
Multi-GeV normalization
Hadron simulation
CC ντ xsec.
FC/PC separation
FC reduction
Fiducial volume
Non-ν bkg. e-like
Non-ν bkg. mu-like
Ring separation
PID 1-ring
PID multi-ring
Energy scale
Up/down energy scale asym.
Decay-e tagging eﬀ.
2Rπ 0 eﬀ.
Sub-GeV FSI parameter #00

Table A.1: List of systematic parameters. Continues to Table A.2.

180

Appendix A. List of Systematic Parameters

51
52
53
54
55
56
57
58
59
60
61
62
63

Sub-GeV FSI parameter #01
Sub-GeV FSI parameter #02
Sub-GeV FSI parameter #03
Sub-GeV FSI parameter #04
Sub-GeV FSI parameter #05
Sub-GeV FSI parameter #06
Sub-GeV FSI parameter #07
Sub-GeV FSI parameter #08
Sub-GeV FSI parameter #09
Sub-GeV FSI parameter #10
Sub-GeV FSI parameter #11
Multi-GeV FSI parameter #00
Multi-GeV FSI parameter #01

64
65
66
67
68
69
70
71
72
73
74
75
76

Multi-GeV FSI parameter #02
Multi-GeV FSI parameter #03
Multi-GeV FSI parameter #04
Multi-GeV FSI parameter #05
Multi-GeV FSI parameter #06
Multi-GeV FSI parameter #07
Multi-GeV FSI parameter #08
Multi-GeV FSI parameter #09
Multi-GeV FSI parameter #10
Multi-GeV FSI parameter #11
Multi-GeV FSI parameter #12
Multi-GeV FSI parameter #13
Multi-GeV FSI parameter #14

Table A.2: List of systematic parameters. Continues from Table A.1.

181

Appendix B

Atmospheric Neutrino
Selection Improvement by
fiTQunIn the previously published Super-K atmospheric neutrino oscillation analyses, APFIT was used for event selection instead of fiTQun. The impact of
employing fiTQun for event selection is discussed in this appendix.
The event selection criteria for APFIT-based analysis, as described indetail in [106], are very similar to the fiTQun-based event selection introduced in Section 8.1.1, with some diﬀerences in the sample definitions. In
particular, the events selected as sub-GeV one-ring e-like with 0 decay electron by APFIT are further subdivided into two categories using the fiTQun
π 0 fit variables discussed in Section 6.9.1: out of the selected events, theones which satisfy ln(Lπ0 /Le ) > 130 and 95 < mπ0 (MeV/c2 ) < 185 arecategorized as “Sub-GeV one-ring π 0 ” sample, and the rest are categorized
as “Sub-GeV one-ring e-like 0 decay” sample. The additional cut usingfiTQun was introduced to reduce the rather large NCπ 0 background contamination in the events selected as one-ring e-like by APFIT. Other thanthe π 0 cut above, all selection cuts are done using APFIT information onlyin the APFIT-based analysis. Such additional π 0 rejection cut is not employed for the fiTQun-only selection since the one-ring e-like selection by
fiTQun reduces the π 0 background to a suﬃciently low level.
Table B.1 shows the expected event rates and the interaction mode
breakdown of each event sample selected by the APFIT-based event selection. MC expectations are calculated using the same oscillation parameters
and normalization as described in Section 8.1.4. Comparing this to Table 8.1 for fiTQun-only selection, it can be seen that fiTQun event selection
achieves higher fraction of the targeted event category and lower fractionof backgrounds across all event samples. In particular, νµ contamination in
the e-like samples in multi-GeV is significantly reduced by fiTQun.
In order to see the impact of the above improvement in event selection on

182

Appendix B. Atmospheric Neutrino Selection Improvement by fiTQun%
1Re 0dcy 1Re≥1dcy 1Rµ 0dcy 1Rµ 1dcy 1Rµ≥2dcy
νe CC
72.77
81.20
4.36
0.11
0.05
ν̄e CC
23.92
1.80
1.32
0.00
0.00
νµ CC
0.19
8.24
68.07
68.21
94.75
ν̄µ CC
0.04
2.31
8.03
29.11
3.29
NC
3.08
6.44
18.23
2.57
1.91
Total 4899.5
563.0
909.0
4185.9
367.0
%
νe CC
ν̄e CC
νµ CC
ν̄µ CCNC
Total

1Re νe
63.70
9.44
8.98
1.55
16.33
345.8

1Re ν̄e
54.96
37.68
0.72
0.20
6.44
1040.4

1Rµ
0.27
0.12
62.19
37.26
0.15
1358.7

MRe νe
53.43
10.03
14.70
1.98
19.86
431.9

MRe ν̄e
53.66
26.60
4.73
0.58
14.43
427.7

1Rπ 0
9.97
3.25
1.23
0.23
85.32
234.6

2Rπ 0
7.04
2.44
0.99
0.08
89.44
889.9

MRµ MRe oth.
2.74
29.31
0.42
3.26
70.50
31.72
21.25
3.54
5.09
32.17
1190.2681.5

Table B.1: Interaction mode breakdown and the total MC event rates foreach event sample categorized by the APFIT-based selection, where the upper and the lower table is for sub-GeV and multi-GeV samples respectively.
MC expectations are calculated with two-flavour oscillations at sin2 2θ=1.0
and ∆m2 =2.5 × 10−3 eV2 and is normalized to 2519.89 days exposure.
the sensitivity to oscillation parameters, a sensitivity study was performed
following the same procedures described in Section 8.5. The Asimov datasets
for this study were produced by taking sin2 θ23 = 0.55 and ∆m232 = +2.39 ×
10−3 eV2 (normal hierarchy), and the rest of the oscillation parameters set
to the values listed in Table 8.3. The exposure was taken to be 2519.89 days.
Figure B.1 compares resulting 90% HPD credible regions for ∆m232 and2sin θ23 between the cases of using APFIT or fiTQun for event selection.
The posterior probability for normal hierarchy is 61.5% for APFIT selection
and 63.0% for fiTQun selection.

183

Appendix B. Atmospheric Neutrino Selection Improvement by fiTQun∆m232 (10-3 eV2)

Normal Hierarchy
4
APFIT
fiTQun

3

2

1

0.3

0.4

0.5

0.6

0.7
sin2θ23∆m232 (10-3 eV2)

Inverted Hierarchy
−1

−2

−3
APFIT
fiTQun

−4

0.3

0.4

0.5

0.6

0.7
sin2θ23

Figure B.1: 90% HPD credible regions for ∆m232 and sin2 θ23 compared
between the cases of using APFIT or fiTQun for event selection, on Asimov
datasets. The black contours are for APFIT selection and the red contours
are for fiTQun. The upper and the lower plots are the cases of assuming
normal and inverted hierarchy respectively.
184