BOSTON UNIVERSITYGRADUATE SCHOOL OF ARTS AND SCIENCES

Dissertation

DISCERNING THE NEUTRINO MASS ORDERINGUSING ATMOSPHERIC NEUTRINOS IN
SUPER-KAMIOKANDE I–V

byTHOMAS WESTER
B.A., University of Chicago, 2017

Submitted in partial fulfillment of therequirements for the degree ofDoctor of Philosophy2023

© 2023 by
THOMAS WESTERAll rights reserved

Approved byFirst Reader
Edward T. Kearns, Ph.D.
Professor of Physics

Second Reader
Robert M. Carey, Ph.D.
Professor of Physics

To my Dear Ol’ Dad!

iv

Acknowledgments
I have been incredibly fortunate to meet and work with a host of fantastic physicists
who, through their combined mentorship, set me on a path to write this dissertation.
In approximate chronological order, thanks to Stephen Pordes for giving me my firstsummer research job at Fermilab, and to Thomas Alexander for helping me destroy
my first PMT. Thanks to my University of Chicago mentors, Professor Luca Grandi
and Dr. Richard Saldanha, for connecting me to the world of experimental particle
physics, for the opportunity to build and run our own detector, and for teaching me
to think harder about my plots before I show them at group meetings. Thanks to
my mentors at Fermilab and MIT, Professors Janet Conrad, Taritree Wongjirad, andDr. Daniel Winklehner. I would not have decided to study neutrino physics without
your encouragement and support.
I owe so much of my scientific training to my mentors at Boston University, Dr.
Linyan Wan, and my advisor, Professor Ed Kearns. Linyan’s vast knowledge of SuperK, her ability to find new and interesting analysis techniques using our 27-year-old
experiment, and her clear and comprehensive presentations make her an exemplary
physicist that I aspire to be like. Ed’s emphasis on using basic principles of detectors
as the building blocks for complicated analyses has helped me to develop my intuition
as an experimental particle physicist. I am also grateful to Ed for his perspectives onpast and future experiments, which provided much-needed context when first entering
the world of neutrino oscillation analyses. I am further indebted to Ed’s keen eye foraesthetics, which has saved me many times from presenting inscrutable plots to ourcollaborators. Finally, I am most grateful for Ed’s guidance in helping me to distill thestory of the measurement presented in this dissertation from all its technical aspects.
The analysis presented in this dissertation is only possible thanks to the hard
work of so many Super-Kamiokande collaborators. I was fortunate to meet many ofvthem in person throughout my first two years of grad school during trips to Japan.
In 2018, I got to witness the entire collaboration mobilize to refurbish the 10-storey
SK detector from head-to-toe—an incredible feat of coordination that highlighted
the unique expertise of each collaborator. Through my small roles managing theouter detector PMT high voltages, and tuning the SK outer detector simulation, I
learned the fundamentals of SK calibration from Dr. Shunichi Mine, who I hope toaccompany on another ski trip one day.
For the analysis presented in this dissertation, I am especially thankful to theoscillation working group members, Pablo Fernández, Seungho Han, and Magda
Posiadala-Zezula, for their direct contributions to many aspects of this analysis, andinsights during our regular meetings. Thank you for fielding my requests for obscureanalysis inputs, and for all your critical work needed to prepare the new expandedfiducial volume and SK V data for analysis.
A special thanks is in order to the OscMaestro himself, Professor Roger Wendell.
I have learned so much about neutrino oscillations and assembling the myriad components required for this analysis under Roger’s guidance. Roger gave me the tools I
needed to perform this analysis, and then encouraged me to question conventions andto look for improvements. Roger also fully convinced me that I’d rather know theneutrino mass ordering than the value of δCP . After this dissertation is submitted,
I am excited to continue working with Roger to finish writing the next SK neutrino
oscillation paper.
Physics is no fun without friends. I am thankful for all my friends at Boston
University, especially Chris Cosby, Brian Dawes, Diana Gooding, Paige Kunkle, andDavid Long. Aside from being accomplished physicists, they are kind, funny, andinsightful colleagues who have made my time at Boston University so enjoyable. No
matter if we were working on problem sets, discussing literature1 , or just chatting in
1

Spiderlight doesn’t qualify.

vithe office, their upbeat attitudes and support during our time in grad school havemade it a joy to come to campus every day.
I would be remiss to not mention my friends who remind me that there’s more
to life than neutrinos. From my time at the University of Chicago (UC), Jonathan
Gerstell, Isaac Krone, Aidan Lilienfeld, Ted Litchfield, and Avery Robinson haveremained the best of friends despite our distance. Our regular calls and perpetualstream-of-consciousness chats are a constant reminder for me to find humor in everyday things, and that the lessons of particle physics only provide one out of many
ways to approach complicated problems. I also thank my fellow members of Vacuist,
Stavros Aganis, Henry “Mica” Goldsmith, and Iain Standing, for giving me an outlet
to focus on music instead of neutrinos for a few hours every week here in Boston.
The people who deserve the most credit are the ones who have sustained me
through the marathon that is obtaining a physics Ph.D. Moving to Boston put meon the East Coast for the first time, and I have been so lucky to have another homebase here with the Kapadia family. Quaid, Carolyn, Andrew, Alistair, Grandmom,
Dadima, and Caesar have always made me feel at home, provided me with delicious homemade Indian food, and, occasionally, a healthy amount of chaos! Back in
Chicago, I have had the lifelong support of my parents, Barbara and William, and mybrother Stephen. My father, a particle physicist himself, showed me by example whata fulfilling career pursuing interesting questions about the universe can look like. Last
but not least, thank you to my partner of 8 years (time flies!), Alex Kapadia. I owe
who I am as a person today to her profound thoughtfulness, intellect, and kindness.
I am so happy to have built our life together in Chicago and Boston, and can’t wait
to start our next adventure.

vii

DISCERNING THE NEUTRINO MASS ORDERINGUSING ATMOSPHERIC NEUTRINOS IN
SUPER-KAMIOKANDE I–V
THOMAS WESTER
Boston University, Graduate School of Arts and Sciences, 2023
Major Professor: Edward T. Kearns, Ph.D., Professor of Physics

ABSTRACT
Neutrino oscillation experiments have demonstrated evidence for three distinct
neutrino masses. However, whether there are two light neutrinos and one heavy
neutrino (normal), or the other way around (inverted), known as the neutrino massordering, remains undetermined. This thesis presents a search for indications of theneutrino mass ordering in 6511 live-days (484 kiloton-years) of atmospheric neutrino
data collected with the Super-Kamiokande (SK) detector between 1996 and 2020.
The data set is a 30 % increase in exposure since the previous published analysis,
and the analysis methodology includes improvements to the separation of neutrino
and anti-neutrino data. This thesis also presents an analysis of the SK data withconstraints on neutrino oscillation parameters from reactor neutrino experiments andthe T2K long-baseline experiment. The constraints from the T2K experiment include,
for the first time, an anti-neutrino-enhanced data sample. The atmospheric neutrino
analysis favors the normal neutrino mass ordering, rejecting the inverted ordering
at the 92.3 % (1.43 σ) level. The inclusion of external constraints from T2K data
increases the rejection to the 97.9 % (2.03 σ) level.

viii

Contents
Title Pagei

Acknowledgementsv

Abstract

viii

List of Tables

xiii

List of Figures

xvi

List of Abbreviationsxxii

1 Introduction

1

1.1

The Neutrino Mass Ordering . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Brief History of Neutrinos . . . . . . . . . . . . . . . . . . . . . . . .

3

1.3

Neutrinos in the Standard Model . . . . . . . . . . . . . . . . . . . .

5

1.4

Neutrino Mass & Oscillations . . . . . . . . . . . . . . . . . . . . . .

8

1.4.1

Neutrino Oscillation Theory . . . . . . . . . . . . . . . . . . .

8

1.4.2

Matter Effects . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

Neutrino Oscillation Experiments . . . . . . . . . . . . . . . . . . . .

14

1.5.1

Solar Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . .

14

1.5.2

Reactor Neutrinos. . . . . . . . . . . . . . . . . . . . . . . .

18

1.5.3

Atmospheric Neutrinos . . . . . . . . . . . . . . . . . . . . . .

20

1.5.4

Long Baseline Neutrinos . . . . . . . . . . . . . . . . . . . . .

23

Present Status of Oscillation Measurements . . . . . . . . . . . . . .

26

1.6.1

27

1.5

1.6

Octant of θ23 . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix

1.7

1.6.2

Value of δCP . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

1.6.3

Neutrino Mass Ordering . . . . . . . . . . . . . . . . . . . . .

28

Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

2 The Super-Kamiokande Detector

31

2.1

Detection Principles . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

2.2

Detector Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

2.2.1

Inner Detector . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

2.2.2

Outer Detector . . . . . . . . . . . . . . . . . . . . . . . . . .

36

2.2.3

Water System . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

2.2.4

Electronics & Data Acquisition . . . . . . . . . . . . . . . . .

40

Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

2.3.1

ID PMT Calibration . . . . . . . . . . . . . . . . . . . . . . .

45

2.3.2

OD PMT Calibration . . . . . . . . . . . . . . . . . . . . . . .

49

2.3.3

Water Parameter Measurements . . . . . . . . . . . . . . . . .

49

Account of SK V Open Tank Work . . . . . . . . . . . . . . . . . . .

51

2.3

2.4

3 Data Reduction & Reconstruction
3.1

3.2

55

Data Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

3.1.1

FC Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

3.1.2

PC Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

3.1.3

Up-µ Reduction . . . . . . . . . . . . . . . . . . . . . . . . . .

69

3.1.4

Post-reduction Eye-scanning . . . . . . . . . . . . . . . . . . .

72

3.1.5

ID-OD Cross Talk . . . . . . . . . . . . . . . . . . . . . . . .

72

Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

3.2.1

Event Vertex . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

3.2.2

Ring Counting . . . . . . . . . . . . . . . . . . . . . . . . . .

76

3.2.3

Particle Identification . . . . . . . . . . . . . . . . . . . . . . .

77

x

3.2.4

Event Vertex, Revisited . . . . . . . . . . . . . . . . . . . . .

81

3.2.5

Momentum Reconstruction . . . . . . . . . . . . . . . . . . . .

81

3.2.6

Ring Counting, Revisited . . . . . . . . . . . . . . . . . . . . .

82

3.2.7

Decay Electron Tagging . . . . . . . . . . . . . . . . . . . . .

82

3.2.8

Neutron Tagging . . . . . . . . . . . . . . . . . . . . . . . . .

84

4 Simulation of Atmospheric Neutrinos
4.1

4.2

4.3

Atmospheric Neutrino Flux Calculation . . . . . . . . . . . . . . . . .

92

4.1.1

Corrections to the Atmospheric Neutrino Flux . . . . . . . . .

93

Neutrino Interaction Models . . . . . . . . . . . . . . . . . . . . . . .

95

4.2.1

Quasi-elastic . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96

4.2.2

Single Pion Production . . . . . . . . . . . . . . . . . . . . . . 102

4.2.3

Deep Inelastic Scattering . . . . . . . . . . . . . . . . . . . . . 105

4.2.4

Final State Interactions . . . . . . . . . . . . . . . . . . . . . 106

Detector Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5 Atmospheric Neutrino Oscillation Analysis
5.1

5.2

91

111

Neutrino Sample Selection . . . . . . . . . . . . . . . . . . . . . . . . 112
5.1.1

FC Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.1.2

PC Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.1.3

Up-µ Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.1.4

Zenith & Momentum Binning . . . . . . . . . . . . . . . . . . 122

5.1.5

Expanded Fiducial Volume . . . . . . . . . . . . . . . . . . . . 123

Systematic Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.2.1

Atmospheric Neutrino Flux Uncertainties . . . . . . . . . . . . 130

5.2.2

Neutrino Interaction Model Uncertainties . . . . . . . . . . . . 133

5.2.3

Detector Performance & Reconstruction Uncertainties . . . . . 140

5.2.4

Oscillation Uncertainties . . . . . . . . . . . . . . . . . . . . . 150
xi

5.3

Analysis Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
5.3.1

Oscillation Probability Calculation . . . . . . . . . . . . . . . 151

5.3.2

Fij Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

5.3.3

χ2 Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

5.4

Atmospheric Neutrino Oscillation Results . . . . . . . . . . . . . . . . 156

5.5

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

6 Adding External Constraints
6.1

Constraints on θ13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.1.1

6.2

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

Constraints from T2K . . . . . . . . . . . . . . . . . . . . . . . . . . 172
6.2.1

6.3

167

The T2K Experiment . . . . . . . . . . . . . . . . . . . . . . . 173

T2K Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
6.3.1

Flux Re-weighting . . . . . . . . . . . . . . . . . . . . . . . . 182

6.3.2

Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

6.3.3

Cross Section Re-weighting . . . . . . . . . . . . . . . . . . . . 186

6.3.4

Systematic Uncertainties . . . . . . . . . . . . . . . . . . . . . 199

6.3.5

T2K Model Performance & Limitations . . . . . . . . . . . . . 206

6.4

SK+T2K Model Results . . . . . . . . . . . . . . . . . . . . . . . . . 209

6.5

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

7 Conclusions

215

A Best-Fit Atmospheric Neutrino Samples

218

B Best-Fit Systematic Uncertainties

225

List of Journal Abbreviations

232

Bibliography

234

Biography

241xii

List of Tables
1.1

The masses and electric charges of the three generations of spin-1/2
fermions in the Standard Model. . . . . . . . . . . . . . . . . . . . . .

1.2

6

Listing of averaged PMNS parameters and mass splitting measurements from various experiments [51]. . . . . . . . . . . . . . . . . . .

27

2.1

Operating conditions for the six data-taking phases of the SK experiment. 33

2.2

Triggers used in the different SK data-taking phases. . . . . . . . . .

44

3.1

Variables used in the SK IV neutron tagging neural network selection.

89

4.1

Solar minimum and maximum fractions used for the simulation of atmospheric neutrinos in each phase of SK data taking in this analysis.

4.2

96

Probability scaling factors applied to the neut FSI processes used in
the simulation of atmospheric neutrino interactions. . . . . . . . . . . 108

5.1

Momentum bin definitions for atmospheric neutrino sub-samples in thisanalysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5.2

Nominal MC and data events in each sub-sample for this analysis. . . 127

5.3

Fractional change for DIS events due to a 1σ change in the PDF Difference uncertainty. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

5.4

Parameter sets used for computing the change in event weights due touncertainty in FSI processes. . . . . . . . . . . . . . . . . . . . . . . . 138

5.5Reduction uncertainties for the different neutrino samples used in thisanalysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
xiii

5.6

Energy scale uncertainties for each SK phase and fiducial volume region.149

5.7

Neutrino propagation layers and corresponding densities used for calculating neutrino oscillation probabilities in this analysis, based on asimplified PREM [44]. . . . . . . . . . . . . . . . . . . . . . . . . . . 151

5.8

The oscillation point grid used for the atmospheric neutrino analysis.

5.9Best-fit neutrino oscillation parameters from the atmospheric neutrino

157

oscillation analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.1

The oscillation point grid used for the SK analysis with sin2 θ13 constrained. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

6.2

Best-fit neutrino oscillation parameters from the SK analysis withsin2 θ13 constrained. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

6.3

Nominal oscillation parameters used by the T2K model. . . . . . . . . 183

6.4

Un-oscillated event rates at SK, used by the T2K model for normalizing
the SK MC, following Equation 6.7. . . . . . . . . . . . . . . . . . . . 185

6.5

Summary of reduction steps for the five single-ring samples in the T2K
analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

6.6

Nominal T2K model MC e-like event counts after each reduction step. 196

6.7

Nominal T2K model MC CC 1π-like event counts after each reduction
step. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

6.8

Nominal T2K model MC µ-like event counts after each reduction step. 198

6.9

List of T2K flux re-weighting and uncertainty parameters implemented
in the T2K model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

6.10 Cross section parameter values used for re-weighting and calculating
systematic uncertainties in the T2K model. . . . . . . . . . . . . . . . 204
6.11 The oscillation point grid used for the SK+T2K model analysis withsin2 θ13 constrained. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
xiv6.12 Best-fit neutrino oscillation parameters in the combined SK+T2K model
analysis with sin2 θ13 constrained. . . . . . . . . . . . . . . . . . . . . 211
B.1 Flux-, cross section-, and oscillation-related fitted systematic uncertainty parameters (epsilons) at the best-fit point for each of the three
analyses presented in this thesis. . . . . . . . . . . . . . . . . . . . . . 228
B.2 Detector-related fitted systematic uncertainty parameters (epsilons) at
the best-fit point for the atmospheric-only analysis. . . . . . . . . . . 229
B.3 Detector-related fitted systematic uncertainty parameters (epsilons) at
the best-fit point for the SK analysis with θ13 constrained. . . . . . . 230
B.4 Detector-related fitted systematic uncertainty parameters (epsilons) in
the SK+T2K model analysis. . . . . . . . . . . . . . . . . . . . . . . 231

xv

List of Figures
1·1 NOvA and T2K measurements of neutrino oscillation parameters sin2 θ23
and δCP in each of the two neutrino mass ordering scenarios. . . . . .

2

1·2 Feynman diagrams of example (a) CC and (b) NC processes. . . . . .

7

1·3 Feynman diagrams of the two processes exclusive to (a) electron neutrinos and (b) electron anti-neutrinos when propagating through a medium. 13
1·4 Measurements of electron neutrino survival probabilities from the BOREXINO experiment in 2018, adapted from [29]. . . . . . . . . . . . . . .

18

1·5 Atmospheric neutrino path length correspondence with zenith angle. .

21

1·6 Electron neutrino appearance probabilities for atmospheric neutrinos
and anti-neutrinos in each neutrino mass ordering scenario. . . . . . .
1·7 Schematic of the unknown neutrino mass ordering.

24. . . . . . . . . .

29

2·1 Illustration of the SK detector, half-filled with water. . . . . . . . . .

32

2·2 Illustration of Cherenkov radiation detection in SK. . . . . . . . . . .

35

2·3 The 20-inch ID PMT. . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

2·4 The quantum efficiency of a 20-inch PMT as a function of wavelength.

37

2·5 Pictures of the SK OD. . . . . . . . . . . . . . . . . . . . . . . . . . .

38

2·6 The water system used beginning with the SK V period in 2018, reproduced from [64]. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

2·7 Overview of the high voltage tuning using the reference PMT. . . . .

46

2·8 Sum of relative gain-corrected ID PMT single-photoelectron distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xvi

47

2·9 The TQ-map of one ID PMT. . . . . . . . . . . . . . . . . . . . . . .

48

2·10 Overview of the water transparency measurement. . . . . . . . . . . .

50

2·11 The water level during the SK V open tank work. . . . . . . . . . . .

53

2·12 The leak assessment after the refurbishment work. . . . . . . . . . . .

53

2·13 Scenes from SK V open tank work. . . . . . . . . . . . . . . . . . . .

54

3·1 The three neutrino event toplogies at SK. . . . . . . . . . . . . . . . .

56

3·2 Labeled regions of the OD, and the segmentation Tyvek, which arereferenced in several parts of the PC reduction. . . . . . . . . . . . .

63

3·3 The cut applied based on the number of OD hits in each region as partof the PC2 reduction step. . . . . . . . . . . . . . . . . . . . . . . . .

65

3·4 Average number of FC, PC, and Up-µ events per day during the SK I-V
data-taking phases. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

3·5 Graphical demonstration of a Hough transform. . . . . . . . . . . . .

76

3·6 Ring counting likelihood, L2 − L1 , from Equation 3.8, for atmospheric
neutrino data and simulation for the SK IV-V phases. . . . . . . . . .

78

3·7 Neutrino candidate events from SK V data. . . . . . . . . . . . . . .

78

3·8 PID likelihood from Equation 3.12 for atmospheric neutrino data andsimulation for the SK IV-V phases. . . . . . . . . . . . . . . . . . . .

80

3·9 Distribution of the time differences between the first hit in a candidate
residual time PMT hit cluster and the true neutron capture time forsimulated neutron captures. . . . . . . . . . . . . . . . . . . . . . . .

87

3·10 Two example variable distributions used as input to the neutron tagging neural network. . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

3·11 Neural network performance in SK IV for neutron candidates with
N10 g 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

4·1 Atmospheric neutrino fluxes for different neutrino flavors. . . . . . . .

93

xvii

4·2 Neutron monitor counts during the SK I-V data-taking phases. . . . .

95

4·3 A Feynman diagram for the CCQE process. . . . . . . . . . . . . . .

97

4·4 Drawings of 1p1h and 2p2h processes. . . . . . . . . . . . . . . . . . .

98

4·5 A Feynman diagram for a resonant pion production process. . . . . . 103
4·6 Comparison of neut predicted π + −C cross sections for the absorption(ABS) and charge exchange (CX) FSI processes to those measured by
the DUET pion scattering experiment. . . . . . . . . . . . . . . . . . 107
4·7 Total electron neutrino cross sections divided by neutrino energy, versus neutrino energy, for a water target. . . . . . . . . . . . . . . . . . 109
4·8 True neutrino energy distributions of simulated atmospheric neutrino
MC events by interaction mode. . . . . . . . . . . . . . . . . . . . . . 110
5·1 Overview of FC event selection. . . . . . . . . . . . . . . . . . . . . . 113
5·2 Area-normalized SK IV-V MC distributions of the different input variables used for the Multi-Ring BDT selection. . . . . . . . . . . . . . . 118
5·3 νe and ν̄e separation using the Multi-Ring BDT score for SK IV-V data
and MC events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5·4 The modified FC single-ring event selection for the SK IV-V phases,
using the number of tagged neutrons. . . . . . . . . . . . . . . . . . . 120
5·5 Comparison between the zenith angle bins used in this analysis (solid
black lines) with the evenly-spaced bins used in previous SK oscillation
analyses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5·6 Non-neutrino backgrounds determined by eye-scanning during the SK IIV phases in the conventional and additional fiducial volume regions.

126

5·7 Zenith angle or log10 [p (MeV)] distributions for atmospheric neutrino
events used in this analysis, including events from an expanded fiducial
volume. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
xviii

5·8 Zenith angle distributions for the 10 SK IV-V FC single-ring samples
using the number of tagged neutrons. . . . . . . . . . . . . . . . . . . 129
5·9 Change in event weight for a 1σ change in the absolute flux normalization and flux flavor ratio systematics. . . . . . . . . . . . . . . . . 131
5·10 Neutron multiplicity measurement with T2K data versus neut MC
prediction for neutrinos (left) and anti-neutrinos (right). . . . . . . . 139
5·11 Ring counting systematic uncertainty estimation via the “scale-andshift” method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5·12 Absolute energy scale measurements for all SK phases and separately
measured in the conventional (solid lines) and additional (dashed lines)
fiducial volume regions. . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5·13 Demonstration of the ντ CC cross section uncertainty, applied to themulti-GeV single-ring e-like sample. . . . . . . . . . . . . . . . . . . . 155
5·14 1D ∆χ2 profiles of the fitted oscillation parameters for the atmospheric
neutrino oscillation analysis. . . . . . . . . . . . . . . . . . . . . . . . 158
5·15 Sensitivity to the mass ordering, ∆χ2I.O.−N.O. , as a function of sin2 θ23 .

161

5·16 The Up/Down asymmetry parameter for the eight multi-GeV e-likesamples, where the sensitivity to the neutrino mass ordering is expected.162
5·17 Distribution of ∆χ2I.O.−N.O. from toy data sets and p-values in the SK
atmospheric neutrino oscillation analysis with θ13 constrained. . . . . 165
6·1 1D ∆χ2 profiles of the fitted oscillation parameters for the SK analysis
with sin2 θ13 constrained. . . . . . . . . . . . . . . . . . . . . . . . . . 169
6·2 68 % and 90 % confidence level allowed regions of ∆m232 or ∆m231 andsin2 θ23 for the SK analysis with sin2 θ13 constrained. . . . . . . . . . . 170

xix6·3 2D 90 % allowed regions of sin2 θ23 and ∆m232 neutrino oscillation parameters from contemporary neutrino oscillation experiments, analyzed assuming sin2 θ13 is constrained. . . . . . . . . . . . . . . . . . . 172
6·4 Overview of the Tokai to Kamioka (T2K) experiment. . . . . . . . . . 174
6·5 Electron neutrino appearance probabilities versus neutrino energy for
the T2K baseline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
6·6 Neutrino energy spectrum from pion decay in flight from Equation 6.3. 177
6·7 Neutrino flux prediction at SK from the T2K neutrino beam, before
oscillations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
6·8 Expected EνRec. bias for the different neutrino interactions present in
the oscillated T2K FHC flux. . . . . . . . . . . . . . . . . . . . . . . 182
6·9 T2K model flux re-weighting demonstration, using atmospheric neutrino MC events generated according to the Honda flux at a fixed
direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
6·10 Ratios of double-differential 1p1h cross sections computed using theT2K nominal RFG model and the neut default LFG model. . . . . . 187
6·11 LFG to RFG re-weighting demonstration from the T2K model. . . . . 189
6·12 The “BeRPA” parametrization of the RPA correction used in the T2K
analysis and re-implemented as part of the T2K model. . . . . . . . . 190
6·13 Number of rings distribution for T2K model MC events in Left: FHCand Right: RHC modes. . . . . . . . . . . . . . . . . . . . . . . . . . 193
6·14 Number of decay electrons distribution for T2K model MC events in
Left: FHC and Right: RHC modes. . . . . . . . . . . . . . . . . . . 193
6·15 fitqun e-µ discriminator distribution for T2K model MC events in
Left: FHC and Right: RHC modes. . . . . . . . . . . . . . . . . . . 194

xx

6·16 fitqun µ-π + discriminator distribution for T2K model MC events in
Left: FHC and Right: RHC modes. . . . . . . . . . . . . . . . . . . 195
6·17 fitqun e-π 0 discriminator distribution for T2K model MC events in
Left: FHC and Right: RHC modes. . . . . . . . . . . . . . . . . . . 197
6·18 EνRec. distributions for the final T2K model e-like MC events, and corresponding T2K data counts. . . . . . . . . . . . . . . . . . . . . . . 200
6·19 EνRec. distributions for the final T2K model µ-like MC events, and corresponding T2K data counts. . . . . . . . . . . . . . . . . . . . . . . 202
6·20 1σ allowed oscillation parameter ranges from the T2K model and T2K
analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
6·21 1D ∆χ2 profiles of δCP from the T2K model and T2K analysis. . . . . 207
6·22 1D ∆χ2 profiles of the fitted oscillation parameters for the SK+T2K
model analysis with sin2 θ13 constrained. . . . . . . . . . . . . . . . . 212
6·23 Distribution of ∆χ2I.O.−N.O. from toy data sets and p-values in the
SK+T2K model analysis with θ13 constrained. . . . . . . . . . . . . . 213
6·24 1D ∆χ2 profiles of δCP from the SK+T2K model analysis, fit as a
combined data set and separately summed. . . . . . . . . . . . . . . . 214
A·1 SK I-III Sub-GeV single-ring samples, and Sub-GeV samples with onlya single zenith angle bin. . . . . . . . . . . . . . . . . . . . . . . . . . 219
A·2 SK IV-V Sub-GeV single-ring samples. . . . . . . . . . . . . . . . . . 220
A·3 SK I-III Multi-GeV single-ring samples. . . . . . . . . . . . . . . . . . 221
A·4 SK IV-V Multi-GeV single-ring samples. . . . . . . . . . . . . . . . . 222
A·5 SK I-V Multi-GeV multi-ring samples. . . . . . . . . . . . . . . . . . 223
A·6 SK I-V PC and Up-µ samples. . . . . . . . . . . . . . . . . . . . . . . 224

xxi

List of Abbreviations
1p1h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . One-particle one-hole
2p2h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two-particle two-hole
ADC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analog-to-digital converter
BDT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boosted decision tree
CC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Charged current
CCQE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Charged current quasi-elastic
CP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Charge-parity
DIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deep inelastic scattering
DONUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Direct Observation of the Nu Tau
DUNE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deep Underground Neutrino Experiment
ES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electron scatter
FC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fully-contained
FGD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Fine-grained detector
FHC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Forward horn current
FSI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Final state interaction
GALLEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gallium Experiment
GNO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gallium Neutrino Observatory
ID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inner detector
IMB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Irvine-Michigan-Brookhaven
K2K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . KEK to Kamioka
KamLAND . . . . . . . . . . . . . . . . . . Kamioka Liquid Scintillator Antineutrino Detector
KS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kolmogorov-Smirnov
LEP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Large Electron-Positron Collider
LFG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Local Fermi gas
MC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Monte Carlo
MEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Meson exchange current
xxii

MINOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Main Injector Neutrino Oscillation Search
MSW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mikheyev–Smirnov–Wolfenstein
NC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Neutral current
NOvA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .NuMI Off-axis νe Appearance
NRE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nucleon removal energy
OD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Outer detector
PC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partially-contained
PDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parton distribution function
PID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Particle identification
PMNS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pontecorvo-Maki-Nakagawa-Sakata
PMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Photomultiplier tube
POT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Protons on target
PREM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preliminary Reference Earth Model
QAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Charge-to-analog converter
QBEE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . QTC-based electronics with ethernet
QTC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Charge-to-time converter
RENO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reactor Experiment for Neutrino Oscillation
RFG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relativistic Fermi gas
RHC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reverse horn current
RMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Root-mean-square
RPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Random phase approximation
SAGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Soviet–American Gallium Experiment
SI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Secondary interaction
SK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Super-Kamiokande
SNO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sudbury Neutrino Observatory
SSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Standard Solar Model
T2K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Tokai to Kamioka
TDC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time-to-digital converter
TPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time projection chamber
Up-µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Upward-going muon
UV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ultra-violet
xxiiiChapter 1

Introduction
1.1

The Neutrino Mass Ordering

Is there one heavy neutrino and two light neutrinos, or the other way around? Thisis a simple question to pose in 2023, but its simplicity encodes decades of theoreticaland experimental results which have enabled us to ask it. This question presupposes
the following statements: There are three neutrinos. Neutrinos have mass. Two of
the neutrinos are close in mass, while a third is not. We have strong evidence for these
individual statements, but somehow, we ended up short of putting the neutrinos inorder. This is the neutrino mass ordering question.
The outlook for resolving this problem is optimistic: New experiments on the horizon will produce, detect, and measure more neutrinos than ever before. Additionally,
combined measurements of neutrino properties from multiple, currently-operating experiments have the potential to converge on a single answer—either “normal” for one
heavy neutrino, or “inverted” for two—possibly by the end of the decade [1]. In 2023,
our current best measurement of the neutrino mass ordering primarily comes fromthree operating experiments: The Tokai to Kamioka (T2K) experiment, the NuMI
Off-axis νe Appearance (NOvA) experiment, and the Super-Kamiokande (SK) experiment. Figure 1·1 summarizes recent measurements of the neutrino mass orderingfrom T2K and NOvA: The horizontal and vertical axes show possible combinationsof parameters of nature which, when combined with an assumption about the massordering, give better or worse agreement with observed data. Their uncertainties are10.7

Normal Orderingsin2θ23

0.6
0.5
0.4
0.3
0

0.7

(a)

T2K :
Nat. 580

BF≤ 90% CL

≤ 68% CL

NOvA:

BF

≤ 90% CL

≤ 68% CL

0.5

1

1.5

2

Inverted Orderingsin2θ23

0.6
0.5
0.4
0.3
0

(b)

T2K :
Nat. 580≤ 90% CL

NOvA :≤ 90% CL

π
2

π

δCP

≤ 68% CL3π
22π

Figure 1·1: NOvA and T2K measurements of neutrino oscillation
parameters sin2 θ23 and δCP in each of the two neutrino mass ordering
scenarios. Figure is reproduced from [2].
shown by the filled regions for NOvA and as black outlines for T2K. The top panel
shows the parameter measurements assuming the normal ordering, while the bottom
panel shows the same parameters assuming the inverted ordering.
We can make several observations about the figure. First, the normal ordering
contains the parameters which best describe both data sets. This is indicated by theblack cross and square in the top panel. Second, the parameter regions in the bottom
panel are smaller than those in the top panel. This indicates that a larger amountof the parameter space is ruled out in the inverted ordering. Finally, we note that,2

in the normal ordering, the filled region from NOvA and the outline from T2K do
not overlap. This tells us that, at present, their data sets are in tension, and favordifferent parameters of nature.
The tension between T2K and NOvA creates the following situation for the neutrino mass ordering: Both experiments are better described by the normal ordering,but their preferences for other parameters only overlap in the inverted ordering. Disentangling these conflicting statements will require more data from the T2K and
NOvA experiments. In the meantime, we can turn to the other experiment, SK, toweigh in on the mass ordering.
Unlike the T2K and NOvA experiments, which make their measurements using
artificial neutrino beams, the SK experiment observes neutrinos produced in the atmosphere. These atmospheric neutrinos offer a unique perspective on the neutrinomass ordering. This thesis will describe the historical context for how we have arrived
at the neutrino mass ordering question, how the SK experiment is sensitive to measuring it, and then present the current best measurement of the neutrino mass ordering
using atmospheric neutrinos. The thesis will conclude with a combined analysis of
the SK and T2K data sets to further improve the measurement.

1.2

Brief History of Neutrinos

Neutrinos were originally proposed by Wolfgang Pauli in 1930 as an explanation forthe continuous distribution of electron energies observed from beta decays [3]. In abeta decay, it was known that a neutron at rest decays into an electron and a proton,
which, through conservation of momentum and energy, should always result in anelectron and proton with fixed energies,?

n → p + e− .

3

(1.1)

While many explanations, including violations of energy conservation [4], were considered to explain this phenomenon, Pauli proposed the presence of a third particle,
which carried away part of the energy of the decay, as the solution. The beta decay
reaction, including a neutrino, is written asn → p + e− + ν,

(1.2)

where ν represents the neutrino. Two properties of the neutrino, that it was extremely
light and electrically neutral, were inferred from the beta decay spectrum, and thatany charged particles would have been detected.
While the proposed neutrino solved the beta decay issue in theory, experimental
detection of a neutrino would not be accomplished until 1956. A neutral particle,
perhaps massless, would necessarily have a minuscule chance of interaction. Hans
Bethe and Rudolf Peierls, using Enrico Fermi’s theory of the weak nuclear force,
calculated this chance in 1934, estimating the interaction cross section of a neutrino
produced from beta decay as σ < 10−44 cm2 [5, 6]. Thus, only a large detector and ahuge quantity of neutrinos from a known source would suffice to make an observation.
Twenty years later, Arthur Cowan and Frederick Reines devised a plan to observeanti-neutrinos from a nuclear reactor via the inverse beta decay process,ν̄e + p → n + e+ .

(1.3)

The Cowan-Reines experiment hoped to identify these inverse beta decays using thecoincidence of the energy emission from the anti-electron, e+ , annihilation with atomic
electrons in the detector, followed by the additional energy release from the captureof the neutron by cadmium molecules. The experiment observed excess coincidences
when the nuclear reactor was operational, implying the presence of neutrinos, andalso confirmed Bethe and Peierls cross section prediction [7].
4More neutrino experiments followed which established three distinct neutrino“flavors.” The neutrinos observed by Cowan and Reines always produced an antielectron, which later became understood as the anti-particle of one of three flavorsof charged leptons. In 1962, Leon Lederman, Melvin Schwartz and Jack Steinberger
observed neutrinos which produced the next-heaviest charged lepton, the muon, at
Brookhaven National Laboratory [8]. In contrast to the Cowan-Reines experiment,
the Brookhaven experiment used a source of stopped pions which decayed into muonsand neutrinos. When these neutrinos interacted in a downstream detector, muons
were detected instead of electrons. The third neutrino associated with the tau lepton
was observed by the Direct Observation of the Nu Tau (DONUT) experiment in 2000
[9] using high-resolution emulsion detectors to observe any short-lived tau particles
produced in neutrino interactions. Apart from these direct observations, detectors at
the Large Electron-Positron Collider (LEP) observed decays of the 91 GeV Z bosonat a rate in precise agreement with predictions which assumed exactly three neutrino
flavors.

1.3

Neutrinos in the Standard ModelIn the 1970s, Sheldon Glashow, Steven Weinberg, and Abdus Salam created a formulation of particle interactions that included the electromagnetic, weak, and strong
forces, three generations of quarks and leptons, and a mechanism to explain particle masses, which became known as the Standard Model [10, 11]. In the Standard
Model, particles are classified by their quantum spin. Matter consists of spin-1/2
particles called fermions, and the interactions between the fermions are mediated by
the exchange of spin-1 force-carrying particles called bosons. The Standard Modelfurther distinguishes two classes of fermions, quarks and leptons. Quarks interact via
all three forces, while the charged leptons interact via the weak and electromagnetic

5

forces. Neutrinos are leptons, but, since they are electrically neutral, do not haveelectromagnetic interactions. The charges and experimentally-measured masses ofthe fermions are listed in Table 1.1. The final component of the Standard Model isthe Higgs boson, a spin-0 particle which gives rise to the masses of the W and Z
bosons, the massless photon, and the masses of the charged fermions.

Iu

Quarks

2.3 MeV/c2

d

4.8 MeV/c2

eLeptons

GenerationIIc1270 MeV/c2s

95 MeV/c2

µ

III
t

Charge172700 MeV/c2+2/34180 MeV/c2

b

−1/3

0.51 MeV/c2

106 MeV/c2

1780 MeV/c2

τ

−1

νe
−6
< 10 MeV/c2

νµ−6< 10 MeV/c2

ντ−6< 10 MeV/c2

0Table 1.1: The masses and electric charges of the three generations ofspin-1/2 fermions in the Standard Model. Anti-particles have the opposite charge. The electron neutrino mass is bounded above at 0.9 eV/c2
by direct measurements from KATRIN in 2022 [12], and the sum of all
three neutrino masses is bounded above at ∼ 0.2 eV/c2 by data fits tocosmological models [13].
In the Standard Model, neutrinos only experience the weak interaction. There are
two variants of the weak interaction: charged current (CC) and neutral current (NC).
In a CC interaction, particles exchange a W boson and change flavor, either from aneutrino into the corresponding charged lepton, or from an “up,” charge +2/3, type
quark into a “down,” charge -1/3, type quark. The flavor change means that, fora neutrino to undergo a CC interaction, it must be energetic enough to produce its
corresponding charged lepton. In an NC interaction, a particle exchanges a Z boson,and never changes flavor. Example Feynman diagrams depicting weak interaction
processes are shown in Figure 1·2. Anti-particle interactions proceed similarly, butwith opposite charges.
6νe,µ,τ

νµµ−νe,µ,τ

ν̄e

Z
W−

e−

e−
(a)

e−
(b)

Figure 1·2: Feynman diagrams of example (a) CC and (b) NC processes. In (a), a muon decays weakly into a muon neutrino, electron
anti-neutrino and an electron. In (b), a neutrino of any flavor scatterswith an electron via the exchange of a Z boson, leaving the final stateparticles unchanged.
In addition to spin and charge, the quantum fields underlying the particles of theStandard Model also possess either an intrinsic left-handed or right-handed chirality.
Weak interactions depend on a particle’s chirality, as only left-handed particles orright-handed anti-particles may interact via the weak force. Charged quarks and leptons exist as both left-handed and right-handed variants, however, as neutrinos only
have weak interactions, there is no experimental evidence for right-handed neutrinos
or left-handed anti-neutrinos.
By extension, neutrino mass is not present in the Standard Model. In the Standard
Model, the quarks and charged leptons acquire their masses through interactions of
their left- and right-handed chiral variants with the Higgs field. Since neutrinos haveno right-handed particle or left-handed anti-particle, this interaction does not occur
in a minimal Standard Model, leaving neutrinos as massless particles. While adding
a right-handed neutrino with no interactions to the Standard Model would provide
a mechanism for neutrino mass, it still leaves open the question of the smallnessof neutrino mass relative to the other particles. For this reason, neutrino mass is
considered to be an indication of physics processes beyond the Standard Model.

7

1.4

Neutrino Mass & Oscillations

While neutrinos successfully explained beta decay, experiments observing neutrinos
from the Sun in the 1960s, and later using neutrinos produced in the atmosphere in
the 1980s, observed deficits compared to the expected rate of neutrino interactions.
These became known as the solar and atmospheric neutrino problems. Proposedsolutions to these problems varied, and included neutrino decay, or simply that thepredicted rate of neutrinos from these sources were incorrect. However, ultimately thephenomenon of neutrino oscillations would successfully explain both issues. Crucially,neutrino oscillations require that the neutrinos have mass.
1.4.1

Neutrino Oscillation Theory

To see how massive neutrinos oscillate, we propose that the three neutrino flavors,
νe , νµ and ντ , are mixtures of neutrinos with different masses, ν1 , ν2 and ν3 . This
mixture is represented via a matrix:
 
  
ν1
Ue1 Ue2 Ue3νeνµ  = Uµ1 Uµ2 Uµ3  ν2  ,ν3Uτ 1 Uτ 2 Uτ 3
ντ

(1.4)

where the matrix U is called the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix[14, 15]. The elements of U mix the flavor states and mass states. The mixture ofmass states, 1, 2, and 3, for any flavor state, α, can be written compactly as|να ð =

3X
i=1

Uαi |νi ð.

(1.5)

In this framework, we can use Equation 1.5 to compute the probability of observing
a neutrino created as one flavor as another flavor after some time t. The equation8

describing the propagation of a neutrino with flavor α is|να (t)ð =

3
X
i=1

Uαi e−iEi t |νi (t = 0)ð

(1.6)

where Ei is the energy eigenvalue of the νi mass state. For a near-massless neutrino
with energy, E, traveling in a vacuum, the neutrino momentum, p, may be approxip
mated as E, and so Ei = m2i + p2 ≈ E + m2i /2E. We compute the probability ofobserving a neutrino of one flavor, α, as another flavor, β by computing the overlapof Equation 1.6 and squaring. The probability is
P (να → νβ ) = |ïνβ (t)|να (0)ð|2
3
X

=

j=1

=|
=

3
X

i=1
3
X
i=1

U∗βj eiEj t ïνj (t)|

!

3
X
i=1

Uαi e−iEi t |νi (0)ð!2

|U∗βi Uαi e−iEi t |2|U∗βi Uαi |2 +X

Uαi U∗βj U∗αj Uβj e−i(Ei −Ej )t

(1.7)

i̸=j

In the third line, we have used ïνi |νj ð = 0 for i ̸= j and 1 otherwise. Equation 1.7 tells
us that the probability of a neutrino switching flavors by the time we observe it is afunction of t and Ei − Ej . If the neutrinos are massless, then Ei − Ej ∝ m2i − m2j = 0,
and the time-dependent component vanishes. On the other hand, if at least one
neutrino has a mass, the time-dependent component is nonzero, predicting that theprobability of observing the neutrino as a different flavor changes as it travels. Thisphenomenon is known as neutrino oscillation.
To see how neutrino oscillations can produce measurable effects, we consider the
structure of U. By requiring that U is unitary, we can express it as a product of
three rotation matrices which each mix two out of the three mass states via an angle,

9θij , and one complex phase, δ. Then U becomes:cos θ130 sin θ13 e−iδ
1
0
0

0
1
0
U = 0 cos θ23 sin θ23  iδ− sin θ13 e 0
cos θ130 − sin θ23 cos θ23
cos θ12 sin θ12 0

× − sin θ12 cos θ12 0 .
0
0
1

(1.8)

To understand how each parameter affects the probability of observing a neutrinoof a particular flavor, we consider the simpler case of two-neutrino oscillations. In
this case, there are only two flavors, and the mixing matrix only contains a single
angle:
 
  
ν1
να
cos θ sin θ=νβν2− sin θ cos θ(1.9)

In this scenario, Equation 1.7 now becomes:P (να → νβ ) = 2 cos2 θ sin2 θ − cos2 θ sin2 θ e−i(E1 −E2 )t + ei(E1 −E2 )tsin2 2θ
1 −i(E1 −E2 )ti(E1 −E2 )t
=
1−
e
+e
2
2


2
2 (E1 − E2 )t.
= sin 2θ sin
2(1.10)

By substituting Ei = E + m2i /2E, we obtain:2

P (να → νβ ) = sin 2θ sin

2



∆m2 t
4E



,

(1.11)

in natural units. Here, ∆m2 = m21 − m22 , and is called the squared mass difference ormass splitting. In an experiment, we observe neutrinos traveling from a production
point at a distance away from our detector, L. For a near-massless neutrino, we
can substitute t for L. Then, using powers of ℏ and c, we can convert L to units of10

kilometers, ∆m2 to units of eV2 /c4 , and E to units of GeV:
∆m2 /c4 L · c4
∆m2 (eV2 )L(km)∆m2 t≈≈ 1.274E
4E · ℏc
E(GeV)(1.12)

which gives the two-flavor probability formula:2

P (να → νβ ) ≈ sin 2θ sin

2



∆m2 L
1.27
E



.

(1.13)

It is also straightforward to repeat the above calculation for the two flavor survival
probability, i.e., the probability of a neutrino remaining as its initial flavor:
2

P (να → να ) ≈ 1 − sin 2θ sin

2



∆m2 L
1.27
E



.

(1.14)

From Equation 1.13 and Equation 1.14, we see that the various components influence
the oscillation pattern as follows: The mixing angle, θ, controls the overall amplitudeof the oscillation, while the squared mass difference, ∆m2 , controls the frequency ofthe oscillation as a neutrino with energy, E, travels along a baseline, L.
The two flavor approximation is useful to explain the leading-order oscillation
effect when the mass of one neutrino state is sufficiently different from the othertwo. For the three-flavor case, the squared mass differences are expressed as ∆m2ij ≡m2i − m2j . Then the two-flavor approximation holds when |m21 − m22 |k |m23 − m22 | or|m21 − m22 |j |m23 − m22 |. Experiments have determined that this condition is satisfied
to a good approximation, and have measured |∆m232 | to be approximately 30 times|∆m221 |. This also implies that |∆m232 |≈ |∆m231 |.
Note that for the oscillation framework we have considered, the sign of ∆m2ij
does not play any role for neutrino oscillations; ∆m2ij only appears inside the sine
squared term of Equation 1.13 and Equation 1.14. For this reason, the leading-orderneutrino oscillation effects only allow us to measure the mixing angles θij and theabsolute values of the squared mass differences. We can understand the neutrino
11

mass ordering question as the unknown sign of ∆m232 or ∆m231 .
1.4.2

Matter Effects

In matter, electron flavor neutrinos experience a modified potential due to an increased forward scattering amplitude with atomic electrons. Figure 1·3 shows Feynman diagrams of the processes that contribute to this amplitude for electron flavor
neutrinos. These processes are not present for muon and tau flavor neutrinos, so themodified potential changes the neutrino oscillation probabilities asymmetrically fromthe vacuum case. We can compute the modification by adding a potential term to
the electron neutrino component of the neutrino Hamiltonian:
a 0 0
HMatter = HVacuum + U 0 0 0 U
0 0 0
(1.15)

where we have written the Hamiltonian HMatter in the mass basis, i.e., the eigenvaluesof HVacuum are the Ei from Eq. 1.6. The electron potential is written as a 3 × 3
matrix in the flavor basis, and U is the PMNS mixing matrix which rotates thismatrix into the mass basis. The potential, a, depends on the material’s electron
density, Ne , and the energy scale of the weak interaction given by Fermi’s constant,√
GF : a = ± 2GF Ne . The sign of a is positive for neutrinos and negative for antineutrinos [16].
We may inspect the general impact of the matter potential on neutrino oscillations
by again simplifying to the two-flavor case. Starting with the two-flavor mixing matrixof Equation 1.13, the Hamiltonian for massive neutrinos propagating in matter is now

12

νe

e−
ν̄e

W

ν̄e
W−

+

e−

e−

νe

e−
(a)

(b)

Figure 1·3: Feynman diagrams of the two processes exclusive to (a)electron neutrinos and (b) electron anti-neutrinos when propagating
through a medium.
given by:
HMatter =
=

m21
2E

0

0
m22
2E

!



cos θ − sin θ+
sin θ cos θm21+ a cos2 θ2Ea sin θ cos θa sin θ cos θ
cos θ sin θa 0− sin θ cos θ
0 0
!

m22
+ a sin2 θ
2E.

(1.16)

We can interpret Equation 1.16 as the Hamiltonian for neutrino states with effective
masses in a new basis, ν̃1 and ν̃2 . This implies there is an effective mixing angle, θ̃,which mixes the effective mass states into flavor states νe and νx (a mixture of νµ andντ ), such that
  
 
 
νe
ν̃1
ν̃
cos θ̃ sin θ̃=≡ Ũ 1νxν̃2
ν̃2− sin θ̃ cos θ̃(1.17)The new mixing matrix Ũ from Equation 1.17 can be used to diagonalize the matter
Hamiltonian and solve for the effective quantities relevant to neutrino oscillations. It
can be shown that:2∆m̃ = ∆msin2 2θ̃ =

2

s

2

sin 2θ +sin2 2θ

sin2 2θ +

2aE− cos 2θ
∆m2

2
2aE
−
cos
2θ
2
∆m
13

2

(1.18)(1.19)

Equation 1.18 and Equation 1.19 imply that the νe → νx oscillation probability is
modified in matter, and that the modification depends on the amount of mixing in a
vacuum, θ, the matter density, a, and the signed mass splitting, ∆m2 . In contrast, thetwo-flavor vacuum oscillation probabilities given in Equation 1.13 and Equation 1.14
only depend on the magnitude of the mass splitting, |∆m2 |. As we will see, thisdependence of the modified oscillation probabilities in matter on the sign of the mass
splitting provides a way to measure the neutrino mass ordering.
The theoretical results presented in this section have important implications forexperimental tests of neutrino oscillations. The next section show how these results
successfully describe many neutrino phenomena, and discuss how we can use them toprobe remaining questions about neutrino oscillations.

1.5

Neutrino Oscillation Experiments

The goal of neutrino oscillation experiments is to measure the values of the PMNS
parameters and the mass splittings by observing neutrino oscillations. While theoscillations proceed as a function of the neutrino travel distance, L, and energy,
E, experimental considerations restrict observations to a limited range of L and E
combinations. Therefore, different experiments are sensitive to different combinationsof oscillation parameters.
Neutrinos observed by oscillation experiments manifest as an excess, appearance,or deficit, disappearance, of a particular flavor relative to the no-oscillation scenario.
This section will describe experimental techniques used to observe neutrino appearance and disappearance from different neutrino sources.
1.5.1

Solar Neutrinos

Electron neutrinos with energies 0.1 MeV ∼ 10 MeV are produced in the Sun aspart of nuclear fusion and decay processes. Many reactions contribute to the total
14

solar neutrino flux, but the most numerous by several orders of magnitude are fromproton-proton fusion, the pp-chain,p + + p + → 2 H+ + e + + ν e .

(1.20)

While these pp neutrinos are the most numerous, they are also the lowest energy
solar neutrinos, with a maximum energy of < 0.5 MeV. Neutrinos of higher energies
are produced from decays of heavier isotopes which are created from the fusion ofhydrogen into helium, then helium into beryllium and lithium. In particular, the 7 Be
produced in these fusion chains occasionally capture protons, forming boron-8, 8 B.
The 8 B isotopes then decay into neutrinos with energies up to ∼ 15 MeV:8

B → 8 Be + e+ + νe

(1.21)

The first observation of solar neutrinos came from the Homestake experiment[17], led by Ray Davis, using these 8 B neutrinos. The Homestake experiment began
in 1965, and consisted of a chlorine tank and filtration system. Davis hoped toobserve argon atoms produced in the reaction νe + Cl → Ar + e− . This reaction
has a threshold energy of 0.81 MeV, low enough to be induced by 8 B neutrinos. By
1964, the theoretical prediction of the solar neutrino flux from what would become theStandard Solar Model (SSM) [18–20] had calculated the expected flux of 8 B neutrinos[21], thereby directly predicting the number of argon atoms the Homestake experimentshould observe. The Homestake experiment eventually reported a measurement of
the solar neutrino flux of about 1/3 of the SSM prediction in 1969 [22]. While theseresults were puzzling, the results were not widely accepted due to doubts around theexperiment and the SSM itself.
The Homestake measurements of the solar neutrino flux were corroborated by
the Kamiokande-II detector in 1989. Kamiokande-II also observed 8 B neutrinos,
15

and, unlike the Homestake experiment, demonstrated the neutrinos it observed camefrom the Sun by recording the energies and directions of the particles produced inneutrino interactions. Separate efforts by the Soviet–American Gallium Experiment(SAGE) [23] and Gallium Experiment (GALLEX)1 [24] experiments, which operated
from 1990-2007 and 1991-2003 respectively, used gallium detectors with lower detection thresholds to detect pp neutrinos. Both the Kamiokande-II and the gallium
experiments observed a deficit compared to the SSM. Together with Davis’ original
measurements, these deficits became collectively known as the solar neutrino problem.
Neutrino oscillations proved to be the answer to the solar neutrino problem. The
definitive measurement would come from the Sudbury Neutrino Observatory (SNO)
experiment in 2001, which utilized heavy water, D2 O, as a detector target. The
deuterium in the heavy water allowed SNO to detect both CC and NC solar neutrino
interactions. A third interaction type, electron scatters (ES) with atomic electrons,
was also observed. The interactions observed by SNO are summarized as:
CC : νe + d+ → p+ + p+ + e−

(1.22)

NC : νe,µ,τ + d+ → p+ + n + νe,µ,τ

(1.23)

ES : νe,µ,τ + e− → νe,µ,τ + e−

(1.24)

While νµ and ντ neutrinos do not have CC interactions in SNO, all three flavors
have NC and ES interactions. By detecting the electron produced in the CC and
ES interactions, and the neutron associated with the NC interactions, SNO could
effectively measure the CC and NC interactions separately. SNO data showed thatthe deficit in solar νe interactions was only present in the CC interactions, but that
the NC and ES channels agreed with the SSM [25, 26]. This result provided direct
evidence for solar neutrinos oscillating to different flavors, while also reaffirming the1

Later, Gallium Neutrino Observatory (GNO)

16

flux predictions of the SSM.
Solar neutrino oscillations are sensitive to the θ12 mixing angle and the ∆m221 mass
splitting through electron neutrino disappearance. The vacuum probability from
Equation 1.14 tells us that, for long distances, the survival probability approaches
P (νe → νe ) → 1 − 21 sin2 2θ, which is, at minimum, 1/2. This is the leading-ordereffect in solar neutrino oscillations.
Experimentally, the measured probability is closer to 1/3. We obtain this smaller
value of the survival probability by invoking matter effects. Equation 1.19 contains
a resonance condition, 2aE/∆m221 = cos 2θ12 , which maximizes the effective mixing
angle in matter, consequently increasing the probability of νe oscillations to otherflavors. The resonance condition is satisfied for neutrinos with energies E ∼ 1 MeV,and ∆m221 ∼ 10−5 eV2 . Additionally, the resonance condition only occurs if ∆m221 > 0.
Applying this information to the observed solar neutrino deficit of 1/3 the prediction
measures the θ12 mixing angle, the size of the ∆m221 mass splitting, and the order of
two of the neutrino masses, m2 > m1 . The precise treatment of matter effects in solar
neutrino oscillations summarized here was developed in the 1970s-1980s, and became
known as the Mikheyev–Smirnov–Wolfenstein (MSW) effect [27, 28].
Current measurements from solar neutrino experiments place sin2 θ12 ≈ 0.31 and∆m221 ≈ 7.5 × 10−5 eV2 2 . Contemporary solar neutrino experiments aim to more
precisely determine the energy dependence of the solar neutrino oscillation pattern,
and to observe matter effects of solar neutrinos passing through the Earth, leading to “day-night asymmetry.” A thorough treatment of solar neutrino oscillations,
including sub-leading effects, may be found in [30].
2With input from reactor neutrino experiments. See Section 1.5.2

17

0.8
0.7

pp

P(νe→νe)

0.6

7Be

pep

0.5

Vacuum Osc.
8B

0.4
0.3
MSW
0.2

1
Neutrino energy (MeV)

10

Figure 1·4: Measurements of electron neutrino survival probabilities
from the BOREXINO experiment in 2018, adapted from [29]. The
dashed grey line shows the expected survival probability with vacuum
oscillations, while the pink curve shows the expected oscillations accounting for the MSW effect. BOREXINO is sensitive to pp, 8 B andother solar neutrinos at intermediate energies.
1.5.2

Reactor Neutrinos

Nuclear reactors provide a source of electron anti-neutrinos with 10s of MeV of energy.
In a reactor, a 235 U nucleus captures a thermal neutron, either converting it to 236 U,
or, more frequently, initiating fission into lighter elements and more neutrons. The
fission products undergo successive beta decays which produce more neutrons, antielectrons, and electron anti-neutrinos.
The Kamioka Liquid Scintillator Antineutrino Detector (KamLAND) experiment,
which has been operating in Japan since 2002, made subsequent measurements of reactor neutrinos at long baselines. The KamLAND experiment uses a liquid scintillator
detector target to precisely measure the energies of particles produced following an
inverse beta decay interaction, cf. Equation 1.3). Neutrino interactions in the liq18

uid scintillator create light proportional to the energy of the anti-electron produced,
followed by a delayed burst of light from the capture of the neutron, which enables
low-background neutrino identification. The energy resolution of KamLAND is approximately 2 % for 10 MeV neutrinos.
KamLAND observes neutrinos from multiple reactors across Japan, at an average baseline of 180 km. At this average baseline and reactor neutrino energy, KamLAND provides sensitivity to, and is an independent measurement of, ∆m221 values∼ 10−5 eV2 . In 2002, the KamLAND experiment first reported significant electron
anti-neutrino disappearance as a function of neutrino energy, consistent with themeasured mixing from solar neutrino experiments [31].
While KamLAND is sensitive to ∆m221 and θ12 , a detector placed at a much shorter
distance from a reactor, ∼ 1 km away, can observe neutrino oscillations sensitive tothe θ13 mixing angle. The dependence on this angle can be seen from the leading-orderoscillations given by the form of Equation 1.14:
2

P (ν̄e → ν̄e ) ≈ 1 − sin 2θ13 sin

2∆m231 L
1.27
E



(1.25)The short baseline reactor experiments Chooz [32] and Palo Verde [33] were among
the first to search for evidence of nonzero values of θ13 . Both experiments placed upperbounds on the mixing angle, but did not see significant electron anti-neutrino disappearance. Larger reactor experiments followed which could provide higher statistics,
and in 2012, the Reactor Experiment for Neutrino Oscillation (RENO) [34], DayaBay [35] and Double Chooz [36], a successor experiment to Chooz, published results
indicating reactor anti-neutrino disappearance, excluding θ13 = 0.
Reactor neutrino experiments offer the most precise measurements of θ13 . Current
reactor neutrino measurements place sin2 θ13 ≈ 0.0220 with ∼ 2 % uncertainty. Future
reactor experiments, which plan to improve the energy resolution of inverse beta decay

19events and acquire more statistics, have the potential to probe three flavor, subleading effects with electron anti-neutrino disappearance. The next-most sensitive
oscillation effects for reactor neutrinos depend on θ12 and ∆m221 , and, with highenough energy resolution, modifications in the oscillation pattern due to the small
differences between |∆m232 | and |∆m231 | may be observable with reactor neutrinos [37].
1.5.3

Atmospheric Neutrinos

Neutrinos are produced through the interaction of cosmic rays with nuclei in Earth’s
atmosphere. These interactions result in hadron showers of mostly pions and kaons,
which decay into neutrinos and muons. The muons also decay into two neutrinos,i.e.,
π + → µ+ + ν µ ,µ+ → e+ + ν̄µ + νe

(1.26)

π − → µ− + ν̄µ ,µ− → e− + νµ + ν̄e(1.27)

Both the positively- and negatively-charged hadrons are produced at approximately
equal rates, and the number of neutrinos from Equation 1.26 and Equation 1.27
indicates that the flavor ratio present in the atmospheric neutrino flux is (νµ + ν̄µ ) :(νe + ν̄e ) ∼ 2 : 1. Atmospheric neutrinos have energies spanning from a few MeV toseveral TeV, and travel on baselines to an observer on the Earth’s surface ranging
from ∼ 15 km if they are produced directly overhead, to ∼ 13 000 km if they are
produced on the opposite side of the Earth and traverse the Earth’s interior before
arriving at a detector.
A convenient quantity for describing atmospheric neutrino baselines is the zenith
angle which measures the neutrino angle with respect to the normal vector on Earth’s
surface. The relationship between zenith angle and baseline is demonstrated in Figure 1·5. The dependence can be derived for a sphere of radius r and a production

20

height h, and isL(cos θz ) = −r cos θz +p
2rh + h2 + r2 cos2 θz(1.28)

104

L [km]

103
102
101

-1

-0.5

0

0.5

1

cosθz

Figure 1·5: Atmospheric neutrino path length correspondence withzenith angle. Left: a diagram of atmospheric neutrino paths throughthe Earth, adapted from [38]. Neutrinos νA and νB travel on different
baselines and pass through different layers of the Earth, e.g., νA passes
through the crust, mantle, inner core, mantle, and crust before arriving
at an exit point on the surface. The zenith angles are indicated as
θz,A and θz,B respectively. Right: Approximate dependence of theatmospheric neutrino baseline L on the cosine of the zenith angle θz
from Equation 1.28. Negative values of cos θz correspond to neutrinos
arriving from below the horizon, where their baselines are thousandsof kilometers. Positive values of cos θz correspond to neutrinos arrivingfrom above the horizon, with a minimum baseline of ∼ 15 km. The
baseline changes rapidly around the horizon, cos θz ∼ 0.
Atmospheric neutrinos were first detected in 1965 [39, 40]. Two decades later,
hints of the atmospheric neutrino deficit began to emerge with measurements by
the Irvine-Michigan-Brookhaven (IMB) [41] and Kamiokande-II [42] experiments.
Kamiokande-II observed a deficit of muon-flavor neutrinos which could not be explained, even with large uncertainties on the atmospheric neutrino flux.
21

TheKamiokande-II successor experiment, SK, published a definitive observation of a
deficit of atmospheric neutrino events in 1998 [43]. The deficit was found to occur
for muon neutrinos arriving from below the detector, while muon neutrinos arrivingfrom above, and electron neutrinos from all directions, agreed with the predictions of
the atmospheric neutrino flux. Neutrino oscillations could explain both the deficit for
upward-going muon neutrinos and lack of deficit elsewhere: the expected leading-orderdisappearance and survival probabilities for atmospheric neutrinos may be written as∆m231 LP (νe → νe ) ≈ 1 − sin 2θ13 sin 1.27
E
2∆m32 L2
2
P (νµ → ντ ) ≈ sin 2θ23 sin 1.27
E
2

2





≈1

(1.29)(1.30)

for a small θ13 and ∆m231 ∼ 10−3 eV2 . Equation 1.29 predicts minimal disappearance
for the electron neutrinos, while Equation 1.30 predicts that muon neutrinos withenergies ∼ 1 GeV traveling along baselines greater than ∼ 100 km “disappeared”
through νµ → ντ oscillations.
Atmospheric neutrinos experience matter effects as they travel through the Earth.
Atmospheric neutrinos arriving at a detector from below necessarily cross the dense
inner layers near Earth’s core, where the electron density can modify neutrino oscillation probabilities through the mechanism described in Section 1.4.2. Following
Equation 1.19, the effective mixing angle θ13,M , which governs νµ → νe oscillations inmatter, is given bysin2 2θ13,M =sin2 2θ13
2

2aE
sin2 2θ13 + ∆m
2 − cos 2θ13

(1.31)

31

Assuming small vacuum mixing such that cos 2θ13 ≈ 1, the resonance condition issatisfied for ∆m231 /E ≈ a. This is attainable with atmospheric neutrinos. For exam22

ple, choosing round numbers, for ∆m231 ∼ 10−3 eV2 , E ∼ 1 GeV, and the density near
the Earth’s core ρ ∼ 10 g cm−3 [44], then ∆m231 /E ≈ 10−21 GeV and a is similar:√√
a = 2 2GF Ne = 2 2GF ρNA (ℏc)3√≈ (2 2) · 1.17 × 10−5 GeV−2 · 10 g cm−3 · 6.02 × 1023 N mol−1
3
× 1.97 × 10−14 GeV cm≈ 10−21 GeV

The above calculation applies to neutrinos if ∆m231 > 0, and equivalently, theresonance condition is also present for anti-neutrinos if ∆m231 < 0. Therefore, thesign of ∆m231 predicts an enhancement of either νµ → νe oscillations or ν̄µ → ν̄eoscillations in atmospheric neutrinos. Thus, current and future atmospheric neutrino
experiments can look for excess upward-going electron neutrinos or electron antineutrinos to determine the sign of ∆m231 . This is illustrated in Figure 1·6, whichshows νµ → νe oscillations probabilities for atmospheric neutrinos and anti-neutrinosin the normal and inverted neutrino mass ordering scenarios. The resonant region
occurs for cos θz < 0 and Eν ∼ 3 GeV for neutrinos in the normal ordering (top left)
and for anti-neutrinos in the inverted ordering (bottom right).
1.5.4

Long Baseline Neutrinos

Beams of neutrinos can be produced artificially by accelerating protons onto fixed
targets. The protons interact in the target material, producing hadron showers ofmostly pions. The desired negatively- or positively-charged hadrons are then focusedby a series of electro-magnets called horns which simultaneously deflect hadrons of thewrong charge. The focused hadrons pass into a decay volume where they decay into
muons and muon neutrinos, i.e., following Equations 1.26 and 1.27. Any remaining
hadrons and most of the muons are absorbed by dense material placed after the decay
23

Figure 1·6: Electron neutrino appearance probabilities for atmospheric neutrinos and anti-neutrinos in each neutrino mass ordering
scenario. The top row shows the probabilities in the normal ordering
while the bottom row shows the probabilities in the inverted ordering.
The left column shows the probabilities for neutrinos while the rightcolumn shows the probabilities for anti-neutrinos. A resonance region
can be seen for cos θz < 0 and Eν ∼ 3 GeV in the top left and bottom
right panels.

24

volume, leaving just the highly-focused, highly-pure νµ or ν̄µ neutrinos, to propagate
as a beam.
A typical long baseline experiment consists of a neutrino beam with peak energy ∼1 GeV, a near detector, and a far detector. The near detector is positioned to observe
the un-oscillated neutrino beam at a short distance, usually hundreds of meters, while
the far detector measures the neutrino beam after oscillations at distances hundreds ofkilometers away from the production point. The near and far detector, by observing
the same neutrino beam, make simultaneous measurements of the beam that constrain
many of the systematic uncertainties in the experiment.
Since long baseline experiments only observe neutrinos at one distance, preliminary measurements of |∆m232 | by atmospheric neutrino experiments were needed toestablish likely values of |∆m232 | and sin2 θ23 . The first long baseline experiment,
KEK to Kamioka (K2K), sent a beam of neutrinos along a 250 km baseline from the
KEK accelerator complex towards the existing SK detector. K2K published its first
oscillation results in 2002, in agreement with measurements of θ23 and |∆m232 | by SK[45]. Subsequent long baseline experiments followed: The Main Injector Neutrino
Oscillation Search (MINOS) experiment [46], which operated at Fermi National Accelerator laboratory from 2005 to 2016, utilized magnetized calorimeters in its far
detector to separate particles based on charge, suppressing contamination of neutrinos or anti-neutrinos in the opposite beam configuration, and improving the energy
determination of neutrino events. MINOS also confirmed the atmospheric neutrino
measurements.
Long baseline experiments are able to probe θ13 and δCP through νe appearance.
Here, a full three-flavor framework is required. Leading-order oscillations are controlled by combinations of θ13 and θ23 mixing angles, while the next-largest amplitude
involving all three mixing angles and δCP is approximately 20% of the leading-order25

term [47]. The maximum νe appearance corresponds to the L/E value that predicts
a minimum νµ → νµ disappearance. Since L is fixed, long baseline experiments tune
their neutrino beams to have the highest flux near this L/E value. The T2K experiment, the successor to K2K, refined the long baseline technique by sending its
neutrino beam at an off-axis angle with respect to SK, narrowing the observed energy
spectrum [48]. Although at the time, it was unclear if θ13 was nonzero, T2K observed
hints of electron neutrino appearance in 2011, suggesting a nonzero value of θ13 andthe possibility of measuring δCP [49]. The T2K measurement was confirmed andmeasured precisely by reactor experiments shortly after (see Section 1.5.2). In 2015,
the NOvA experiment also observed electron neutrino appearance in a long baseline
experiment [50].
The T2K and NOvA long baseline experiments have made the most precise measurements of |∆m231 |, sin2 θ23 and δCP . Current measurements place sin2 θ23 ≈ 0.55,|∆m231 |≈ 2.45 × 10−3 eV2 , and δCP ≈ 1.3π radians. Future long baseline experiments will employ more powerful neutrino beams, resulting in higher statistics, andhigher-resolution detectors to enable better neutrino energy measurements.

1.6

Present Status of Oscillation Measurements

Neutrino experiments using the various sources described in Section 1.5 have measured all PMNS oscillation parameters and mass splittings. The global average ofmeasurements are summarized in Table 1.2. Despite the high precision of many
PMNS parameter measurements, the uncertainties still permit several fundamentallydifferent neutrino oscillation scenarios. This section will describe several of the open
questions in neutrino oscillation searches.

26

Parameter

Value

Fractional
Uncertainty (%)

sin2 θ12
sin2 θ13
sin2 θ23∆m221 (eV2 )∆m232 (eV2 )δCP /π (rad)0.307 ± 0.013
0.0220 ± 0.0007
0.546 ± 0.021(7.53 ± 0.18) × 10−5
(2.453 ± 0.033) × 10−3
1.36 ± 0.20

4
3
4
2
1
15

Table 1.2: Listing of averaged PMNS parameters and mass splitting
measurements from various experiments [51]. Note that the sign of∆m231,32 is unknown, and results are shown assuming ∆m232 > 0.
1.6.1

Octant of θ23

It is unknown whether or not θ23 is in the upper octant, θ23 > π/4, or the lower octant,θ23 < π/4. Maximal 2-3 mixing, θ23 = π/4, or sin2 θ23 = 0.5, is also allowed. Values ofθ23 near π/4 imply an approximate µ-τ lepton flavor symmetry in the PMNS matrix.
The symmetry holds exactly, e.g., |Uτ i |= |Uµi |, if δCP = ±π/2 and θ23 = π/4, as
can be see from Equation 1.8. The typical implication of a symmetry in a physicalsystem is a conserved quantity, so the proximity of θ23 to π/4 could be a hint of a
new conservation law. A review of the flavor symmetries in neutrino mixing may befound in [52].
1.6.2

Value of δCP

Charge-parity (CP) violation is a difference in particle interactions under the simultaneous interchange of a particle for its anti-particle (charge) and its direction (parity).
CP violation is a necessary condition for explaining the large matter–anti-matterasymmetry of the universe [53]. While CP violation has been observed for quarks
as early as 1964 [54], the amount of CP violation observed in quark interactions is
too small to fully account for all of the matter-anti-matter asymmetry in the uni27

verse. Neutrinos offer another possible channel for CP violation. The phase δCP fromthe PMNS mixing flips sign between neutrinos and anti-neutrinos. Values of δCP
which are non-integer multiples of π, therefore, predict different amounts of mixing
for neutrinos and anti-neutrinos, a clear indication of CP violation.
Current neutrino oscillation experiments are in tension over observed values of
δCP . In 2020, T2K reported a rejection of the δCP scenario at an over 99% confidence
level [55], while in 2021, NOvA reported a measurement more consistent with δCP = π[56], disfavoring the T2K result and implying minimal CP violation. These results
are shown in Figure 1·1.
1.6.3

Neutrino Mass Ordering

Having explored the theory and historical context of neutrino oscillation measurements, we return to the original question of this work: there are two distinct scenarios
for the neutrino mass ordering, depicted in Figure 1·7. In the normal ordering, there
is one heavy neutrino, ν3 , relative to the ν1 and ν2 . This implies ∆m232,31 > 0. The
situation is reversed in the inverted ordering, where ν2 and ν1 are heavier than ν3 .
The neutrino mass ordering has implications for both neutrino oscillations andother areas of physics. The absolute sum of neutrino masses is an input to cosmological models describing the evolution of large scale structure in the universe [57].
The neutrino masses also control the expected rate of the neutrino-less double beta
decay process, with larger masses predicting higher rates [58]. This process, if observed, would have profound implications for our understanding of neutrino masses.
Additionally, as the neutrino mass ordering is sensitive to matter effects, the energy
spectrum of neutrinos produced in supernova is expected to depend on the mass ordering [59]. Neutrinos are a powerful probe of supernova dynamics, since 99 % of asupernova’s energy is radiated in neutrinos.
The neutrino mass ordering remains unknown because the leading-order neutrino28

m2NormalInverted
3

2
m21

2
m32

2
1

2
m31
2

2
m21

1

3

e

e

e

Figure 1·7: Schematic of the unknown neutrino mass ordering. The
neutrino mass states, ν1 , ν2 and ν3 can be ordered in two distinct ways:
with ν3 as the heaviest (normal) or as the lightest (inverted) neutrino.
The flavor content of the mass states, given by the squared row elementsof the reciprocal PMNS matrix, are shown as different colors. The
PMNS parameters are the central values from Table 1.2, with δCP = 0.
oscillation probabilities are independent of the sign of the mass splittings. While matter effects allowed solar neutrino experiments to determine ∆m221 > 0, the same analysis is more difficult with terrestrial neutrinos whose oscillations are instead governedby ∆m232 . Further, we have seen already seen how the T2K and NOvA experiments
have made measurements of the mass ordering that are difficult to reconcile.
Atmospheric neutrinos are sensitive to the mass ordering through resonant oscillations induced by matter effects. However, this approach comes with challenges:
Atmospheric neutrino experiments must contend with the intrinsic electron neutrinos
already present in the atmospheric neutrino flux. Long baseline experiments do nothave this issues, although the matter effect for neutrino beams traveling near Earth’s
surface is smaller than for atmospheric neutrinos passing near the Earth’s core, result-29

ing in less sensitivity to the mass ordering. Future efforts to determine the neutrinomass ordering using detectors with superior energy resolution and reactor neutrinos
are also planned [60].

1.7

Thesis Overview

This thesis will present a measurement of the unknown neutrino mass ordering and ofneutrino oscillation parameters sin2 θ23 , |∆m232 |, and δCP using atmospheric neutrinos.
The atmospheric neutrino data are collected in the Super-Kamiokande detector over
6511.3 live-days, corresponding to a total exposure of 484 kiloton-years. The SK
detector is described in Chapter 2, and the simulation of atmospheric neutrino eventsused in the analysis is described in Chapter 4. Results of an analysis using only SK
atmospheric neutrino data are presented in Chapter 5. Chapter 6 then presents ananalysis of the SK data using external constraints, namely on sin2 θ13 from reactor
neutrino experiments, and on δCP , |∆m231 |, and sin2 θ23 using a model of the T2K long
baseline experiment.

30

Chapter 2

The Super-Kamiokande Detector
The Super-Kamiokande (SK) detector [61] is a 41.4 m tall, 39.3 m diameter, cylindricalwater tank, containing 50 kt of ultra-pure water, illustrated in Figure 2·1. The SK
detector is located in Gifu prefecture, Japan, underneath Mount Ikenoyama. The
mountain provides shielding from cosmic rays, approximately equivalent to a water
depth of 2700 m. A ground-level, kilometer-long tunnel provides access to the SK
site, which includes a network of experimental halls, water circulation systems, andcontrol rooms for monitoring detector operation. Excavation of the SK cavern began
in 1991, and construction of the detector finished in 1996. Data taking officially began
on April 1, 1996.
The original goals of the SK experiment were to search for nucleon decay and toconclusively resolve the hints of the solar and atmospheric neutrino disappearance
observed by the IMB and Kamiokande experiments [62]. Members of both the IMB
and Kamiokande collaborations worked together to build SK. The volume of the SK
detector is approximately 17 times that of the original Kamiokande detector.
The SK detector has operated nearly continuously since its construction. The
SK data-taking periods are divided into phases, SK I-V and SK Gd, summarized inTable 2.1. There were three significant gaps in SK data taking. At the end of the
SK I phase, the detector was drained to replace photomultiplier tubes (PMTs) which
had failed during its first five years of operation. During re-filling, one of the PMTs
imploded, creating a shock wave and chain reaction which destroyed nearly half of all

31

Figure 2·1: Illustration of the SK detector, half-filled with water. The
SK detector resides in an excavated pit underneath Mount Ikenoyama,accessed via tunnel. The figure shows the tunnel which connects thecontrol room and top dome areas, which are accessible during operation. The tank is divided into two regions, the inner detector (ID) andouter detector (OD). Both regions are instrumented with photomultiplier tubes (PMTs) See text.
PMTs in the detector. The experiment proceeded: acrylic covers were installed on
the remaining PMTs, and the PMTs were re-distributed across the detector. The detector operated in this configuration during the SK II phase until replacement PMTs
could be installed. Installation of replacement PMTs occurred in 2006, restoring thedetector to full operation for the SK III phase. The detector then operated stably
for 12 years. In 2008, SK underwent an electronics upgrade which did not require
the tank to be drained. The electronics upgrade marked the beginning of the SK IV
phase. In 2018, the detector was drained a third time for major refurbishment work,
including the installation of a new water system capable of recirculating gadolinium
in the detector’s water, PMT replacement, cleaning, and leak repair. Data taking

32resumed after the work in 2019, initiating the pure-water SK V phase. In 2020,
gadolinium was dissolved into the detector’s water, marking the beginning of the
SK Gd phase. The phases of SK are summarized in Table 2.1.
Livetime Photo(Days) coverage

Phase

Dates

SK I
SK II
SK III
SK IV
SK V

1996–2001
2002–2005
2006–2008
2008–20182019–2020

1489.2798.6
518.1
3244.4
461.0

SK Gd

2020-Present—Electronics

Target

40%
19%
40%
40%
40%

ATM
ATM
ATM
QBEE
QBEE

H2 O
H2 O
H2 O
H2 O
H2 O

40%

QBEE

H2 O + Gd

Table 2.1: Operating conditions for the six data-taking phases of the
SK experiment. SK II had reduced photocoverage following the lossof approximately half of the PMTs at the end of SK I. At the time ofwriting, SK Gd is ongoing. Data from SK Gd is not presented in thisthesis.
This chapter will discuss the SK detector concept, outline the different detector
systems, and present the different methods for calibrating the response of the detector
to neutrino interactions.

2.1

Detection Principles

The SK detector uses Cherenkov radiation [63] to detect charged particles passing
through the detector’s water. Charged particles moving faster than the speed of lightin a medium cause the electric dipoles of molecules in the medium to coherently align,
creating an outward emission of radiation. The detection of Cherenkov radiation in
SK is shown schematically in Figure 2·2. Cherenkov radiation has several characteristics which depend on the index of refraction n of the medium. The minimum

33momentum required for particles to emit Cherenkov radiation is given by
pThresh. = √mc,
n2 − 1(2.1)

and the Cherenkov radiation is emitted at a fixed angle, given by
θC = cos−1 (1/βn),(2.2)

where β is the particle’s velocity expressed as a fraction of the speed of light. The
emitted Cherenkov photon spectrum per unit length is given by the Frank-Tamm
formula [63],2παZ 2
d2 N
=
dxdλ
λ2



1
1− 2 2
nβ



(2.3)

where λ is the wavelength, Z is the charge of the particle, and α is the fine structure
constant. In water, n is wavelength-dependent, but typical values are n ≈ 1.33 withvariations of only a few percent for visible wavelengths. This gives threshold momenta
for particles of pThresh. ≈ 1.14m, e.g., about 120 MeV/c for muons. The Cherenkov
angle for particles with β ≈ 1 is 42◦ in water. In water, a charged particle produces
about 3000 photons per cm in the 300 nm to 550 nm wavelength range.
Cherenkov photons in SK propagate through the water to the PMTs on the detector walls. Photons can knock out electrons on a PMT’s surface via the photoelectriceffect. These “photoelectrons” are then accelerated onto a series of conductive platescalled dynodes using electric fields inside the PMT, knocking out additional electrons
at each dynode. The electrons eventually reach the last dynode, which produces a
few-nanosecond wide current pulse with a voltage proportional to the number of electrons. The typical amplification factor, or gain, of an SK PMT is 107 , and the timing
resolution is 2 ns. PMTs with pulses above a specified threshold are considered hit.
In aggregate, the Cherenkov photons form ring patterns of hit PMTs due to the fixedemission angle given by Equation 2.2. The times, charges, and known positions of34

Figure 2·2: Illustration of Cherenkov radiation detection in SK.
Charged particles such as muons (µ− ) with momentum greater than
the threshold given by Eq. Equation 2.1 emit Cherenkov radiation atan angle, θC , given by Eq. Equation 2.2. Note that neutral particles likeneutrinos (νµ ) and particles below the Cherenkov threshold, indicatedby dashed tracks, do not emit Cherenkov radiation. The Cherenkov
radiation is detected by PMTs on the detector walls (yellow ellipses).
each hit PMT are used to determine properties of the particle, including its energy,
direction, and type.

2.2

Detector Systems

2.2.1

Inner Detector

While the total volume of the SK tank is 50 kt, particles are reconstructed within the
32 kt inner detector (ID), an optically-isolated region within the main tank. The ID
contains over 11 000 20-inch PMTs, mounted to a steel structure situated between
2 to 3 m from the tank walls. The PMTs are arranged orthogonally, providing an
average photocoverage of 40 %.
A schematic view of the Hamamatsu R3600 20-inch ID PMT is shown in Figure 2·3. Following the accident in 2001, acrylic covers were installed around all 20-inch
35

ID PMTs to prevent future chain reactions in the event of another PMT implosion.
The installation of a 20-inch PMT in a cover during 2018 is also shown in Figure 2·3.
The cover is not water-tight, but, in the event of an implosion, reduces the flow rateof water able to enter the PMT. The photocathode is hemispherical bialkali glass,chosen for its high quantum efficiency for Cherenkov radiation wavelengths and low
chance of spontaneous photoelectron emission. The typical quantum efficiency for a
20-inch PMT is shown versus the Cherenkov emission spectrum, from Equation 2.3,
in Figure 2·4. The light collection of the 20-inch PMTs also depends on the incident
angles of photons on the photocathode. This dependence is measured using dedicatedcalibration sources.
~ 70000

cable length
φ 116

φ 82 2glass multi-seal

φ 254 10

photosensitive area > φ 460< φ 520

cable

water proof structure( 610 20 )

~ 720

(mm)

Figure 2·3: The 20-inch ID PMT. Left: Schematic diagram of a 20inch ID PMT from [61]. Right: Installation of a 20-inch PMT in theacrylic shock-proof cover.

2.2.2

Outer Detector

The 2.2 m region between the tank walls and the steel support structure of the ID
PMTs defines the 18 kt cylindrical outer detector (OD) volume. The OD primarily
enables SK to separate cosmic muons from neutrino events: Cosmic muons entering
the detector deposit light in both the ID and OD, while neutrinos may interact in the
ID without leaving a visible signature in the OD. The OD is also used to determine
36

25

3.5
3
2.5

15

2
1.5

10

1

d2N/dxdλ (cm-1nm-1)

Quantum Ef昀椀ciency (%)

20

5
0.5
0
0
250 300 350 400 450 500 550 600 650 700
Wavelength (nm)Figure 2·4: The quantum efficiency of a 20-inch PMT as a function of wavelength. The spectrum of photos produced per cm from
Equation 2.3 is also shown using the right vertical axis. The quantum
efficiency curve is taken from [61].
if a particle stopped within the ID or exited the ID. Particles which are visible in
the ID and then exit into the OD create hits in both regions, with OD hits occurring
later in time. Due to the steel support structure, there is a 55 cm un-instrumented
region between the ID and OD.
Because the OD’s primary role is to veto events rather than reconstruct them,light collection is its highest priority over precise timing and charge determination.
Approximately 1800 8-inch PMTs are mounted to the inner wall of OD. Each PMT has
a 60 × 60 cm2 wavelength-shifting acrylic plate optically coupled to its photocathode,which guides light into the PMT, effectively increasing its area. The walls of the OD
are covered in reflective white Tyvek sheets which increase the chance of any lightbeing detected. Several OD PMTs are shown in-situ in the OD in Figure 2·5. The top
and bottom end-cap regions of the OD also contain the steel beams used to support
37

Figure 2·5: Pictures of the SK OD. Left: The outer detector, viewed
from below & looking up. The OD PMT mounting configuration isshown. PMTs are approximately 2 m apart. The semi-transparent
square plates around each PMT are the wavelength-shifting plates. The
white material is the reflective Tyvek. Right: The OD top end-cap
region. The segmentation Tyvek forming the wall optically separates
the top end-cap region from the rest of the OD. The steel support
beams and the water system plumbing in the top end-cap region are
also visible.
the ID1 . After the SK II phase, additional Tyvek, extending from the steel structuresides to the top and bottom of the SK tank, was installed to optically separate the top
and bottom OD volumes from the barrel volume. This Tyvek is called segmentation
Tyvek. Both the steel support structure and the segmentation Tyvek in the top OD
regions can be seen in Figure 2·5.
While the OD PMTs all share similar geometries, they are comprised of different
models. The oldest OD PMTs are Hamamatsu R1408s recycled from the IMB experiment, which operated between 1981 and 1991. At the end of the SK IV phase,approximately 30% of the OD PMTs were IMB type. In 2018, many were replaced,
lowering the number to 20% IMB type starting with SK V. The remaining OD PMTs
are a newer Hamamatsu R5912 model, which were installed after the SK I, SK II,
1

The steel structure’s impact on light collection in these regions was studied, and found to reduce
the travel distance of light in the outer detector. In simulations of the detector, this is accounted
for as a penalty term to each photon’s chance of absorption. See Chapter 4.

38and SK IV phases.
2.2.3

Water System

The SK detector uses ultra-pure water as its detection target. The water’s purity
affects the attenuation length of light, so regular purification of the water is essential todetector operation. The SK I-IV phases all operated with the water system describedin [61]. Starting with the SK V phase, the water system was upgraded to allow
Gadolinium to be recirculated into the water. A detailed description of the newwater system may be found in [64], and a summary is provided here.
The purpose of the water system is to continuously re-circulate and purify thewater. The SK I-IV water system processed 30 t to 60 t of water per hour during
SK IV, fully re-circulating the water approximately every month. In both the SK IIV and present water systems, the water passes through a series of filters and purifiers
which remove particulate matter. If not removed, the particulate matter can reduce
the transparency of the water to Cherenkov light, or emit radioactive backgrounds
such as radon.
For the SK I-IV water system, the water was filtered though a passive 1 µm meshto remove larger particulates. Then, the water was cooled to 13 ◦ C. The coolinghelps both to maintain a uniform level of dark noise in the PMTs and also to prevent
bacterial growths. The water then passed through a cartridge polisher which removed
ion contaminants, and an ultra-violet (UV) light which killed any remaining bacteria.
Radon-reduced air was then dissolved into the water to improve the efficiency of thenext step, passage through a vacuum de-gassifier, used to remove any dissolved gasses
in the water. Finally the water passed into an ultra-filter and membrane de-gassifier
which removed sub-µm particulates and any remaining radon gas, respectively. The
number of particulates < 0.2 µm in size entering the ultra-filter was estimated tobe 1000 cm−3 , which was reduced to 6 cm−3 after. Radon concentration for water39

returning to the SK tank after filtration was estimated to be 0.4 mBq m−3 .
This thesis will not describe the gadolinium loading of SK, however, the water
system designed for use with gadolinium was implemented starting with the SK V
pure-water phase. Data from the SK V phase is included in the analysis presented inthis thesis. The main differences of the SK V water system compared with the SK IIV system are the installation of various resin-based purifiers which remove heavier
ion contaminants associated with gadolinium sulfate, heat exchangers, which allow
more precise control of the water temperature, buffer tanks to prevent water losses,
and a redundant system capable of recirculating an additional 60 t of water per hour,for a total of 120 t per hour. Figure 2·6 shows a schematic view of the present SK V+
water system. In the new system, water from the SK tank passes through UV filters
and a passive mesh filter, similarly to the SK I-IV system. The water then passes
through new cation- and anion-exchange resins which remove positive and negativeion contaminants. The water then passes through ultra-filters and membrane degassifiers, as in the SK I-IV system. Water is not lost from the SK V water system:
Any water rejected by a filtration process is re-circulated into secondary buffer tanks
which can then pass through the filtration system again.
2.2.4

Electronics & Data Acquisition

Dedicated electronics enable the readout of precise charge and timing informationfrom the PMTs. The SK electronics are also responsible for using the quantity andtime distribution of PMT hits to trigger the detector to record data during particle
interactions.
High voltage and PMT signals from the approximately 11 000 ID PMTs and 1800
OD PMTs are carried by cables between the inside of the tank and the top of thedetector. Cables from each quadrant of the detector enter one of four electronics huts,
small rooms which house high voltage supplies and readout electronics for each set
40

Gd2 (SO4)3 8H2O

Dissolving system
Weighing hopper
Circle feeder

UV

Solution tank Dissolving Solvent tank
tank

2400L

0.2 m

2400L

Shear blender

1 m
1 m Dissolving system bypass line
A-Ex C-Ex TOC
Resin Resin

Pretreatment system

Returnwater ﬁlter

Main return line

1st buﬀer tank

2400L

4600L

C-Ex A-ExUVResinResin
1 m

Pump

2nd buﬀer tank Temp. control

unit A

UF

Supply HE
pump

4600L

2400L

TOC HETOC HE C-Ex A-Ex 1 m UVResin Resin

Temp. control
unit BMD
SK tankUF

Re-circulation systemHEbottomReturnpumps

Figure 2·6: The water system used beginning with the SK V period
in 2018, reproduced from [64]. Gadolinium was not present during
SK V, so the only elements relevant to this thesis are the re-circulation
loop beneath the bold black line, i.e., excluding the dissolving andpretreatment systems. The cation- and anion- exchange resins (C-Ex
and A-Ex) were not present in the original SK I-IV water system, andremove contaminants associated with gadolinium sulfate. The TOC(total organic carbon reduction) and UV represent types of UV filters,
the UF represents an ultra-filter, the HE represents a heat-exchanger,
and the MD represents a membrane de-gassifier.
of PMTs. The primary difference between the high voltage systems between the ID
and OD is the per-PMT adjustable high voltage for each ID PMT provided by CAEN
SY527 high voltage main frames. In contrast, the high voltage for the OD PMTs is
distributed across custom “paddle cards”, which send one high voltage to 12 PMTs
at once. The OD PMTs also use a combination cable which carries both high voltage
and the light detection signal.
41

The SK detector has used two distinct sets of readout electronics. The first set
was in use for the SK I-III phases, and was subsequently replaced beginning with
SK IV.
SK I-III Electronics
For SK I-III, PMT signals were processed by 12-channel ATM boards, which recorded
the total charge and arrival time of each PMT pulse via a combined analog-to-digitalconverter (ADC) and time-to-digital converter (TDC) circuit. The charge signal
was then split into several other readout systems. One signal was sent to a global
trigger system, which tracked the total integrated charge of all PMTs above a specified
threshold. One of the other split PMT signals was sent to a charge-to-analog converter(QAC), and another was used to start a constant current integrator. When the
global trigger system crossed a total charge threshold, it output a signal to the QAC
and current integrator. The signal height of the integrated constant current wasconverted to the PMT’s timing offset relative to the trigger, and the PMT charge
and time information from the QAC was converted and stored digitally. A redundant
integration and charge readout system was in place for each PMT so that triggers in
rapid succession could be stored without dead time. The QACs had to be read out
after accumulating 1.3 µs of data, so ID PMT data was stored in 1.3 µs segments. For
a comprehensive review of SK I-III ID electronics, see [61, 65].
The OD readout electronics for SK I-III were fundamentally similar to the ID,
but used different hardware. The OD PMT signals were read out via custom chargeto-time converter (QTC) chips which converted the signal into a fixed-width pulse
for easier digitization. The rising edge of this signal could be interpreted as the hit
time, and the width was proportional to the total charge. The OD PMT signals
were similarly fed into the global trigger system, and, in the event of a trigger, weredigitized and saved into memory buffers for offline processing. A larger window, 16 µs,
42

of OD hits could be stored than ID hits, which was useful for vetoing triggers when
activity in the ID was preceded by background-like activity in the OD.
The global trigger system issued different triggers based on the number of hit
PMTs. Separate low-energy and high-energy trigger thresholds were set to selectphysics events for different analyses. The number of hits from the outer detector
PMTs could also be used as an independent trigger. The triggers used for SK I-III
are summarized in Table 2.2.
SK IV+ Electronics
The SK I-III electronics were replaced in 2008 with both new physical readout circuitboards for both the ID and OD PMTs and a software trigger [66, 67]. The newboards are QTC-based electronics with ethernet (QBEEs), which, similar to the OD
electronics of SK I-III, output a single pulse, encoding the time and total charge ofeach signal. The QBEEs have several advantages over the ATM boards of SK I1. EachIII. First, QBEEs have multiple gain settings, set in a ratio of 1 : 71 : 49

QBEE can process a signal with the three gain settings simultaneously, and return
only the lowest setting which is not saturated. Thus, higher-charge PMT signals
can be recorded along side lower-charge signals while still retaining acceptable charge
resolution. The dynamic range of PMT signals recorded by QBEEs in an event isfive times that of the ATM system. Another advantage is the length of the readout:
ATM boards were cleared of their charges after 1.3 µs, but QBEEs can record 40 µsof data before needing to be read out. Each QBEE receives signals from 20 PMTs.
The PMT signals from each QBEE are then sent to merger computers which apply
trigger logic in software.
A major goal of the SK IV hardware upgrade was to enable a variable-length
event time window using a software trigger. Longer event windows help to containdelayed particle processes, such as muon or charged pion decays and neutron captures,
43

following the primary interaction (see Section 3.2). In particular, the AFT, or “after,”trigger was implemented to save several hundreds of microseconds of data following
activity in the ID with no activity in the OD. A summary of the triggers used during
SK data taking is presented in Table 2.2.
SK I-III

Trigger
Logic

SLE

LE
HE
SHE
OD
AFT

5.7 → 3.5 MeV
equivalent ID
hits, and vertex
fit within ID
29 ID hits in
200 ns
31 ID hits in
200 ns—19 OD hits in
200 ns—SK IV+

Event Window
(µs)

Logic

Event Window
(µs)

1.3

34 → 31 ID hits
in 200 ns[−0.5, 1.0]

47 ID hits in
200 ns50 ID hits in
200 ns70 → 58 ID hits
in 200 ns
22 OD hits in
200 ns
SHE, no OD

1.3
1.3
—
16
—

[−5, 35][−5, 35][−5, 35][−5, 35][35, 535]

Table 2.2: Triggers used in the different SK data-taking phases. Hit
requirements reflect the approximate numbers used, because the definitions were updated throughout data taking in response to changes in
PMT dark rates and failed PMTs. SLE refers to “super low energy,”
and was implemented in the SK I-III periods using a dedicated external computer to filter events based on the charge in units of MeV and
fit event vertices in real time. LE refers to “low energy,” HE refers to“high energy,” SHE refers to “super high energy,” and “AFT” refers to“after.” AFT triggers were introduced by means of a software trigger
implemented at the start of the SK IV phase.
For all SK phases, digitized PMT hit charge and time information in a time
window around each trigger is assembled into “events” on dedicated computers inside
the mine. Event data is periodically transferred out of the SK detector area via
optical fiber to nearby off-site computers for offline processing and storage. Events44

are stored in the ZEBRA file format [68], and also as ROOT [69] files.

2.3

Calibration

The response of each SK PMT is tuned to controlled calibration sources to ensure an
accurate and uniform response to physics events. In addition, the SK water is studied
to characterize the light propagation within the detector.
2.3.1

ID PMT Calibration

The calibration of the ID PMTs begins with high voltage tuning for a subset of 420
ID PMTs designated as “reference” PMTs before installation in the detector. The
high voltages of the reference PMTs are tuned to light from a xenon lamp connected
to a scintillator ball via optical fiber, so that each PMT’s charge output is identical.
These reference PMTs are then installed in the SK tank in four vertical strip patterns,
one along each quadrant of the detector. The scintillator ball, connected to the same
xenon lamp, is then lowered into the tank center, and serves as an isotropic lightsource, illuminating all PMTs. Since the charge output of the reference PMTs to
the light intensity is known in advance, the high voltages of the other, non-reference,
PMTs can be adjusted to match the charge response of reference PMTs at the equivalent distance from the scintillator ball. The geometric pattern of the reference PMTs
and the scintillator ball tuning method is demonstrated in Figure 2·7.
The high voltage tuning results in a coarse calibration of PMT response, so following the tuning, the relative gain of each PMT is precisely determined. The relative
gains are assessed using separate high- and low-intensity light sources. The highintensity source sends multiple photons to each PMT, such that the charge responseof each PMT, Qi , may be written as:
Qi ∝ IHigh × ai × QEi × Gi ,
45

(2.4)

Groupऀ

Scin%llator耠 ballऀGroupऀ

Figure 2·7: Overview of the high voltage tuning using the reference
PMT. Left: Location of the reference (red circles) PMTs in the SK
detector, shown in an unrolled cylinder view. Right: Illustration ofgain setting using a scintillator ball for the non-reference PMTs. Bothfigures are reproduced from [70].
where i indexes each PMT, a is the geometric acceptance factor, QE is the quantum
efficiency, and G is the gain. The low-intensity source is then used so that any hits
are likely to be due to single photoelectrons. The number of hits for the low-intensity
source is:Ni ∝ ILow × ai × QEi .

(2.5)

The relative gain is set for each PMT using that Gi is proportional to Qi /Ni . Because
the geometric factor and quantum efficiency cancel in the ratio, the proportionality
is the same for all PMTs.
Once the relative gains are known, another calibration method measures the av46

erage gain of the entire detector. A nickel source, which isotropically emits 9 MeV
photons, is placed at the center of the detector. The photons from the nickel source
are sufficiently low-energy such that virtually any hit can be attributed to a single
photoelectron emission. The nickel-induced single-photoelectron distribution of each
PMT, corrected by each PMT’s relative gain, is then added into a single distribution,
shown in Figure 2·8. A fit to this whole-detector single-photoelectron distribution
establishes a single picocoulombs-to-photoelectrons conversion factor.

Figure 2·8:
Sum of relative gain-corrected ID PMT singlephotoelectron distributions. The peak shows the average charge, inpicocoulombs, corresponding to one photoelectron.
PMT signal times are calibrated to account for differences in the photoelectrontransit times within each PMT, and the differences in signal transit time due to each
PMT’s cable length. The timing calibration uses a fast-pulsing nitrogen laser, fed intoa diffuser ball at the center of the SK detector. The times of the hits induced by thelaser are time-of-flight subtracted based on the PMT’s distance to the light source,47

so that each PMT hit is expected to arrive at the same time. Any nonzero “residual
times” then correspond to the offsets which need to be added to each PMT’s signal
times for proper calibration. Another consideration for the timing calibration is thatlarger signals tend to arrive earlier than small signals, a phenomenon known as time
walk. To address time walk, the timing offsets are calculated as a function of charge.
The resulting table of offsets for a given PMT charge is referred to as a TQ-map. A
TQ-map is created for each PMT, and fits to these maps are applied as a correction
function to all raw data hits prior to any other offline processing. An example of a

T (ns)

TQ-map is shown in Figure 2·9.
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
0.5

1200

0.4
1190
0.3
0.2

1180

0.1
1170
0

20

40

60

80

100

120

140

160

180

QBin

Figure 2·9: The TQ-map of one ID PMT. Each entry of the histogramis a time-of-flight subtracted hit on this PMT induced by a calibration
laser. The lower x-axis shows the charge in units of an internal binningscheme, while the upper x-axis shows the charge converted to picocoulombs. The y-axis shows earlier times as larger values and latertimes as smaller values, so the time-walk effect, with higher-charge hits
arriving earlier, is visible. The black line represents a polynomial fit to
the data. Figure is adapted from [70].

48

0.0

2.3.2OD PMT Calibration

The outer detector PMTs do not have per-channel control of high voltage settings,
and, due to the amount of reflected light in the OD, do not have as precise timing
information as the ID PMTs. Still, OD PMTs can be coarsely calibrated. First, darkhits, i.e., noise, observed in the window before an event trigger are collected. Dark
hits are nearly all due to single-photon detections, so they form single-photoelectron
distributions for each OD PMTs. The relationship between high voltage and thecharge response from a single-photoelectron is roughly known for each PMT type, so
an ideal target high voltage can be calculated. The high voltage for each OD PMTs
can then be lowered by adding a zener diode jumper directly to the paddle card.
Timing calibration of the OD PMTs is performed using cosmic muons. The trajectories of cosmic muons which pass through the OD and ID can be fit with ID PMTs
to establish an entry point and direction in the ID. Then, the muon’s path can beextrapolated backwards into the OD region. The distance of each hit OD PMT tothe muon track is calculated assuming fixed-angle Cherenkov emission, allowing fora time-of-flight subtraction. The time-of-flight subtracted hits are used to calculate
offsets which correct for any observed timing bias.
2.3.3

Water Parameter Measurements

Light intensity due to water transparency in SK is modeled with an exponentialfunction,I(l, λ) = I0 (λ) exp [−l/L(λ)] ,(2.6)

where I is the intensity, l is the distance from the source, λ is the wavelength ofthe light, and I0 (λ), is the source intensity. Equation 2.6 also include a wavelength-49

dependent scattering coefficient, L(λ), which is decomposed into three components:
L(λ) = (αabs. + αsym. + αasym. )−1 .

(2.7)

Equation 2.7 contains an absorption term, αabs. , a symmetric scattering term, αsym. ,which includes Rayleigh and symmetric Mie scattering, and an asymmetric term,
αasym. , which includes the asymmetric component of Mie scattering. Each component
is measured using a multi-wavelength laser placed at different locations in the SK tank.
Measurements of scattered light from the laser constrain αsym. and αasym. , while thedrop in intensity, measured using PMTs near the laser location and at the opposite
side of the tank, establishes αabs. . Figure 2·10 illustrates the water transparency
measurement.
10

01!234456-457- *+$'"#,-.'/%&!
8976-887854-(:!

-1

Total
Absorption
Symmetric
Asymmetric

10

-2

;<!
;=!
10

;4!

-3

!"#$$%&'()!

;8!
;7!

10

-4

300

325

350

375

400

425

450

475

Figure 2·10: Overview of the water transparency measurement. Left:
Setup of the water transparency measurement with a laser. The labels
B1-B5 indicate regions at different depths within in the ID. Right:
Example results of the laser calibration. Data are shown as points,
while the intensity predicted by a tuned simulation is shown as solidlines. Both figures are reproduced from [70].

50

500

The water transparency can be measured separately by cosmic muons. Most cosmic ray muons deposit a near-constant 2 MeV/cm of energy in water, corresponding
to a fixed amount of Cherenkov radiation per unit track length as they traverse thedetector. The ID PMTs are used to reconstruct the muon track through the detector so that the expected hits on each PMT due to the muon’s trajectory may be
calculated. The ratio of observed charge to the expected charge on each PMT determines the rate of scattering and absorption. The transparency measurement withmuons is insensitive to wavelength, but has the advantage that it can be continuously
performed during normal detector operation.

2.4

Account of SK V Open Tank WorkIn 2018, the SK IV data-taking phase was ended in order to conduct maintenancework. The main goals of the work were:• Install a new water system, capable of recirculating gadolinium in the detector
water. This is described in Section 2.2.3.
• Fix a leak, to prevent any future Gadolinium-loaded water from leaking into
the environment.
• Replace PMTs which had failed during the SK III and SK IV phases, a periodof 12 years.
• Clean the detector, including removing rust which was thought to potentiallyreact with gadolinium, affecting the water transparency.
The SK “open tank work” involved draining the detector in stages, so that work
could be conducted on detector components at varying depths. When open, workers
entered the SK detector via gondola, which lowered them and their equipment onto

51

“floating floors” resting at the current water level. In the ID, additional rafts were
deployed from the floating floor for traversal away from a primary platform, while in
the OD, a floating floor was deployed in a ring, allowing workers to access all sides.
Planning the open tank work involved an assessment of which PMTs and water
system components would be accessible on each day, given the water level. Replacing
PMTs in both the ID and OD involved cutting PMT cables, splicing new connectors,
and covering the splices with heat-shrink tubing to create water-tight connections.
Approximately 150 ID and 220 OD PMTs throughout the detector were replaced
during the open tank work.
Concurrent with the PMT replacement, workers cleaned the steel structure of the
SK tank. Because the steel structure and the tank walls are nominally wrapped in
Tyvek (see Figure 2·5), the Tyvek in the outer detector was removed at each layer.
This was also necessary for accessing the cables for the ID PMT replacement work.
Cleaning the steel structure involved removing rust with electrodes, vacuuming, andretrieving any remaining pieces of PMT glass leftover from the 2001 accident.
To address the leak, workers painted the weld joints and bolt anchors connecting
the SK water tank to the cavern walls with MineGuard, a polyurea-silica paste developed for moisture protection in industrial settings. First, the area was cleaned ofany rust or debris. Then, MineGuard was applied in two coats with a 24-hour set-intime after the first coat.
After work was finished on an 2 m tall section of the detector, new Tyvek was
wrapped around the steel structure, and the water level of the tank was drained an
additional 2 m so that work could proceed on the next layer. Figure 2·11 shows thewater level as a function of time. The step pattern corresponds to the three days ofwork on each layer can be seen as the water level decreases. Refilling the detector up
to the top level occurred continuously. Additional work was required near the top to52

re-seal the tank before filling to the full capacity. Figure 2·13 shows scenes from theopen tank work. The images show work in the ID and OD at various water levels.

Figure 2·11: The water level during the SK V open tank work. The
step pattern reflects times where workers performed various tasks inside
the tank.

Figure 2·12: The leak assessment after the refurbishment work. Data
from SK V is compared to an equivalent period from SK IV, showing
no decreases in the water level after the tank open work.
Once the tank open work had completed, the detector water system and waterlevel was monitored precisely. Figure 2·12 shows a measurement of a week-long period
from SK IV and at the beginning of SK V. The figure shows that the decrease in waterlevel from the original leak was no longer observed following the open tank work.
Calibrations of PMT hit charge and timing, similar to those described in Section 2.3, were carried out at the beginning of the SK V data-taking phase. In particular, new PMTs installed were calibrated, and the OD high voltages were re-adjusted
to improve the consistency of OD PMT single photoelectron responses.
53

Figure 2·13: Scenes from SK V open tank work. Top: Interior viewof the ID floating floors during the SK V open tank work. Workers canbe seen with replacement PMTs in boxes on one floating floor, while
another worker takes measurements from a raft. Bottom: Cleaning
work in the OD. In the foreground, a worker removes rust using anelectrode, while another worker vacuums the steel structure behind an
OD PMT. MineGuard from the previous layer can be seen along thejoints of the tank wall at the top of the frame.

54

Chapter 3

Data Reduction & Reconstruction
The SK detector records approximately one million triggers per day, primarily fromlow-energy radioactive backgrounds at a rate of ∼ 11 Hz, and from cosmic muons
at a rate of ∼ 3 Hz. Only about 10 of these triggers per day correspond to atmospheric neutrino interactions. The process of selecting likely neutrino interactions
from background triggers is called reduction.
Once atmospheric neutrino candidates are identified in data, properties about theparticles observed in the time window around the trigger are inferred from patternsof hit PMTs. This process is called event reconstruction.

3.1

Data Reduction

Atmospheric neutrino candidates have three broad classifications at SK: fully-contained(FC), partially-contained (PC), and upward-going muon (Up-µ). The distinguishing
characteristics between these classifications are:
• FC events have an interaction vertex reconstructed within the ID, and no OD
activity. Hence, all final state particles are fully contained.
• PC events have an interaction vertex reconstructed within the ID, and no entry
point in the OD, but do have an exit point in the OD. The exiting particle istypically a muon produced by a CC νµ interaction.
• Up-µ events are induced by neutrinos interacting in the rock around SK or in
55

FC

PC

Up-µ

Figure 3·1: The three neutrino event toplogies at SK. The cylinders
represent the ID within the SK tank. FC and PC events have a vertex
within the ID. PC events have a particle that exits the ID, creating anexit point in the OD. Up-µ events have a vertex outside the ID and an
upward-going muon track.
the OD water. These events look like cosmic muons, but pointed upward, i.e.,coming from below the horizon. Up-µ neutrino events well-below the horizon
have a low contamination from cosmic muons, since the Earth acts as a shield
against cosmic muons coming from below.
The three neutrino classifications are shown in Figure 3·1. The SK reduction
procedure identifies neutrinos of each class by applying a series of successive cuts.
The cuts either remove or accept events, collection of PMT hit charges and times
recorded within a time window around a trigger. Earlier cuts remove obvious nonneutrino events using simple calibrated detector information, while later cuts, whichremove more uncommon backgrounds, tend to be more computationally expensive.
This section describes the selection process for neutrino events in each of the threeclasses. Timing information used in the cuts is relative to the event trigger, defined
to be at 0 ns. Additional details of the reduction process may be found in other theses[71, 72].

56

3.1.1FC Reduction

The FC reduction consists of five steps, FC1-FC5.
FC1
The first step of FC reduction removes obvious radioactive backgrounds and other
low-energy triggers, and cosmic muons, which produce many hits in the OD. FC1
requires that: The number of ID photoelectrons in a 300 ns sliding window must exceed 200(100 for SK II)
 The number of hits in the OD between −500 ns and +300 ns around the event
time must not exceed 50 (SK I-III) or 55 (SK IV+)
This cut is almost 100 % efficient for selecting neutrinos. Approximately 3500 events
pass FC1 per day, a 99 % reduction from the initial one million triggers. Note thatevents which fail the second cut of FC1 may still be classified as PC or Up-µ events.
FC2
FC2 removes high-charge noise events coming from a single ID PMT, and applies astricter cut to remove additional lower-energy events which may have passed FC1.
The cuts are:
 No single ID PMT may account for more than 50 % of the charge in the event If the number of photoelectrons is less then 100 000 (50 000 for SK II), then thenumber of OD hits between −500 ns and +300 ns around the estimated event
time must not exceed 25 (30 for SK IV+)
After FC2, approximately 900 events remain per day.
57

FC3
FC3 removes cosmic ray muons with certain characteristics not covered by FC1 andFC2, particular configurations of low-energy events, and for removing “flasher” events
caused by spontaneous discharge of light from PMTs. The two types of muons addressed by FC3 are• High-energy muons which lose energy due to bremsstrahlung and pair production, instead of ionization. These energy loss processes result in short bursts oflight in the OD, which sometimes evade the FC1 and FC2 cuts.
• Muons which produce concentrated hits in the OD. These muons do not produce
enough OD hits to be cut by FC1 and FC2, but produce a cluster of hits in the
OD around an entry or exit point.
FC3 addresses the high-energy muons with the following cut:
 There must be fewer than 40 OD hits in a 500 ns sliding window
Muons with visible entry points in the OD are found via a dedicated muon trackfitting procedure. The fit assumes the earliest saturated ID PMT to be the muon entry
point. If the muon passes through the detector and exits the ID, an exit point, andtherefore direction, is found using the center of all saturated ID PMTs. Otherwise,
the fit calculates the muon direction such that it maximize the expected charge in
the ID. The fit also computes a goodness score, i.e., the likelihood that the time ofeach PMT hit originated from a Cherenkov cone along the muon’s track. The entry
and exit point are used in the following cuts to identify and remove muons: (Entry & exit point): The event must not match the following conditions: The
maximum charge on a single PMT exceeds 230 photoelectrons, there are morethan 1000 ID PMT hits, and the number of OD hits within an 8 m radius of
the entry or exit point is 10 or more.
58 (Entry point only): The event must not match the following conditions: The
number of OD hits within an 8 m radius of the entry point is 10 or more, or(SK I only) the number of OD hits within an 8 m radius of the entry point ismore than four, and the muon’s fitted goodness score is more than 0.5.
Next, due to gaps in the OD PMT coverage from cables and plumbing, scintillator
paddles are installed as vetoes at four positions around the detector. FC3 requires
 There must be no scintillator veto activity within 4 m of the fitted entry point.
FC3 also attempts to remove additional low-energy events that can evade the FC1
step due to multiple random coincidences. For a true neutrino event, a single vertex
should be present, while multiple coincident low energy events will not have a single
best-fit vertex. FC3 chooses a vertex as the point which gives the smallest residuals
for time-of-flight subtracted hits. This time-of-flight vertex is used for the following
cut, and in subsequent cuts: The number of time-of-flight subtracted hits in a 50 ns sliding window must
exceed 50 (25 for SK II)
This cut alone is not sufficient to remove all low-energy events. If a cosmic muon
arrives after the low energy event triggers the detector, it can still pass this and thepreceding cuts. Therefore, another cut is applied to search for these late-arrivingmuons:
 If the number of photoelectrons in the ID is greater than 5000 (2500 for SK II),
then there must be fewer than 20 OD hits in a +300 ns to +800 ns window.
Finally, FC3 removes flasher events, where a single PMT undergoes electricaldischarge which produces light detectable by neighboring PMTs. A characteristicof flasher events are PMT hits occurring at later times after a trigger than in true
particle interactions, which motivates the following cuts:59 (SK I only) The minimum number of ID hits within a 100 ns residual timesliding window between +100 ns and +700 ns must be less than 10. If there are
more than 800 ID hits, the number of ID hits in found by the sliding window
must be less than 15.
 (SK II+) The minimum number of ID hits within a 100 ns sliding window between +100 ns and +700 ns must be less than 20.
 The goodness score of the time-of-flight vertex must be greater than 0.4.
Approximately 80 events per day remain after FC3. Events which pass FC3 arefully reconstructed, with a much more sophisticated algorithm, described in Section 3.2.
FC4
FC4 implements a data-driven cut to remove additional flasher events. Experts selectflasher candidates for inclusion in a database of reference events, and new events arecompared to these references. FC4 implements a likelihood function consisting of
two variables constructed using the hit patterns of the events: a correlation, and theKolmogorov-Smirnov (KS) test statistic. The correlation, r, between two events, A
and B, is defined as:N
1 X 1  A
B
(Qi − ïQA ð)(QB
−
ïQ
ð)
r=
i
N i σA σB(3.1)

where i indexes groups of nearby ID PMTs (about 4 m2 ), Qi is the summed charge ofthe ith region, ïQð is the average change of the event, and σ is the root-mean-square(RMS) of the distribution of charges in the event. Similar events will have large valuesof r. The KS statistic is computed using the distributions of charges within each ID
PMT region. Smaller values of the KS statistic indicate better agreement. The 10
60

patches with the highest r values and 10 smallest KS values are used to compute thelikelihood. Events which produce a high likelihood when compared to known flasher
patterns are cut by FC4.
FC5
A final set of quality cuts are applied to events passing FC4 to remove remaining
flashers, low-energy backgrounds, and muons which pass near additional gaps in the
OD which are un-instrumented.
First, flashers are addressed by applying a more sophisticated vertex fit, using themethod described in Section 3.2. Hit times are newly time-of-flight subtracted fromthis fitted vertex, resulting in new residual times. FC5 applies two flasher cuts using
the new vertex and residual times:
 If the minimum number of hits in a 100 ns residual time sliding window is six
or more, the vertex goodness score must be greater than 0.4.
 (SK II+ only) If the minimum number of hits in a 100 ns residual time sliding
window is less than six, the vertex goodness score must be greater than 0.3.
FC5 attempts to remove low-energy backgrounds caused by invisible muons, i.e.,muons with momentum below the Cherenkov threshold. An invisible muon may not
produce enough hits in the OD to be removed by earlier reduction steps, but can still
trigger the detector when it decays into an electron. FC5 introduces a cut based on
the maximum energy of an electron from a muon decay at rest, estimated to be 1000
photoelectrons (500 for SK II). Events below this maximum energy are searched forusing time-clustered hits in the OD inside a 200 ns sliding window from −9000 ns to−200 ns. A second cluster is searched for from −200 ns to 200 ns. The goal of theseclusters is to detect entering activity from the OD, e.g., an energetic muon in the61

OD which produces a secondary muon below Cherenkov threshold that subsequently
enters the ID. Based on these OD clusters, two cuts are introduced:
 If the OD hit cluster locations are 500 cm apart, the number of hits in the firstcluster must be less than five, and the number of hits in both clusters must be
less than 10.
 Otherwise, the number of hits in the first OD cluster must be less than 10.
While FC3 addressed gaps in the OD using scintillator paddles, FC5 addresses
several remaining un-instrumented gaps. The following cut is applied: If the muon fitter’s goodness score is greater than 0.4, there are more than 1000
photoelectrons in the ID, and the fitted muon direction is larger than 37◦ above
the horizon, the distance between the OD entry point and the nearest gap mustbe larger than 2.5 m.
Finally, FC5 applies one cut based on the reconstructed vertex (see Section 3.2)
to further remove entering muons: There must be fewer than 4 OD hits within 8 m of the reconstructed entry point
in a 200 ns sliding window between −500 ns and +300 ns.
3.1.2

PC Reduction

PC events are less common than FC events, occurring at a rate of 0.6 per day. Dueto the segmentation Tyvek installed at the start of SK III (see Section 2.2.2 andFigure 3·2), the PC reduction was overhauled for SK III onward. This thesis will
describe the current method for selecting PC events. The SK I-II PC reduction
technique is documented in other theses [73].

62BARRELTOP

segmentation
Tyvek

BOTTOM

Figure 3·2: Labeled regions of the OD, and the segmentation Tyvek,
which are referenced in several parts of the PC reduction. The top andbottom of the OD cylinder are collectively called the end-cap regions,and are optically separated from the barrel region by the segmentation
Tyvek. Figure is adapted from [70].
PC1
PC events ideally have one exit cluster of hits in a single OD region. PC1 enforcesthat no two OD regions, top end-cap, barrel, and bottom end-cap, illustrated in
Figure 3·2, have obvious OD clusters. PC1 also ensures that the energy deposition in
the ID is consistent with a particle exiting into the OD. The first quality cuts are There must be at least 1000 photoelectrons in the ID.
 There must be fewer than 11 OD hits in the top end-cap region or fewer than
10 hits in the bottom end-cap region.
 There must be fewer than 84 OD hits in the barrel region or fewer than 29 hits
total in the top end-cap and bottom end-cap regions.

63PC1 also calculates the average distance between pairs of hit OD PMTs, a quantity
called ODRMean , defined as
ODRMean =

1
NPair

XX
i

|⃗xi − ⃗xj |

(3.2)

j̸=i

where NPair is the number of pairs of hits, and ⃗xi is the position of the ith PMT.
ODRMean is large for events with multiple OD clusters, and small for events with onlya single cluster. PC1 also requires
 ODRMean must be less than 2140 cm.
PC2
PC2 implements a more sophisticated algorithm to identify OD hit clusters than in
the FC reduction. The algorithm used for PC reduction divides the OD into regions,then merges hits in each region into the neighboring region with the most hits. PC2
requires There must be at most one region with more than 10 OD hits found by theclustering algorithm.
PC2 also excludes events based on the number of hits in the end-cap regions as afunction of the number of hits in the barrel region. This function, f (NBarrel ), plottedin Figure 3·3, is a piecewise exponential which decreases in the number of OD barrel
hits, NBarrel . PC2 requires that
 The number of OD end-cap hits must be less than f (NBarrel ).

PC3PC3, similarly to FC3, removes flasher events by searching for ID hits arriving after
the primary event trigger. PC3 applies a sliding window cut:64End-cap Hits

350
300
250
200
150
100
50
0

Excludedf(NBarrel)

0

50

100

150

200

NBarrelFigure 3·3: The cut applied based on the number of OD hits in eachregion as part of the PC2 reduction step.
 The minimum number of ID hits within a 100 ns residual time sliding window
between +200 ns and +700 ns must be less than 10 if there are fewer than 800
ID hits, otherwise the number must be less than 15.
PC4
PC4 removes events with improbable vertices. As in the FC reduction, PC4 estimatesa vertex using time-of-flight subtracted hits. PC4 also applies a dedicated fitter,muboy, which classifies muons in one of four categories• Stopping: The muon enters the ID and stops• Through-going: The muon enters the ID, then exits the ID• Multi-muon: There are multiple muons with different entry points in the sameevent
• Corner clipper : The muon entered and exited through the corner of the end-cap
region, resulting in a thin Cherenkov ring in the ID.

65

muboy also assigns an overall goodness score based on an event’s likelihood of beinga true cosmic muon. The muboy algorithm assumes the vertex of any event is outside
the ID, so true PC events will have low goodness scores according to muboy.
With the two vertices, and direction and classification information from muboy,
PC4 creates several optional conditions. Depending on the muboy classification, a
subset of these conditions must be passed. The conditions are• The difference between the angle from the muboy vertex and the angle from thetime-of-flight vertex to the highest-charge OD hit cluster (calculated as part ofPC2) must be less than 90◦ .
• The difference between the angle from the muboy vertex and the angle fromthe time-of-flight vertex to the earliest saturated ID PMT must be less than
143.13◦ .
• The length of the fitted muon track from muboy must be less than 1750 cm.
• The goodness of the muboy fit must be less than 0.52.
• The distance from the muboy vertex to the corner of the tank must be larger
than 300 cm.
PC4 assumes that Multi-muon and corner-clipper events are more difficult toreconstruct, and so applies looser cuts than for stopping and through-going muons.
PC4 applies the conditions
 If the muboy class is stopping or through-going, the event must not fail morethan one of the five PC4 optional conditions
 If the muboy class is Multi-muon or corner-clipper, the event must not fail morethan three of the five PC4 optional conditions66

Finally, PC4 places additional restrictions on events identified as stopping muons
by muboy: If the muboy class is stopping, the goodness of the muboy fit must be less than0.5, or there must be fewer than 10 OD hits within 8 m of the muboy entrance
point in a −200 ns to +300 ns window.
There are approximately 900 events per day that which pass PC4.
PC5
PC5 further removes muon backgrounds from the PC sample. Muons which enternear the top edge of the detector and exit through the bottom edge can produce asingle large cluster of hits in OD, and, depending on the proximity to the detector
wall, a small cluster of hits in the ID. Such events can evade PC1-PC4. PC5 applies
a cut using OD PMTs contained by 8 m spheres along the top and bottom detector
edges. The spheres on the top and bottom which contain the maximum charge areused for the following cut: At least one sphere must contain fewer than seven OD hits, or at least one
sphere must contain fewer than 10 photoelectrons, or the average hit times inthe spheres must be outside the interval given by [0.75c/40 m, 1.5c/40 m], i.e.,a range of plausible transit times across the SK tank.
The final check of this cut ensures that the distance between hits in the selected
spheres is consistent with a muon traversing the height of the detector.
Events which pass PC4 are fit to identify a Cherenkov ring and vertex using thealgorithms described in Section 3.2. PC5 applies a series of cuts to further mitigate
stopping muons, corner clippers, and events near the OD gaps using the informationfrom this fit. These are67 The angle between the fitted Cherenkov ring and the center of the largest OD
hit cluster must be less than 90◦ .
 The fitted vertex must be greater than 150 cm away from the ID top and bottom
edges.
 If a scintillator veto paddle is triggered, the angle between the line from theveto paddle to the fitted vertex and the line from the fitted vertex to the fitted
ring direction must be greater than 37◦ .
As in PC4, PC5 implements a cut which requires events to satisfy a subsets ofoptional conditions. Each condition addresses a specific background category. The
conditions are listed below with the intended background they address in parentheses:• (Through-going muons) If the fitted entrance point and exit point, determinedby extrapolating the vertex and fitted ring direction, contain OD hits which
are separated in time by [0.75c/d, 1.5c/d], where d is the distance between the
points, there must be fewer than five OD hits within an 8 m radius of either
point.
• (Through-going muons) There must not be a cluster of OD PMT hits containing
17 hits, and a second cluster containing 10 or more hits.
• (Stopping muons) There must not be 10 or more hits within 8 m of the fitted
entrance point.
• (Stopping muons) A fit assuming the entrance point is the earliest hit ID cluster
must not contain more than 60 % of the photoelectrons in a 42◦ cone aroundthe fitted track, and there must not be more than six OD hits within 8 m of theassumed entrance point.

68• (Stopping muons) The angle between the fitted Cherenkov ring and the largest
OD hit cluster must be less than 90◦ .
• (Corner-clippers) The track length determined from the vertex cannot be morethan 15 m longer than the track length estimated from the total ID charge
divided by 2 MeV/cm. Corner clipper muons typically have short tracks in theID, so there can be large disparities in track length estimated from the ring andfrom the charge.
• (Non-neutrino) There must be 1 decay electron in the event.
The final optional condition targets muons which pass through the detector without producing hadrons. Such muons would create events with no decay electrons,
while neutrino events at PC energies are likely to produce hadrons which decay into
visible electrons near the vertex. This is discussed in Chapter 4. The corresponding
PC5 cut based on the above optional conditions is: Events must not fail more than one of the PC5 optional conditions.
3.1.3

Up-µ Reduction

Up-µ events constitute about 1.5 atmospheric neutrino events per day. Backgrounds
for Up-µ events well-below the horizon are much less than for FC and PC due tothe shielding of cosmic muons by the earth. However, events with fitted directions
near the horizon can result from downward-going cosmic rays just above the horizon
which produce upward-scattered muons, creating an irreducible background for theUp-µ sample. The reduction steps presented here involve quality checks for muonlike events and comparisons of fit directions using different methods. Since the Up-µselection relies on the direction of the events, requirements on zenith angle, cos ¹z ≲ 0,
appears in the selection. See Section 1.5.3.
69

First Reduction
The first Up-µ reduction cut corresponds to a minimum momentum cut for a 1 GeV/cmuon, and a maximum momentum cut, above which muon reconstruction is not
feasible.
 The number of ID photoelectrons in a 300 ns sliding window must be between
6000 and 1 750 000 (3000 and 800 000 for SK II)
Second Reduction
This step successively applies seven different muon vertex and direction fitting routines optimized for different types of muons, e.g., through-going, stopping, or muons
which lose energy due to radiative losses instead of ionization. While high goodness
scores according to various muon fitters were used to remove backgrounds in the FC
and PC reductions, they are used here to accept neutrino events. The conditions forthe second reduction step are The muon must be fit as upward-going or horizontal with a goodness scoreabove a threshold by at least one fitter The muon must not be fit as downward-going with a goodness score above athreshold by any of the fitters
A full list of the fitters and the associated thresholds, as well as additional logic about
the fitting order, is presented in other theses [74].
Third Reduction
Events which pass second reduction are fit by a computationally-intensive fitted called
precisefit. Using the precisefit information, a cut is applied on the direction
and on minimum allowed momentum:70 The fitted direction must be upward-going If the muon is classified as stopping, the momentum must be greater than
1.6 GeV/c.
 If the muon is classified as through-going, the fitted track length must be greaterthan 7 m.
Fourth Reduction
The Up-µ fourth reduction step was implemented beginning with SK IV, and utilized
the segmentation Tyvek in the OD (see Section 2.2.2). As with PC events, the fourth
step uses the number of OD PMT clusters, calculated via a clustering algorithm.
 The number of OD hit clusters must be exactly one, or the direction between
the two clusters with the most hits must be upward-going or horizontal withcos ¹z < 0.035.
Fifth Reduction
In 2019, and enacted starting with the SK V data set, the Up-µ reduction was further
automated with a fifth reduction step. The new step places additional cuts whichwere previously evaluated manually by experts manually on an event-by-event basis.
 The angle between the muboy (see Section 3.1.2) fitted direction and the dedicated Up-µ muon fitter fitted direction must be less than 20◦ .
 The muboy goodness score must be > 0.25.
 The muboy fitted direction must be cos ¹z < 0.02.

71

3.1.4Post-reduction Eye-scanningBecause SK observes ∼ 10 atmospheric neutrino events per day, it is feasible forexperts to visually inspect all events, known as eye-scanning. The purpose of eyescanning is not to remove events, rather, it is to estimate the background contamination after the final reduction steps in each sample. Each sample, FC, PC, and Up-µ,is eye-scanned following the sample’s final reduction step.
3.1.5

ID-OD Cross Talk

In 2018, it was discovered that large amounts of charge deposited on ID PMTs could
induce small amounts of charge on nearby OD PMTs, which could then register as
hits, known as cross talk. In 2019, a new preprocessing step was implemented toremove OD hits induced by cross talk. All SK data was re-processed using the above
reduction steps, but excluding any OD hits with times greater than 50 ns after thetrigger and charges less than 0.2 photoelectrons. The development of this cut isdocumented in the thesis by C. Kachulis [75].
Figure 3·4 shows the average number of FC, PC, and Up-µ events remaining after
all reduction steps per day during the SK I-V data-taking phases. The figure shows
the stability of these reduction criteria over the lifetime of the experiment.

72

Average Events / Day

10

FC

Up-µ
1

PC

I
0

II
2000

III

IV

4000
6000
Days since April 1, 1996

V
8000

Figure 3·4: Average number of FC, PC, and Up-µ events per day
during the SK I-V data-taking phases. The events shown are the final
sample used in this analysis, and have passed all reduction steps, including additional criteria described in Section 5.1.

3.2

Reconstruction

Reconstruction involves counting and classifying the particles in each event which
passes the reduction stage for physics analyses. The primary reconstructed object
at SK is a Cherenkov ring, which contains information about the particle’s vertex,
direction, and type. SK uses a set of algorithms to identify, count, and categorize
Cherenkov rings, collectively called apfit. This section will provide an overview ofapfit functionality. Further documentation is also available in [76].
A separate algorithm, fitqun, has also been used to reconstruct SK events [77].
fitqun will be discussed in Chapter 6 as part of the analysis with external constraints
from T2K.

73

3.2.1

Event Vertex

When reconstructing neutrinos events, assuming a single vertex is a powerful constraint on the expected timing distribution of PMTs. Reconstruction for FC and PC
events begins with a preliminary vertex fit in the ID based on the residual hit times
after subtracting the time-of-flight to the vertex. apfit refines this time-of-flightvertex later using additional information. The residual times from a trial vertex, ⃗x,
are computed astResid.
(⃗x) = ti −in|⃗x − ⃗xPMT
|
i
c

(3.3)

where n is the index of refraction in water, ti represents the calibrated hit time relative
to the trigger, and ⃗xPMTis the position, of the ith PMT. The best vertex is taken asithe point which minimizes the width of the distribution of tResid.
. This is accomplishediusing a goodness of fit parameter, G, based on a Gaussian likelihood,G(⃗x) =Xi(tResid.
− t0 ) 2
1i
exp −Ã(qi )
2[1.5Ã(ïqð)]2(3.4)

where qi is the charge of the ith hit PMT, Ã(q) is the timing resolution of the readout
electronics for a pulse of charge q, and t0 is a free parameter for the residual time
distribution peak position. The vertex which minimizes G becomes the starting point
for the next step.
The time-of-flight vertex can be improved by considering just the hits produced bya single Cherenkov ring. While an event might contain multiple rings, the brightest
ring contains the most information for finding a vertex. This ring can be identified
by charge-weighting each hit, correcting for water attenuation and the incident angle.
The corrected charge is defined as
qiCorr. (⃗x) = qi exp|⃗x − ⃗xPMT
|
i
−
L

74



cos ¹
f (¹)

(3.5)

where, the vertex, position, and charge are the same as in Equation 3.4, L is theattenuation length of the water, ¹ is the angle of incidence from the vertex to the hit
PMT, and f is the acceptance of the PMT as a function of ¹. The vector sum of hit⃗
PMTs, weighted by the corrected charge, establishes a preliminary ring direction, d.
The ring direction is refined by testing various Cherenkov angles. The Cherenkov
angle (see Equation 2.2) for a particle moving close to the speed of light in water is
42◦ , but this can be smaller for particles with lower momenta. A new goodness offit parameter is defined which attempts to accomplish three goals: (i) penalize small,
non-Cherenkov-like angles, (ii) match the opening angle to the expected opening
angle from a particle which would produce the contained charge, and (iii) maximize
the charge contained in a ring of the assumed angle. The measure is2 Zθc  Corr.

2
(¹
−
¹
)
dq
1
c
Exp.
⃗ ¹c ) =q Corr. (¹)d¹
×× exp −
|θ=θc
G(d,sin ¹c
2Ãθ2
d¹

(3.6)

0

where ¹Exp. is one of three angles: 42◦ , or the angle assuming a muon, or the angle
assuming an electron, produced the observed charge. Ãθ is the angular resolution, andthe differential term gives the change in the total corrected charge contained versus
opening angle of the ring, i.e., to measure how suddenly the charge changes outside
the ring edge. The first term addresses goal (i), the second term addresses goal (ii),and the third term addresses goal (iii).
Minimizing Equation 3.6 establishes a refined ring direction and Cherenkov angle.
The initial time-of-flight vertex from Equation 3.3 may now be refined by shifting
the residual times of hits within the Cherenkov ring by the difference from the initialvertex, ⃗x, to a vertex, ⃗y , constrained to be on the line given by the fitted angle andthe fitted ring direction:
tResid.
(⃗x, ⃗y ) = ti −i1n|⃗y − ⃗xPMT|− |⃗y − ⃗x|
i
c
c
75

(3.7)

The vertex goodness is once again calculated via Equation 3.4, and separate goodness scores are constructed for hits inside and outside the Cherenkov ring. A minimization routine finds the new best-fit vertex based on the new residual time distribution.
3.2.2

Ring Counting

For events containing multiple rings, apfit transforms the ring fitting problem into apeak-finding problem using the method of Hough transformations [78]. The method
is illustrated in Figure 3·5. In the figure, circles of the same radius are drawn aroundeach hit PMT position, white dots in the left panel. For a true circular arrangement of
PMT hits, the rings drawn around each PMT will overlap at a single point, creatinga region of highest intensity, shown in the right panel. apfit refines the Houghtransformation technique by applying the transformation to the expected charge fromrings at a particular angle, weighted by the observed charge. The hit positions of
PMTs are mapped from Cartesian coordinates into the altitudinal and azimuthal
directions within the SK ID cylinder for performing the transformation. The positionsof any peaks found in the transformation space then correspond to a ring direction.

Figure 3·5: Graphical demonstration of a Hough transform. Left:
Points are arranged in rings. Right: Rings are drawn around eachpoint from the left panel. Due to the ring arrangement of the points,
the drawn rings overlap, forming bright points in the Hough space.
76

Rings are added to the event iteratively. apfit implements a likelihood variable
for N rings. Additional rings are added only if they cause the likelihood to increaseover the hypothesis for the original number of rings. The likelihood is

LNRing =

NX
PMT
i

 

log P qi ,

NRing

X
j



Exp. 
³j qi,j(3.8)

where P (q1 , q2 ) is the Poisson (if qiExp. < 20 photoelectrons) or Gaussian probability ofobserving a charge q1 given an expected charge q2 , ³j is a weight applied to the j th ring,Exp.
and qi,jis the expected charge contribution to the ith PMT from the j th ring. apfit

will add up to five rings per event. Figure 3·6 demonstrates how Equation 3.8 may
be used to separate single and multi-ring events. The difference between the single
ring hypothesis, L1 , and the hypothesis with more than one ring, L2 , for atmospheric
neutrino data from the SK IV-V phases and simulation (see Chapter 4) is shown.
The distributions reflect the expectation that lower-energy neutrino interactions are
more likely to produce only one ring.
3.2.3

Particle Identification

Particle identification (PID) in apfit involves classifying ring candidates as originating from electromagnetic showers, e-like or muons, µ-like. Electrons tend to radiate
energy via bremsstrahlung, or through Compton scattering, producing energetic photons. These photos then undergo pair production, producing additional electrons andpositrons which produce their own Cherenkov rings. These processes tend to deflect
the direction of electrons, resulting in ring patterns with blurred edges. In contrast,
muons, which are heavier than electrons, tend to travel without significant deflection,
resulting in sharp ring edges. Figure 3·7 shows example e-like and µ-like rings fromcandidate neutrino interactions in SK.
apfit constructs a likelihood function based on the expected distribution of hits77

FCFV Sub-GeV

FCFV Multi-GeV
1400

Events2000

SK IV-V
Single Ring Multi-Ring
Data
CC QE
CC 1π
CC DISCC Other
NC

1200
1000
Events2500

1500

SK IV-V
Single Ring Multi-Ring
Data
CC QE
CC 1π
CC DISCC Other
NC

800
600

1000
400
500

200

0
−15

−10

−5
0
5
Ring Counting Likelihood

10

0
−15

15

−10

(a)

−5
0
5
Ring Counting Likelihood

10

15

(b)

Figure 3·6: Ring counting likelihood, L2 − L1 , from Equation 3.8,
for atmospheric neutrino data and simulation for the SK IV-V phases.
Positive values indicate that the hypothesis with at least two rings waspreferred over the single-ring hypothesis. The case for sub-GeV events(visible energy less than 1330 MeV) is shown in (a), and multi-GeV
events (visible energy greater than 1330 MeV) is shown in (b).

Super-Kamiokande V

Super-Kamiokande V

Run 81972 Sub 221 Event 227792241

Run 81968 Sub 259 Event 267180161

19-09-08:01:13:10

19-09-04:02:34:55

Inner: 3004 hits, 7420 pe

Inner: 1724 hits, 4716 pe

Outer: 0 hits, 0 pe

Outer: 1 hits, 1 pe

Trigger: 0x10000007

Trigger: 0x10000007

D_wall: 804.8 cm

D_wall: 1011.6 cm

Evis: 767.1 MeV

Evis: 542.0 MeV

e-like, p = 767.1 MeV/c

mu-like, p = 675.0 MeV/c

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
700

330
0 mu-edecays560

1 mu-e
decay

264

420

198

280

132

140

66

0

0
0

500

1000

1500

2000

0

Times (ns)

500

1000

Times (ns)

(a)

(b)

Figure 3·7: Neutrino candidate events from SK V data. Bright dots
represent hit PMTs, with color and radius corresponding to the observed charge. (a) shows an e-like event with blurred ring edges, while(b) shows a µ-like event with sharp ring edges.

78

1500

2000

and charges assuming a muon or electron parent particle for the ring. This pattern
likelihood for the nth ring is
LPattern(e, µ) =nNYPMTθi <1.5θc"

X Exp.
Exp.
P qi , qi,n(e, µ) +
qi,j
j̸=n

#

(3.9)

where the product is taken over all PMTs contained within a ring with opening
angle given by 1.5 times the Cherenkov angle, ¹C , and the probability P (q1 , q2 ) has
the same definition as in Equation 3.8. The contribution to the total charge fromother rings in the event is added via the summation term within the probability,
without assuming a particle type. For this likelihood, the expected charge from thenth ring, qiExp. (e, µ), is calculated from a combination of already-fitted quantities andpre-computed simulations. Both the e and µ scenarios account for the fitted vertex
and ring direction, and use the average light produced in simulations of probableparticle trajectories produce estimates for the expected charge. The key differences
between the muon and electron expectations are that the photon emission points
are constrained to be on the muon’s estimated track, and that the muon may also
produce extra light from energetic electrons knocked out as it ionizes the water. A
full description of the expected charge distributions for this step is presented in [79].
apfit applies an additional correction to events with a single ring, based on
the expected Cherenkov angle for an electron or muon with a specified momentum,¹Exp. (e, µ):P Angle (e, µ) =






i
h
(θ−θExp. (e,µ))2
NRing = 1
exp −
2σθ



1

,

(3.10)

NRing > 1

where ¹ is the reconstructed Cherenkov angle, and Ãθ is the estimated Cherenkov
angle fitting resolution. Using Equation 3.9 and Equation 3.10, apfit outputs asingle number to indicate e- or µ-like PID preference. First, the pattern likelihood is79

converted to a probability assuming a Ç2 distribution,
Ç2n (e, µ) = −2 log LPattern(e, µ)
n=⇒

PnPattern (e, µ) = exp[Ç2n (e, µ) − min(Ç2n (e), Ç2n (µ))]2−
4NPMT(3.11)Such that Pn (e, µ) = PnPattern (e, µ) × P Angle (e, µ). Then the preference for e- or µ-likefor the nth ring may be expressed as
LPID
=
n

p
p
− log Pn (µ) − − log Pn (e)(3.12)The distributions of Equation 3.12 for SK IV-V data and simulation (see Chapter 4)
are shown in Figure 3·8.
FCFV Sub-GeV

2500

Events2000

SK IV-Ve-likeDataν e and ν e CCν µ and ν µ CCν τ and ν τ CCNCµ-like600
500
Events3000

FCFV Multi-GeV
700

1500

SK IV-VData
ν e and ν e CCν µ and ν µ CCν τ and ν τ CCNC

e-like µ-like

400
300

1000

200

500
0
−15

100

−10

−5

0
5
PID Likelihood

10

0
−30

15

(a)

−20

−10
0
10
PID Likelihood

20

(b)

Figure 3·8: PID likelihood from Equation 3.12 for atmospheric neutrino data and simulation for the SK IV-V phases. Positive values
indicate that the leading ring is µ-like, while negative values indicate elike. The case for Sub-GeV events (visible energy less than 1330 MeV)
is shown in (a), and Multi-GeV events (visible energy greater than
1330 MeV) is shown in (b).

80

30

3.2.4

Event Vertex, Revisited

For single-ring events, apfit re-integrates the PID information from the previous
reconstruction step to improve the vertex fit. In an iterative process, the vertex anddirection are moved in small steps to maximize the pattern likelihood from Equation 3.9. The procedure is repeated for seven iterations, or until the vertex is stable
to within 5 cm and the direction is stable to within 0.5◦ between subsequent iterations.
3.2.5

Momentum Reconstruction

Each Cherenkov ring corresponds to a particle with a particular momentum. In thecase of single-ring events, the momentum may be computed from the sum of theobserved charge within the Cherenkov ring. In the case of multi-ring events, rings
may overlap, and so the observed charge on must be divided among each ring. The
observed charge on the ith PMT from the nth ring is defined using the fraction of the
expected charge from the nth ring,
qi,n = qi

Exp.
qi,n

PNRing
j

Exp.
qi,j

!

(3.13)

apfit uses the divided charges (with acceptance corrections from Equation 3.5) tocompute an intermediate quantity for each ring, RnTot. ,RnTot. =

GMC
GData

X

Corr.
qi,n
−

i

X
j

qjScatter!

i ∈ (¹i,n < 70◦ and − 50 ns < tResid.
< 250 ns)ij ∈ (¹j,n < 70◦ )(3.14)

where GMC and GData are the relative gains in the simulation and data respectively, iand j index PMT hits satisfying the listed angular and residual time conditions, ¹i,nis the angle of the ith PMT relative to the nth ring vertex and direction, and qjScatter is81the expected charge scattered onto the j th PMT not originating from the Cherenkov
ring. RnTot. is converted to a momentum for each ring using a lookup table whichmatches RnTot. values for each ring to simulated particles with known momenta.
3.2.6

Ring Counting, Revisited

apfit uses the momentum of each ring, pi , to merge lower-energy rings into higherenergy ones, and remove spurious rings. The two conditions are• Two rings i and j are merged if ¹i,j < 30◦ and pi cos ¹i,j < 60 MeV/c.
• A ring i is removed if pi < 50 MeV/c and pi /3.2.7

Decay Electron TaggingP

j pj < 0.05.

In addition to Cherenkov rings, SK also sees electrons from the decay of muons following the primary interaction. The observation of decay electrons provides information
about the parent muon track and the charge, positive or negative, of the muon. The
charge information is statistical: negatively-charged muons, and pions, which decay
into negatively-charged muons (see Equation 1.27), are more likely to be absorbed in
the water before decaying. Therefore, decay electrons are most often observed associated with positively-charged muons. The usefulness of this fact for distinguishing
between neutrino and anti-neutrino interactions will be discussed in Chapter 5.
The mean lifetime of the muon is 2.2 µs, so many of these electrons are contained
within the primary SK I-III event windows, and all are contained within the SK IV+
event windows, listed in Table 2.2. In the case of SK I-III, decay electrons appearing
after the primary event window may trigger the detector again, resulting in a second“sub-event.”
The simplest decay electrons to search for in SK I-III are decay electrons within
sub-events, and, in SK IV+, decay electrons 450 ns after the primary trigger. A82

dedicated algorithm searches for signatures of these decay electrons. While energetic
decay electrons can have enough momentum to produce Cherenkov rings, the numberof hits is often too low to attempt a full reconstruction. Instead, decay electrons arefound with timing information. The algorithm scans for peaks in a 30 ns residual
time window above background. Here, the background, NBkg , is estimated from thenumber of hits in a small window before the peak. A decay electron must satisfy thefollowing conditions:• NPeak − NBkg g 50 (25 for SK II)• (NPeak − NBkg )/p
NBkg > 6.63 (5.17 for SK II)

Following the identification of a peak, the hits in the peak are used to estimate avertex, once again by minimizing Equation 3.4. The fitted t0 parameter from thevertex fit estimates the decay time. The decay electron is counted if the number ofhits in a 50 ns window around t0 is greater than 50 (25 for SK II) and the goodness
parameter G > 0.5.
In some cases, a decay electron can be identified, but its vertex cannot be reliably
established. One such case is when a decay happens too soon after the primary
event, such that a hit peak is visible but the decay electron hits are too difficult
to distinguish from the primary event hits. Decay electron tagging in SK I-III was
further complicated by an impedance mismatch which occasionally caused spurious
hits due to reflected signals. Decay electrons were not searched for in the reflected hit
region. Additionally, due to the shorter event window during SK I-III, decay electrons
could be split between multiple events if not all hits were contained within the same
event window. The identification of these decay electrons is described in [80]. For the
SK I-III data used in this thesis, these split decay electrons are counted, but theirvertex is considered unreliable. This is relevant to the Multi-Ring event classification
discussed in Section 5.1.1.
83

3.2.8

Neutron Tagging

Neutrons produced following a neutrino interaction may be captured by hydrogen in
the detector’s water. The captures create excited deuterium, which decays, emitting
photons with an average combined energy of 2.2 MeV,n + p+ → d + + µ(3.15)The 2.2 MeV energy deposition corresponds to approximately seven ID PMT hits.
This is enough to be detected in SK. Observing neutrons has several benefits for SK
physics: The energy of neutrino interactions correlates with the number of neutrons
produced, the presence or absence of neutrons can be used to identify particle interaction processes, and the average number of neutrons produced for neutrino andanti-neutrino interactions of the same energy differs. This last consideration will be
discussed in relation to distinguishing between neutrino and anti-neutrino interactionsin Chapter 5.
The average neutron capture time on hydrogen is Ä = 204.8 ± 0.4 µs, so detecting
neutrons using the SK I-III electronics was not feasible. As described in Section 2.2.4,
the software trigger implemented for SK IV allowed variable event time windows
extending out to 535 µs after the trigger, enabling the detection of neutron captures
for the SK IV phase and later.
A dedicated neutron tagging algorithm was developed for SK IV based on a two
step process [75, 81]. First, clusters of hits were identified as possible candidates,
and these hits were fit to find a capture vertex. Next, the hits in the clusters andthe vertex were used to define variables which were input into a neural network. The
neural network classified the event as either a neutron capture or background basedon weighted combinations of each variable.
Beginning with SK V, the neutron tagging algorithm was modified to disregard
84the fitted neutron capture vertex1 . The present algorithm uses a modified set ofvariables calculated assuming the primary event vertex is the neutron capture vertex,
and uses a different neural network structure from the original SK IV algorithm. This
thesis will provide a summary of the SK IV approach and note the differences with
the SK V method.
Step 1: Candidate Selection
Neutron capture candidates are found using a sliding 10 ns residual time window,
where the residual times are calculated assuming the primary event vertex2 . Candidate selection is based on the number of hits in, and around, the window:• The number of hits in the 10 ns sliding window must be seven or more and lessthan 50.
• The number of hits in a 200 ns window around the time of the first hit of thecandidate hit cluster must be less than 200
The first requirement bounds the expected hit count from a true neutron capture
to reduce spurious coincidences from both PMT noise and higher-energy radioactive
backgrounds or decay electrons. The second requirement ensures the neutron captures
are not coincident with, e.g., a muon passing through the detector. This candidate
selection also establishes a capture time, t0 , taken to be the time of the first hit in
the candidate window.
For SK IV only, once the candidates are selected, three fits for the neutron capture
vertex were performed using the hits within the 10 ns window. The first fit, BONSAI
1

Seungho Han, NTag software and analysis for SK-V. Super-Kamiokande Collaboration Meeting,
December 2, 2022
2
This first step is the same for SK IV and SK V, except for the hit threshold, changed to five,and time window, extended to 14 ns. The prompt vertex assumption follows from simulation studies
showing that 70 % of neutron captures occur within 200 cm of the primary event vertex, so theresidual times are minimally affected by the assumption.

85

[82], checks for vertices at points which produce near-zero residual times for subsets of
hit PMTs, then successively adds additional hits. Vertices which decline in goodness
as more hits are added are dropped from consideration. The second fit searches for a
vertex on a fixed grid within a sphere centered at the primary event vertex and witha radius of 2 m. The third fit is a re-application of the previous fit, but considers thewhole SK detector. The results of the fits are used as inputs in the next section.
Step 2: Neural Network Classification
Neural network classifiers add weighted combinations of information from an event,and output a score reflecting the similarity of the event to pre-classified, “labeled”
events. The neural network weights are obtained through a training process. Training
consists of applying the neural network to labeled events, assessing the classification
decision as correct or incorrect, and adjusting the network weights based on the assessment. For sufficiently realistic training data, the network will eventually optimize
the event information to make statistically accurate classification decisions.
In order for a neural network to classify neutron captures, it must be trained ontrue neutron capture signals and realistic backgrounds. Simulated neutron captures
within a realistic description for the SK detector (see Section 4.3) are used as training
set for the neutron tagging neural network. However, not all true neutron captures will
make an observable signal within the SK detector. For training the neural network,“true” neutron captures are defined as captures identified using the initial candidate
selection step, and have a t0 within 100 ns of the true capture time from simulation.
This requirement produces ∼ 1 % fake events labeled as “true” and ∼ 0.1 % true
events labeled as background in the simulated training data. The candidate selection
step also produces false positives, leading to a flat background which has no correlation
with true neutron captures. The distribution of differences between the calculated t0
and the nearest true simulated neutron capture is shown in Figure 3·9. A peak of86

6

Neutron Candidates / 2 ns

10

5

10

104

103

102
0

100
200
300
400
500
600
700
800
900 1000Time Between Neutron Candidate and Nearest True Neutron Capture (ns)Figure 3·9: Distribution of the time differences between the first hit ina candidate residual time PMT hit cluster and the true neutron capture
time for simulated neutron captures. The capture is labeled as “true”
if the difference is less than 100 ns, i.e., events to the left of the dashedline. The flat background, red line, is extended to the left of the 100 ns
line as an estimate of the background rate for this labeling process.
Reproduced from [81].
true neutron captures is visible below 100 ns. The flat background component can beseen extending to large time differences. This component has been extrapolated into
the “true” region below 100 ns as an estimate of the background.
Once simulated true candidates and background events have been labeled, measurable quantities of potential neutron captures which are statistically different fortrue neutron captures and backgrounds must be chosen. The variables used and abrief description for SK IV are listed in Table 3.1. Each variable’s inclusion in the
network is motivated by physical characteristics of neutron captures. For example,
radioactive backgrounds occur most frequently near the detector walls, due to contaminants present in the detector components. Therefore, backgrounds are more
likely to occur near a wall than true neutron captures. Additionally, PMT hits from87

# of events# of events

Data

106

Total MC105

Signal MC
Bckg MC

40000

Data

35000

Total MC

30000

Signal MC

104

Bckg MC

25000
103

20000
15000

102

10000
10
1
5

5000
6

7

8

0
−0.2

9 10 11 12 13 14 15
N10

0

0.2

0.4

0.6
β

0.8

1

1.2

3

Figure 3·10: Two example variable distributions used as input to theneutron tagging neural network. Left: N10 distribution for simulated
signal and background events versus SK IV data. Signal events tend tohave higher N10 than background events, as can be seen from the green
histogram extending to higher values. Right: Isotropy parameter, ´3 ,distribution for simulated signal and background events, versus SK IV
data. Signal events tend to have lower isotropy than background events,
as can be seen from the narrow distribution of signal events near 0. Bothfigures are reproduced from [81].
true neutron captures tend to have narrow residual time distributions, small isotropy(they are spatially clustered), and have high consistency with Cherenkov patterns.
Background events may be different in one or more of these areas. Two variables usedin both the SK IV and SK V selection, N10 and ´l for l = 3, are shown in Figure 3·10.
Distributions for the other variables may be found in [81].
For SK V, the following variables from Table 3.1 are kept: N10 , N300 , tRMS , ϕRMS ,´l , NFWall and Ltowall . The exact values of the window sizes for several variables, in
ns, were also adjusted. All other variables were removed. Two new variables werealso added: a data-driven likelihood of each hit originating from PMT dark noise,and the ratio between the number of hit PMTs with a preceding hit within 10 µs to88Variable

Description

N10
N300
tRMS

The number of hits within a 10 ns residual time window
The number of hits within a 300 ns time window
The RMS of the residual times of hits
The smallest RMS of residual time hits for combinations of threehits within the 10 ns residual time window
The difference of N10 using the primary event vertex and the reconstructed neutron vertex
The difference of tRMS using the primary event vertex and thereconstructed vertex

min(tRMS )
∆N10
∆tRMS
¹Mean
ϕRMS
NC
PAccept.
LC
´l
NLow
BSE
BSWallNFWall(NF − AP)Dist.
(NF − BS)Dist.
Ltowall

Average opening angle from the reconstructed vertex to hit PMTs
The RMS of the azimuthal angles of PMT hits
The number of clustered hits found by merging hits in 14◦ cones
centered at the fitted vertex
Calculated probability of hits based on PMT geometrical acceptance for the fitted vertex.
Likelihood of hits originating from a single Cherenkov cone
Isotropy of the hits viewed from the residual time fitted vertex (5parameters)
Number of hits on low-probability PMTs.
Reconstructed energy from BONSAI fit
Distance from BONSAI vertex to the nearest wall
Distance from residual time fitted vertex to the nearest wall
Distance from the residual time fitted vertex to the apfit primaryvertexDistance from the residual time fitted vertex to the BONSAI vertex
Distance from the residual time fitted vertex to the wall in the
direction of the hits from the neutron capture

Table 3.1: Variables used in the SK IV neutron tagging neural network
selection. The variables are grouped into three qualitative categories:
hit counts & times, hit spatial distribution, and vertex information.
Table is adapted from [81].

89the total number of hit PMTs in the residual window, N10 .
Figure 3·11 shows the performance of the neural network algorithm for identifying
neutrons in SK IV data. The neural network outputs a classification score basedon the degree of similarity between an event and trained signal and background
events. Cutoff scores for different values of the N10 variable were chosen to accept or
reject neutron capture candidates based on the estimated contamination rate of false
neutron captures. For neutron capture candidates with N10 g 7, the chosen cutoff
values corresponds to an average false tagging rate of 0.016 neutrons per event, andan efficiency of selecting true neutron captures of 26 %.
Number of events

Neural Net Output(7hits or more)

Data

104

Total MC
3

10Signal MC
102

10

1
−0.2

0

0.2

0.4

0.6

0.8
1
1.2
Neural Net Output

Figure 3·11: Neural network performance in SK IV for neutron candidates with N10 g 7. The green histogram shows the distribution ofneural network scores for simulated “true” neutron captures (see text),
while the red line shows the distribution for all simulated candidates.
The black points show the neural network score distribution for neutron
capture candidates following neutrino interactions in SK IV data. The
vertical line represents the cutoff value on the neural network output
for accepting neutron candidates. Reproduced from [81].
.

90Chapter 4

Simulation of Atmospheric Neutrinos
Extracting neutrino oscillation parameters from SK atmospheric neutrino data requires a robust prediction of atmospheric neutrino interactions at SK. Due to thecomplexities of the atmospheric neutrino flux, neutrino cross sections, and detector
effects, an analytic prediction is not feasible. Instead, simulated neutrino interactions
drive the prediction.
Simulated neutrino interactions are generated by first assigning each neutrino a
direction, energy, and flavor based on the predicted flux of atmospheric neutrinos.
The particles produced from the possible interaction processes involving a neutrinoof a particular energy are then generated according to a set of cross section models. These secondary particles are stepped through a realistic simulation of the SK
detector to produce simulated events. Simulated events are reconstructed with thesame algorithms as data events, so that properties of the simulation and data may
be compared directly.
The analysis in this thesis uses the flux model of Honda et al. [83], and thecross section models of the neut [84] version 5.4.0 neutrino interaction generator.
Propagation of particles within the SK detector is performed using the skdetsim
program, developed internally by the SK collaboration.

91

4.1

Atmospheric Neutrino Flux Calculation

The first step of simulating atmospheric neutrinos is to estimate the energy- anddirection-dependent atmospheric neutrino flux observed at SK. The necessary components of the atmospheric neutrino flux calculation are explained in detail in [85].
Calculating the atmospheric neutrino flux begins with measurements of the primarycosmic ray flux. Cosmic rays with energies relevant for atmospheric neutrinos aremeasured directly at heights of ∼ 30 km by balloon experiments [86], or at evenhigher altitudes, ≳ 100 km, by experiments on spacecraft [87]. These measurements
are performed at multiple sites around the world and during different seasons tocapture the variations due to atmospheric conditions.
Next, the primary cosmic ray flux measurements are convolved with models of
proton-nuclei interactions to generate the number and energy distributions of secondary hadrons. The primary hadrons relevant to neutrino production are pions
and kaons. While pions nearly always decay into muon neutrinos, kaons decay into
different neutrino flavor combinations, e.g., listed with branching ratios,
KL0 → Ã ± + e∓ + ¿e

(41 %)(4.1)

KL0 → Ã ± + µ∓ + ¿µ

(27 %)(4.2)

There are more secondary pions than kaons produced per primary cosmic ray interaction, but at high energies, pions tend to interact before decaying, increasing thecontribution to the neutrino flux from kaons. Further, high-energy muons from pionand kaon decays tend to reach the Earth’s surface before decaying, leaving kaons as
the primary source of electron neutrinos at the highest energies. These effects resultin an increase of the muon-to-electron neutrino flavor ratio, discussed in Section 1.5.3,
from about 2 : 1 below a few GeV to 4 : 1 to near 10 GeV. The dependence of theflavor ratio as a function of neutrino energy is visualized in Figure 4·1.
92

Figure 4·1: Atmospheric neutrino fluxes for different neutrino flavors. Left: 1D Atmospheric neutrino fluxes calculated for the Kamiokasite. Right: 1D atmospheric neutrino flavor ratios calculated for the
Kamioka site. Both figures are adapted from [83].
The most precise inputs for the hadron production models come from experiments
which accelerate protons on to fixed targets [88]. However, these experiments do notcover the entire phase-space of incident proton energies and outgoing hadron kinematics required to fully predict the atmospheric neutrino flux. Instead, measurementsof secondary cosmic muons are used as an estimate for the hadron production model
inputs where no direct measurements are available.
4.1.1

Corrections to the Atmospheric Neutrino Flux

The atmospheric neutrino flux calculation has time-varying and anisotropic corrections due to additional effects. These effects mean that the flux must be calculated in“4D”: as a function of neutrino energy, zenith angle, azimuth angle, and for different
time conditions. The fluxes provided by Honda et al. are also calculated using theterrain profiles of different detector sites, e.g., for SK underneath Mt. Ikenoyama, toreflect the mediation of the cosmic ray flux based on the shape of the overburden.
93

Because primary cosmic rays are positively charged, the Earth’s magnetic field
preferentially deflects cosmic rays in an anisotropic way. The deflection additionallyimplies that cosmic rays below certain energies cannot enter the atmosphere and interact, since they would have had to travel on nonphysical trajectories to enter theatmosphere. This is known as the East-West Effect [89], and produces an azimuthal
dependence in the atmospheric neutrino flux. The East-West effect results in a deficitof neutrinos arriving from the eastern direction. This is accounted for in the atmospheric neutrino flux calculation by computing the negative-time trajectories of eachprimary cosmic ray in the presence of the Earth’s magnetic field. If the trajectoryintersects the Earth in the past, i.e., the cosmic ray was not of astrophysical origin,
the ray is rejected.
Cosmic rays entering Earth’s atmosphere also must pass through the solar wind,the magnetized plasma emitted by the Sun. The solar wind prevents low-energy,∼ 1 GeV, cosmic rays from entering into the solar system, while higher-energy cosmic
rays enter diffusely as they lose energy [90]. During periods of high solar activity, or
solar maximum, the turbulence of the solar wind suppresses the lower-energy cosmic
rays. On the other hand, during periods of solar minimum, more of these lower-energy
cosmic rays can pass into the solar system. The solar wind has modulations due tosolar activity every 11 years, and therefore produces differences in the atmospheric
neutrino flux during SK data taking. The solar modulation effect can change the fluxof the lowest-energy neutrinos used in the present analysis by up to 10 %. The sizeof this effect decreases rapidly with increasing neutrino energy.
The atmospheric neutrino fluxes used for simulating events in this analysis arecalculated for three levels of solar activity: minimum, maximum, and “middle.” Thesimulated neutrino events are generated assuming the flux of primary cosmic rays at
solar middle, then re-weighted by the fraction of live time spent in solar minimum94

Neutron Monitor Counts

II

III

IV

V

700

100

600

90

500

80

400

70

300

60

200

Sunspot Number

I

110

50
100
40
0
1997

2000

2003

2006

2009

2012

2015

2018

Figure 4·2: Neutron monitor counts during the SK I-V data-taking
phases. The neutron monitor data is from [92]. Each SK phase isindicated by a filled grey region. The solar minimum and maximum
fractions for each SK phase are calculated as the neutron count ratio
during periods of solar minimum and maximum activity. The calculated
fractions are tabulated in Table 4.1.
and solar maximum conditions. These fractions are calculated using counts from aneutron monitor [91], which counts the number of cosmic neutrons as a proxy for thenumber of primary cosmic rays. The fractions are formed by linearly extrapolating the
average neutron monitor counts over the each SK phase between the counts during thenearest solar minimum and solar maximum, as defined by the nearest local minimum
and maximum sunspot number. The neutron monitor data from [92] and sunspot
numbers are visualized in Figure 4·2. Table 4.1 lists the computed maximum andminimum fractions used for the flux re-weighting.

4.2

Neutrino Interaction Models

Neutrino interactions can be broadly categorized based on the final state particles
produced. The relevant processes for the analysis presented in this thesis are quasielastic, single pion production, and deep inelastic scattering (DIS). neut implements
several models which predict the interaction cross sections and the kinematics of the95

PhaseSK I
SK II
SK IIISK IV
SK VSolar Activity (%)
Min.

Max.

70
30
100
44
94

30
70
0
56
6

Table 4.1: Solar minimum and maximum fractions used for the simulation of atmospheric neutrinos in each phase of SK data taking in thisanalysis.
outgoing particles in each process.
At atmospheric and beam neutrino energies, neutrinos primarily interact withthe protons and neutrons (nucleons) bound within nuclei. For interactions in water,
relevant for SK physics, neutrino interactions are separately modeled for interactions
with the hydrogen nuclei, nearly free of nuclear effects, and the bound nucleons within
the oxygen nucleus.
4.2.1

Quasi-elastic

Quasi-elastic processes are the primary interaction process for neutrinos with energies
between ∼ 100 MeV − 1 GeV. Quasi-elastic processes in nuclei involve a neutrino
interacting with a nucleon to produce an outgoing particle and ejected nucleons, andcan be charged current (CC) or neutral current (NC), e.g.,(CC): ¿l + n → p + l−

(4.3)(CC): ¿¯l + p → n + l+

(4.4)(NC): ¿l + p, n → ¿l + p, n(4.5)(NC): ¿¯l + p, n → ¿¯l + p, n

(4.6)

96

These interactions are quasi -elastic because there is only one outgoing lepton which
carries away a majority of the momentum compared to the ejected nucleon, andthe nucleus has a negligible recoil. A Feynman diagram of the most relevant quasielastic process for this analysis, charged current quasi-elastic (CCQE), is shown in
Figure 4·3. The figure depicts a neutrino interacting with one of the valence downquarks in a neutron, converting it to an up-quark, and producing an outgoing lepton.
Quasi-elastic processes are further categorized into one-particle one-hole (1p1h) andtwo-particle two-hole (2p2h) processes, referring to the number of nucleons ejectedfrom the nucleus. Illustrations of the 1p1h and 2p2h processes are shown in Figure 4·4.

νl

l−
W+

n

d
u
d

u
u
d

p

Figure 4·3: A Feynman diagram for the CCQE process. The neutrino
with lepton flavor l interacts with a quark inside a nucleon, changing
its flavor. An outgoing lepton is produced, and the nucleon remains
in-tact after the interaction.

1p1h
The 1p1h interaction refers to both the CC and NC variants of the quasi-elastic
process involving a neutrino interaction freeing a single nucleon. Because the ejectednucleon is often invisible, e.g., below Cherenkov threshold, or electrically-neutral if it
is a neutron, the double-differential cross section in outgoing lepton energy and angle

97

μ-

νμ

μ-

νμ

p

n

n

p

p

Figure 4·4: Drawings of 1p1h and 2p2h processes. Left: An example
1p1h process in a nucleus, in which a muon neutrino converts a neutron
into a proton and ejects it from the nucleus. Right: An example 2p2h
process, in which a muon neutrino interacts with a neutron in a strongly
correlated nucleon pair, ejecting both nucleons from the nucleus.
is used:|⃗kLep | G2d2 Ã
L W µσ=2 µσ⃗
dELep d cos ¹Lep
4Ã| kν |(4.7)

where ¹Lep is the angle of the outgoing lepton in the lab frame, ELep is the lepton
energy, ⃗k are the four momenta of the particles in the lab frame. The final two terms,
Lµσ and W µσ , are the leptonic and hadronic tensors which encode the relativistic
kinematics of the process. The leptonic tensor is calculated asαLµσ = kLep,µ kν,σ + kLep,σ kν,µ − gµσ (kν · kLep ) ∓ iϵµσαβ kLep
kνβ

(4.8)

where ϵijkl is the Levi-Civita tensor, and gµσ is the Minkowski metric. The sign for the
final term is negative for neutrinos and positive for anti-neutrinos. The exchange ofthe W or Z boson, and the momentum transfer to the recoiling hadronic system is thenencapsulated in the hadronic tensor. The hadronic tensor is not easily computed; itis typically an effective function of nucleon masses, nuclear effects, quark distribution
98

functions, and various corrections.
The authors of [93] calculate the hadronic tensor of Equation 4.7 for the quasielastic process. In the calculation, the key contribution comes from the combination ofvector and axial-vector current terms, V α −Aα , characteristic of the weak interaction,αV = 2 cos ¹C

F V (q 2 )F1V (q 2 )µ α + iÃ αβ qβ 2



2M


2M
α
2
α
α
A = cos ¹C GA (q ) µ µ5 + q µ5 2mπ − q 2(4.9)(4.10)

where ¹C is the Cabibbo angle from quark mixing, µ are the Dirac gamma matrices,
M is the nucleon mass, mπ is the pion mass, and q is the four-momentum transfer,⃗kν − ⃗kLep . Equation 4.9 contains three form factors, F V , F V and GA . Of these, F V
1
2
1
and F2V are measured via electron-nucleon scattering experiments, and are consideredwell-constrained. However, the GA form factor is only present for neutrino-nucleus
scattering, making external measurements difficult. GA has an assumed dipole form:
2

GA (q ) = gA

q2
1 − QE 2
MA

!−2

,

(4.11)

where the proportionality, gA ≡ GA (q 2 = 0), is measured precisely to be ∼ 1.27 frombeta decay [94]. Equation 4.11 contains a parameter, the axial mass, MAQE , which isconsidered experimentally uncertain. Measured values of MAQE differ by as much as∼ 25 % [95]. In addition to modifying the q 2 dependence of the quasi-elastic crosssection, larger values of MAQE increase the total cross section while smaller values
decrease the total cross section.
For interactions with nucleons bound within a nucleus, inter-nuclear effects canmodify the momentum of a nucleon within a nucleus, as well as the energy required tofree it, known as the nucleon removal energy. An example of one such nuclear effectis Fermi Motion, referring to the non-zero momentum of each nucleon within the99

nucleus. This, and other effects, change the total phase-space available for neutrino
interactions to produce outgoing particles with particular kinematics. As a result,the description of the nucleons bound within a nucleus affects the interaction crosssection.
There is currently no one universally accepted model for bound nucleon momentum distributions and nucleon removal energies. neut implements several models,
but for this thesis, two are relevant: relativistic Fermi gas (RFG) and local Fermi
gas (LFG). The RFG treatment in neut follows the LLewellyn-Smith formalism [96],and assumes a single nucleon momentum distribution and nucleon removal energy for
all nucleons in the nucleus. Nucleons in the RFG model occupy momentum states
up to the Fermi Momentum of the nucleus, pF , adhering to the Pauli exclusion principle. The LFG model implements the momentum distributions of the RFG model,
but postulates radially-dependent pF and nucleon removal energies. The LFG modeltends to predict broader distributions of outgoing lepton energies and angles thanthe RFG model. The differences between the RFG and LFG models will be further
discussed in Section 6.3.3.
neut implements the random phase approximation (RPA) correction, an additional effect for 1p1h process due to weak-charge screening effects from other nucleons
within the nucleus [97]. In neut, RPA corrections are pre-tabulated as a functionof the four-momentum transfer, q 2 , and can be applied to both RFG and LFG crosssections. In general, the RPA correction decreases the cross section at smaller valuesof q 2 . RPA corrections are not applicable to neutrino interactions with free nucleons.
For the analysis presented in this thesis, simulated bound 1p1h interactions use
the default neut 5.4.0 configuration, including the “BBBA05” [98] vector form factor
parametrization and the LFG model with RPA corrections based on the Valencia
model [99]. The axial mass is also set to the neut 5.4.0 default value, MAQE =100

1.05 GeV/c2 .
2p2h
2p2h, and, more broadly, multi-nucleon interactions, involve a neutrino interactingwith a strongly-correlated nucleon pair, such that more than one nucleon is freed fromthe nucleus. The 2p2h contribution to quasi-elastic processes has been estimated to
be as high as 20 % [99], although no direct measurements are available for a watertarget. Models of 2p2h-like interactions include nucleon-nucleon correlations andmeson exchange currents (MECs). Nucleon-nucleon correlations refer to short-range
strong and electromagnetic forces between the quarks in the nucleons, while MEC
refers the longer-range forces due to multi-quark (meson) exchange. The kinematicsof the outgoing particle can differ between the different 2p2h mechanisms: MEClike 2p2h often results in a ∆ baryon resonance excitation of one of the nucleons,
without decay to a pion. Thus, the cross section for 2p2h is thought to be higher for
momentum transfers q near the ∆ baryon mass. The implications of this effect on
the analysis presented in this thesis is mentioned in Section 6.3.5.
In SK, 1p1h and 2p2h have identical experimental final states, since the ejectednucleons are not observed, leaving the outgoing lepton (if CC) as the only visible particle. Since 1p1h and 2p2h processes have different distributions of lepton kinematics,
a realistic neutrino interaction simulation needs to separately account for the 2p2h
contribution.
The analysis presented in this thesis uses the neut 5.4.0 default implementationof 2p2h cross sections based on the Valencia model [99]. This model is an extensionof the 1p1h model from the previous section, but includes multi-nucleon potentials in
the hadronic tensor calculation. The model also applies RPA corrections to the 2p2h
cross section.

101

4.2.2Single Pion Production

Processes producing a lepton, recoiling nucleus, and single pion in the final state
constitute a significant portion of the total neutrino cross section at intermediate
energies, ∼1 GeV to 3 GeV. This energy range is especially relevant for atmosphericand beam neutrinos. There are two main mechanisms for single pion production,
resonant and coherent.
Resonant Pion Production
Resonant pion production refers to single pion production from the decays of excited
baryon resonances. Resonant pion production is estimated to account for ≳ 90 % of
the single pion production process. The resonant pion production processes include
CC channels,¿l + p → l − + p + Ã + ,¿¯l + p → l+ + p + Ã −(4.12)¿l + n → l − + n + Ã + ,¿¯l + n → l+ + n + Ã −(4.13)¿l + n → l − + p + Ã 0 ,¿¯l + p → l+ + n + Ã 0(4.14)¿l + p → ¿l + p + Ã 0 ,¿¯l + p → ¿¯l + p + Ã 0(4.15)¿l + p → ¿l + n + Ã + ,¿¯l + p → ¿¯l + n + Ã +(4.16)¿l + n → ¿l + p + Ã − ,¿¯l + n → ¿¯l + p + Ã −(4.17)¿l + n → ¿l + n + Ã 0 ,¿¯l + n → ¿¯l + n + Ã 0(4.18)and NC channels,

A Feynman diagram for an example CC resonant pion production process is shownin Figure 4·5.
To calculate the resonant pion production cross section, one needs to add the

102

νl

l−
W+

p

d
u
u

u
u
u
∆++

u
d̄

π+

d
u
u

p

Figure 4·5: A Feynman diagram for a resonant pion production process. A neutrino with lepton flavor l interacts with a nucleus, changing
the flavor of one of the quarks. A ∆++ baryon is temporarily createdin an excited state, which then radiates a gluon, producing a protonand pion.
contributions from each baryon resonance that produces the desired final state. The
largest contribution to these resonances at few-GeV energies is the ∆(1232), which is
the lowest-mass resonance above the pion production threshold. However, there are
18 such resonances below 2 GeV. neut simulates resonant single pion processes in
two steps. In the first step, neut calculates the cross section for producing a baryon
resonance, accounting for interference between the multiple possible resonance states
up to q < 2 GeV. In the second step, neut computes decays of the excited baryon
into pions, considering branching fractions and the kinematic dependence on angularmomentum of the parent baryon state.
neut uses the Rein-Sehgal model [100] for resonant pion production. As in thecase of quasi-elastic cross sections, the resonant pion production cross sections contains an axial form factor,
CA5 (q 2 ) = CA5 (0)q2
1 − Res 2MA−2(4.19)In Equation 4.19, both the resonant pion production axial mass, MARes , and the coeffi-103cient, CA5 (0) (simply referred to as CA5 ), are considered unconstrained by experiments.
Resonant single pion production also produces identical final states as some nonresonant channels, e.g., excited protons and neutrons. The authors of [100] note thata contribution from these isospin-1/2 states, added incoherently, is both theoreticallymotivated and better describes data. The size of this contribution is a free parameterin neut, referred to as the isospin-1/2 background, I 1 .
2

Resonant single pion processes are simulated for the present analysis using theRein-Sehgal [100] model, which is the default model in neut 5.4.0. The value of theaxial mass for resonant single pion production, MARes , is set to 0.95 GeV/c2 , the CA5form factor coefficient is set to 1.01, and the isospin-1/2 background scaling parameter, I 1 , is set to 1.3.
2

Coherent Pion Production
Coherent pion production refers to neutrino interactions with the whole nucleus instead of a nucleon. The cross section for this process is much smaller than for resonant
production, but cannot be distinguished from resonant pion production at SK. Because coherent pion production only occurs at low q 2 , the nuclear recoil is small, suchthat the outgoing pion is forward-scattered.
Coherent single pion processes are simulated for this analysis using two different
models in neut. Below 10 GeV, the Berger-Sehgal model [101] is used, while above
10 GeV, the Rein-Sehgal [100] model is used. The main difference between the models
is the inclusion of non-zero lepton mass in the Berger-Sehgal calculation. The use of
the two models was newly introduced as the default configuration for neut 5.4.0 toreflect recent scattering experiment data [102].

1044.2.3

Deep Inelastic Scattering

Deep inelastic scattering (DIS) becomes the dominant interaction process for neutrinos with energies ≳ 5 GeV. DIS interactions are characterized by a neutrino interacting with a quark directly, breaking apart the nucleon, and creating multiple hadrons
in the final state. In neut, an interaction which produces two or more hadrons in the
final state is considered to be DIS. DIS is especially relevant for atmospheric neutrinos,
as a large fraction of the atmospheric neutrino flux extends into the DIS-dominatedenergy region.
Due to the potentially high multiplicity of outgoing particles, the DIS cross section
is most conveniently calculated in terms of the Bjorken scattering variables x and y.
Here, x represents the fraction of the initial nucleon momentum carried by the struck
quark, and y represents the fraction of momentum transferred from the neutrino to
the hadronic system. Higher values of x correspond to more elastic-like collisions, andhigher values of y imply more energy available for the hadronic system. Following[103], the double-differential CC DIS cross section in terms of x and y is



i
h 
y2
y
G2F M Ed2 Ã1−y++ C2 xF3 (x)=+ C1 F2 (x) ∓ y 1 −
dxdy
Ã
2
2
M
m2 y
m2
m2
−
xy
+
−
C1 =
4E 2 2E
4M Ex 2M Ex
m2
C2 = −
4M Ex(4.20)

where M is the nucleon mass, m is the outgoing lepton mass, E is the neutrinoenergy, and F2 and F3 are parton distribution functions. The sign of Equation 4.20 isnegative for neutrinos and positive for anti-neutrinos. The sign difference implies thatneutrino and anti-neutrino interactions will produce differing amounts of hadrons forthe same incident neutrino energy. This will be discussed later in Section 5.1.
neut simulates DIS events based on the calculated invariant mass of the hadronic105system, W . For W < 2 GeV/c2 , neut uses a custom multi-pion production model
tuned to hadron multiplicities measured from bubble chamber data [104, 105]. For
W > 2 GeV/c2 , PYTHIA v5.72 [106] is used. PYTHIA implements its own hadronmultiplicity models, which sometimes result in single pion production. To avoid
overlap with the single pion production cross sections separately computed by neut,these are discarded.
The analysis presented in this thesis uses the default DIS model of neut. This
includes GRV98 parton distribution functions (PDFs) [107] with corrections in the
low-q 2 region from Bodek and Yang [108].
4.2.4

Final State Interactions

Final state interactions (FSIs) refer to nuclear effects that modify the outgoinghadrons produced in neutrino interactions. In neut, four such processes are considered,• Scattering: A hadron produced in an interaction scatters, changing its momentum and direction.
• Absorption: A hadron produced in an interaction is absorbed before being
detected.
• Production: An additional hadron is produced through inelastic interactions
with other nucleons.
• Charge exchange: A hadron interaction results in the conversion of a hadron to
one with a different charge, e.g. Ã + + n → Ã 0 + p.
A related class of effects are secondary interactions (SIs), which refer to the same FSI
processes occurring in the detector medium instead of within a nucleus.

106Figure 4·6: Comparison of neut predicted Ã + − C cross sections forthe absorption (ABS) and charge exchange (CX) FSI processes to thosemeasured by the DUET pion scattering experiment. Figure is adaptedfrom [110].
FSI and SI processes are implemented in a statistical way in neut. For FSI,
each pion is stepped through the nuclear medium according to estimates of the mean
free path. At each step, the pion may randomly undergo one of the FSI processes.
neut has six parameters which scale the probabilities of each FSI process: The
charge exchange and scattering without additional hadron production processes haveseparate scale parameters for pions with momentum above and below 500 MeV/c.
These parameters are tuned to pion scattering data, see Figure 4·6, [109]. The valuesof the parameters used for simulated interactions in this thesis are listed in Table 4.2.
The total cross sections as a function of incident neutrino energy, calculated withneut 5.4.0, for the processes described in this section are shown in Figure 4·7 forelectron neutrinos. The cross sections for muon and tau-flavored neutrinos behave
similarly, except for the different threshold energies for CC interactions.

107

FSI Process

neut Name

Value

Elastic scattering, pπ < 500 MeV/cElastic scattering, pπ > 500 MeV/c
Charge exchange, pπ < 500 MeV/c
Charge exchange, pπ > 500 MeV/c
Absorption
Production

FEFQE
FEFQEHFEFCX
FEFCXHFEFABS
FEFINEL

1.069
1.824
0.697
1.800
1.404
1.002

Table 4.2: Probability scaling factors applied to the neut FSI processes used in the simulation of atmospheric neutrino interactions. The
values are the default values of neut 5.4.0.

4.3

Detector Simulation

Neutrino interactions simulated by neut ultimately produce lists of final state particles and their momenta. To be compared with SK data, these particles still need tobe simulated within the SK detector. This is accomplished with the skdetsim program, based on GEANT3 software [111]. skdetsim is responsible for simulating particle
traversal through the SK water and the detection of photons by SK hardware.
Particles produced in neutrino interactions must propagate in the detector water,where they can be affected by SIs. skdetsim uses a combination of hadronic interaction models to simulate SIs: For hadrons with momenta below 500 MeV/c, skdetsim
uses the interaction simulation from the neut FSI model (see Section 4.2.4). Above
500 MeV/c, skdetsim uses the GEANT3 library GCALOR [112].
skdetsim simulates Cherenkov radiation emission for charged particles using a
combination of GEANT models with additional custom features. Among these are
the Rayleigh and Mie scattering components of the water transparency model (seeSection 2.3.3). skdetsim also implements an additional “top-bottom asymmetry”
parameter which corrects for differences in the water quality at the bottom versus
top of the SK tank.
108

Total
CC 1p1hσ/E ν [× 10-38

CC DIS
CC 1π
NC
CC 2p2h010−1

CC Other

1
Eν [GeV]

0.5

-38

1

cm2/GeV/Nucleon]νeσ/E ν [× 10cm2/GeV/Nucleon]

νe

10

0

10−1

1
Eν [GeV]10

Figure 4·7: Total electron neutrino cross sections divided by neutrinoenergy, versus neutrino energy, for a water target. The cross sections arecalculated with neut 5.4.0. The different lines show the contribution to
the total cross section for each interaction mode. “CC Other” refers tosingle, non-pion hadron production. Left: Cross sections for neutrinos.
Right: Cross sections for anti-neutrinos, including the anti-neutrino
CC contribution with free protons in water. The line colors match thelabels in the left plot.
Cherenkov photons are propagated by skdetsim within an approximate geometry of the SK tank. skdetsim also contains tuning parameters which adjust theeffective reflectivity of Tyvek (see Section 2.2). For photons reaching SK PMTs,
skdetsim implements a data-driven model of PMT responses and dark noise. The
PMT response model in skdetsim considers the incident angle and wavelength ofincident photons, and the relative quantum efficiency of each PMT. Each photon
which reaches the PMT photocathode may additionally reflect, absorb, or transmit
through the photocathode without producing a photoelectron. An overall quantum
efficiency correction, COREPMT, is applied to all PMTs. COREPMT is determined fromstudies of through-going cosmic ray muons for analyses using neutrinos with energies> 100 MeV. Additional details on the detector simulation, and details relevant toanalyses with neutrino energies < 100 MeV, may be found in [113].
For the analysis presented in this thesis, 500 years of atmospheric neutrino events109

CC QE

800

CC 1π

600

CC 2p2h

400

CC DISCC Other

200

NC

0
10−1 13

10 102 10Eν (GeV)5

104 10

90
80
70
60
50
40
30
20
10
0
10−1

80

PC
Events/1000 DaysFC
Events/1000 Days

Events/1000 Days

1000

Up-µ

70
60
50
40
30
20
10

1

3

10 102 10Eν (GeV)5

104 10

0
10−1

1

3

5

10 102 10 104 10
Eν (GeV)Figure 4·8: True neutrino energy distributions of simulated atmospheric neutrino MC events by interaction mode. “CC Other” refers tosingle, non-pion hadron production. Distributions are shown for Left:
FC events, Center: PC events, and Right: Up-µ events.
are simulated according to the detector conditions in each phase of SK data taking.
By using Monte Carlo (MC) events from each phase, the MC accounts for the average
detector conditions, e.g., the number of failed PMTs and average PMT gains. The
true neutrino energy distributions for the SK IV atmospheric neutrino MC events areshown in Figure 4·8. The distributions are separated by true interaction process toshow the energy dependence of the interaction cross sections. The panels in the figure
correspond to the distribution for events selected as FC, PC and Up-µ.

110

Chapter 5

Atmospheric Neutrino Oscillation
Analysis
This chapter describes the analysis procedure to determine neutrino oscillation parameters from SK atmospheric neutrino data. Neutrino events identified in the SKdata by the procedure in Section 3.1, and simulated MC neutrino events, are separated by reconstruction information into sub-samples which are sensitive to different
oscillation effects. Neutrino oscillations are then applied to the MC events to produce
a prediction for the data. In addition to oscillation parameters, systematic uncertainties on the flux and cross section models, and on reconstruction efficiency, vary the
MC prediction. Oscillation parameters are measured by quantifying the degree ofdata-MC agreement for a set of oscillation parameters, after accounting for variations
in systematic uncertainties.
SK has performed several atmospheric neutrino oscillation analyses in the past
[38, 114, 115]; the analysis presented here is a continuation of these analyses utilizing
new data and new techniques. The major updates for this analysis are the inclusion ofdata from an expanded fiducial volume, neutron tagging information used in the event
selection, a Multi-Ring classification scheme using a boosted decision tree (BDT), andthe inclusion of SK V data.

111

5.1

Neutrino Sample Selection

As discussed in Chapter 1, the most relevant parameters for atmospheric neutrinos
oscillations are θ23 , |∆m232,31 |, θ13 , δCP , and the neutrino mass ordering. Of these,
atmospheric neutrinos are uniquely sensitive to the neutrino mass ordering throughmatter effects, resulting in an excess of either upward-going νe or ν̄e events with several
GeV of energy. Consequently, identifying these events within all the atmospheric
neutrino candidates is a major first step for the analysis. At the same time, other types
of atmospheric neutrinos which are not as sensitive to the neutrino mass ordering
constrain other oscillation parameters and systematic uncertainties. As described in
Chapter 3 atmospheric neutrino candidates are classified as either FC, PC, or Upµ. For this oscillation analysis, these neutrino candidates are further separated into
different sub-samples based on their likely flavors and energies.
Atmospheric neutrino events with large uncertainties on reconstructed quantities
do not provide useful oscillation information, and tend to introduce backgrounds. For
this reason, FC, PC, and Up-µ neutrino candidates are subject to additional quality
cuts. These cuts, and later selections, use the reconstructed visible energy, EVis ,defined as the sum of the reconstructed momenta from all reconstructed rings in an
event assuming the ring was produced by an electron. The cuts are: (FC): EVis must be greater than 30 MeV, the number of OD hits must be lessthan 16 (10 for SK I), and the reconstructed vertex must be greater than 100 cm
from the detector walls.
 (PC): The ID charge must be greater than 3000 photoelectrons (1500 photoelectrons for SK II), the number of OD hits must be less than 16 (10 for SK I),
and the reconstructed vertex must be greater than 200 cm from the detector
walls.

112

 (Up-µ Stopping): The reconstructed muon momentum pµ must be greater than
1.6 GeV/c.
 (Up-µ Through-going): The fitted track length must be greater than 700 cm.
These cuts suppress any remaining low-energy events, entering cosmic muon backgrounds, and events with poor reconstructed fit information.
5.1.1

FC Samples

Numberof Rings
EVis

Single Ring
(SK I-III)

Sub-GeV
Multi Ring
Fully ContainedSingle Ring
(SK I-III)Multi-GeV
Multi Ring

PID

e-like0 decay e
1+ decay e
Pass 0 Likelihood-like

0 decay e
1 decay e
2+ decay ee-likeBoth e-like, in 0 mass range-like

Leading ring has p > 600 MeV/c

Multi-Ring -likee-like0 decay e
1+ decay e

Multi-GeV e-like
Multi-GeV e-like
Multi-GeV -like

Highest BDT score is e
Pass BDT Highest BDT score is eHighest BDT score is
Fail BDT

Multi-Ring e-like
Multi-Ring e-like
Multi-Ring -like
Multi-Ring Other-like

Sub-GeV e-like, 0 d.e.
Sub-GeV e-like, 1 d.e.
Sub-GeV Single-Ring 0-like
Sub-GeV -like, 0 d.e.
Sub-GeV -like, 1 d.e.
Sub-GeV -like, 2 d.e.
Sub-GeV Two-Ring 0-likeFigure 5·1: Overview of FC event selection. For this analysis, subGeV and multi-GeV single ring events from the SK IV-V phases are
separated out into additional samples. Note that the Multi-Ring µlike sample contains both sub-GeV and multi-GeV events. Figure is
inspired by Figure 7.1 from [116].
FC events are divided into 14 sub-samples for SK I-III, and 16 sub-samples for
SK IV-V. The sub-samples are based on EVis , the number of rings, PID information,and the numbers of decay electrons and neutrons. Figure 5·1 shows an overview of
the FC sub-sample selection. The first step categorizes FC events as either sub-GeV
113

or multi-GeV using EVis , either below or above 1330 MeV1 . Then, sub-GeV andmulti-GeV events are separated further based on the number of rings, either single
or multiple, and into likely e- and µ-like samples.
This analysis uses the numbers of decay electrons and neutrons to enhance thepurity of neutrinos and anti-neutrinos in the FC sub-samples. Because decay electrons
are more likely to be observed with positively-charged muons, the following neutrino
and anti-neutrino interactions can be statistically separated based on the number ofdecay electrons,νµ + n → µ− + p,ν̄µ + p → µ+ + n(→ e)(5.1)

νl + p → l− + p + π + (→ e),ν̄l + p → l+ + p + π −(5.2)

νl + n → l− + n + π + (→ e),ν̄l + n → l+ + n + π −(5.3)

and similarly, the number of neutrons produced can help separate neutrinos fromanti-neutrinos: At low energies, the CCQE interaction produces free neutrons foranti-neutrino events but not for neutrino events,ν̄l + p → l+ + n

νl + n → l− + p,

(5.4)

and at high energies, anti-neutrino interactions are also more likely to produce multiple neutrons than neutrinos due to the structure of the DIS cross section, discussed
in Section 4.2.3.
Since neutron tagging was not possible until the SK IV phase, the event selection
for neutrinos observed during the SK I-III phases only uses the number of decay
electrons. Section 5.1.1 and Section 5.1.1 will discuss the event selection without
neutron tagging, then Section 5.1.1 will discuss the incorporation of tagged neutron
1

The division at 1330 MeV has historical origins in SK analyses searching for proton decay, where
the expected signal is EVis ∼ mp ≈ 931 MeV, and atmospheric neutrinos are the main background.

114

information for the event selection used for data collected during the SK IV-V phases.
Sub-GeV Samples
Sub-GeV events are more numerous than multi-GeV events due to the exponentiallyfalling flux of atmospheric neutrinos with neutrino energy. However, sub-GeV events
also have worse direction and energy resolution compared to multi-GeV events, resulting in smeared oscillation effects. To remove poorly-reconstructed events, sub-GeV
events must also pass a minimum momentum cut based on the PID classification of
the most energetic ring: (e-like): pe > 100 MeV
 (µ-like): pµ > 200 MeV
Sub-GeV events which pass the minimum momentum requirements are classified by
the number of rings and PID information. For sub-GeV events with a single e-like ring,
three sub-samples are defined. Two samples are formed based on the number of decay
electrons, one or more. A third sample was introduced to separate NC interactions
where a π 0 is produced, but only one of the rings from its decay is reconstructed. SubGeV single-ring e-like events are passed through a dedicated fitter which forces twoe-like rings to be reconstructed. If the dedicated fitter result better agrees with theobserved light pattern than the original single-ring hypothesis, the event is categorized
as single-ring π 0 -like.
Sub-GeV events with a single µ-like ring are separated based on the number ofdecay electrons. There are separate categories for zero, one, and two or more decay
electrons. CCQE interactions are expected to produce up to a single decay electron,
so the zero and one decay electron categories are purer in true CCQE interactions.
Events with two decay electrons are separated from the CCQE-enhanced samples

115

since these likely result from additional pions produced below Cherenkov threshold,leading to a larger reconstructed momentum bias than for CCQE interactions.
Sub-GeV events with multiple rings are only included in two instances: First,
if there are exactly two e-like rings, and the reconstructed momentum is consistent
with the π 0 mass, the event is classified as two-ring π 0 -like. Next, if there are multiple
rings, and the most energetic ring is µ-like, the event is only kept if the reconstructed
momentum is > 600 MeV/c, and it is classified as multi-ring µ-like. Other sub-GeV
multi-ring events are not used in this analysis, as the neutrino flavor determination
is typically less reliable, adding minimal sensitivity to oscillation effects.
Multi-GeV Samples
Multi-GeV events with a single e-like ring are separated into sub-samples with zeroand one or more decay electrons corresponding to ν̄e -like and νe -like, respectively. No
decay electron separation is performed for multi-GeV single-ring µ-like events.
In previous analyses, multi-GeV events with multiple rings were classified according to a likelihood-based selection. The likelihood of an event being e-like or µ-like
was computed as the likelihood ratio for four reconstruction input variables,
L=

4
X
i

ΓSi (xi )log B
Γi (xi )




(5.5)

where i indexes each variable, xi is the observed value of the variable, and ΓS,Bi
is the MC distribution of each variable using true signal and background events,
respectively. Here, the signal is defined as true νe and ν̄e CC events, while otherevents are considered to be background. The four variables used in the likelihood
calculation were: the number of decay electrons, the distance of the furthest decay
electron from the primary event vertex, LDecay e 2 , the PID likelihood of the most2
The calculation of LDecay e for SK I-III events excludes decay electrons with unreliable vertices.
See Section 3.2.7.

116

energetic ring, and the fraction of momentum carried by the most energetic ring,
Fmom . This approach was able to improve the separation of e-like from µ-like multiring events versus simply using the PID likelihood of the most energetic ring from∼ 50 % to ∼ 70 %.
The likelihood methodology was extended to a two-stage process in [38]. A second
likelihood, designed to separate νe from ν̄e , was applied to the e-like candidates.
This likelihood used three variables: the number of decay electrons, the transverse
momentum of the most energetic ring, Tmom , and the number of rings.
This analysis replaces the multi-ring likelihood-based classification with a BDT
classifier based on the same input variables used in both likelihood steps. The development of the BDT is documented in [117]. Compared to the likelihood selection,
the BDT has the advantage of weighting each input variable’s importance for theclassification decision based on learned outcomes from training data. MC events withtrue, known interactions channels are used to train the multi-ring BDT. Each MC
event is classified into one of four samples: νe -like CC, ν̄e -like CC, µ-like (for both νµand ν̄µ CC), or “other.” The “other” category removes NC and ντ CC events fromthe CC samples which increases the purity of the CC samples.
The BDT performance is sensitive to the number of true events of each sample. For
example, because there are fewer true ν̄e events than νe in the atmospheric neutrinoMC, the BDT can correctly classify more events overall by classifying all ν̄e eventsas νe . To circumvent this problem, the events used for training are weighted suchthat the correctly classifying a ν̄e event counts more towards the BDT’s classification
performance.
The distributions of the input variables used for the Multi-Ring BDT are shownfor the SK IV-V phases in Figure 5·2, and the output classification decision, “BDTscore” is compared for νe -live and ν̄e like in Figure 5·3.

117

Events (arb. units)

0.6

0.12

0.5

0.1

0.4

0.08

0.3

0.06

0.2

0.04

0.1

0.02

Events (arb. units)01

2 3 4 5 6
Number of rings

0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0

5000
10000
E Vis. (MeV)

0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

0
−2000−1000 0 1000 2000
Most energetic ring PID
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0

0.2 0.4 0.6 0.8
T mom.

1

1
10− 1
10− 2
−3

10

10− 4
−5

10
0 1 2 3 4 5 6 7 8 9 10
Number of decay electrons

0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0

0

500 1000 1500 2000
L decay e. (cm)Super-Kamiokande IV-V MC
ν e CCν e CCν µ & ν µ CCν τ & ν τ CCNC
0.5
F mom.

1

Figure 5·2: Area-normalized SK IV-V MC distributions of the different input variables used for the Multi-Ring BDT selection. True CC
neutrino interactions are shown as solid lines while the NC contribution
is shown as a filled area. Tmom is the transverse momentum of the mostenergetic ring, and Fmom is the fraction of the energy carried by themost energetic ring.
Neutron Tagged Samples
The number of tagged neutrons is used as an additional handle when classifying FC,
single-ring events from the SK IV-V phases. Figure 5·4 shows an overview of themodified selection compared to the SK I-III selection. The modification is the samefor both sub-GeV and multi-GeV events based on the following principles:• νe CC interactions will produce decay electrons more often than ν̄e interactions
due to π + production.
• νµ CC interactions will produce exactly one decay electron less often than ν̄µ
CC interactions, due to µ− capture.
• Neutrino CC interactions will produce fewer neutrons than anti-neutrino CC
118

300
250

Events200

SK IV-VData
MC totalν e CCν e CCν e-Likeν e-Like

150
100
50
0
−1 −0.8 −0.6 −0.4 −0.2 0
0.2 0.4
ν e-ν e BDT Score

0.6

0.8

1

Figure 5·3: νe and ν̄e separation using the Multi-Ring BDT score for
SK IV-V data and MC events. The events shown have a maximum
score for either the νe or ν̄e -like category, and the final classification
decision is the maximum score of any category. The distributions of
νe and ν̄e scores are preferentially peaked to the right and left of 0
respectively, indicating that the BDT is able to statistically separate
these events. The contribution of νµ , ντ , and NC backgrounds to theνe and ν̄e -like samples is reflected in the MC total line.
interactions.
The corresponding event selection criteria correspond to five sub-samples each for
sub-GeV and multi-GeV:• For e-like events, there are three sub-samples:– If the event has at least one decay electron, it is classified as νe -like.
– If the event has no decay electrons and no tagged neutrons, it is classified
as a νe -ν̄e -like mixture.

119

PID
EVis

e-like0 decay

0 neutronse1+ decay

1+ neutronseSub-GeV

e-like Mix

Sub-GeVe -like

Sub-GeVe -like

Sub-GeV-like

Sub-GeV-like

Sub-GeV
1 decay-likee and 1+ neutronse or 0 neutrons

1 decay
Fully ContainedSingle Ring
(SK IV-V)

e-like0 decay

0 neutronse1+ decay

1+ neutronseMulti-GeV

e-like Mix

Multi-GeVe -like

Multi-GeVe -like

Multi-GeV
1 decay-likee and 1+ neutronse or 0 neutrons

1 decay

Multi-GeV-like

Multi-GeV-like

Figure 5·4: The modified FC single-ring event selection for the SK IVV phases, using the number of tagged neutrons. The motivations forthe selection criteria for each sub-sample are described in the text.
– If the event has no decay electrons at least one tagged neutron, it is classified as ν̄e -like.
• For µ-like events, there are two sub-samples:– If the event has exactly one decay electron, and at least one tagged neutron,
it is classified as ν̄µ -like.
– Otherwise, it is classified as a νµ -like.
5.1.2

PC Samples

PC events are further divided into two sub-samples based on the estimated stopping
point of the produced muon, either in the OD, “PC stopping,” or completely outside
the SK tank, “PC Through-going.” Since PC stopping events have an end point of
the OD, their energies may be estimated using the total muon track length, i.e., using
the portion of the track in the ID extrapolated to a stopping point in the OD. For
PC Through-going events, the momentum of the muon can only be estimated using120

the portion of the track inside the tank, so the total energy is unknown.
The separation between PC stopping and Through-going sub-samples is performed
using the ratio of observed charge to expected charge for a muon with the observed
track length. The observed charge consists of the charge of the primary ring reconstructed in the ID, and the maximum charge in a 500 ns sliding window from −500 ns
to 500 ns in the OD. The expected charge is the dE/dx for a relativistic muon with apath length equal to the ID path length extrapolated to the SK tank wall, accounting
for the un-instrumented region between the ID and OD.
Two additional corrections are applied to the expected charge: For PC events,
the high interaction energies often result in multiple outgoing particles with similar
directions, which can appear as a single ring. A correction is applied if the charge
estimated from the ID track length is greater than the expected dE/dx energy lossof a single particle. The expected charge calculation also applies a correction factor
for the different configurations of the OD Tyvek and PMTs during the different SK
phases. The factor is determined from the agreement between cosmic muon data andMC, and is applied to the expected charge based on the exit point region in the OD,
i.e., top, barrel, or bottom.
5.1.3

Up-µ Samples

Up-µ events are divided into three sub-samples based on each event’s expected energy. From lowest to highest average energy, Up-µ events which stop within the ID
are categorized as “Up-µ stopping,” while Up-µ through-going events are separated
into non-showering and showering. The non-showering and showering sub-samples
are distinguished based on whether or not the reconstructed muon track is consistent
with the expected charge from dE/dx. Showering muons typically have a catastrophic interaction with a nucleus as they travel, creating additional particle showers
at one or more sites along the track. The full algorithm which performs the non121

showering/showering separation is described in [74].
The event selections described above for FC, PC, and Up-µ events results in 19
sub-samples for SK I-III data and 21 sub-samples for SK IV-V data.
5.1.4

Zenith & Momentum BinningBecause neutrino oscillations depend on the neutrino oscillation baseline, L, andneutrino energy, E, events in each sub-sample are binned into a 2D binning scheme
based on their reconstructed zenith angle and observed momentum as correlates of L
and E, respectively. FC and PC events are divided into 10 zenith angle bins spanning−1 ≤ cos θz ≤ 1. There are four exceptions for FC samples, which are only assigned
a single zenith angle bin due to limited statistics and poor directional information:
single- and two-ring π 0 -like, sub-GeV e-like with one decay electron, and sub-GeV µlike with two decay electrons. Up-µ events, which, by definition, come from below thehorizon, are divided into 10 evenly-spaced zenith angle bins spanning −1 ≤ cos θz ≤ 0.
This analysis updates the zenith angle bins for FC and PC events compared topast SK oscillation analyses. In previous SK analyses, the 10 zenith angle bins were
evenly-spaced on the [−1, 1] interval, i.e., with a constant width of 0.2 [38]. However,these bin edges do not align with the mass ordering signal region, defined by thezenith angles where the matter-induced resonant oscillations are expected to occur.
This can lead to a potential ambiguity when assessing the data counts in the bins
nearest to the signal region. This analysis has updated the zenith angle bin definitions
to better capture the resonance region using variable-width zenith angle bins. The
comparison of the bins used in this analysis with the evenly-spaced bins is shownin Figure 5·5. The zenith angle bins for the Up-µ samples are not updated for thisanalysis; they remain evenly-spaced on the interval [−1, 0].
The second axis of the 2D bins separates events by momentum information. This
analysis does not attempt to reconstruct the neutrino energy, rather, the observed
122

Figure 5·5: Comparison between the zenith angle bins used in thisanalysis (solid black lines) with the evenly-spaced bins used in previous
SK oscillation analyses. The updated bins are symmetric about cos θz =0. The color scale shows the νµ → νe oscillation probability calculated
for the normal mass ordering. Using the evenly-spaced bin scheme, theresonance region is partially divided between two bins which include
significant fractions of non-resonance regions. In the updated scheme,
the resonance region near cos θz ≈ −0.8 is better separated from thenon-resonance region at smaller zenith angles.
momenta of reconstructed rings for FC events, or of muon tracks for PC and Up-µevents, is used as a correlated quantity. The atmospheric neutrino sub-samples utilize
up to five momentum bins, listed in Table 5.1. The definition of “momentum” changes
for each sub-sample based on the assumed best correlate of the neutrino energy.
The bin definitions result in 140 FC multi-ring bins, 10 FC π 0 -like bins, 60 PC
bins, and 50 Up-µ bins, which are used for all SK phases. The remaining FC singlering events are binned into 260 bins for SK I-III and 410 bins for SK IV-V. The grand
total is 930 bins used for the atmospheric neutrino events in this analysis.
5.1.5

Expanded Fiducial Volume

Previous SK atmospheric neutrino analyses only included FC events with a reconstructed vertex at least 200 cm away from any ID wall which defined a 22.5 kt fiducial
volume. This atmospheric neutrino oscillation analysis includes, for the first time, FC
events with reconstructed vertices between 100 cm to 200 cm from the detector walls
for all SK phases, adding a 4.7 kt additional volume. The conventional and additional
volumes are collectively called the expanded fiducial volume which contains 27.2 kt
123SampleNumber of Bins

Edges, log10 [p (MeV)]

5 e± momentum

2.0, 2.4, 2.6, 2.8, 3.0, 3.2

Sub-GeV two-ring π 0 -like

5 π 0 momentum

2.0, 2.2, 2.4, 2.6, 2.8, 3.2

Multi-GeV single-ring e-like

4 e± momentum

3.0, 3.4, 3.7, 4.0, 5.0

Multi-GeV single ring µ-like

2 µ± momentum

3.0, 3.4, 5.0

Multi-Ring νe -like and ν̄e -like

3 visible energy

3.0, 3.4, 3.7, 5.0

Multi-Ring µ-like

4 visible energy

2.0, 3.12, 3.4, 3.7, 5.0

Multi-Ring Other

4 visible energy

3.0, 3.4, 3.7, 4.0, 5.0

2 visible energy

2.0, 3.4, 5.0

4 visible energy

2.0, 3.12, 3.4, 3.7, 5.0

3 visible energy

3.2, 3.4, 3.7, 8.0

FC
Sub-GeV single-ring
single-ring π 0 -likeandPC
Stopping
Through-going
Up-µ
Stopping
Through-going non-showering& showering

Single bin

2.0, 8.0

Table 5.1: Momentum bin definitions for atmospheric neutrino subsamples in this analysis. Each row in the Edges column lists the lower
bin edges and the upper edge of the highest-momentum bin. The edges
labeled 3.12 are evaluated as exactly log10 (1330 MeV/c).

124

of water, a 20 % increase over previous analyses.
To expand the fiducial volume, several aspects of the reconstruction were updated,first reported in [118]. Notably, the expected charge distributions from simulated
neutrino interactions, used in the calculation of the ring counting and PID likelihoods(see Section 3.2), were separately re-computed using events within the additional
region. The updated charge tables account for the decreased number of hits forevents closer to the detector walls. Using these new tables in the reconstruction
process reduces the bias in reconstructed quantities versus using the same charge
tables as in the conventional fiducial volume.
Accepting events closer to the detector walls also means accepting more nonneutrino backgrounds. Close to the walls, the reduction steps which remove flasher
PMTs and cosmic muons are less efficient. Experts eye-scanned events in the additional fiducial volume region to newly estimate the non-neutrino backgrounds separately from the conventional region. The distribution of non-neutrino backgrounds
estimated for the SK I-IV phases is shown in Figure 5·6. While the non-neutrino
backgrounds are higher, they are still tolerable. As demonstrated in the figure, thenumber of non-neutrino background events increase from approximately 0.1 % to only
0.5 % in the additional region, which is still deemed acceptable. Extending the fiducial
volume even closer to the walls is prohibited by the sharp increase in backgrounds.
Figure 5·7 and Figure 5·8 show the atmospheric neutrino data and MC events,
including FC events from the expanded fiducial volume, categorized into the different
analysis sub-samples. Table 5.2 lists the total number of data and MC events foreach sub-sample. The table also shows the MC purity, i.e., the fraction of the true
neutrino flavor of MC events in each sub-sample. Note that the SK IV-V multi-GeVν̄e sample, which uses neutron tagging information, has a higher purity of true ν̄e CC
events than the equivalent SK I-III sample.

125

Number of External
Background Events [/10 cm]

50
45

SK-I to -IV
Cosmic-Ray Muon
PMT Flasher

40
35
30
25
20

Additional
RegionConventional
Region

15
10
5
0
0

50
100 150 200 250 300 350
Distance to the nearest ID Wall [cm]Figure 5·6: Non-neutrino backgrounds determined by eye-scanningduring the SK I-IV phases in the conventional and additional fiducial
volume regions. Figure is adapted from [118].

SampleMC Purity (%)νeν̄eνµ + ν̄µντ + ν̄τ

Fully contained (FC), single ring, Sub-GeV
SK I-IIIe-like0 decay-e
73.3 22.7
0.4
0.0
1 decay-e
79.6 1.6
10.6
0.1µ-like0 decay-e
2.7 0.8
85.3
0.1
1 decay-e
0.1 0.0
96.8
0.0
2 decay-e
0.1 0.0
97.7
0.1
SK IV-Vνe -like
79.4 1.6
11.5
0.1
νe or ν̄e -like
78.9 17.6
0.4
0.0
ν̄e -like58.2 36.7
0.5
0.1
νµ -like
1.1 0.3
92.9
0.0
ν̄µ -like
0.1 0.0
95.0
0.1

Events
NC

MC

Data

3.6
8.1

6399.6
612.0

6647
682

11.2
3.0
2.2

2153.64241.1
330.6

2419
4476
336

7.4
3.1
4.4
5.7
4.9

943.7
5951.42264.86595.62150.0

1093
6669
1668
7879
1793

Continued on next page126

Table 5.2 – Continued from previous pageSampleMC Purity (%)νeν̄eνµ + ν̄µντ + ν̄τ

Fully contained (FC), single ring, Multi-GeV
SK I-IIIνe -like
56.9 8.6
11.5
3.9
ν̄e -like55.6 34.1
1.6
1.2
νµ -like
0.2 0.1
99.2
0.3
SK IV-Vνe -like60.8 8.7
11.3
3.3
νe or ν̄e -like
63.8 28.7
1.0
0.7
ν̄e -like43.5 46.0
1.1
1.5
νµ -like
0.2 0.1
99.2
0.3
ν̄µ -like
0.1 0.0
99.4
0.4
SK I-V common samples
Fully contained (FC) Sub-GeV NC π 0 -likeSingle-ring
21.9 6.4
2.0
0.1
Two-ring
9.6 2.8
1.6
0.0
Fully contained (FC) Multi-GeV, multi-ringνe -like49.5 6.6
18.8
3.5
ν̄e -like
52.0 26.0
5.8
2.5µ-like2.8 0.3
91.4
0.6
Other
20.4 2.3
27.1
7.4
Partially-contained (PC)
Stopping
8.9 3.4
82.1
1.1
Through-going 0.6 0.2
97.9
0.7
Upward-going muons (Up-µ)Stopping
0.8 0.3
98.6
0.0
Non-showering
0.2 0.1
99.7
0.0
Showering
0.1 0.0
99.8
0.0

Events
NC

MC

Data

19.0
7.5
0.2

360.1
1361.6
1587.3

383
1339
1564

15.9
5.8
7.9
0.1
0.2

584.9
867.6
737.0
1462.9
592.7

643
986
616
1619
503

69.6
86.0

747.9
2095.2

868
2494

21.5
13.8
5.0
42.9

2152.31211.63255.2
838.0

2411
1131
3427
982

4.5
0.6

641.3
3308.0

689
3397

0.3
0.1
0.1

1571.85314.71051.41753.86423.91110.6

Table 5.2: Nominal MC and data events in each sub-sample for thisanalysis. Columns two to six list the fractions of true neutrino flavor
in the MC. The MC events are computed with neutrino oscillations
and without fitted systematic uncertainties. Oscillations are calculated assuming the normal mass ordering and ∆m232 = 2.4 × 10−3 eV2 ,sin2 θ23 = 0.5, sin2 θ13 = 0.0220, and δCP = 4.71. Up-µ data events areshown after background subtraction.

127

200

Sub-GeV e-like 1 d.e.*

1000

100

500

0
1.5
1
0.5

0
1.5
1
0.5
1-ring π0-like

400

100

200

50

0
1.5
1
0.5

0
1.5
1
0.5

150

Sub-GeV e-like 0 d.e.*Multi-GeV e-like νe*

800
600
400
200

300

150

400

50

500

0
1.5
1
0.5

0
1.5
1
0.5

100
0
1.5
1
0.5

0
1.5
1
0.5
2

2.5
3
log [p (MeV/c)]
10

600

200

400

100

200

0
1.5
1
0.5
−1

0
1.5
1
0.5
−1

−0.5

0
cosθz

0.5

1

1500

0
1.5
1
0.5

0
1.5
1
0.5

500

PC Stopping

100

200

300

0
1.5
1
0.5

1000

0
1.5
1
0.5

1000

Multi-GeV e-like νe*1000

Multi-Ring e-like νeUp-µ Stopping

200

0
1.5
1
0.5

200

Multi-GeV µ-like*

400

100

50

2-ring π0-like

Sub-GeV µ-like 1 d.e.*

200

300

Multi-Ring e-like νe

Sub-GeV µ-like 2 d.e.*

100

400 Sub-GeV µ-like 0 d.e.*
300
200
100
0
1.5
1
0.5

PC Through-going300

Up-µ Non-showering

Up-µ Showering

200

500

100
0
1.5
1
0.5
Multi-Ring µ-like

200

0
1.5
1
0.5
−1 −0.8 −0.6 −0.4 −0.2
cosθzMulti-Ring Other

Super-Kamiokande I-V
Data
νe CC
νe CC
νµ CC
νµ CC
ντ & ντ CC
NC

100

−0.5

0
cosθz

0.5

1

0
1.5
1
0.5
−1

−0.5

0
cosθz

0.5

1

Figure 5·7: Zenith angle or log10 [p (MeV)] distributions for atmospheric neutrino events used in this analysis, including events from an
expanded fiducial volume. The 19 samples without tagged neutron information are shown. Black data points show the observed number ofcounts in each bin, while the filled histograms show the contribution to
the expected number of counts from the MC prediction. The MC includes neutrino oscillations. The data-to-MC ratio is plotted beneatheach sub-sample. All error bars are statistical. Asterisks represent
samples containing only SK I-III FC single ring events, i.e., where the
SK IV-V events have been separated out.

128

0

200
Sub-GeV νe-like

150

1000

100

300

Sub-GeV e-like

500

50
0
1.5
1
0.5

0
1.5
1
0.5

150

Multi-GeV νe-like

500

100

0
1.5
1
0.5

0
1.5
1
0.5

0
1.5
1
0.5

Multi-GeV νe-like

300

50

100

0
1.5
1
0.5
−1

0
1.5
1
0.5
−1

0
1.5
1
0.5
−1

0
1.5
1
0.5
−1

1

−0.5

0
0.5
cosθz

1

Multi-GeV νµ-like

100

Multi-GeV νµ-like

50

100

0
0.5
cosθz

Sub-GeV νµ-like

200

200

50

−0.5

300

100

100

100

Sub-GeV νµ-like

1000

150

Multi-GeV e-like
200

1500
Sub-GeV νe-like

200

−0.5

0
cosθz

0.5

1

−0.5

0
0.5
cosθz

1

0
1.5
1
0.5
−1

−0.5

0
cosθz

0.5

1

Figure 5·8: Zenith angle distributions for the 10 SK IV-V FC singlering samples using the number of tagged neutrons. The meaning of
the data points, error bars, and filled histograms is identical to what isshown in Figure 5·7.

5.2

Systematic Uncertainties

This analysis considers multiple sources of systematic uncertainties which affect theoscillation parameters extracted from the MC fit to data. Systematic uncertainties
are treated as Gaussian fluctuations in the nominal flux, cross section, and detector
response parameters which alter the nominal MC prediction. This section lists thecorresponding change in the nominal MC prediction for a 1Ã fluctuation in eachsystematic uncertainty source. The largest uncertainties, and those with the largest
effect on the fitted oscillation parameters, are related to the flux and cross section
inputs to the simulation. However, the detector performance can induce some nonnegligible bias on certain oscillation parameters, discussed below.
This atmospheric neutrino oscillation analysis includes a total of 193 independentsources of systematic uncertainties. Of these, 48 account for effects common to all SK
phases, e.g., inputs to flux and cross section models. The remaining 145 uncertainties
129result from 29 × 5 sources of detector and temporal effects relevant to individual
SK phases, e.g., the performance of the detector reconstruction during a particular
data-taking phase.
Many of the systematic errors are identical to those documented in detail in [71].
However, several uncertainty sources have been updated or removed entirely, andseveral new uncertainty sources have been added. This thesis will provide an overview
of each source used in the present analysis.
5.2.1

Atmospheric Neutrino Flux Uncertainties

The atmospheric neutrino flux uncertainties are described briefly here. Several of
the uncertainties are provided by [119], and compare the predictions of the Honda
atmospheric neutrino flux model with two models produced by other groups [120,121], referred to as “the three flux models” below.
Flux normalization: An overall flux normalization uncertainty, estimated from the
combined uncertainties on the hadron production models, air density, and hadroninteraction models used as inputs to the atmospheric neutrino flux calculation, varies
the event rate of all samples as a function of the true neutrino energy. There are
independent errors for neutrinos with energies above and below 1 GeV. Fluctuationsin this uncertainty correspond to changes in the normalization of the MC events,
visualized in Figure 5·9.
Flavor ratios: There are three sets of uncertainties on the overall muon-to-electron
flavor ratio, (¿µ + ¿¯µ )/(¿e + ¿¯e ), and the individual neutrino-anti-neutrino ratios for¿e /¯¿e and ¿µ /¯¿µ . The overall muon-to-electron flavor ratio uncertainty is calculated
as the percent difference in flavor ratio between the three flux models, averaged overall directions. The neutrino-anti-neutrino ratios are calculated by comparing theratios of Ã + /Ã − and K + /K − production in the three flux models, also averaged130

over all directions. Each uncertainty has three independent errors for energy ranges:
Eν < 1 GeV, 1 GeV < Eν < 10 GeV, and Eν > 10 GeV. For these uncertainties, a1Ã change corresponds to an increase in the weight by +1/2 of the percent difference
for the numerator neutrino type, and a simultaneous −1/2 change by the percent
difference for the denominator neutrino type. The change in event weight as a function

Change in Event Weight at 1σ (%)

of neutrino energy for the flux uncertainties is visualized in Figure 5·9.

25
20

_
νμ+νμ
_
νe+νe

Norm.

15

_
νμ/νμ

10

_
νe/νe

5
0
10-1

100

101
Eν (GeV)

102

103

Figure 5·9: Change in event weight for a 1Ã change in the absolute
flux normalization and flux flavor ratio systematics. The weights arecomputed as linear functions of log10 [Eν (GeV)]. The absolute flux
normalization has two independent systematics in the analysis, which
are split at the vertical bar at 1 GeV. The flavor ratio uncertainties
have three separate components, split at the vertical bars at 1 GeV
and 10 GeV. The normalization uncertainty affects all neutrinos in theMC, while the flavor ratio uncertainties only apply to neutrinos of thespecified flavor. Note that the change in event weight for the flavor in
the denominator of the flavor ratios has the opposite sign.

K/Ã ratio: Above 10 GeV, kaons become an appreciable fraction of the neutrinoproducing particles in the atmospheric neutrino flux. Uncertainty on the ratio of131pions to kaons translates to uncertainty on the atmospheric neutrino flux prediction.
To estimate the uncertainty, the Honda flux model is computed assuming an alternate
ratio of kaons and pions. The energy-dependent ratio between the alternate calculation and the nominal model is taken as the 1Ã effect. The effect is largest for thehighest energies; it is approximately 3 % for the FC and PC samples, and extends up
to 10 % for the Up-µ samples.
Neutrino path length: Variations in the atmospheric density can change the averageatmospheric neutrino production height, and therefore the oscillation baseline. This
has a few-percent effect on the path length of neutrinos coming from above the horizon, and is negligible for neutrinos coming from below the horizon. The 1Ã variation
is taken as a change in the atmospheric density by 10 %, estimated from the comparison of two atmospheric density models [122, 123]. The corresponding event weights
are computed as the ratio of oscillation probabilities after this change to the nominal
oscillation probabilities.
Up/Down Ratio: Earth’s magnetic field produces an asymmetry in the zenith angle
distributions for low-energy atmospheric neutrinos. The uncertainty on this effect is
calculated by comparing the zenith angle, ¹z , distributions predicted by the three flux
models. The change in event weight for a 1Ã fluctuation of this uncertainty ranges
from 0.02 % to 3.4 % depending on the neutrino direction, energy, and sub-sample.
Horizontal/Vertical Ratio: Near the horizon, the distance traveled by atmosphericpions and muons increases, allowing for more time to interact or decay. This makes
the flux near the horizon sensitive to atmospheric neutrino flux model inputs notassessed by the up/down ratio uncertainty. The horizontal/vertical ratio systematicquantifies the variation in the flux predictions in the horizontal (0 < cos ¹z < 0.1)
and vertical (0.9 < cos ¹z < 1.0) directions from the three flux models. The change132

in event rate from a 1Ã fluctuation of this uncertainty is approximately 1 % to 3 %.
Solar Activity: This uncertainty modifies the dependence of the low-energy atmospheric neutrino flux due to the solar wind time variation, discussed in Section 4.1.1.
The ±1Ã effect is found by re-calculating the minimum and maximum solar activity
fractions (see Table 4.1) for a shift in the dates of each SK phase by ±1 year. Events
are re-weighted by the ratio of the flux after the shift to the nominal flux. The change
in event rate from this uncertainty is < 1 % for all bins in the analysis, and is largest
for low-energy neutrinos. The largest effect among the SK phases is for the SK II
phase, which occurred mostly in-between a solar minimum and maximum.
Relative Normalizations: While the absolute flux normalization accounts for the uncertainty on the Honda flux inputs, differences in the absolute flux normalization between the three flux models are not accounted elsewhere. These differences primarily
occur above 10 GeV, which is the energy range most relevant for the multi-GeV FC,
PC, and Up-µ stopping samples. Separate 5 % uncertainties on the normalizations of
the multi-GeV FC and PC+Up-µ samples are introduced to cover these differences.
5.2.2

Neutrino Interaction Model Uncertainties

This analysis implements 26 uncertainties for various aspects of the cross section
models in neut.
Quasi-elastic Uncertainties
Quasi-elasitc uncertainties modify aspects of the 1p1h and 2p2h cross section models.
These uncertainties primarily affect CC 1p1h, i.e., CCQE, events. For these, the neutdefault model, LFG with RPA corrections and MAQE = 1.05 GeV/c2 , is compared toan alternative RFG model without RPA corrections, and MAQE = 1.21 GeV/c2 . These
are referred to as the “nominal LFG model” and “RFG model” below.
133

Quasi-elastic Axial Mass: The uncertainty on MAQE from Equation 4.11 modifies theaxial form factor, which in turn affects the total cross section and q 2 dependence of
the CCQE cross section model. This uncertainty re-weights CCQE events by the
double-differential cross section ratio d2 Ã/dELep d cos ¹Lep between a 1Ã fluctuationin MAQE and the nominal value. The 1Ã fluctuation is set to MAQE = 1.21 GeV/c2 .
Currently, the double-differential cross section ratio is computed assuming the simpler
RFG model for both the fluctuated and nominal double-differential cross section. The
next set of uncertainties compare the differences between the LFG and RFG models.
CCQE Shape: This uncertainty compares the relative differences in neutrino energy
dependence between the RFG and LFG model. The 1Ã change is taken to be thefractional difference in the normalized total cross sections, as a function of Eν , between
the RFG model and the nominal LFG model.
CCQE Normalizations: This uncertainty compares the total cross section prediction
between the RFG and LFG model. The 1Ã change is taken to be the fractional
difference in the total cross section between the RFG model and the nominal LFG
model. Two independent uncertainties are implemented for the normalizations aboveand below 1330 MeV. For sub-GeV ¿e and ¿µ CCQE events, the uncertainties are 5 %and 1 %, respectively. For multi-GeV CCQE events, the uncertainties for ¿e and ¿µflavors are 25 %.
CCQE Flavor Ratios: Additional uncertainties are placed on ratio of neutrino andanti-neutrino CCQE total cross sections, (¿e + ¿µ )/(¯¿e + ¿¯µ ), and the ratio of electronto-muon flavor CCQE cross sections, (¿µ + ¿¯µ )/(¿e + ¿¯e ), as a function of neutrinoenergy. In each case, the 1Ã uncertainty re-weights CCQE events by the fractionalchange in the total cross section ratio between the RFG model and the nominal LFG
model.
134

2p2h: This analysis assigns a 100 % uncertainty on the 2p2h process, due to theabsence of direct measurements.
Single Pion Production Uncertainties
MARes , CA5 , I 1 : Independent uncertainties vary the three primary inputs to the Rein2

Sehgal resonant pion production model. In each case, the input parameter is shifted by
its +1Ã uncertainty, and the ratio of the double-differential cross section, d2 Ã/dq 2 dW ,with respect to the nominal values is taken as the 1Ã effect. Here, q 2 is the squared
4-momentum transfer, and W is the invariant mass of the pion-nucleon system. The
uncertainties for each parameter are MARes = 0.95 ± 0.15 GeV/c2 , CA5 = 1.01 ± 0.12,
and I 1 = 1.30 ± 0.20.
2

Neutrino/Anti-neutrino Ratio in Single Pion Production: This uncertainty accounts
for differences in the neutrino and anti-neutrino resonant pion production cross sections between the nominal Rein-Sehgal model and the Hernández model [124]. The
1Ã effect re-weights resonant pion events by the energy-dependent total cross section
ratio between the two models. The weights are separately computed for the three CC
modes and the four NC modes listed in Equations 4.12–4.18.
Charged/Uncharged Ratio in Single Pion Production: This uncertainty compares theratio of charged pion, Ã ± , production to neutral pion, Ã 0 , production between the
Hernández model and the nominal Rein-Sehgal model. The 1Ã effect increases theweight of the resonant Ã 0 production modes by 20 % and decreases the weights of theresonant Ã ± modes by 20 %.
Coherent Pion Production Normalization: This uncertainty produces a 100 % change
in the normalization of the CC ¿µ coherent pion production modes, and a 50 % change
in the normalizations of the CC ¿e and NC modes at 1Ã.

135

DIS Uncertainties
DIS Normalization: This uncertainty compares the neut nominal DIS total crosssection to the world-average DIS total cross section [51]. The uncertainty is calculated from the difference in the total cross section for neutrino energies ranging from30 GeV to 200 GeV. The 1Ã variation corresponds to a 3.5 % change for DIS neutrino
interactions and a 6.5 % difference for DIS anti-neutrino interactions.
PDF Difference Below 10 GeV: While neut computes the DIS cross section using
the GRV PDFs, other PDFs are available. This systematic treats the DIS total crosssection ratio between the total DIS cross section computed with PDFs from [125] andthe nominal neut DIS total cross section as a 1Ã effect. The ratio is computed as afunction of energy, and for neutrinos and anti-neutrinos. The corresponding change
for DIS events is listed in Table 5.3.
Energy
(GeV)
<2
2–3
3–5
5–10

Change at 1Ã (%)
¿

¿¯

30
10
5
5

50
40
20
10

Table 5.3: Fractional change for DIS events due to a 1Ã change in the
PDF Difference uncertainty.

DIS q 2 Dependence: There are four independent uncertainties implemented to capture variations of the Bodek-Yang corrections applied to the neut DIS cross section
models. The first handles cases where the invariant mass of the hadronic system, W ,is above 2 GeV/c2 . This uncertainty uses the change in DIS cross section with andwithout the Bodek-Yang corrections applied, as a function of q 2 , as the 1Ã uncertainty.

136

For W < 2 GeV/c2 , three independent uncertainties vary the vector and axial
components, and overall normalization, of the Bodek-Yang correction. For the vector
and axial component uncertainties, the 1Ã change re-weights events by the ratio of thecross section with and without additional features described by the authors in [126].
The vector uncertainty considers the effect of computing the next-to-leading-order
QCD contribution to the F2 and xF3 form factors. The axial uncertainty considers
the effect of allowing the axial form factors to diverge from the vector components,
which are, in the nominal Bodek-Yang correction, assumed to be the same. The
normalization uncertainty uniformly scales the size of the correction by ±3 %.
DIS Hadron Multiplicity: This uncertainty considers the uncertainty in the numberof hadrons produced in DIS interactions for a given invariant mass of the hadronicsystem, W . The calculation involves computing the ratio of cross sections as a function of W between the GENIE [127] event generator and neut, multiplied by thechange in hadron multiplicity at the specified W value. The 1Ã effect of this uncertainty does not modify the total number of DIS events; rather, it increases anddecreases the frequency of DIS events with particular numbers of outgoing hadrons,
before accounting for FSI effects.
Final State Interaction Uncertainties
FSI+SI: Different FSI and SI processes can result in final states that are difficult or
impossible to distinguish. As a result, the six FSI and SI parameters in neut, listed
in Table 4.2, tend to have high correlations when fit to charged pion scattering data.
While these correlated fluctuations are not implemented in the fit, this analysis applies
a conservative uncertainty on FSI and SI processes: The six FSI+SI parameters arevaried based on their uncertainties, taking into account correlations, to produce sets ofpossible parameter combinations. The two parameter sets which produce the largest

137increase and largest decrease in total event rate are taken as the 1Ã uncertainty. The
parameter sets used are listed in Table 5.4. Note that the minimum set has thehighest scaling of the pion absorption process, while the maximum set has the lowest.
neut Name

Min. Set

Max. Set

1.6
1.1
1.6
2.3
1.6
1.5

1.4
2.3
0.6
1.3
0.6
0.5

FEFQE
FEFQEHFEFCX
FEFCXHFEFABS
FEFINEL

Table 5.4: Parameter sets used for computing the change in event
weights due to uncertainty in FSI processes.

Neutron Production Uncertainties
Neutron Multiplicity Generator Comparison: This uncertainty takes the probability
ratio of producing a given number of neutrons between the GENIE event generator
and the nominal neut neutron multiplicity model as the 1Ã effect. The change in
event weight based on this ratio is calculated as a function of the neutrino energy.
Neutron Multiplicity versus Transverse Momentum: Studies of neutron multiplicities
in T2K data from [128] revealed a difference in the number of neutrons predicted by
the MC versus data, shown in Figure 5·10. The difference was measured for ¿µ and¿¯µ interactions, and has a dependence on the transverse momentum of the outgoing
muon. The 1Ã effect of this uncertainty changes the number of neutrons by thefractional difference between the MC and data on average, then re-classifies the event
according to the procedure in Section 5.1.1.
Previous versions of this uncertainty calculated the change in the number of neutrons by multiplying the number of tagged neutrons by the fractional difference be138

Figure 5·10: Neutron multiplicity measurement with T2K data versus
neut MC prediction for neutrinos (left) and anti-neutrinos (right). The
uncertainty values used in this analysis are the fractional differences
between the MC and data points as a function of the lepton transverse
momentum. Figures are reproduced from [128].
tween MC and data as the 1Ã effect. However, because the event selection procedurein this analysis only relies on the difference between zero and one tagged neutron,
simply applying a multiplicative factor to the number of tagged neutrons may notaccurately reflect the average change in classification outcome. For example, if thefractional difference for an event is 50 %, an event with two neutrons would moveto 1 tagged neutron at −1Ã, and leave the event classification unaffected. Thus,
the change at 1Ã for the event is 0, and scaling the effect would not reflect a larger
fluctuation as expected.
Instead, this analysis calculates the change in the number of neutrons by dividing
the total event weight into multiple events with any possible shift in the number ofneutrons. The probability of increasing or decreasing the number of tagged neutrons
by k is calculated from a one-sided Gaussian distribution with a width set such thatthe expected value of the number of neutrons after the shift is n × (1 ± r), where n is
the original number of tagged neutrons in the event, and r is the fractional difference
between MC and data. If n = 0, the width is set to r. In the original example,the −1Ã decrease in this systematic splits the weight of the event with two tagged139

neutrons between three events with zero, one or two tagged neutrons, such that the
expected value is 2 × 0.5 = 1 tagged neutron. This method then accounts for shifts
from 2 → 0 and 2 → 2 tagged neutrons.
Other Interaction Uncertainties
Tau Neutrino Cross Section: This uncertainty changes the rate of CC ¿τ interactions
by 25 % as a 1Ã effect. The number is estimated from direct measurements of the ¿τcross section. In 2018, SK published its own measurement with a 20 % uncertainty[129]. It is a future effort to incorporate this measurement, or the ¿τ identification
techniques used in that measurement, into this oscillation analysis.
CC/NC Ratio: The 1Ã effect of this uncertainty scales the weight of all NC events by
20 %. The size of this error is a conservative estimate from a synthesis of world NC
measurements in comparison with theoretical predictions [130].
5.2.3

Detector Performance & Reconstruction Uncertainties

Detector systematic uncertainties are evaluated separately for each SK phase. While,
in principle, the uncertainty on various aspects of the reconstruction performance is
not fully independent between SK phases, treating them as independent is a conservative estimate of an ultimately minor source of uncertainty, and vastly simplifies thetreatment of these sources in the analysis.
Reduction Uncertainties
FC, PC, and Up-µ Reduction Efficiencies: Separate uncertainties are estimated basedon MC studies of the various cuts described in Section 3.1. For FC events, thereduction uncertainty primarily comes from data versus MC disagreement in the
flasher probability distribution used in FC4, and is highest for the SK V phase. The

140

uncertainty on the efficiency of the cuts is estimated to be < 1 % for the Up-µ sample
during all SK phases. The values are listed in Table 5.5.
Reduction Uncertainty (%)

SK Phase

SK I
SK II
SK III
SK IV
SK V

FC

PC

Up-µ
Stopping

Up-µThrough-going

0.2
0.2
0.8
1.3
1.7

2.4
4.8
0.5
1.0
1.0

0.7
0.7
0.7
0.5
0.7

0.5
0.5
0.5
0.3
0.5

Table 5.5: Reduction uncertainties for the different neutrino samples
used in this analysis. Uncertainties are separately estimated for Up-µstopping and through-going events, but a single uncertainty varies thenormalization of both samples simultaneously.

FC/PC Separation: A small number of FC and PC events are estimated to be wrongly
classified in the alternative sample. The uncertainty on this estimate is calculated foreach SK period, and increases the normalization of FC multi-GeV µ-like events while
decreasing the normalization of PC events. The change in FC and PC normalizations
from this uncertainty at 1Ã is ≲ 1 % for all SK periods.
Fiducial Volume/Vertex Resolution: Due to the vertex resolution of apfit, FC andPC events can be incorrectly reconstructed inside or outside the fiducial volume. A
2 % uncertainty on the normalization of FC and PC events is assumed from this effectat 1Ã. This uncertainty does not apply to Up-µ events since these do not have event
vertices within the fiducial volume.

141

Reconstruction Uncertainties
Many detector systematic uncertainties are estimated using a “scale-and-shift” procedure. The procedure fits the MC distribution of a reconstruction quantity, x, used
to separate signal and background in the event selection process, to atmospheric neutrino data. First, MC events are labeled as true signal or background events. Next,
the signal and background distributions of x are fit to data by applying a linear
transformation to each MC event’s x: x′ = ´1 x + ´0 . The signal and background
distributions are transformed independently, so there are four ´ parameters. Oncethe best-fit ´ parameters are found, they are randomly varied assuming Gaussian
fluctuations around their best-fit values, where the fluctuations have widths given by
the fitted uncertainties. These variations produce a distribution of signal and background event rates, since different ´ combinations move events above or below the xvalue used to select signal events. The maximum observed variation in signal event
rate from this process is taken as the 1Ã uncertainty.
Uncertainties in the ring counting likelihood, ring PID likelihood, and Ã 0 likelihood
variables are estimated with the scale-and-shift procedure. For this analysis, thesereconstruction-related uncertainties were separately estimated for events inside theconventional fiducial volume and for events in the additional fiducial volume region.
A demonstration of the scale-and-shift fit to the ring counting likelihood distribution
using SK V data in each fiducial volume region is shown in Figure 5·11.
Ring Separation: The number of rings is a basic handle for separating neutrinos events
into the different analysis sub-samples. The level of disagreement in the distribution ofring counting likelihood (Equation 3.8) between data and MC is used as an estimateof the uncertainty on the ring counting reconstruction. The maximum fractionalchange in the number of true single- or multi-ring events using the scale-and-shift
procedure is taken as the 1Ã effect. For this analysis, this systematic is computed
142

18
100

16

SK-V Conventional FV
Data

80

EventsBest fit: total

EventsData

14

Nominal MC: total

Best fit: 1-ring

60

SK-V Additional FVBest fit: multi-ring

40

Nominal MC: total

12

Best fit: total

10

Best fit: 1-ring
Best fit: multi-ring

8
6
4

20

2

Data/Best fit

1.4
1.2
1
0.8
0.6

Data/Best fit

1.4
1.2
1
0.8
0.6
− 10

−5

0

Ring couting LLR

5

10

− 10

−5

0

Ring couting LLR

5

10

Figure 5·11: Ring counting systematic uncertainty estimation via the“scale-and-shift” method. The figures show the ring counting likelihoodfor SK V data and MC events classified as sub-GeV µ-like, with fitted
muon momentum between 200 MeV/c to 400 MeV/c. The uncertainty
estimation is performed in each fiducial volume region, Left: conventional and Right: additional. Before fitting, the data (black points)
and nominal MC distribution (grey histogram) have some degree of
disagreement. True single-ring (blue histogram) and multi-ring (green
histogram) events are identified as signal and background in the MC,then both distributions are varied to fit the data points, resulting in an
improved total fit (red histogram). The vertical dashed line shows thecut value used to classify the event as single- or multi-ring.
for each SK phase and each fiducial volume region. In each case, separate signal andbackground distributions are fit for six event types: Sub-GeV events are split into 4types based on the PID of the most energetic ring, electron- or muon-like, and thereconstructed momentum, above or below 400 MeV/c. Multi-GeV events are split
into 2 types based on the PID of the most energetic ring.
Particle Identification: Similar to the ring counting uncertainty, an uncertainty on
the PID likelihood (Equation 3.12) used to identify the ring-producing particles in an
event is estimated through a scale-and-shift procedure. The uncertainty is estimatedfor each SK phase and separately using data in the conventional and additional fiducial
143

volume regions. In each case, 4 signal classes are used: sub-GeV and multi-GeV,
with either a single e- or µ-like ring. An equivalent, independent uncertainty is also
implemented for multi-ring events, using the PID of the most energetic ring to define
the signal.
Non-¿e Contamination: The multi-GeV e-like samples receive an additional uncertainty to account for the normalization of the ¿µ contamination. The uncertainty
is estimated using the scale-and-shift procedure between data and MC for the ringcounting likelihood, as in the ring separation uncertainty. Then, the uncertainty istaken as the ratio between the fitted and nominal fraction of ¿µ events classified as
e-like.
NC Contamination in Single-Ring µ-like 3 : Because NC events which produce acharged pion cannot typically be distinguished from muons, a background of NC
events is expected in the muon-like samples. This uncertainty places a 10 % uncertainty on NC events which are classified as single-ring µ-like.
Two-Ring Ã 0 : Uncertainty on the likelihood quantity used to select NC Ã 0 events issimilarly implemented to the ring separation and PID likelihood uncertainties. Here,the signal event category used in the scale-and-shift procedure is true NC Ã 0 eventswith two visible e-like rings.
Non-¿ Background : The uncertainty in the non-neutrino background contamination
is estimated using the counts of flasher and cosmic ray muon events found via eye-scanafter all reduction steps. The uncertainty is separately estimated in each SK phasefor sub-GeV, multi-GeV single-ring, and multi-GeV multi-ring events. Independent
systematic uncertainties are implemented for e-like and µ-like events. The contamination of these backgrounds in the PC samples is also included in the uncertainty
3This is referred to as the “Hadron Simulation” uncertainty in other SK references.

144

for µ-like events. This uncertainty modifies the normalization of the corresponding
event type, and the 1Ã effect changes the normalizations by < 1 % for all SK phases
and event types.
PC Uncertainties
PC Stopping/Through-Going Separation: This analysis places an uncertainty on theefficiency of the expected charge ratio cut used to separate PC stopping and throughgoing events. This uncertainty increases the normalization of PC stopping events andproportionally decreases the normalization of PC through-going events, such that the
total PC event counts remain the same. There are three independent uncertainties
implemented for PC events based on their exit point, either through the ID cylinderside, top, or bottom. The 1Ã effect of this uncertainty corresponds to a typically∼ 20 % change in the number of PC stopping events.
Up-µ Uncertainties
Energy & Path Cut Efficiencies: The minimum energy requirement for Up-µ stopping
events and the minimum path requirement for Up-µ through-going events both have
uncertainties on their efficiencies. The uncertainties for each cut are independent,
and the estimated change in the normalization of the corresponding sub-samples is
estimated to be 1 % to 3 % for all SK phases.
Up-µ Stopping/Through-Going Separation: Up-µ events are separated into stopping
and through-going sub-samples based on the number of hits within 8 m of the projected exit point in the OD. The separation uses a simple hit-based cut, and is estimated to have an ∼ 1 % uncertainty, depending on the SK phase. The 1Ã effectof this uncertainty proportionally increases the normalization of the Up-µ stopping
sub-sample and decreases the normalization of the Up-µ through going sub-samples,both non-showering and showering.
145

Up-µ Showering/Non-showering Separation: The algorithm used to separate showering and non-showering Up-µ events is based on the ratio of expected charge fromsimulation to measured charge. The degree of disagreement in this distribution between simulation and data is propagated to a total uncertainty on the showering/nonshowering separation. This uncertainty changes the relative normalization of nonshowering and showering events, such that the total number of events is preserved.
At 1Ã, the non-showering event rate changes by 3 % to 4 %, depending on the SK
phase.
Up-µ Background Subtraction: The irreducible cosmic muon background in the Upµ sample due to the up-scattering of cosmic muons from downward-going cosmic
rays is not simulated in the MC. Therefore, to compare Up-µ data and MC events,
the estimated cosmic muon background must be subtracted from SK data. The
subtraction removes the estimated number of background events from the two zenith
angle bins nearest to the horizon, and as a function of momentum, for the Up-µstopping sample, and in single zenith angle bin nearest to the horizon for the Up-µthrough-going samples. The uncertainty on the background rate is taken as the 1Ã
effect; typically 10 to 20 events are subtracted. More details on the estimation of the
background rate and uncertainty may be found in [74].
Energy Scale Uncertainties
Absolute Energy Scale: SK uses several standard sources to estimate its reconstruction performance at energies relevant to atmospheric neutrino oscillations: Below
100 MeV, stopping cosmic muons which decay at rest produce decay electrons witha measurable energy distribution. Around 100 MeV, the momenta of the two e-likerings from NC Ã 0 decays can be summed to reconstruct the Ã 0 invariant mass. At
several GeV, stopping cosmic muons, with a path length determined by the ID en-

146

trance point and the vertex of a subsequent decay electron, leave calculable energy
depositions. Each of these three sources is simulated and compared to data. The maximum fractional difference between MC and data between any of the three sources
is taken as an overall uncertainty on the reconstructed momenta of fitted particles.
The energy scale uncertainty is estimated for each SK phase, shown in Figure 5·12.
For this analysis, the energy scale uncertainty is also estimated for the decay electron
spectrum and Ã 0 invariant mass sources using only events in the additional fiducial
volume region.
While other systematic uncertainties modify the normalization of different event
types, the absolute energy scale systematic, for a 1Ã change, shifts the reconstructed
momentum of all events by the estimated uncertainty. This means that this uncertainty can cause events to move between different momentum bins within their own
samples.
Up/Down Energy Scale: The decay electrons from stopped cosmic muons, if energetic
enough to produce a Cherenkov ring, can be further separated based on the ring directions. The deviation of the fitted decay electron energy peak between MC and data,binned according to ring direction, provides an estimate of the energy scale uniformity
throughout the detector. This uncertainty varies the normalization of upward-goingand downward-going FC and PC events by the largest observed deviation. Both theabsolute and up/down energy scale uncertainties used in this analysis are listed inTable 5.6.
Other Detector Uncertainties
Decay-e Tagging: The decay electron efficiency uncertainty proportionally changes thenormalization of events in the FC sub-GeV and multi-GeV samples which utilize thenumber of decay electrons as part of their sample selection. The estimated uncertainty

147

6

SK-I

SK-II

SK-III

SK-IV

SK-V

103
102
104
Momentum [MeV/c]4(MC - Data) / Data [%]

2
0
−2
−4
6
4
2
Conventional FV Michel-e0Expanded FV Michel-e
Conventional FV π0 Mass

−2

Expanded FV π0 Mass

−4
−6

Sub-GeV Stopping Muon
Multi-GeV Stopping Muon

102

103

104

102

103

104

Momentum [MeV/c]

Figure 5·12: Absolute energy scale measurements for all SK phases
and separately measured in the conventional (solid lines) and additional (dashed lines) fiducial volume regions. The measurements spana range of energies relevant to atmospheric neutrinos, and are found to
be within a few percent for all SK phases & regions.
on the efficiency is 1.5 % for SK I-III and 0.8 % for SK IV-V, and corresponds to a
few-percent change in the decay-electron-selected samples for a 1Ã variation.
Neutron Tagging: The uncertainty on the neutron tagging algorithm’s efficiency isestimated from a combination of two smaller systematic effects: the dependence of
the algorithm’s efficiency on the neutron’s distance from the primary event vertex,
and on changing detector conditions. An overall data versus MC neural network
efficiency uncertainty is also included. The estimated overall uncertainty is 8 % to
16 %, and is separately estimated for sub-GeV and multi-GeV e-like and µ-like singlering samples. A 1Ã variation in this uncertainty increases the normalization of thesamples with no tagged neutrons, and decreases the normalization of the samples148

SK Phase

SK I
SK II
SK IIISK IV
SK V

Energy Scale Uncertainty (%)
Absolute
Up/Down
Conv.

Add.

Conv.

Add.

3.3
2.0
2.4
2.1
1.8

3.3
3.9
2.4
2.2
2.0

0.6
1.1
0.6
0.5
0.7

1.4
1.5
1.3
0.5
0.1

Table 5.6: Energy scale uncertainties for each SK phase and fiducial
volume region. “Conv.” refers to the conventional fiducial volume while“Add.” refers to the additional fiducial volume.
with one tagged neutron proportionally. This systematic is assumed to be correlated
between the SK IV-V phases.
Multi-Ring BDT Efficiency & Sample Migration: Uncertainties related to the MultiRing BDT selection are calculated using a scale-and-shift procedure. The seven input
variables are scaled-and-shifted to fit the MC distributions to the data. Next, pseudoevents, i.e., combinations of the seven input variables, are drawn from the fitted
distributions and classified according to the BDT. This procedure is repeated for
different values of the fitted ´ parameters to form a distribution of efficiencies for
combinations of ´ parameters. The change in the average efficiency for events drawn
from the post-fit distribution from the nominal efficiency, added in quadrature withthe width of the post-fit efficiency distribution, is taken as the systematic uncertainty.
An additional systematic uncertainty is computed using the scale-and-shift procedure on the distribution of BDT scores themselves. The BDT score distribution foreach event type is fit to data assuming a true signal component, e.g., “true ¿e CC”,
and a background component. The change in signal event rate for each event typefor variations of the fitted ´ parameters, added in quadrature with the width of the149distribution of signal event rates, is taken as an additional uncertainty.
Of the two Multi-Ring BDT uncertainties, the uncertainty computed by varying
the input variable distributions is the larger effect, and is approximately 2 % to 10 %,depending on the SK phase and event type.
5.2.4

Oscillation Uncertainties

1–2 Mixing Parameters: Atmospheric neutrinos are relatively insensitive to the ¹12and ∆m221 oscillation parameters. In this analysis, these are taken to be fixed to theglobal best fit point, with 1Ã uncertainties listed in Table 1.2. The corresponding
change in event weight is calculated from a full three-flavor oscillation probability
framework, in which the numerator is the probability after the ±1Ã change and thedenominator is the oscillation probability assuming the central value. Chapter 6 will
discuss the impact of constraining ¹13 in a similar way.
Matter Effect: A conservative 6.8 % uncertainty is assumed on the Earth matterelectron density, corresponding to the difference between the electron-to-proton ratioof a pure iron core, Ne /Np ≈ 0.467, and a core of lighter elements, Ne /Np ≈ 0.5. A+1Ã change in this uncertainty weights events by the ratio of oscillation probabilities
calculated with +6.8 % electron density and the nominal density.

5.3

Analysis Procedure

Neutrino oscillation parameters are extracted from a fit of binned oscillated MC events
to atmospheric neutrino data. The fit involves two steps: First, oscillation probabilities are calculated for each MC event given a set of oscillation parameters. Next,
the oscillated MC events are fit to the SK data by varying systematic uncertainties
to minimize a Ç2 statistic. Neutrino oscillation parameters are scanned over a fixed
grid, and the resulting map of best-fit Ç2 values at each grid point establishes a global
150

LayerAtmosphere
Crust
MantleOuter core
Inner core

RMin. (km)RMax. (km)Density(g cm−3 )

6371
5701
3480
1220
0

–
6371
5701
3480
1220

0
3.3
5.0
11.3
13.0

Table 5.7: Neutrino propagation layers and corresponding densities
used for calculating neutrino oscillation probabilities in this analysis,
based on a simplified PREM [44]. RMin. and RMax. refer to the radii ofthe spherically-symmetric Earth model.
best-fit point and allowed regions of neutrino oscillation parameter space.
5.3.1

Oscillation Probability Calculation

Section 1.4.1 discussed the theoretical treatment for calculating neutrino oscillation
probabilities, both in a vacuum and in constant-density matter. These techniques
form the basis of calculating the oscillation probabilities of atmospheric neutrinos.
First, atmospheric neutrinos do not, in general, traverse constant-density matter;they pass through layers of varying density, with the highest densities being near the
Earth’s core. The extension of oscillations along non-constant density baselines ispresented in [131]. The authors compute the oscillation probability along the entire
baseline by multiplying the oscillation probabilities for constant-density steps. Thiscan be written in a computationally-tractable form using the general matrix form of
the propagated mass eigenvectors, X, for neutrinos passing through a fixed matter
density,#"
X Y 2EHMatter − Mj2 I
Mk2 L
exp −i
X=
2
2
M
−
M
2E
j
k
k
j̸=k

(5.6)

where HMatter is given by Equation 1.15, and Mi2 /2E are its eigenvalues. This definition allows the neutrino probability along a path of changing matter density to be151written as
P (¿α → ¿β ) =UY

X(E, Äi , Li )U

i

!

2

,

(5.7)

αβ

where U is the PMNS matrix, and Li and Äi are the baseline and density of the ith step
respectively. This analysis assumes a spherically-symmetric Earth, such that the neutrino oscillation baseline, L, only depends on the zenith angle and production height,
i.e., it does not depend on the azimuth angle. The matter densities assumed for each
Earth layer are taken from a simplified Preliminary Reference Earth Model (PREM),
listed in Table 5.7. Neutrinos with zenith angles above the horizon propagate in air,with negligible density.
The contribution of an individual muon or electron neutrino MC event to thetotal number of predicted events after oscillations is calculated as the sum of theevent’s survival probability and the probability for a neutrino of the other flavor,
either electron or muon, to to oscillate to the event’s original flavor. Note that theflavor ratio of muon and electron neutrinos in the atmospheric neutrino flux differs.
The flux difference must be corrected when computing the total oscillation probability
for an individual event, e.g.,
P (¿x → ¿e ) = P (¿e → ¿e ) +Φνµ
P (¿µ → ¿e )
Φν e

(5.8)

where Φν is the atmospheric neutrino flux, evaluated for each neutrino’s energy anddirection. Tau neutrinos are not intrinsic to the atmospheric neutrino flux; they
are simulated assuming the same flux as muon neutrinos and their contribution is
calculated as purely due to ¿µ → ¿τ and ¿e → ¿τ oscillations.
Path Length and Nearest-Neighbor Averaging
The oscillation probability calculated for low-energy neutrinos can change significantly for small changes in path length and energy due to the L/E dependence of152neutrino oscillations. Such rapid oscillation effects are not visible to SK due to thefinite energy and direction resolution of the reconstruction. However, if oscillation
probabilities are computed assuming exact knowledge of an event’s L and E, these
rapid oscillations can still be visible in the MC due to limited MC statistics. Thiscan create artifacts in the fitted allowed oscillation parameter regions where the MCdoes not change smoothly, or even bias the fit result. While producing larger MC
sets is one solution, it is not computationally efficient. Instead, an averaging scheme
is developed to smooth out oscillation effects for low-energy neutrinos.
The averaging method considers both neutrino path length and energy averaging.
First, “neighbor” events are defined as MC events with the same neutrino flavor,
within the same analysis sub-sample, and with the same zenith angle bin. For each
event, the 20 neighbors with the nearest true neutrino energies are used to compute an RMS energy, ERMS . The RMS energy defines five energy sampling points:
E − ERMS , E − ERMS /2, E, E + ERMS /2, E + ERMS . Next, because the neutrino production height in the atmosphere depends on the neutrino energy, for each of thefive energies, 20 production heights are calculated and averaged over to determine anaverage production height. Finally, each event’s oscillation probability is calculated
as the average of the five probabilities computed using the averaged energies andcorresponding averaged path lengths. More details are discussed in the thesis by R.
Wendell [132].
5.3.2

Fij Method

The Fij method incorporates the effect of systematic uncertainties by computing ascale factor for the fractional change in the ith bin for a 1Ã change in the j th systematic
uncertainty source, referred to as an Fij . To calculate the Fij s, MC events are rebinned after applying the +1Ã and −1Ã effect of each systematic uncertainty, then153

the Fij for each bin i is computed as:
+1σj

Fij =
±1σj

where Ni

Ni

−1σj

− Ni
2Ni0

,

(5.9)

is the number of events in the ith bin after a ±1Ã change in the j thsystematic, and Ni0 is the nominal number of events in the ith bin. Except for few
systematic uncertainty sources, the number of events after the change is found by
applying a weight to each MC event, then summing the weights in each bin. The
weights are calculated as:
Weight = 1 ± g(Eν , cos ¹z , . . . ),(5.10)

where g(. . . ) is the fractional increase or decrease due to a 1Ã change in the systematic. Each systematic uncertainty described in Section 5.2 implements a different g
depending on the true neutrino kinematics, interaction mode, reconstructed quantities, or analysis sample. The Fij ’s impact on the number of events is linear, e.g., a2Ã change in the j th systematic uncertainty will change the number of events in theith bin by (2Fij ) %. Fij s may not decrease a bin’s event count below zero.
Regardless of the how the systematic uncertainty modifies the events, the Fij iscalculated using Equation 5.9. Figure 5·13 shows an example of the ¿τ CC crosssection uncertainty Fij calculation and its application. True ¿τ events in the MC
are re-weighted based on the ±1Ã values of the uncertainty, g = 25 % for ¿τ , and 0
otherwise. The MC event counts in each bin are re-calculated after the re-weighting
to evaluate the Fij . The Fij can then be used to scale the counts in each analysis bin,visualized for one analysis sub-sample in the right panel. Thus, the Fij s can modify
the number and distribution of the nominal MC events.

154

SK I Multi-GeV Single-Ring e-likeSK I Multi-GeV Single Ring e-like

Nominal MC

10ν τ CC Xsec. 1σ variation

1.5
1
0.5

30

3.4 < Log [p (MeV)] < 3.7
10

1.5
1
0.5

3.7 < Log [p (MeV)] < 4.0
10

20

10

1.5
1
0.5
−1

40

3.0 < Log [p (MeV)] < 3.4

EventsTrue ντ CC Events

1.5
1
0.5

4.0 < Log [p (MeV)] < 5.0
10

− 0.5

0
cosθz

0.5

0
−1

1

−0.5

0
cosθz

0.5

1

Figure 5·13: Demonstration of the ¿τ CC cross section uncertainty,
applied to the multi-GeV single-ring e-like sample. Left: The systematic uncertainty varies the number of true ¿τ CC events, which populate
the various reconstructed momenta and zenith angle bin in the sample.
Right: The corresponding change in event rate in the whole sample
for a ±1Ã variation in the ¿τ CC component is shown as a function of
zenith angle, summing over all momentum bins.
5.3.3

Ç2 Calculation

The Ç2 statistic for comparing MC and data is derived assuming Poisson fluctuationsin the number of events in each bin. The probability of observing O counts from a
Poisson distribution with mean E is
P (O|E) =e−E E OO!

(5.11)The likelihood for observing a particular configuration of counts, Oi , in n bins, eachwith expected values, Ei , is therefore
L(E|O) =nY
e−Ei E Oi
i

i=1

Oi !

(5.12)The Ç2 statistic may then be formed from the log-likelihood ratio between a particularconfiguration of Ei and the maximum-likelihood estimator, the Oi themselves, using
155

Equation 5.12:
n 
X
L(E|O)Oi
Ç = −2 log=2Ei − Oi + Oi log
L(O|O)
Eii2(5.13)

To incorporate systematic uncertainties, the Fij s from each systematic uncertainty
are used to modify the expected counts in each bin,nSyst.

Ei → Ei1+X

ϵj Fijj!

(5.14)

where j indexes each systematic uncertainty parameter, and ϵj (with units of Ã) scales
the 1Ã effect of the j th systematic uncertainty. The ϵj s are added as penalty terms
to Equation 5.13 to form the complete Ç2 statistic:
2

Ç =2

n
X
i

"

nSyst.

Ei

1+

X
j

ϵj Fij!

− Oi + Oi log

Ei 1 +Oi
PnSyst.
jϵj Fij# +nSyst.

Xϵ2jj(5.15)

Equation 5.15 may be minimized by observing that ∂Ç2 /∂ϵj = 0 for all ϵj s at thebest fit point. This creates a system of equations which is solved numerically. Moredetails may be found in [133]. For this analysis, the Ç2 statistic is minimized at eachpoint in a grid of neutrino oscillation parameters, listed in Table 5.8. At each point,
the Ei s in Equation 5.15 are the number of MC events in each bin after applying
oscillation probabilities.

5.4

Atmospheric Neutrino Oscillation Results

Fit results are presented using the ∆Ç2 with respect to the global best fit across all
oscillation parameters in both mass orderings. Figure 5·14 shows the 1D ∆Ç2 profilesof each fitted oscillation parameter, where the value at each point is the minimum ∆Ç2among all other oscillation parameters. The solid lines indicate the data fit result,while the dashed lines show the sensitivity at the best-fit oscillation point, i.e., the156

Parameter

Min.

Max.

Steps

sin2 ¹13
sin2 ¹23∆m232 or ∆m231 (10−3 eV2 )
¶CP

0.0
0.3
1.2
0

0.075
0.775
3.6
2Ã

16
20
25
21

Table 5.8: The oscillation point grid used for the atmospheric neutrino
analysis. Oscillation parameters are scanned in equally spaced steps,
including the minimum and maximum points listed in the table. The
grid is scanned twice, once for each mass ordering.
∆m232,312

sin2 ¹23

sin2 ¹13

Normal2.40+0.07
−0.09

+0.06
0.45−0.03

0.020+0.016−0.011Inverted+0.05
2.40−0.33

0.48+0.07−0.05

0.010+0.021−0.008Ordering(10−3 eV )

¶CP

Ç2

Ç2

(−π, π)930 bins

Syst.

−1.89+0.87−1.18

1022.0653.50

1027.29

53.57−1.89+1.32−1.97

Table 5.9: Best-fit neutrino oscillation parameters from the atmospheric neutrino oscillation analysis. The uncertainties on each oscillation parameter are the ±1Ã allowed regions assuming a Ç2 distribution
with one degree of freedom. The second-to-last column shows the total
Ç2 , while the last column shows the contribution to the Ç2 from the
193.00 systematic pull terms, cf. the final summation in Equation 5.15.
expected result from a fit to the nominal MC expectation. The dotted lines indicate
the critical values of the Ç2 distribution for one degree of freedom, corresponding to
the probability of obtaining a particular result.
The mass ordering preference is expressed as the ∆Ç2 between the best fits in the
inverted and normal orderings, ∆Ç2I.O.−N.O. . For this atmospheric neutrino analysis,
the data fit prefers the normal ordering, with ∆Ç2I.O.−N.O. = 5.23.

5.5

Discussion

The oscillation parameters measured using atmospheric neutrinos are in good agreement with existing measurements from other experiments. The value of sin2 ¹23 is in

157

16

16
SK I-V expanded FV

14
12

14

Data fit

Inverted

MC expectation

Normal

12

∆χ2

10

8

6

4

4

2

2

0
−π

∆χ2

8

6

−π/2

0
δCP

π/2

0

π

16

16

14

14

12

12

10

10

8

∆χ2

∆χ2

10

99%

6

0.01 0.02 0.03 0.04 0.05 0.06 0.07
sin2θ13

8
6

4 95%

4

2 68%

2

90%

0

0

1.5

2 2 2.5 -3
3∆m32,31(10 eV2)03.50.35 0.4 0.45 0.5 0.55 0.6 0.65
sin2θ23

Figure 5·14: 1D ∆Ç2 profiles of the fitted oscillation parameters forthe atmospheric neutrino oscillation analysis. The ∆Ç2 values are takenwith respect to the best-fit in the normal ordering, listed in Table 5.9.
For each parameter profile, the value of ∆Ç2 is minimized over all otheroscillation parameters. Solid line shows the result of the fit to SK
data, while the dashed lines show the fit to the nominal MC, assuming
the best-fit oscillation parameters. The dotted lines correspond to thecritical values of the Ç2 distribution with one degree of freedom, as ameasure of the probability of obtaining a particular result.

158

the lower octant, sin2 ¹23 < 0.5, while the global average value is in the upper-octant.
However, upper-octant values are allowed at the 1Ã level, indicating no strong preference. The fitted value of sin2 ¹13 in the normal ordering agrees well with the global
measurement, which is dominated by reactor neutrino experiments. In the invertedordering, sin2 ¹13 is fit somewhat smaller. Both fits exclude sin2 ¹13 = 0 at the 1Ã level.
Recall that large values of sin2 ¹13 enhance the matter effect, which, in the normal
ordering, causes an increase ¿e appearance at multi-GeV energies. In the invertedordering large values of sin2 ¹13 predict an increase of ¿¯e . Therefore, the SK data fit
indicates ¿e appearance and relatively suppressed ¿¯e appearance.
The SK measured value of ¶CP is near −Ã/2 in both orderings, implying an excessof ¿e appearance relative to ¿¯e appearance for neutrinos at both sub-GeV and multiGeV energies. However, ¶CP = Ã is allowed at the 1Ã level, which is consistent withno vacuum neutrino-anti-neutrino oscillation differences. Note that the constraints
on ¶CP are weaker in the inverted ordering than in the normal ordering, as can beseen in the top left panel of Figure 5·14. This is consistent with fewer anti-neutrinosin the atmospheric neutrino sample: ¶CP induces smaller changes in the predicted
event rate by modulating ¿¯e appearance than ¿e appearance, so constraints on thisparameter are weaker.
The expected sensitivity to the neutrino mass ordering at the best-fit oscillation
parameters in the normal ordering is ∆Ç2I.O.−N.O. = 1.8, visible as the height differenceat the smallest-∆Ç2 points between the dashed normal and inverted lines in any ofthe panels in Figure 5·14. That the SK data result is in excess of the sensitivity calls
for additional interpretation. Several considerations to contextualize the results aredescribed below.
Mass ordering sensitivity for other parameters: The mass ordering sensitivity is highly
dependent on the values of sin2 ¹23 , sin2 ¹13 , and ¶CP . Figure 5·15 shows the sensitivity
159

for the mass ordering calculated assuming different values of oscillation parameters
allowed by the fit at 1Ã. Higher values of sin2 ¹23 , higher values of sin2 ¹13 , and valuesof ¶CP near −Ã/2 predict the largest ¿e appearance signal, and therefore the highest
sensitivity to rejecting the inverted mass ordering. Conversely, small values of sin2 ¹23
imply a small ¿e appearance signal, and values of ¶CP near +Ã/2 enhance the ¿¯µ → ¿¯eindependent of the mass ordering, further suppressing the normal ordering signal.
The range of sensitivity values includes the SK result for values of sin2 ¹23 allowed
at the 90 % level, indicating that the difference from fitted data and sensitivity, whilein tension, could be resolved through better constraints on sin2 ¹23 .
Normal mass ordering preference in the data: Recall that the mass ordering signature
is expected to be an excess of upward-going ¿e or ¿¯e events with a few GeV of energy.
For a true mass ordering signature, this pattern should be visible in the data. The
flux of multi-GeV ¿e and ¿¯e atmospheric neutrinos is approximately symmetric aboveand below the horizon, which motivates defining an asymmetry parameter to quantify
any excesses from upward-going events. Define:
Asymmetry =Up − Down,
Up + Down(5.16)

where “Up” and “Down” refer to the counts in each zenith angle bin below cos ¹z f
−0.4 and above cos ¹z g 0.4, respectively. Figure 5·16 shows the asymmetry parameter for the eight FC multi-GeV e-like samples, plotted as a function of the momentum
estimator for each sample. It can be seen that, in nearly all bins where the MC predicts a higher asymmetry in the normal ordering, the SK data agrees with or is inexcess of the prediction. The full zenith angle distributions for each momentum bin
are provided in Appendix A. In summary, the excess mass ordering preference compared to the expectation originates from an unlikely configuration of observed counts
in the signal bins of this analysis.
160

90% Allowed Sensitivity
7
6Δχ2I.O.-N.O.

5
4
3
2
1
0
0.35

0.4

0.45

0.5 0.55
sin2θ23

0.6

0.65

0.7

Figure 5·15: Sensitivity to the mass ordering, ∆Ç2I.O.−N.O. , as a function of sin2 ¹23 . The width of the band corresponds to the minimum andmaximum sensitivity for different combinations of the ¶CP and ∆m232
oscillation parameters contained at the 90 % confidence level for thenominal sensitivity with ¹13 constrained, cf. Section 6.1. The width of
the band is primarily set by variations in ¶CP .
Note that the ¿¯e -enhanced samples still contain a background of ¿e events. This
is noticeable where, for example, in the SK I-III multi-GeV ¿¯e sample, the normal
ordering predicts a larger asymmetry than in the inverted ordering. The equivalent
SK IV-V sample, which has a smaller background of ¿e contamination due to utilizing
neutron tagging information, does not show this feature as strongly. This is anadditional verification that the neutron tagging information is improving the purity
in those samples.
Distribution of ∆Ç2I.O.−N.O. : When comparing two hypotheses, the significance of a161

Up - Down / Up + Down

0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
Data
0.2
0.2
0.2
Normal best-fit
0
00
0.2
0.2
−0.2
Inverted best-fit
0.4 SK I-III Multi-GeV ν -like
0.4 SK I-III Multi-GeV ν -like
−0.4
e
e
0.6
0.6
−0.6
0.8
0.8
0.8
3
3.5
4
4.5
53
3.5
4
4.5
5
0.6
0.6 SK IV-V Multi-GeV e-like0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
0
00
0.2
0.2
0.2
−0.2
0.4 SK IV-V Multi-GeV ν -like
0.4
0.4
−0.4
SK IV-V Multi-GeV νe-like
e
0.6
0.6
0.6
−0.6
0.8
0.8
0.8
3
3.5
4
4.5
53
3.5
4
4.5
35
3.5
4
4.5
5
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
0
0
0
0.2
0.2
0.2
−0.2
0.4 SK I-V Multi-Ring ν -like0.4
0.4 SK I-V Multi-Ring Other−0.4
SK I-V Multi-Ring νe-like
e
0.6
0.6
0.6
−0.6
0.8
0.8
0.8
3
3.5
44
4.5
53
3.5
44
4.5
35
3.5
44
4.5
5
3.5
4.5
3.5
4.5
3.5
4.5
log [Energy (MeV)]
10

Figure 5·16: The Up/Down asymmetry parameter for the eight multiGeV e-like samples, where the sensitivity to the neutrino mass ordering
is expected. The asymmetry is calculated as a function of energy bin,
defined in Table 5.1. The black points are the observed counts fromSK data, while the solid lines show the predicted asymmetry from thebest-fit normal and inverted oscillation parameters.

162∆Ç2 statistic can be estimated by taking its square root, i.e., the number of stanp
dard deviations is Ã ≈ ∆Ç2 . This originates from Wilks’ Theorem [134] which
postulates that, for large sample sizes, the log-likelihood ratio between two hypotheses approaches a Ç2 distribution, with degrees of freedom equal to the difference indegrees of freedom between the hypotheses. Crucially, this theorem only applies tohypotheses which are “nested,” i.e., one hypothesis contains additional parameters
relative to the other. A naı̈ve application of Wilks’ Theorem to the atmospheric neu√trino fit results in 5.23 = 2.29 Ã, but this is generally not valid because the normal
and inverted mass orderings are not nested hypotheses.
A more robust method for estimating the significance of the observed ∆Ç2I.O.−N.O.
is to simulate its distribution directly by randomly fluctuating the bin counts, known
as “toy” data sets. This approach has the following steps:
1. Generate toy data sets by fluctuating the statistics in each bin and fluctuating
the nominal values of the systematic uncertainties.
2. Fit each data set to the normal and inverted mass ordering scenarios to generatea distribution of ∆Ç2I.O.−N.O. statistics.
3. Compute the p-value of the SK data, assuming the measured ∆Ç2I.O.−N.O. isdrawn from this distribution.
While the above procedure is straightforward, it is also computationally-intensive.
In general, if the p-value is expected to be small, which is likely for a data result
which exceeds its median sensitivity, it will depend on the tails of its underlying distribution. This is the case for the mass ordering analysis: A previous SK analysis[38] used several thousand toy data sets to estimate the mass ordering p-value froma toy data ∆Ç2I.O.−N.O. distribution. Further, for each data set, the SK analysis software calculates a Ç2 at each point in the full multi-dimensional oscillation parameter163

space. The number of points is further multiplied by a factor of two when computing∆Ç2I.O.−N.O. , since oscillation parameters in both mass ordering scenarios need to bescanned. Fitting thousands of toy data sets at each point on a multi-dimensionalparameter grid quickly becomes infeasible.
To speed up the calculation of ∆Ç2I.O.−N.O. , toy data sets are fit using a procedure
which utilizes that the ∆Ç2I.O.−N.O. calculation only requires knowledge of the best-fit
point in each ordering and not the fit result at every point in the oscillation grid.
The fitting procedure calculates the Ç2 at several points on the oscillation grid as
before, then extrapolates to estimate which point will have a smaller Ç2 value basedon the gradient (slope) of the ∆Ç2 values between the fitted points and the points’
distances in the grid. This algorithm converges on the best-fit point much faster thanthe exhaustive search of the entire grid. To avoid cases where the gradient estimation
leads to a local minimum best fit, the algorithm employs a stochastic annealing [135]routine which re-tries the fit at a randomly-selected nearby point after finding aminimum. A stop condition ensures a finite number of re-tries. This approach isfound to be 20 to 30 times faster than the full oscillation parameter grid scan forcomputing ∆Ç2I.O.−N.O. .
Figure 5·17 shows the distribution of ∆Ç2I.O.−N.O. from the fitted toy data sets
generated assuming a true normal and true inverted mass ordering4 . The SK data
fit result is shown as a solid line at ∆Ç2I.O.−N.O. = −5.69. The probability thatthe observed SK data fit result would be more extreme, given the distribution of∆Ç2I.O.−N.O. from the toy data fits, is the p-value: For the normal ordering, the pvalue is the area to the right of the data result, i.e., the fraction of toy data setswith larger ∆Ç2I.O.−N.O. values. Similarly, for the inverted ordering, the p-value isthe area to the left of the data result. The data result corresponds to a p-value of
p = 9.1 × 10−3 ≈ 2.36 Ã assuming the inverted ordering, while the p-value calculated
4This toy study was performed for the analysis with θ13 constrained, see Section 6.1.

164

from the measured ∆Ç2I.O.−N.O. value assuming Wilks’ Theorem is√

5.69 ≈ 2.39 Ã.

Data=-5.69
True N.O.

Fraction of Toy MCs

10True I.O.

1

I.O.

10

2

10

3

10

4

p = 0.88
p = 0.0091

N.O.

20

10

0
2
N. O.

10

20

I. O.

Figure 5·17: Distribution of ∆Ç2I.O.−N.O. from toy data sets and pvalues in the SK atmospheric neutrino oscillation analysis with ¹13 constrained. The blue and orange histograms represent the distributionsof ∆Ç2I.O.−N.O. extracted from fits to toy data sets generated assuming
the normal and inverted mass orderings, respectively. The filled area tothe right of the blue histogram, and to the left of the orange histogram,
represent the probability of obtaining a less likely result than the datain each ordering.
Once the p-value is obtained from the simulated ∆Ç2I.O.−N.O. distribution, there
is one additional correction which is relevant when testing mutually-exclusive hypotheses such as the neutrino mass ordering. Even though the ∆Ç2I.O.−N.O. statistic
indicates a preference for the normal mass ordering over the inverted ordering in the
SK data, the excess preference over the expected sensitivity indicates that it does
not necessarily agree with the normal ordering either. One way of incorporating thelevel of agreement from both hypotheses is to correct the p-value used to reject the165

inverted ordering by the p-value of accepting the normal ordering, known as the CLs
statistic [136],
CLs =

pI.O.
1 − pN.O(5.17)The CLs statistic is widely used throughout particle physics experiments to ensurethat results are not overstated when comparing two hypotheses. The CLs value fromthe p-values shown in Figure 5·17 is CLs = 0.077, i.e., a rejection of the inverted
mass ordering at 92.3 % ≈ 1.43 Ã. This result is better aligned with the expected
sensitivity of the experiment than the significances obtained from Wilks’ Theorem
and the uncorrected p-value.

166

Chapter 6

Adding External Constraints
The neutrino mass ordering measurement presented in Chapter 5 is inherently limited
by SK’s ability to simultaneously determine all neutrino oscillation parameters. This
chapter considers two extensions to the atmospheric-only analysis: constraints fromreactor neutrino experiments on the value of sin2 ¹13 , and constraints on sin2 ¹23 ,∆m232 , ¶CP , and the mass ordering itself, from T2K data.

6.1

Constraints on ¹13

As discussed in Section 1.5.2, reactor neutrinos experiments measure ¹13 by observing¿¯e disappearance. The stringent constraint on sin2 ¹13 from reactor experiments stems
from high statistics, pure ¿¯e fluxes, and minimal dependence on other oscillation
effects. The current best measurements are provided by the Daya Bay [137], RENO
[138], and Double-Chooz [139] experiments, and give an average measurement ofsin2 ¹13 = 0.0220 ± 0.0007 [51].
To incorporate the measurement of sin2 ¹13 into the SK atmospheric neutrino oscillation analysis, oscillation probabilities are computed assuming sin2 ¹13 = 0.0220,
and a new systematic uncertainty is introduced. The systematic uncertainty varies
sin2 ¹13 at its ±1Ã value, and calculates the corresponding change in oscillation probabilities for all MC events. The change in event rate is then used to form an Fij (seeSection 5.3.2) which can be varied in the fit in the same way as the other systematic
uncertainties.
167

Parameter

Min.

Max.

Steps

sin2 ¹23∆m232 or ∆m231 (10−3 eV2 )
¶CP

0.3
1.0
0

0.725
4.9
2Ã

35
40
37

Table 6.1: The oscillation point grid used for the SK analysis withsin2 ¹13 constrained. Oscillation parameters are scanned in equally
spaced steps, including the minimum and maximum points listed in
the table. The grid is scanned twice, once for each mass ordering.
The ¹13 -constrained analysis analysis uses a new oscillation parameter grid when
fitting to increase the number of points scanned for ∆m232 , sin2 ¹23 , and ¶CP . The
new grid definition is shown in Table 6.1. All other aspects of the analysis remainunchanged.
6.1.1

Results

The 1D profiles of the fitted oscillation parameters in the ¹13 -constrained analysis areshown in Figure 6·1. The 2D profiles of ∆m232 or ∆m231 versus sin2 ¹23 are shown in
Figure 6·2. The best-fit oscillation parameters are listed in Table 6.2. The normal
mass ordering preference is ∆Ç2I.O.−N.O. = 5.69, an increase of 0.46 compared to theatmospheric-only analysis. The constraint on ¶CP has improved, and unlike in theatmospheric-only analysis, ¶CP has similar constraints in both orderings, due to theremoved degree of freedom. Constraints on both sin2 ¹23 and ∆m232 are unchanged,
since the constraints on these parameters are largely driven by ¿µ disappearance,
which is not as sensitive to sin2 ¹13 as ¿e appearance.
An increase is mass ordering preference is expected, given that the atmosphericonly best-fit in the inverted ordering preferred a smaller value of sin2 ¹13 than 0.0220.
The same caveats discussed in Section 5.5 apply to the ∆Ç2I.O.−N.O. value obtainedin the ¹13 -constrained analysis: The number of standard deviations estimated from√Wilks’ Theorem, 5.69 ≈ 2.39 Ã, is generally an overestimate of the significance, and
168

16

16
SK I-V expanded FV
Data fit
MC expectation

14
12

14

Inverted
Normal

12

∆χ2

10

8

8
99%

6

6

4

4

2

2

0
−π

−π/2

π/2

0
δCP

π

0

95%
90%
68%

1.5

2
2.5
3
-3
2
∆m32,31(10 eV2)

16
14
12
10
∆χ2

∆χ2

10

8
6
4
2
0

0.35 0.4 0.45 0.5 0.55 0.6 0.65
sin2θ23

Figure 6·1: 1D ∆Ç2 profiles of the fitted oscillation parameters for
the SK analysis with sin2 ¹13 constrained. The ∆Ç2 values are takenwith respect to the best-fit in the normal ordering, listed in Table 6.2.
The meaning of the colors, solid and dashed curves, and dotted lines is
the same as in Figure 5·14.

169

3.5

∆m232 (10-3 eV2)

3.2
3.0

3.4

Normal ordering
90%
68%

3.2
∆m231 (10-3 eV2)

3.4

2.8
2.6
2.4
2.2

3.0
2.8
2.6
2.4
2.2

2.0

2.0

1.8

1.8
0.35 0.40 0.45 0.50 0.55 0.60 0.65
sin2θ23

Inverted ordering
90%68%0.35 0.40 0.45 0.50 0.55 0.60 0.65
sin2θ23

Figure 6·2: 68 % and 90 % confidence level allowed regions of ∆m232
or ∆m231 and sin2 ¹23 for the SK analysis with sin2 ¹13 constrained. The
regions are drawn assuming a Ç2 distribution with two degrees of freedom for the Left: normal and Right: inverted mass ordering scenarios.
The ∆Ç2 values are taken with respect to the best-fit in each ordering,indicated by a cross, and listed in Table 6.2. Note that no values of theoscillation parameters in the inverted ordering are allowed at the 90 %confidence level with respect to the best-fit in the normal ordering.

170∆m232,312

sin2 ¹23

Normal2.40+0.07−0.09

0.45+0.06−0.03

Inverted

2.40+0.06−0.12

0.45+0.08−0.03

Ordering(10−3 eV )

¶CP

Ç2

Ç2

(−π, π)930 bins

Syst.

−1.75+0.76−1.25

1022.06

53.55

1027.75

53.54−1.75+0.89−1.22

Table 6.2: Best-fit neutrino oscillation parameters from the SK analysis with sin2 ¹13 constrained. The uncertainties on each oscillation
parameter are the ±1Ã allowed regions assuming a Ç2 distribution with
one degree of freedom.
neglects the mutual exclusivity of the two mass ordering scenarios. Recalling theresults of Section 5.5, the CLs value obtained from the p-values shown in Figure 5·17
is CLs = 0.077, corresponding to a rejection of the inverted mass ordering at 92.3 % ≈1.43 Ã.
Figure 6·3 shows a comparison of the measurements of ∆m232 and sin2 ¹23 from the¹13 -constrained analysis with measurements from other oscillation experiments. The
figure compares contemporary neutrino oscillation experiments which are sensitive to¿µ disappearance with similar L/E combinations as SK. Of the experiments shown,both SK and IceCube use atmospheric neutrinos, although IceCube reconstructs neutrinos with higher average energies than SK, ∼ 10 GeV to 300 GeV. The NOvA andT2K experiments measure ∆m232 , sin2 ¹23 , and ¶CP using neutrino beams, where theneutrino baseline is known exactly, and the energy spectrum is much narrower thanfor atmospheric neutrinos. These features allow NOvA and T2K to set more precise
measurements on ∆m232 than SK and IceCube. Measurements of these two parameters are consistent between the various experiments. Notably, all experiments have abest-fit oscillation in the normal ordering, although the other experiments are not as
sensitive as SK to the mass ordering. Of the four experiments, only SK has a best-fit
value of sin2 ¹23 in the lower octant (sin2 ¹23 < 0.5), but its 90 % allowed region spans
both octants, and contains all other best-fit points.

171

3.2

∆m232 (10-3 eV2)3
2.8

Normal ordering, 90% C.L.
IceCube
NOvA
T2K
SK

2.6
2.4
2.2
2
0.35

0.4

0.45

0.5 0.55
sin2θ23

0.6

0.65

Figure 6·3: 2D 90 % allowed regions of sin2 ¹23 and ∆m232 neutrino
oscillation parameters from contemporary neutrino oscillation experiments, analyzed assuming sin2 ¹13 is constrained. The best-fit values ofsin2 ¹23 and ∆m232 are indicated with markers. The IceCube contours
are from [140], the NOvA contours are from [141], and the T2K contours are from [142].

6.2

Constraints from T2K

As demonstrated by the ¹13 -constrained analysis, we can improve the SK mass ordering measurement further by introducing constraints on other oscillation parameters.
The oscillation parameters sin2 ¹23 , ∆m232 , and ¶CP have all been measured more precisely by the NOvA and T2K experiments than by SK, suggesting possible improvements. While additional systematic uncertainties could be introduced to implement
these constraints, similarly to the ¹13 -constrained fit, the T2K experiment provides
a unique opportunity to obtain an even better measurement. Because T2K records
neutrino interactions within the SK detector, variations in the neutrino interaction
uncertainties produce a simultaneous effect in both the atmospheric neutrino prediction and in the T2K beam neutrino prediction. This means that a combined analysis

172

of SK and T2K data has the added benefit of improved constraints on neutrino interaction uncertainties, in additional to complementary sensitivity to oscillation effects.
The SK and T2K collaborations have historically analyzed their data independently, with each choosing simulation tools, cross section models, and fitting techniques to best suit their experimental signatures. This thesis will discuss the implementation of T2K data into the atmospheric neutrino analysis using SK softwareand simulation tools, and publicly-available information from T2K publications only.
This has the advantage of using the most recent SK analysis techniques and data,
but also has limitations which will be discussed in Section 6.3.5. Separately from thework presented in this thesis, collaborators from both SK and T2K are working toformally analyze SK and T2K data using the resources of both collaborations. The
formal SK-T2K joint analysis will unify the reconstruction software, interaction models used in simulation, and perform its fit using the full reconstruction information ofboth SK and T2K data events, all of which are beyond the scope of this thesis.
6.2.1

The T2K ExperimentThis section will discuss the T2K experiment’s sensitivity to neutrino oscillation effects and the components of the T2K experiment. An overview of the T2K experimentis shown in Figure 6·4. There are three main components: the neutrino beam production site at the J-PARC accelerator complex in Tokai, Japan, a series of neardetectors which monitor the beam and perform cross section measurements, and thefar detector, SK, described in detail in Chapter 2.
Neutrino Oscillations with T2K
The T2K experiment uses a 99 %-pure muon neutrino beam to precisely measure∆m232 and ¹23 through muon neutrino disappearance, and ¹13 and ¶CP through electron neutrino appearance. The neutrino beam travels along a 295.3 km baseline near173^ƵƉĞƌͲ<ĂŵŝŽŬĂŶĚĞ

EĞĂƌĞƚĞĐƚŽƌƐ

:ͲWZ

Dƚ͘/ŬĞŶŽͲzĂŵĂϭ͕ϳϬϬŵďĞůŽǁƐĞĂůĞǀĞů

EĞƵƚƌŝŶŽĞĂŵϮϵϱŬŵ

Figure 6·4: Overview of the T2K experiment. The neutrino beam
produced at J-PARC passes through a series of near detectors, thenpropagates through the Earth towards Super-Kamiokande (SK). Figure
is adapted from [143].
Earth’s surface. Neglecting matter effects, which are small for T2K, the muon neutrino survival probability, including the sub-leading effects of nonzero ¹13 , is approximately
4

2

2

2



P (¿µ → ¿µ ) ≈ 1 − cos ¹13 sin 2¹23 + sin 2¹13 sin ¹23 sin

2∆m232 L
1.27
E



(6.1)

and is equivalent for anti-neutrinos. The electron neutrino appearance probability,
including the effects of ¶CP , is approximately[144]:P (¿µ → ¿e ) ≈ 4|T3µe |2 sin2 ∆31 + 8|T2µe ||T3µe |sin ∆31 sin ∆21 cos (∆32 + ¶CP ) ,(6.2)

where T3µe = 12 sin 2¹13 sin ¹23 eiδCP , T2µe ≈ 12 sin 2¹12 cos ¹13 cos ¹23 . The squared massdifferences are absorbed into ∆ij ≡ 1.27∆m2ij L/E. For T2K baseline and typicalneutrino energies, oscillation effects due to sin2 ∆21 are small, and have been neglected.
The largest effect is given by the sin2 ∆31 term. The second term, referred to as
the “interference” term, has a relative amplitude of ∼ 20 %. The interference term
crucially depends on ¶CP and the sign of ∆31 . Because |∆m232 |≈ |∆m231 |, there isa degeneracy in the interference term under interchange of ¶CP → Ã − ¶CP and thenormal, ∆31 > 0, and inverted, ∆31 < 0, mass orderings. The degeneracy is lifted174for certain values of ¶CP by invoking matter effects: Although the T2K neutrino
beam experiences smaller matter effects than atmospheric neutrinos passing near the
Earth’s core, the matter effects can enhance the neutrino appearance probability
further than through ¶CP alone. In particular, the combination of ¶CP = −Ã/2 andthe normal ordering increases the electron neutrino appearance probability higher
than for any value of ¶CP in the inverted ordering.
Examples of the electron neutrino appearance probabilities, calculated as a function of neutrino energy for the fixed T2K baseline and matter density, are shownin Figure 6·5. The curves are drawn with matter effects which, while small, allow
T2K to probe different combinations of ¶CP and the neutrino mass ordering. Some
combinations of ¶CP and the mass ordering are easier to distinguish than others: The
left panel shows a pessimistic case, where the effect of different values of ¶CP is degenerate with the mass ordering. The right panel shows a best-case scenario, where, if¶CP = −Ã/2, the normal and inverted mass ordering predict a few-percent differencein the electron neutrino oscillation probability at the peak T2K beam energy.
Both the muon neutrino disappearance and the electron neutrino appearance channels are crucial for T2K’s ability to constrain neutrino oscillation parameters. The
muon neutrino disappearance channel is largely unaffected by matter effects and theeffects of ¶CP , and so helps to constrain flux and cross section uncertainties somewhat independently of the electron neutrino appearance channel. In addition, themuon neutrino disappearance constrains ∆m232 and sin2 ¹23 better than the electron
neutrino appearance channel due to increased statistics.
Neutrino Beam
The T2K neutrino beam begins at the J-PARC accelerator complex which accelerates
31 GeV protons onto a carbon target. Proton interactions in the target produce showers of pions and hadrons, which are focused through a series of three electromagnetic
175

0.1

0.08

N.O., δ CP =π/2

0.06

I.O., δ CP =π/4

0.04

P(ν µ → ν e)P(ν µ → ν e)

0.1

0.02
0

0.08

N.O., δ CP =3π/2

0.06

I.O., δ CP =3π/2

0.04
0.02

0.5

1
1.5
E ν (GeV)

0

2

0.5

1
1.5
E ν (GeV)

2

Figure 6·5: Electron neutrino appearance probabilities versus neutrino energy for the T2K baseline. Left: If ¶CP is near Ã/2, the effectof the mass ordering is degenerate with similar values of ¶CP . Right:
For ¶CP = 3Ã/2, the appearance probability is maximized in both orderings, but the normal ordering further enhances the probability.
horns. The direction of the current in the horns determines whether positively- or
negatively-charged hadrons are focused, referred to as forward horn current (FHC)for positively-charged hadrons, or reverse horn current (RHC) for negatively-charged
hadrons. The hadrons then propagate in 96 m decay volume where they decay into
neutrinos. The most numerous hadrons are pions, which decay into muon neutrinos
and muons in FHC mode, or anti-neutrinos and anti-muons, in RHC mode. The secondary muons are absorbed by a beam dump placed after the decay volume, resulting
in a 99 % pure muon neutrino or anti-neutrino beam which propagates towards SK.
The T2K beam is directed 2.5◦ off-axis from the SK detector. The off-axis angle
reduces the overall neutrino flux, but enhances the flux in the narrow energy range
most sensitive to oscillation effects for T2K’s baseline. The neutrino energy as a

176

Eν (GeV)3

θ = 0°

2

2.0°

1

2.5°
0

0

2

4
6
Eπ (GeV)810Figure 6·6: Neutrino energy spectrum from pion decay in flight from
Equation 6.3. At off-axis angles (¹ ̸= 0◦ ), a large range of pion energies
produce similar neutrino energies.
function of off-axis angle can be calculated from relativistic kinematics:[(Eπ , p⃗π ) − (Eν , p⃗ν )]2 = (Eµ , p⃗µ )2=⇒ m2π − 2Eπ Eν + 2|⃗pπ ||⃗pν |cos ¹ = m2µ=⇒ Eν =

m2π − m2µ,2(Eπ − 2|⃗pπ |cos ¹)(6.3)

where ¹ is the angle between the pion and neutrino in the lab frame, and |⃗pν |≈ Eν .
The neutrino energy dependence on the pion energy is plotted for different angles
in Figure 6·6. At off-axis angles, the outgoing neutrino energy spectrum is muchnarrower than the on-axis case.
The unoscillated T2K neutrino flux observed at SK is shown for both FHC and
RHC modes in Figure 6·7, per 1021 protons on target (POT). The contaminationof “wrong-sign” neutrinos, e.g., anti-neutrinos in FHC mode and neutrinos in RHCmode, is larger for the RHC flux due to the smaller anti-neutrino cross section. Electron neutrino contamination is also present due to kaons produced from proton in177

T2K RHC Flux at Super-K

106

νµ
νe

105

Flux (/cm2/50MeV/1021 POT)

Flux (/cm2/50MeV/1021 POT)

T2K FHC Flux at Super-K

νµ
νe

104
103

0

2

4
6
Eν [GeV]

8

10

106

νµ
νe

105

νµ
νe

104
103

0

2

4
6
Eν [GeV]810Figure 6·7: Neutrino flux prediction at SK from the T2K neutrino
beam, before oscillations. Left: FHC, neutrino mode and Right:
RHC, anti-neutrino mode.
teractions in the target.
Near Detectors
The T2K neutrino beam passes through a detector complex 280 m downstream of
the T2K target before arriving at SK. These near detectors monitor properties ofthe beam and measure neutrino interaction cross sections with higher statistics thanis available at SK. A key feature of the near detectors is that they observe an unoscillated neutrino flux: Neutrinos observed at the near detectors have not traveleda sufficient distance to have appreciable oscillation effects. The combination of unoscillated near detector data and oscillated far detector data helps to resolve theeffects of flux and cross section models from oscillation parameters. The two primarynear detectors are INGRID, an on-axis beam line monitor, and ND280, an off-axis
high-resolution tracking detector. These detectors are described in detail in [143] andare summarized here.
INGRID monitors CC ¿µ interactions to characterize the T2K beam profile and178intensity. INGRID consists of 14 detector modules arranged in a cross, which viewsections of the beam within a 10 m × 10 m transverse area. The T2K beam center
passes through two modules placed at the center of the cross. Each detector module
consists of alternating iron plates and scintillator panels. Muon neutrino interactionsin the iron plates create muons which leave tracks in the scintillator panels, which canbe reconstructed. Veto panels installed on the outside of each module ensure cosmic
muons entering from outside the modules are rejected. INGRID observes a sufficientnumber of interactions to monitor the beam center with a precision of 10 cm each day.
ND280 is a multi-purpose detector which monitors the flavor composition, energies, and interaction rates of neutrinos from the T2K beam. The ND280 detector isplaced on the same off-axis angle as SK, such that it observes the same flux of neutrinos as SK. The ND280 detector is surrounded by a magnet and consists of several
sub-systems within. First, the T2K beam passes through a detector optimized forobserving NC processes which produce Ã 0 s, i.e., ¿ + N → ¿ + N + Ã 0 + X. The Ã 0
detector consists of alternating water targets and scintillator panels. Next, the beam
passes through two alternating combinations of fine-grained detectors (FGDs) andtime projection chambers (TPCs). This section of the detector identifies the flavor
and energy of neutrino interactions using high-resolution tracking information andthe magnetic field to separate particles based on electric charge and calculate their
momenta. The FGDs provide the interaction target and vertex information, while the
TPCs are used for PID and momentum reconstruction. The TPCs contain argon gas,
and ionization charge from particle traversals is drifted towards wire planes which
produce an image of the track.
The ND280 data is analyzed as additional samples as part of the T2K experiment’s
neutrino oscillation measurement. ND280 events are selected based on the number ofpions in FHC mode, or the number of tracks in RHC mode. The different samples are179

enhanced for different interaction modes, e.g., the zero-pion sample in FHC mode,and one-track sample in RHC mode, are pure in CCQE interactions.
T2K Event Selection & Reconstruction at SK
T2K events at SK must pass a series of reduction cuts to ensure reliable event reconstruction and remove NC events which are backgrounds to the oscillation analysis.
Beam timing information is used to trigger SK at the time of arriving T2K events,
which significantly reduces the number of cosmic muon and low-energy contamination. T2K events must fall within a [−2, 10] µs window of the beam signal, and beam
signals must not occur within 100 µs of cosmic muon triggers at SK.
Selected T2K events are reconstructed using the fitqun algorithm, described indetail in [77]. Like apfit, described in Section 3.2, fitqun reconstructs the number,
momenta, and PID of Cherenkov rings, and counts the number of decay electrons.
Unlike apfit, fitqun reconstructs events by iteratively updating a likelihood function using trial combinations of particle types and kinematics which produce expectedconfigurations of hit PMTs. The likelihood function is expressed as:L(Γ, ¹) =

Unhit
Y
j

Pj (Unhit|µj )

Hit
Y
i

[1 − Pi (Unhit|µi )] × fq (qi |µi )ft (ti |Γ, ¹),(6.4)

where Γ represents a particular particle hypothesis, e.g. “single ring e-like”, and ¹ isa set of the particle’s kinematic variables. The products compare the light patternusing the probabilities of registering a hit, Pi , of both “unhit” and “hit” PMTs. For
hit PMTs, the probability also includes the charge likelihood, fq (qi , µi ), where qi is
the observed charge and µi is the expected charge, and a likelihood for the observed
hit time, ft (ti |Γ, ¹), where ti is the observed hit time.
Unlike the SK analysis, T2K uses the reconstructed particle information to estimate the energy of the neutrino. The neutrino energy estimation relies on two features

180

of T2K beam neutrinos which are not present for atmospheric neutrinos: First, thedirection of T2K neutrinos is known exactly, since all neutrinos arrive from the direction of the beam. Second, the T2K flux has a sharp peak for neutrino energies
near 600 MeV, where the CCQE process has the largest cross section. By knowing
the beam direction and assuming CCQE interactions, the neutrino energy can be
estimated using the outgoing lepton kinematics as
EνRec. =2mN,i El − m2l + m2N,f − m2N,i,2(mN,i − El + |⃗pl |cos ¹l )(6.5)

where mN,i and mN,f are the initial and final nucleon masses, El is the lepton energy,p⃗l is the lepton momentum, and ¹l is the angle of the lepton with respect to theneutrino direction. mN,i is typically the effective nucleon mass after subtracting thenucleon removal energy (NRE), i.e., the average energy required to free the nucleon
from the nucleus.
The distribution of true neutrino energy bias due to EνRec. , computed using thetrue lepton direction and energy, is shown in Figure 6·8. The bias distribution for
CCQE events is peaked around zero, while non-CCQE interactions have broader
distributions with longer tails at negative values. This tails reflect that Equation 6.5
does not account for additional particles in the event.

6.3

T2K ModelThis thesis presents a combined SK and T2K analysis using an effective description
the T2K experiment’s flux, cross section models, event selections, and systematic
uncertainties, referred to as the “T2K model”. The T2K model presented in thisthesis is an upgrade to the previous T2K model described in [38], and is based on a
description of the more recent T2K Runs 1–9 analysis presented in [144]. The T2K
Runs 1–9 analysis uses data collected with 1.494 × 1021 POT in FHC mode, and
181

FHC
90
Events/22.5 kt/1021 POT

80
70
60
50

CCQE
2p2h
Single Pion
DIS
NC

40
30
20
10
0
−1

−0.8 −0.6 −0.4 −0.2
0Rec.
Eν /E ν - 1

0.2

0.4

Figure 6·8: Expected EνRec. bias for the different neutrino interactions
present in the oscillated T2K FHC flux. The distributions show thenominal MC prediction of the T2K model before any event selections.
The oscillation probabilities are computed using the parameters listed
in Table 6.3.
1.635 × 1021 POT in RHC mode. Compared to the previous T2K model, the present
model includes twice as much data in FHC mode, an additional sample in FHC mode,
and, for the first time, a description of two anti-neutrino-enhanced samples in RHC
mode.
The T2K model re-weights the existing SK atmospheric neutrino MC events to
produce T2K MC events distributed according to T2K’s flux and nominal cross section
models. In the final analysis, the T2K model MC, together with binned T2K data,
are added as extra samples in the fit alongside the SK atmospheric neutrino samples.
6.3.1

Flux Re-weighting

To use the atmospheric neutrino MC as a prediction for T2K events, the T2K model
re-weights each atmospheric neutrino MC using the predicted flux ratio between the182

ParameterValue

sin2 ¹12
sin2 ¹13
sin2 ¹23∆m221 (10−5 eV2 )∆m232 (10−3 eV2 )¶CP
Ordering

0.304
0.0212
0.528
7.53
2.509
4.682
NormalTable 6.3: Nominal oscillation parameters used by the T2K model.
The values are the same as in Table III from [144].
T2K flux and the Honda atmospheric neutrino flux. While the T2K flux is a 1D
function of energy, the Honda flux is expressed as a function of both the zenith
angle, ¹z and the azimuthal angle, ϕ. The corresponding weight for each atmospheric
neutrino MC event is:wα (Eν , ¹z , ϕ) ∝

ΦT2K,β (Eν ),ΦHonda,α (Eν , ¹z , ϕ)

(6.6)

where ³ and ´ indicate neutrino flavors. For muon neutrino events, ³ = ´ = ¿µ , while
for electron neutrino events, the weights are computed both for intrinsic, ³ = ´ = ¿e ,and appearance, ³ = ¿e , ´ = ¿µ , oscillation channels. Each electron neutrino MC
event is arbitrarily assigned to either the intrinsic or appearance channel, and therelative contribution of each is corrected in the next step. Because the T2K flux ofelectron neutrinos is small relative to that of muon neutrinos, ¿e → ¿µ oscillations
are a negligible contribution and are not considered. Flux weights are computed for
all atmospheric neutrino MC events with energies from 0.1 GeV to 10 GeV.
An example re-weighting from the Honda flux to the T2K flux is shown in Figure 6·9. The figure shows events generated at a fixed direction according to the Honda
flux, which typically has a falling-exponential shape, while the T2K flux has a peaked
shape near Eν ≈ 600 MeV. The weights change the relative frequency of the events
183

0.1

Re-Weighted Events/T2K Flux

Normalized Flux or Events

Honda Flux
0.08

T2K Flux
Re-weighted Events

0.06

0.04

0.02

1.2
1.1
1
0.9
0.8

0.2 0.4 0.6 0.8

1 1.2 1.4 1.6 1.8
Eν (GeV)2

Figure 6·9: T2K model flux re-weighting demonstration, using atmospheric neutrino MC events generated according to the Honda flux
at a fixed direction. The lower panel shows the ratio between the reweighted events and the T2K flux distribution.
generated according to the Honda flux to match the T2K flux. Note that because theMC events were generated according to the Honda flux, the MC statistics decrease
with increasing energy.
6.3.2

Normalization

As described in Chapter 4, 500 years of atmospheric neutrino MC is generated for
the SK analysis. To re-weight the atmospheric neutrino MC to meaningful units
for T2K, the T2K model applies a normalization to each oscillation channel. The
normalizations are calculated by dividing the expected event rate of neutrinos withenergies between 0.1 GeV to 10 GeV by the total number of T2K flux-weighted MC

184

Flux

Events/22.5 kt/1021 POT

Cross Section

FHC

RHC

Φν µ
Φν µ
Φν e

Ã νµ
Ã νe
Ã νe

1480.45
1562.2028.90

210.24216.80
8.90

Φν̄µ
Φν̄µ
Φν̄e

Ãν̄µ
Ãν̄e
Ãν̄e

45.15
46.71
2.54

344.94362.87
6.25

Table 6.4: Un-oscillated event rates at SK, used by the T2K model fornormalizing the SK MC, following Equation 6.7. The T2K fluxes arefrom [144], and the total cross sections are computed using the neut
version 5.4.0 default configuration.
events in a fixed volume, i.e., the flux-weighted events are normalized toNx ∝10ZGeVÃα (Eν ) × Φβ (Eν )dEν

(6.7)

0.1 GeV

where Ãα (Eν ) is the total interaction cross section for neutrino flavor ³, and Φβ (Eν )is the T2K flux for neutrino flavor ´. For the ¿µ → ¿µ and ¿e → ¿e channels, ³ = ´,
while for the ¿µ → ¿e channels, the cross section index ³ = ¿e , and the flux index´ = ¿µ . The conventional 22.5 kt fiducial volume, computed as Avogadro’s number
multiplied by neut’s cross section units and the fiducial volume mass, 6.022 × 1023 ·10−38 ·22.5×109 , sets the proportionality. The expected event rates for each oscillation
channel are listed in Table 6.4. The event rates were calculated using the T2K fluxes
from [144] and the total cross sections of the default neut 5.4.0 configuration.
After the normalization, the near detector constraints on the T2K flux are applied
as an additional scaling. The constraint changes the normalization of each flavor in
fixed energy ranges of the FHC and RHC fluxes. The values applied as part of theT2K model are listed in Table 6.9.

185

6.3.3

Cross Section Re-weighting

Following the flux re-weighting and normalization, the remaining differences between
the T2K model MC events and the nominal MC prediction of the T2K Runs 1–
9 analysis are due to differences in the neutrino interaction cross section models,
discussed below.
1p1h: For the nominal simulation of 1p1h interactions, the T2K analysis uses an RFG
model with MAQE = 1.13 GeV/c2 and Fermi momentum pF = 205 MeV/c, while the
SK analysis uses an LFG model (see Section 4.2.1). The T2K model implements ascheme to resolve the differences between the RFG and LFG models, as well as otherfeatures of the T2K nominal 1p1h simulation. This is a crucial step of reproducing
the T2K MC because the distribution of lepton kinematics from 1p1h interactions
has the largest impact on the predicted distribution of EνRec. for T2K events.
The T2K model 1p1h re-weighting method is similar to the method used to calculate shifts in MAQE for the purposes of evaluating systematic uncertainties (seeSection 5.2.2). The weight is calculated as the ratio of the double-differential crosssections, d2 Ã/dELep d cos ¹Lep , between the T2K nominal RFG model and the SK nominal LFG model, for each event’s lepton kinematics and neutrino energy. However,this is an imperfect approach: There are some combinations of lepton angles and energies which have non-zero cross sections in one model but are zero, or very small inthe other. Attempting to calculate the cross section ratio for these combinations canlead to large or undefined weights. Further, the neut implementation of the LFG
model relies on a random number generator to calculate the double-differential crosssection; the cross section is calculated for neutrino interactions at multiple, randomly
selected radii within the nucleus, and averaged. The randomization leads to variations in the calculated double-differential cross section which should not be taken
literally.
186

Double-differential Cross Section Ratio

Double-differential Cross Section Ratio

QE

QE

RFG, M =1.13 GeV/c 2 / LFG

>2

A

1

νµ

−0.5

0.5

0.4
0.6
ELep/E ν

0.81

cos θLepcos θLep

1

0.2

0.5

1.5

0

−1
0

>2

A

νµ

Eν =0.6 GeV

0.5

RFG, M =1.13 GeV/c 2 / LFG

1

0

Eν =1.0 GeV

1.5

0

1

−0.5

0.5

−1
0

0.2

0.4
0.6
ELep/E ν

0.8

1

0

Figure 6·10: Ratios of double-differential 1p1h cross sections computed using the T2K nominal RFG model and the neut default LFG
model. The color scale indicates the weight, and weights above 2
are not differentiated. No smoothing or averaging has been applied.
Cross sections are for muon neutrinos. Left: Eν = 600 MeV. Right:Eν = 1 GeV.
Example 1p1h double-differential cross section ratios are shown for fixed neutrinoenergies in Figure 6·10. The figure shows the double-differential cross section ratios in
the areas of phase space where the RFG and LFG models overlap. The x-axis of eachpanel shows the fraction of the neutrino energy carried by the outgoing lepton, andthe y-axis shows the cosine of the lepton direction with respect to the true neutrino
direction. The T2K nominal RFG model predicts smaller energy fractions for theoutgoing lepton, as can be seen by the large weights on the left-hand edges and small
weights on the right-hand edges of either panel. The T2K nominal RFG model also
predicts leptons more aligned with the direction of the neutrino compared to the LFG
model, indicated by the large weights near cos ¹Lep ≳ 0.5. The double-differential crosssection ratios are not smooth due to the random number generator used in computing
the LFG cross sections.
To stabilize the RFG-LFG re-weighting, the T2K model implements the following
corrections when computing the weights: (i) a cutoff weight is defined, such that187

weights above a certain threshold are set to 0. (ii) The double-differential crosssections are averaged over nearby regions of phase space before taking the ratio.
(iii) The double-differential cross section in regions of non-overlapping phase spaceis integrated, and applied as an overall correction to preserve the total cross section
after re-weighting. The corrections from (iii) are ∼ 1 %.
A demonstration of the 1p1h re-weighting is shown as a function of the total
neutrino cross section for muon neutrinos in Figure 6·11. The left panel shows the“out-of-the-box” agreement between the total T2K RFG cross section model after
re-weighting LFG events by the double-differential cross section ratio computed byneut with no further corrections1 . The right panel shows the result after the T2K
model corrections are applied. The level of agreement before the corrections is poorat several energies, while, for the T2K model, the level of agreement in the total crosssection is within a few percent for energies relevant to T2K.
The T2K nominal 1p1h model also includes RPA corrections which modify thecross section as a function of q 2 . In the T2K analysis, the default neut RPA corrections from the Nieves model are replaced with an effective function based on Bernsteinpolynomials, referred to as “BeRPA”. The function is given in [144] as

f (x) =
A(1 − x′ )3 + 3B(1 − x′ )2 x′ + 3p1 (1 − x′ )x′2 + Cx′3 , x < U1 + p2 exp(−D(x − U )),

(6.8)

xgU

where x ≡ q 2 and x′ = q 2 /U . The coefficients p1 and p2 are set to enforce continuity
1

In neut 5.4.0, a software bug results in the LFG total cross section being shifted by 50 MeV in
neutrino energy, cf. the N1p1h::IntegralCrossSection function included in neut. This also needs
to be corrected as part of the re-weighting if events are generated according to this function, andso it is partially responsible for the poor agreement at low energies in the left panel of Figure 6·11.
The bug has been fixed in more recent versions of neut, and never affected the double-differential
cross sections.

188

νµ

νµ

Cross section or scaled events

0.6

0.5
0.4
0.3
RFG re-weighted LFG events

0.2

LFG eventsQE

T2K MC RFG MA =1.13 GeV

0.1

SK MC LFG model0

Events / Xsec.

Events / Xsec.

Cross section or scaled events

0.6

1.05
1
0.95
0

0.5

1

1.5

2

2.5 3
Eν [GeV]

3.5

4

4.5

5

0.5
0.4
0.3
RFG re-weighted LFG events

0.2

LFG eventsQE

T2K MC RFG MA =1.13 GeV

0.1

SK MC LFG model

0
1.05
1
0.95
0

0.5

1

1.5

2

2.5 3
Eν [GeV]

3.5

4

4.5

5

Figure 6·11: LFG to RFG re-weighting demonstration from the T2K
model. Neutrino interactions (grey points) are generated according
to the LFG model (blue line), then re-weighted by the ratio of the
double-differential cross sections between the T2K RFG model and the
LFG model. The black data points show the events after re-weighting.
The red line shows the total cross section for the T2K RFG model.
The lower panels show the ratio of events to the predicted total crosssection before re-weighting (blue points) and after re-weighting (redpoints). Left: The result of the re-weighting using the raw values fromneut. Large weights cause irregularities for some events. Right: The
level of agreement in the total cross section with corrections applied by
the T2K model.
for changes in the other parameters,
p1 = C +U D(C − 1)
,
3

(6.9)

p2 = C − 1.

(6.10)The parameters A, B, C, D, and U are set to nominal values in the T2K analysis,
then constrained by the fit to the near detector data2 . This function is implemented
2

The BeRPA values are listed for “A”, “B”, “D”, “E”, and “U” in Table XXV of [144]. The “D”
value corresponds to the “C” parameter, and the “E” value corresponds to the “D” parameter inEquation 6.8.

189

1.4

RPA Weight

1.2

Pre-ﬁt1
ND280 Post-ﬁt
0.8
0.6
0.4
0

0.5

1

1.5

2

q2 (GeV2)

Figure 6·12: The “BeRPA” parametrization of the RPA correctionused in the T2K analysis and re-implemented as part of the T2K model.
The x-axis is the four-momentum transfer, q 2 , and the y-axis shows thecorrection factor applied to bound-nucleon CCQE MC events.
within the T2K model, and it is applied as a correction to CCQE interactions withbound nucleons. The function is shown for the pre-fit and ND280 post-fit values fromthe T2K analysis in Figure 6·12.
2p2h: Both the T2K analysis and the SK analysis use the same 2p2h model, discussed
in Section 4.2.1. However, the T2K analysis introduces an extra parameter to control
the relative amount of the quasi-elastic and ∆-baryon-like resonance components ofthe 2p2h model. The parameter is then fit in the T2K analysis including the T2K
near detector data. This parameter is not available in the default version of neut,
and its implementation is not yet publicly available. Therefore, the T2K model does
not re-weight 2p2h events.
Resonant pion production: Both the SK and T2K analyses use the same resonant
single pion production models, although there are differences in the nominal values of190

MARes , CA5 , and I 1 . The T2K model re-weights the atmospheric neutrino MC events
2using the double-differential cross section ratio between the two sets of parameters.
This is the same method used to computing the shift in neutrino event weights due
to uncertainties in the resonant pion production parameters, discussed in Section 5.2.
Coherent pion production: The T2K analysis and the SK analysis use the same coherent pion production models. The T2K analysis also implements relative normalization
parameters for the CC and NC coherent pion production processes. The T2K model
implements these parameters by scaling the weight of these processes.
DIS: At T2K energies, DIS events are only a small fraction of the total event rate, sothese are not re-weighted, despite slightly different treatment between SK and T2K.
Event Selection
The T2K oscillation analysis includes five single-ring samples: FHC e-like, FHC µlike, FHC CC 1Ã-like, RHC e-like, and RHC µ-like. The FHC and RHC e-like andµ-like samples target CCQE interactions by selecting a events with a single, e-likering with no decay electrons, or a single µ-like ring with zero or one decay electrons.
The FHC CC 1Ã-like sample selects single-ring e-like events with one decay electron.
This is the expected final state for a neutrino CC interaction which produces a Ã +below Cherenkov threshold3 .
Events in all samples must pass quality cuts, analogous to the SK atmospheric
neutrino reduction. The visible energy, EVis , must be larger than 30 MeV, and the
maximum number of OD hits in a single OD cluster must be smaller than 15. Events
must also be contained within a fiducial volume. While the SK fiducial volume isdefined by a fixed, minimum vertex distance to the ID walls, the T2K fiducial volume
3

In RHC mode, the equivalent anti-neutrino interaction produces a π − which is likely to beabsorbed before decaying.

191

Selection

1-ring µ-like

1-ring e-like
0 d.e.

CC 1Ã
(FHC Only)

Dist. from wall (cm)Dist. to wall (cm)PID
Momentum (MeV/c)
Decay electrons
NC Ã Rejection
EνRec (MeV)

> 50
> 250
e
Λµ < 0.2pe
> 200
f1
π+
Λµ < 0.15pµ
–

> 80
> 170

> 50
> 270
Λeµ > 0.2pe
> 100

0
1
π0
Λe < 175 − 0.875mγγ< 1250

Table 6.5: Summary of reduction steps for the five single-ring samples
in the T2K analysis. The µ-like and e-like with zero decay electron
samples are identical in FHC and RHC mode. The definition of Λαβ
is given in Equation 6.11. This table is reproduced from Table IV of[144].
is defined by both the vertex distance to the wall, “dwall,” and the vertex distance tothe wall along the fitted direction of the ring, “towall.” The combination of dwall
and towall allows events with vertices closer to the wall to be kept in cases where
the Cherenkov ring is pointed away from the nearest wall. The chosen cut values
for the fiducial volume are separately optimized for each sample, and are listed inTable 6.5. Events with multiple rings and two or more decay electrons are not usedin the T2K Runs 1–9 analysis. The distribution of the number of rings, and the
distribution of decay electrons, for events which pass the fiducial volume cuts areshown in Figure 6·13 and Figure 6·14, respectively.
The remaining single ring events are further separated based on their PID characteristics. The T2K analysis separates events using the likelihood ratio between the
different particle hypotheses considered by fitqun, expressed asΛαβ ≡ log

192

Lα
Lβ

(6.11)RHCNC
CC non-QE
400

T2K-Weighted Events

T2K-Weighted Events

FHC

CCQE

200

0

1

2

3
4
5
Number of Rings

200

CCQE

100

0

6

NC
CC non-QE

1

2

3
4
5
Number of Rings

6

Figure 6·13: Number of rings distribution for T2K model MC eventsin Left: FHC and Right: RHC modes. Events with dwall > 80 cm
and towall > 170 cm are shown. Oscillations are applied using theparameters listed in Table 6.3. The figures show the equivalent distributions to Figure 16 from [144].

RHC

NC
CC non-QE
νµ and νµ CCQEνe and νe CCQE

300

T2K-Weighted Events

T2K-Weighted Events

FHC

200

100

0

0

150

100

50

0

1
2
3
4
5
Number of Decay-ElectronsNC
CC non-QE
νµ and νµ CCQEνe and νe CCQE

0

1
2
3
4
5
Number of Decay-Electrons

Figure 6·14: Number of decay electrons distribution for T2K model
MC events in Left: FHC and Right: RHC modes. Events shown are
the same as in Figure 6·13. The figures show the equivalent distributions to Figure 18 from [144].

193

RHC

80

νµ and νµ CC60

νe and νe CC
NC

40
20
0
−2000

−1000
0
1000
e/ µ PID discriminatorνµ and νµ CC

T2K-Weighted Events

T2K-Weighted Events

FHC

40

νe and νe CC
NC

20

0
−2000

2000

−1000
0
1000
e/ µ PID discriminator

2000

Figure 6·15: fitqun e-µ discriminator distribution for T2K model
MC events in Left: FHC and Right: RHC modes. Events with negative scores are considered e-like, while events with positive scores are
considered µ-like. The events shown are the same as in Figure 6·13.
The figures show the equivalent distributions to Figure 17 from [144].
where ³ and ´ are one of e± , µ± , Ã ± or Ã 0 . T2K separates µ- and e-like rings using
a momentum-dependent cut, Λeµ − pe /5 MeV/c > 0 for e-like, and µ-like otherwise,
where pe is the fitqun fitted momentum assuming the electron hypothesis. The e/µ
separation condition is visualized in Figure 6·15. Selected µ-like events then musthave a reconstructed momentum pµ > 200 MeV/c, while e-like events must have areconstructed momentum pe > 100 MeV/c.
The T2K analysis rejects NC backgrounds by separating µ-like events from charged
pions, and e-like events from neutral pions. The separation conditions are based on
the fitqun likelihood ratios,+(µ-like)Λπµ − 6pµ /40 MeV/c < 0(6.12)(e-like)Λπe − 175 + 35mγγ /40 MeV/c2 < 0(6.13)

0where mγγ is the reconstructed invariant mass of two photon rings, assuming a Ã 0
hypothesis. The distributions of the µ/Ã ± and e/Ã 0 discrimination variable are shown
194

RHC

νµ and νµ CC

T2K-Weighted Events

T2K-Weighted EventsFHC

νe and νe CC
NC

40

20

0
−300

−200
−100
0
µ/ π+ PID discriminator100νµ and νµ CC
νe and νe CC
NC

20

10

0
−300

−200
−100
0
µ/ π+ PID discriminator

100

Figure 6·16: fitqun µ-Ã + discriminator distribution for T2K model
MC events in Left: FHC and Right: RHC modes. Events with scores< 0 are selected; events with positive scores are rejected as NC-like.
Events shown have passed all other cuts of the µ-like reduction. The
figures shows the equivalent distributions to Figure 21 from [144].
in Figure 6·16 and Figure 6·17, respectively.
The last selection requires that EνRec. < 1250 MeV for the e-like samples. Electron
neutrinos above this energy do not add sensitivity to the ¿e appearance signal. Whencomputing EνRec. via Equation 6.5, T2K uses the proton and neutron masses, and an
NRE value of 27 MeV, for the initial and final nucleon masses. For the CC 1Ã sample,
since a resonant pion production process is assumed, T2K instead uses the mass ofthe ∆ baryon for the nucleon masses and an NRE value of 0.
Tables 6.6–6.8 list the number of re-weighted atmospheric neutrino MC events
remaining after each reduction step for each of the five T2K samples. The numbersin the penultimate row of each table show the accepted signal and background events
remaining after all reduction steps. For comparison, the final row of each table lists
the number of MC events from the T2K analysis in parentheses.
The final T2K model MC distributions of the five T2K samples are shown in
Figure 6·18 and Figure 6·19. The three e-like samples are have 10 EνRec. bins each from195

FHC e-like¿µ + ¿¯µCC¿e + ¿¯eCC¿ + ¿¯NC¿µ → ¿eCC¿¯µ → ¿¯e
CC

FC and FVSingle particle
Electron likepe > 100 MeV/c
No decay-e
Erec < 1250 MeV
Not Ã 0643.69324.62
6.72
2.76
0.65
0.26
0.11

40.73
23.18
23.12
23.02
19.68
10.23
9.56

199.14
40.50
22.19
14.94
12.68
8.23
4.11

80.72
70.03
69.79
68.84
62.42
60.43
58.30

0.75
0.61
0.61
0.61
0.60
0.44
0.40

T2K

(0.27)(8.79)(4.21)(58.53)(0.38)FC and FVSingle particle
Electron likepe > 100 MeV/c
No decay-e
Erec < 1250 MeV
Not Ã 0311.24168.00
2.14
1.14
0.27
0.11
0.05

21.33
12.08
12.06
12.03
10.80
4.73
4.35

87.64
18.06
9.83
6.74
5.73
3.70
1.71

5.75
4.40
4.39
4.35
3.81
3.17
3.00

10.00
8.94
8.91
8.88
8.83
8.30
7.81

T2K

(0.13)(3.70)(2.40)(2.65)(7.37)

RHC e-like

Table 6.6: Nominal T2K model MC e-like event counts after eachreduction step. The top and bottom portions of the table show thenumbers in the FHC and RHC samples, respectively. Counts are shownwith oscillations applied, calculated using the parameters listed in Table 6.3. The final row, labeled “T2K” shows the numbers from the T2K
analysis, listed in Table V from [144].

196

RHC

20

νµ and νµ CC
νe and νe CC

15

NC

T2K-Weighted Events

T2K-Weighted Events

FHC

10
5
0

−100

NC

3
2
1
0

0
100
200
300
e/ π0 PID discriminatorνµ and νµ CC
νe and νe CC

4

−100

0
100
200
300
e/ π0 PID discriminator

Figure 6·17: fitqun e-Ã 0 discriminator distribution for T2K model
MC events in Left: FHC and Right: RHC modes. Events with scores< 0 are selected; events with positive scores are rejected as NC-like.
Events shown have passed all other cuts of the e-like reduction. The
figures show the equivalent distributions to Figure 20 from [144].

FHC CC 1Ã-like¿µ + ¿¯µCC¿e + ¿¯eCC¿ + ¿¯NC¿µ → ¿eCC¿¯µ → ¿¯e
CC

FC and FVSingle particle
Electron likepe > 100 MeV/c
1 decay-e
Erec < 1250 MeV
Not Ã 0645.95320.52
6.78
2.74
1.32
0.29
0.12

41.48
23.33
23.28
23.17
3.15
1.06
0.97

203.37
40.61
22.78
15.18
1.85
0.66
0.33

80.86
69.91
69.71
68.71
6.34
5.84
5.54

0.75
0.61
0.61
0.61
0.01
0.00
0.00

T2K

(0.16)(0.93)(0.38)(5.64)(0.01)

Table 6.7: Nominal T2K model MC CC 1Ã-like event counts after
each reduction step. The counts are calculated using the oscillation
parameters listed in Table 6.3. The final row, labeled “T2K” shows thenumbers from the T2K analysis, listed in Table VI from [144].

197

FHC µ-like¿e + ¿¯eCC¿ + ¿¯NC¿µ + ¿¯µ
CC non-QE

¿µ
CCQE

¿¯µ
CCQE

FC and FVSingle particle
Muon likepµ > 200 MeV/c
0 or 1 decay-e
Not Ã +

116.84
90.76
0.27
0.27
0.27
0.26

194.59
39.69
18.30
18.10
17.65
8.43

349.91
77.86
73.73
73.70
48.77
46.87

241.05220.76217.62217.41215.41
212.47

15.18
13.94
13.87
13.87
13.79
13.58

T2K

(0.08)(8.31)(36.75)(210.64)(12.18)FC and FVSingle particle
Muon likepµ > 200 MeV/c
0 or 1 decay-e
Not Ã +

34.45
24.25
0.05
0.05
0.05
0.05

85.58
17.68
8.20
8.12
7.90
3.57

191.78
45.53
40.21
40.17
31.52
29.08

50.62
43.97
43.69
43.68
43.04
42.38

82.86
78.26
77.55
77.52
77.19
76.14

T2K

(0.02)(3.89)(24.71)(35.36)(65.00)

RHC µ-like

Table 6.8: Nominal T2K model MC µ-like event counts after each reduction step. The counts are calculated using the oscillation parameterslisted in Table 6.3. The final row, labeled “T2K” shows the numbers
from the T2K analysis, listed in Table VII from [144].

198

[0, 1250] MeV, while the two µ-like samples have 30 EνRec. bins each from [0, 3000] MeV.
Combined, the five T2K samples add 90 bins to the fit. Events with EνRec. > 3000 MeV
in the µ-like samples are consolidated in the last bin. The figures also show the binned
data counts from the T2K Runs 1–9 analysis, Figures 25-27 of [144].
6.3.4

Systematic Uncertainties

The T2K model implements several of the systematic uncertainties from [144]. For
the combined analysis with the SK atmospheric neutrino samples and the T2K model
samples, several of the systematic uncertainties described in Section 5.2 are replaced,and several new uncertainties are added.
Flux Uncertainties: The flux uncertainties in the T2K analysis consist of 25 parameters each for the FHC and RHC fluxes. These parameters change the normalizationof the flux in different energy ranges and for each of the neutrino flavors present in
the T2K beam. The T2K model implements a single flux uncertainty which varies all
50 of the flux parameters in each FHC and RHC mode from the T2K analysis by the1Ã uncertainties simultaneously. The ±1Ã variations in each of the flux parameters
are listed in Table 6.9.
CCQE Uncertainties: The CCQE-related uncertainties described in Section 5.2.2 are
replaced in favor of the equivalent uncertainties from the T2K analysis. The follow
uncertainties from the SK analysis are not used: CCQE shape, CCQE normalizations(both Sub-GeV and Multi-GeV), CCQE neutirno ratios (both ¿e /¿µ and ¿¯/¿). These
are instead replaced by the T2K uncertainties on the RFG Fermi momentum, pF , the
BeRPA parameters, and uncertainties on the ¿e /¿µ and ¿¯e /¯¿µ ratios. In addition, theeffect of a 1Ã change in MAQE on CCQE MC events was re-computed using the T2K
post-fit central value and uncertainty.
2p2h Uncertainties: The T2K analysis implements uncertainties on the normalization
199

T2K FHC e-Like

35
30
25
20

T2K runs 1-9 ModelDataνµ → νe CC
νµ → νe CC
νe → νe and νe → νe CC
νµ and νµ CC
NC

10

15
10
0
0

8
6

T2K runs 1-9 ModelDataνµ → νe CC
νµ → νe CC
νe → νe and νe → νe CC
νµ and νµ CC
NC

4
2

5
0.2

0.4

0.6

0.8

1

0
0

1.2

0.2

0.4

1

1.2

ERec.
(GeV)
ν

0.6
0.8
ERec.
(GeV)
ν

1

T2K FHC CC 1π
12Counts/125 MeVCounts/125 MeV

40

T2K RHC e-Like
12Counts/125 MeV

45

10
8
6

T2K runs 1-9 ModelDataνµ → νe CC
νµ → νe CC
νe → νe and νe → νe CC
νµ and νµ CC
NC

4
2
0
0

0.2

0.4

0.6
0.8
ERec.
(GeV)
ν

Figure 6·18: EνRec. distributions for the final T2K model e-like MC
events, and corresponding T2K data counts. The MC includes theeffect of oscillations, computed with the parameters listed in Table 6.3.
The data are from [144] Figures 25 & 26. For the CC 1Ã sample,
the definition of EνRec. uses the mass of the ∆ baryon and no nucleon
removal energy.

200

1.2

FHC
Flux

Energy
(GeV)

Value

RHC
Flux

Energy
(GeV)

Value

¿µ
¿µ
¿µ
¿µ
¿µ
¿µ
¿µ
¿µ
¿µ
¿µ
¿µ
¿¯µ
¿¯µ
¿¯µ
¿¯µ
¿¯µ
¿e
¿e
¿e
¿e
¿e
¿e
¿e
¿¯e
¿¯e

[0.0 , 0.4]
[0.4 , 0.5]
[0.5 , 0.6]
[0.6 , 0.7]
[0.7 , 1.0]
[1.0 , 1.5]
[1.5 , 2.5]
[2.5 , 3.5]
[3.5 , 5.0]
[5.0 , 7.0]
[7.0 , ∞][0.0 , 0.7]
[0.7 , 1.0]
[1.0 , 1.5]
[1.5 , 2.5]
[2.5 , ∞][0.0 , 0.5]
[0.5 , 0.7]
[0.7 , 0.8]
[0.8 , 1.5]
[1.5 , 2.5]
[2.5 , 4.0]
[4.0 , ∞][0.0 , 2.5]
[2.5 , ∞]1.01 ± 0.06
1.03 ± 0.05
1.02 ± 0.05
0.98 ± 0.04
0.93 ± 0.06
0.95 ± 0.05
1.02 ± 0.04
1.04 ± 0.05
1.03 ± 0.04
0.99 ± 0.04
0.97 ± 0.05
0.98 ± 0.08
0.97 ± 0.05
0.98 ± 0.06
1.03 ± 0.06
1.10 ± 0.07
1.02 ± 0.05
1.02 ± 0.04
1.02 ± 0.04
1.01 ± 0.04
1.03 ± 0.04
1.03 ± 0.04
1.03 ± 0.06
1.04 ± 0.06
1.08 ± 0.12

¿¯µ
¿¯µ
¿¯µ
¿¯µ
¿¯µ
¿¯µ
¿¯µ
¿¯µ
¿¯µ
¿¯µ
¿¯µ
¿µ
¿µ
¿µ
¿µ
¿µ
¿¯e
¿¯e
¿¯e
¿¯e
¿¯e
¿¯e
¿¯e
¿e
¿e

[0.0 , 0.4]
[0.4 , 0.5]
[0.5 , 0.6]
[0.6 , 0.7]
[0.7 , 1.0]
[1.0 , 1.5]
[1.5 , 2.5]
[2.5 , 3.5]
[3.5 , 5.0]
[5.0 , 7.0]
[7.0 , ∞][0.0 , 0.7]
[0.7 , 1.0]
[1.0 , 1.5]
[1.5 , 2.5]
[2.5 , ∞][0.0 , 0.5]
[0.5 , 0.7]
[0.7 , 0.8]
[0.8 , 1.5]
[1.5 , 2.5]
[2.5 , 4.0]
[4.0 , ∞][0.0 , 2.5]
[2.5 , ∞]1.00 ± 0.07
1.01 ± 0.05
0.99 ± 0.05
0.97 ± 0.04
0.97 ± 0.05
0.99 ± 0.05
1.03 ± 0.04
1.06 ± 0.05
1.06 ± 0.07
1.04 ± 0.06
1.00 ± 0.09
0.98 ± 0.07
0.99 ± 0.05
1.00 ± 0.05
1.05 ± 0.05
1.04 ± 0.05
1.01 ± 0.05
1.00 ± 0.05
1.00 ± 0.05
1.01 ± 0.04
1.04 ± 0.05
1.04 ± 0.07
1.08 ± 0.13
1.04 ± 0.05
1.04 ± 0.07

Table 6.9: List of T2K flux re-weighting and uncertainty parameters
implemented in the T2K model. The values scale the flux normalizationfor each flavor in the given energy range. The values are from the“ND280 postfit” columns of Tables XIII-XIV in [144].

201

T2K FHC µ-Like
50

30

T2K runs 1-9 ModelDataνµ CC non-QE
νµ CCQE
νµ CC Non-QE
νµ CCQE
νe and νe CC
NC

30
20

T2K runs 1-9 ModelDataνµ CC non-QE
νµ CCQE
νµ CC Non-QE
νµ CCQE
νe and νe CC
NC

25Counts/100 MeV

40Counts/100 MeV

T2K RHC µ-Like

10

20
15
10
5

0
0

0.5

1

1.5
2
ERec.
(GeV)
ν

2.5

0
0

3

0.5

1

1.5
2
ERec.
(GeV)
ν

2.5

3

Figure 6·19: EνRec. distributions for the final T2K model µ-like MC
events, and corresponding T2K data counts. The MC includes theeffect of oscillations, computed with the parameters listed in Table 6.3.
The data are from [144] Figure 27.
of neutrino and anti-neutrino 2p2h events, as well as a separate parameter on the ∆baryon resonance component of 2p2h interactions. Since this final parameter is not
implemented in the T2K model, the 100 % uncertainty on the normalization of 2p2h
processes from the SK analysis is used instead, as a conservative estimate.
Single Pion Production Uncertainties: The effects of a 1Ã change in MARes , CA5 , and theI 1 parameter on resonant single-pion MC events, and the normalizations of both CC
2and NC coherent pion MC events, were re-computed using the T2K post-fit central
value and uncertainty.
The SK resonant pion uncertainty model includes separate uncertainties on theÃ ± /Ã 0 ratio and the ¿¯/¿ ratio from a comparison of single-pion production models.
The T2K analysis does not include these uncertainties directly in the fit. Instead,the T2K analysis estimates the effect of using an alternative single-pion production
model from [145] on fitted oscillation parameters directly with toy data studies. Thesethe toy data studies involve simulations of both near and far detector data, and so202are not reproducible using the T2K model. Due to this limitation, the original SK
uncertainties are kept in the fit as a conservative estimate of the model differences.
FSI+SI Uncertainties: Both the T2K analysis and the SK analysis use the samemodels of FSI and SI processes implemented in neut, up to differences in the parameters, listed in Table 4.2 for the SK analysis. While it is possible to re-weight MCevents to resolve the parameter differences, the FSI uncertainty treatment between
SK and T2K is not compatible. For the T2K analysis, the FSI parameters are varied
as part of the fit, while in the SK analysis, the largest effect due to variations of fixedparameter sets is used instead (see Section 5.2.2). In addition, the T2K fit treats
the variations of the different FSI parameters as correlated, and the correlations are
not published. As a result, the T2K model does not apply a re-weighting based on
the FSI parameters, and the FSI uncertainties from the SK analysis are used as aconservative estimate.
NC Uncertainties: The T2K analysis places separate uncertainties on the normalization of NC processes, including an independent normalization uncertainty on NC
processes producing a single photon, an important background to the ¿e appearance
search. These uncertainties replace the NC/CC ratio uncertainty used by SK.
Table 6.10 lists the cross section parameter values and uncertainties implemented by
the T2K model.
Reconstruction Uncertainties: The T2K analysis uses atmospheric neutrino events
to estimate reconstruction performance. The estimation is performed similarly tothe scale-and-shift procedure, described in Section 5.2.3. The T2K model implements reconstruction uncertainties on the e/µ discriminator distribution, the e/Ã 0
discriminator distribution, and the µ/Ã + discriminator distributions using the same
scale-and-shift procedure from the atmospheric-only analysis. Additionally, the T2K
203

Parameter

Value

MAQE (GeV/c2 )pF (MeV/c)
BeRPA ABeRPA BBeRPA CBeRPA D
BeRPA U
CA5
MARes (GeV/c2 )I12¿e /¿µ Ratio¿¯e /¯¿µ Ratio
CC Coherent Norm.
NC Coherent Norm.
NC 1µ Norm.
NC Other Norm.

1.13 ± 0.08
205 ± 150.69 ± 0.06
1.60 ± 0.12
0.96 ± 0.13
0.87 ± 0.35
1.20 ± 0.10
0.98 ± 0.06
0.81 ± 0.04
1.31 ± 0.26
1.00 ± 0.03
1.00 ± 0.03
0.87 ± 0.28
0.94 ± 0.30
1.00 ± 1.00
1.00 ± 0.30

Table 6.10: Cross section parameter values used for re-weighting andcalculating systematic uncertainties in the T2K model. The parameters
are a subset of those listed in the “ND280 Postfit” columns of Table
XXV from [144].
model performs the scale and shift procedure separately for events with zero, one,and two decay electrons, and in each of six fiducial volume regions, defined by combinations of wall and towall. The detector regions are visualized in Figure 30 of[144].
The energy scale uncertainty during the SK IV phase uses the same implementation as in the atmospheric analyses, described in Section 5.2.3. A 1Ã change in theenergy scale uncertainty for the SK IV phase simultaneously varies the reconstructed
momentum values of SK IV atmospheric neutrino MC events and T2K model MC
events in the combined fit.
The T2K analysis places an uncertainty on the NRE value used in the calculation204

of EνRec. . However, this uncertainty is not part of the fit. Similarly to the alternative
single pion production model uncertainty, T2K quantifies variations in NRE using a
toy data study, which includes the propagation of its effects from the near detector
data to the far detector data analysis. The result of the study showed that variationsof the NRE value assumed could produce some bias in the fitted value of ∆m232 . To
account for the bias, the allowed range of ∆m232 was smeared after the T2K oscillation
fit. The smearing routine is not implemented within the T2K model, so the effects of
NRE are neglected. The added uncertainty on ∆m232 was reported as 0.041×10−3 eV2 .
¿e Appearance Normalization Uncertainties: The T2K experiment’s sensitivity to ¶CP
and the neutrino mass ordering is driven by the total number of ¿µ → ¿e and ¿¯µ → ¿¯eevents, i.e., largely independent of reconstructed neutrino energy and the other fitted
oscillation parameters, sin2 ¹23 and ∆m232 . An increase or decrease in the nominal MC
prediction of these electron neutrino appearance events can bias the fit result to be
more or less sensitive to ¶CP . Since the T2K model does not implement all featuresof the T2K analysis, differences in the nominal MC predictions are expected.
The difference in the total number of electron neutrino appearance events in thenominal MC predictions of T2K model and the T2K analysis was estimated by implementing two additional features from the T2K analysis. For this uncertainty estimation only, the ratios between the near detector-constrained and default neutvalues of the 2p2h normalization parameters and the FSI parameters were used to
re-weight T2K model events. The differences in nominal MC prediction between theT2K model and T2K analysis after applying these weights is 5 % for the total numberof ¿µ → ¿e events, and 2 % for the total number of ¿¯µ → ¿¯e events. These differences
are taken as the 1Ã effects of two additional systematic uncertainties which independently modify the normalization of electron and anti-electron neutrino appearance
events in the T2K model MC.
2056.3.5

T2K Model Performance & Limitations

The T2K model, fit to the binned 1D T2K Runs 1–9 data, and the T2K analysis,
produce ∆Ç2I.O.−N.O. ∼ 3.6. The T2K model finds a best fit in the normal ordering,with ∆m232 = 2.45 × 10−3 eV2 , sin2 ¹23 = 0.55, and ¶CP = −1.74. These are within
the uncertainties of the fitted values from the T2K analysis, visualized as fractional
differences from the T2K analysis values in Figure 6·20. Note that the smaller uncertainties on ∆m232 measured by the T2K model do not include the additional post-fit
uncertainty due to NRE added in the T2K analysis. Figure 6·21 shows the constraints
on ¶CP in the two neutrino mass ordering scenarios from the T2K analysis, Figure 54of [144], and the T2K model, fit using the SK method, cf. Equation 5.15.
T2K Runs 1–9, Best fit
T2K PRD 2021
T2K Model% Difference

40
20
0
-20
-40
δCP

sin2θ23Δm232

Δχ2I.O.-N.O.

Figure 6·20: 1Ã allowed oscillation parameter ranges from the T2K
model and T2K analysis. The T2K PRD 2021 values are taken fromthe “NO” column of Table XV from [144].
While the T2K model successfully reproduces some aspects of the T2K analysis,
it has three primary limitations:
1. Lack of near detector description

206

∆χ2

T2K Run 1-9 Model
Inverted
20
NormalT2K PRD 2021
15
Data fit
10
5
0
−π

−π/2

0
δCP

π/2

π

Figure 6·21: 1D ∆Ç2 profiles of ¶CP from the T2K model and T2K
analysis. Both ∆Ç2 profiles are drawn with respect to the best-fit pointin the normal ordering. The T2K PRD 2021 contours are taken from
Figure 54 of [144].
2. Differences in uncertainty treatment
3. Reliance on re-weighting

Lack of near detector description: The near detector data set is integral to the T2K
oscillation analysis, as it is jointly fit with the far detector samples to constrain theflux and cross section models. The T2K model does not describe the T2K near detector data. Without a near detector description, the T2K model is unable to reproduce
the T2K toy data uncertainty studies, and cannot perform a simultaneous fit between
the T2K near detector, T2K far detector, and SK atmospheric data samples. Such asimultaneous fit is desirable for a true joint analysis of all available data to assess thedegree of agreement in cross section models between each data set. The T2K model
assumes that the addition of SK atmospheric neutrino data would not significantly
207change the T2K near detector constraint. This assumption is based on the larger
uncertainties in reconstructing the energies of atmospheric neutrino events, originating from the wider energy range of the atmospheric neutrino flux than for the T2K
samples, and the unknown true neutrino direction. However, this assumption has notbeen quantified for the T2K model analysis.
Differences in uncertainty treatment: Although the T2K model implements similar
uncertainties to those in the T2K analysis, the T2K analysis does not implement
its uncertainties using the Fij method of the SK analysis (see Section 5.3.2). In the
SK method, all uncertainty sources are uncorrelated, and the response of the Fij s
is linear. In the T2K analysis, uncertainty sources are correlated, and several have
non-linear responses in the fit. These differences create an interpretation issue whencomparing results between the T2K model and the T2K analysis, i.e., even if the T2K
model was able to perfectly reproduce the effect sizes of each systematic uncertainty,
the differences in fit method could still produce different allowed regions of neutrino
oscillation parameters.
Reliance on re-weighting: Re-weighting from a source distribution to a target distribution is only possible when the source distribution overlaps with the target distribution.
In the case of re-weighting SK events to T2K events, the atmospheric neutrino flux
overlaps the neutrino energy range of the T2K flux distribution, and the cross section
models used by each collaboration have similar phase-spaces. However, as discussed
in Section 6.3.3, the CCQE models do not have complete overlap, and so re-weighting
will necessarily be inaccurate. Further, even if there is overlap between the source
and target distributions, re-weighting events can lead to an undesirable dependence
on MC statistics: If the original MC set has few events in a region where the target
MC has many events, the weights may need to be large, amplifying any statisticalfluctuations present in the original MC. This is visible in Figure 6·9: The re-weighted
208events have statistical MC uncertainties according to the Honda flux, despite being
re-weighted on average to the T2K flux. The re-weighting issue is not severe for thisversion of the T2K model, but the method would not work in general to convert
between other cross section and flux models.
The T2K and SK analyses also use different methods to draw allowed oscillation
parameter ranges. This means that the ∆Ç2 contours are expected to differ, independent of the T2K model’s ability to reproduce the T2K analysis’ MC and systematic
uncertainties. The T2K analysis uses the marginalization technique: The likelihood
after fitting is integrated over all other parameters to draw the 1D projection. The SK
analysis, on the other hand, uses the profiling technique: The likelihood is maximized
over all other parameters, equivalent to taking the minimum ∆Ç2 value over all otherparameters. As pointed out in [146], the marginalization and profiling procedures
produce different contours when the fitted model has a non-linear dependence on theparameters, as is the case for neutrino oscillation analyses. Generally speaking, theprofiling method, used to analyze results in this thesis, is more conservative.
Figure 6·21 shows the comparison of constraints on ¶CP from the T2K analysis,
drawn with the marginalization technique, and the T2K model fit, drawn using the
profiling technique used in the SK analysis. In the figure, the T2K model appears
slightly conservative, indicated by the smaller values of ∆Ç2 for some values of ¶CP .
Since the techniques used to draw the ∆Ç2 profiles differ, the conservative result of
the T2K model fit compared to the T2K analysis cannot be entirely attributed todifferences between the T2K analysis and the T2K model.

6.4

SK+T2K Model Results

In order to treat the cross section uncertainties as correlated between the T2K samples
and the atmospheric neutrino samples in the analysis, all samples must use the same
209

Parameter

Min.

Max.

Steps

sin2 ¹23∆m232 or ∆m231 (10−3 eV2 )
¶CP

0.3
1.5
0

0.725
3.45
2Ã

35
40
37

Table 6.11: The oscillation point grid used for the SK+T2K model
analysis with sin2 ¹13 constrained. Oscillation parameters are scannedin equally spaced steps, including the minimum and maximum points
listed in the table. The grid is scanned twice, once for each massordering.
cross section models. Consequently, the cross section re-weighting aspects of theT2K model are also applied to the MC events used for the SK atmospheric neutrino
samples, and the Fij s are re-computed for the atmospheric neutrino bins using thenominal prediction after the cross section re-weighting.
The SK+T2K model analysis maintains the constraint on ¹13 . The oscillation gridis also updated to increase the number of oscillation points scanned in a narrower
range of ∆m232 values, reflecting T2K’s improved precision in this parameter. The
oscillation parameter grid for the SK+T2K model analysis is listed in Table 6.11.
The 1D profiles of the fitted oscillation parameters are shown in Figure 6·22, andthe allowed 1Ã ranges for each of the parameters are listed in Table 6.12. Note thatthe Ç2 values are computed for 1020 bins, increased from 930 bins in the atmosphericanalyses. The constraints on ¶CP , ∆m232 , and sin2 ¹23 have all improved dramaticallycompared to the atmospheric-only analyses, reflecting both the excellent precision of
the T2K experiment to measure these parameters, and the good complementarity of
the SK data with the T2K data. The normal mass ordering preference is ∆Ç2I.O.−N.O. =8.54, an increase of 2.85 compared to the atmospheric analysis with ¹13 constrained.
This increase is comparable to the T2K model’s ∆Ç2I.O.−N.O. value when fit alone,indicating very good agreement between the SK and T2K model preferences for thenormal mass ordering. The T2K model prefers ¹23 in the upper octant (sin2 ¹23 > 0.5).
210∆m232,312

sin2 ¹23

Normal2.40+0.06−0.02

0.51+0.04−0.04

Inverted

2.40+0.02−0.04

0.53+0.03−0.04

Ordering(10−3 eV )

¶CP

Ç2

Ç2

(−π, π)1020 bins

Syst.

−1.75+0.41−0.59

1111.4156.07

1119.95

56.24−1.57+0.39−0.43

Table 6.12: Best-fit neutrino oscillation parameters in the combined
SK+T2K model analysis with sin2 ¹13 constrained. The uncertainties
on each oscillation parameter are the ±1Ã allowed regions assuming a
Ç2 distribution with one degree of freedom.
This is in slight tension with the atmospheric results, but the T2K constraints aremuch stronger, bringing the SK+T2K model fitted value closer to the T2K model
preferred value of sin2 ¹23 .

6.5

Discussion

The SK atmospheric and T2K beam neutrino samples are largely in agreement: Bothobserve excess ¿e appearance, best explained by the normal ordering and values of¶CP near −Ã/2. Like the atmospheric-only result, the T2K Runs 1–9 result from [144]also exceeds its sensitivity for preferring the normal mass ordering, ∆Ç2I.O.−N.O. ≈ 2.
This is consistent with T2K observing ¿e appearance in excess of the prediction fromthe best-case scenario, ¶CP ∼ −Ã/2 with a further enhancement from matter effects
in the normal ordering.
The ∆Ç2I.O.−N.O. result indicates a strong preference for the normal mass ordering.
Figure 6·23 shows the distribution of ∆Ç2I.O.−N.O. values and corresponding p-values
for the SK+T2K model fit, cf. Section 5.5. The data fit result is shown as a solidline at ∆Ç2I.O.−N.O. = −8.54, and corresponds to a p-value of p = 5.5 × 10−3 ≈ 2.54 Ã
assuming the inverted ordering. In contrast, the p-value calculated from the measured√∆Ç2I.O.−N.O. value assuming Wilks’ Theorem is notably larger: 8.54 ≈ 2.92 Ã. The
SK+T2K model CLs value, following Equation 5.17, is CLs = 0.021. This value
211

50

50
SK I-V + T2K Runs 1-9 Model
Data fit
MC expectation
Inverted
Normal

40
35

40
35
30

25

25

20

20

15

15

10

10

5

5

0
−π

0

−π/2

π/2

0
δCP

π

99%
68%

2

2.2
2.4
2.6
-3
2
∆m32,31(10 eV2)

50
45
40
35
30
∆χ2

∆χ2

30

45

∆χ2

45

25
20
15
10
5
0
0.35 0.4 0.45 0.5 0.55 0.6 0.65
sin2θ23

Figure 6·22: 1D ∆Ç2 profiles of the fitted oscillation parameters for
the SK+T2K model analysis with sin2 ¹13 constrained. The ∆Ç2 values
are taken with respect to the best-fit in the normal ordering, listed inTable 6.12. The meaning of the colors, solid and dashed curves, anddotted lines is the same as in Figure 5·14.

212

2.8

Data=-8.54
True N.O.

10

1

True I.O.

p = 0.74
p = 0.0055

Fraction of Toy MCs

N.O.
I.O.

10

2

10

3

10

4

20

10

0
2
N. O.

10

20

I. O.

Figure 6·23: Distribution of ∆Ç2I.O.−N.O. from toy data sets and pvalues in the SK+T2K model analysis with ¹13 constrained. The meaning of the histograms is the same as in Figure 5·17.
corresponds to a rejection of the inverted mass ordering at 97.9 % ≈ 2.03 Ã.
The benefit of shared systematic uncertainty on the oscillation fit is shown in
Figure 6·24. The figure compares the ¶CP ∆Ç2 profile from the SK+T2K model
fit, i.e., the upper- left panel from Figure 6·22, with the sum of ∆Ç2 profiles fromindividual fits to the SK samples and the T2K samples. The combined analysis places
stronger constraints on ¶CP than the sum, indicating an improvement due to the effectof shared systematic uncertainties.

213

50

∆χ2

SK I-V + T2K run 1-9 model
Inverted
40
NormalCombined fit
SK/T2K ∆χ2 sum
30
20
10
0
−π

−π/2

0
δCP

π/2

π

Figure 6·24: 1D ∆Ç2 profiles of ¶CP from the SK+T2K model analysis,
fit as a combined data set and separately summed. The solid line is thesame as in the upper-left panel of Figure 6·22.

214Chapter 7

Conclusions
The neutrino mass ordering connects our understanding of fundamental neutrino
properties to far-reaching areas of physics. Discerning the mass ordering would fix
the neutrino’s role in cosmology, further motivate searches for neutrino-less double
beta decay experiments, and help us interpret the flux of neutrinos from supernova.
This thesis presented an analysis of the world’s richest atmospheric neutrino data
set for probing the mass ordering using neutrino oscillations. The Super-Kamiokande(SK) experiment’s longevity, size, and detection technique make it uniquely wellsuited to search for the mass ordering signature. At the same time, disentangling
the signatures of the mass ordering from neutrino oscillations is complicated—the SKdata tell the following story:• We observe neutrino oscillations, well-described by the three-flavor PMNS
paradigm, which are consistent with measurements from other sources of neutrinos.
• We observe an excess of upward-going electron neutrino-like events, with a few
GeV of energy.
• We don’t observe any excess of the equivalent anti-electron neutrino-like events.
The normal neutrino mass ordering scenario predicts this situation, while the inverted
neutrino mass ordering does not. It appears the answer is at hand. Not so fast.

215

The neutrino mass ordering is unlike other measurements in physics: Insteadof a mass or coupling, which could take a continuous range of possibilities, the massordering is a discrete choice between two options. When we ask our data to choose oneor the other, unexpected results can happen. As this thesis showed, both the SK and
T2K experiments exceed their sensitivity. These excesses should not necessarily beinterpreted as stronger evidence for the normal ordering. Rather, the excesses result
from unlikely scenarios which should instead motivate us to ask more questions.
At the same time, the analysis of atmospheric neutrino data with the T2K model
demonstrated good agreement between the two data sets. This analysis observed
maximum electron neutrino appearance in both the atmospheric samples and beam
neutrino samples simultaneously, best explained by the combination of ¶CP ≈ −Ã/2
and the normal mass ordering. This result is intriguing because it requires consensus between both naturally-produced atmospheric neutrinos and artificially-produced
beam neutrinos which are subject to different systematic effects. It also showcases
the benefits of performing the mass ordering analysis using atmospheric neutrino andbeam neutrino data collected in a single detector. The planned Deep Underground
Neutrino Experiment (DUNE) [147] and Hyper-Kamiokande experiments [148] will
have the opportunity to perform single-detector combined atmospheric and beam
neutrino analyses, and are recommended to do so!
While neither the atmospheric analysis nor the analysis including T2K constraints
obtain a sufficient 5 Ã (or even 3 Ã, depending on who you ask) statistical significanceto claim a discovery of the mass ordering, the outlook for the neutrino mass ordering
remains optimistic. If we are seeing hints of the normal mass ordering and values of¶CP ≈ −Ã/2, then nature has made our job easy: This scenario in particular predicts
the largest electron neutrino appearance in neutrino oscillation experiments. Observing the electron anti-neutrino excesses expected in the inverted mass ordering would216

require much higher statistics, simply because anti-neutrinos interact less frequently.
Similarly, other values of ¶CP lead to degeneracies in the predicted signal between themass ordering and the other oscillation parameters.
The SK detector is currently operating with gadolinium in its water, which promises
to increase its ability to detect neutrons. As this thesis demonstrated, SK does better at separating neutrinos and anti-neutrinos using neutrons. We expect further
sensitivity improvements to the mass ordering beyond increased statistics in a future
analysis using SK Gd data. There are also worldwide efforts to jointly analyze existingneutrino data now underway. While the T2K model analysis presented in this thesis
is one such way of combining data sets, the SK, T2K, and NOvA collaborations are
already sharing data and jointly developing analysis techniques. These joint analyses
may bring us closer to a deeper understanding of neutrino oscillations and may evendefinitively reveal the neutrino mass ordering within the next decade.

217

Appendix A

Best-Fit Atmospheric Neutrino Samples
The following figures show the data and MC counts in the 930 atmospheric neutrinobins used in the analyses presented in this thesis. The data counts are identical forthe atmospheric-only analysis, the atmospheric analysis with ¹13 constrained, andthe SK+T2K model analysis. The MC counts are shown with best-fit oscillation parameters and systematic uncertainty pulls (epsilons) from the atmospheric analysis
with ¹13 constrained, cf. Section 6.1. The MC counts are shown for both the normal
ordering (blue line) and the inverted ordering (orange line). While the inverted ordering fit is drawn in every figure, it is only visible in bins where the MC prediction
differs between the two orderings. Each figure also shows the 68 % variation of the
MC prediction at the best-fit oscillation parameters from 1000 random throws of thesystematic uncertainties, indicated by grey shaded regions.
Figures either show the zenith angle distribution in each momentum bin for eachsample, or the momentum distribution for samples where there is only a single zenith
angle bin. The Up-µ through-going samples only use a single momentum bin. The
bin definitions are described in Section 5.1.4, and the momentum bin definitions arelisted in Table 5.1.

218

SK I-III Sub-GeV e -like 0 d.e.
2.0<Log10[p (MeV)]<2.4

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Log10[p (MeV)]>3.0

Events

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Log10[p (MeV)]>3.0

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Figure A·1: SK I-III Sub-GeV single-ring samples, and Sub-GeV samples with only a single zenith angle bin.

219

1

SK IV-V Sub-GeV νe-like
2.0<Log10[p (MeV)]<2.4

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Log10[p (MeV)]>3.0

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Figure A·2: SK IV-V Sub-GeV single-ring samples.

220

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0.5
cosθz

0
−1 −0.5

1

0
0.5
cosθz

20

EventsEventsEvents50

Events

60

100

10
0
−1 −0.5

1

0
0.5
cosθz

10
5
0
−1 −0.5

1

0
0.5
cosθz

1

SK IV-V Multi-GeV νe-like
3.4<Log10[p (MeV)]<3.7

3.7<Log10[p (MeV)]<4.0

60

30

40
20
0
−1 −0.5

0
0.5
cosθz

EventsEventsEvents

80
60

40
20
0
−1 −0.5

1

Log10[p (MeV)]>4.0
20

0
0.5
cosθz

1

Events

3.0<Log10[p (MeV)]<3.4

20
100−1 −0.5

15
10
5

0
0.5
cosθz

1

0
−1 −0.5

SK IV-V Multi-GeV νµ-like
3.0<Log10[p (MeV)]<3.4

Log10[p (MeV)]>3.4
100

150

EventsEvents

200

100

50

50
0
−1 −0.5

0
0.5
cosθz

0
−1 −0.5

1

0
0.5
cosθz

1

SK IV-V Multi-GeV νµ-like
3.0<Log10[p (MeV)]<3.4

Log10[p (MeV)]>3.4

EventsEvents100

500−1 −0.5

0
0.5
cosθz

1

40
20
0
−1 −0.5

0
0.5
cosθz

1

Figure A·4: SK IV-V Multi-GeV single-ring samples.

222

0
0.5
cosθz

1

SK I-V Multi-Ring νe-like
3.0<Log10[p (MeV)]<3.4

3.4<Log10[p (MeV)]<3.7

Log10[p (MeV)]>3.7

150

100

100
50
0
−1 −0.5

0
0.5
cosθz

EventsEventsEvents

150
100
50
0
−1 −0.5

1

0
0.5
cosθz

50

0
−1 −0.5

1

0
0.5
cosθz

1

SK I-V Multi-Ring νe-like
3.0<Log10[p (MeV)]<3.4

3.4<Log10[p (MeV)]<3.7

100

100

Log10[p (MeV)]>3.7

50

60

EventsEventsEvents

80

40

50

20
0
−1 −0.5

0
0.5
cosθz

0
−1 −0.5

1

0
0.5
cosθz

0
−1 −0.5

1

0
0.5
cosθz

1

SK I-V Multi-Ring νµ-like
3.1<Log10[p (MeV)]<3.4

150

150

100
50

3.4<Log10[p (MeV)]<3.7

100
50

0
−1 −0.5

0
0.5
cosθz

0
−1 −0.5

1

Log10[p (MeV)]>3.7

150

0
0.5
cosθz

60
Events200
Events200
EventsEvents

2.0<Log10[p (MeV)]<3.1

100
50

20

0
−1 −0.5

1

40

0
0.5
cosθz

0
−1 −0.5

1

0
0.5
cosθz

1

SK I-V Multi-Ring Other
3.0<Log10[p (MeV)]<3.4

3.4<Log10[p (MeV)]<3.7

Log10[p (MeV)]>4.0
60

20
10

40
20

Events30

Events40
EventsEvents400−1 −0.5

3.7<Log10[p (MeV)]<4.0

60
30
20

20

10
0
0.5
cosθz

1

0
−1 −0.5

0
0.5
cosθz

1

0
−1 −0.5

40

0
0.5
cosθz

1

0
−1 −0.5

Figure A·5: SK I-V Multi-GeV multi-ring samples.

223

0
0.5
cosθz

1

SK I-V PC Stopping
2.0<Log10[p (MeV)]<3.4

Log10[p (MeV)]>3.4
60
EventsEvents

60
40

40
20

20
0
−1 −0.5

0
0.5
cosθz

0
−1 −0.5

1

0
0.5
cosθz

1

SK I-V PC Through-Going
2.0<Log10[p (MeV)]<3.1

3.1<Log10[p (MeV)]<3.4

3.4<Log10[p (MeV)]<3.7

Log10[p (MeV)]>3.7

200

20

50

300

150

Events40

100

Events60

EventsEvents

80

100

100

50

0
−1 −0.5

0
0.5
cosθz

0
−1 −0.5

1

0
0.5
cosθz

0
−1 −0.5

1

0
0.5
cosθz

1

SK I-V Up-µ Stopping
3.2<Log10[p (MeV)]<3.4

3.4<Log10[p (MeV)]<3.7

Log10[p (MeV)]>3.7
100

50

Events150
EventsEvents

100

100

50

50
0
−1 −0.8 −0.6 −0.4 −0.2 0
cosθz0−1 −0.8 −0.6 −0.4 −0.2 0cosθzNon-showering

Showering0−1 −0.8 −0.6 −0.4 −0.2 0cosθz

SK I-V Up-µ Through-going

200
EventsEvents

1000

500

0
−1 −0.8 −0.6 −0.4 −0.2 0
cosθz

100
0
−1 −0.8 −0.6 −0.4 −0.2 0cosθzFigure A·6: SK I-V PC and Up-µ samples.

224

200

0
−1 −0.5

0
0.5
cosθz

1

Appendix B

Best-Fit Systematic Uncertainties
This appendix provides listings of the fitted systematic error parameters (epsilons)at the best-fit oscillation points in the normal and inverted orderings for the analyses
presented in this thesis, cf. Section 5.4, Section 6.1.1, and Section 6.4. Table B.1
lists the flux, cross section, and oscillation systematic uncertainties which are, withthe exception of the solar activity uncertainty, common to all SK phases. Table B.2,
Table B.3, and Table B.4 list the detector-related systematic uncertainties which
are fit independently for each SK phase. Dashes indicate that a systematic uncertainty source is not used in a fit. In particular, the θ13 uncertainty is not used in
the atmospheric neutrino fit with no external constraints, and several cross section
uncertainties used in the SK atmospheric neutrino fits are replaced with alternative
uncertainties in the fit with the T2K model. The SK IV absolute energy scale uncertainty simultaneously varies the atmospheric neutrino and T2K samples in the
SK+T2K model fit.

225

Systematic
Uncertainty
Atmospheric ν Flux
Normalization
Eν < 1 GeV
Eν > 1 GeV(νµ + ν̄µ )/(νe + ν̄e ) ratioEν < 1 GeV
1 GeV < Eν < 10 GeV
Eν > 10 GeVνµ /ν̄µ ratioEν < 1 GeV
Eν > 10 GeV
1 GeV < Eν < 10 GeV
νe /ν̄e ratioEν < 1 GeV
Eν > 10 GeV
1 GeV < Eν < 10 GeV
K/π ratio
Up/Down Ratio
Horizontal/Vertical Ratio
Relative Normalization
FC Multi-GeV
PC+UPMU Stopping
Solar Activity
SK I
SK II
SK IIISK IV
SK V
Neutrino path length
CCQE
MAQE
pF
Shape
Norm., Sub-GeV
Norm., Multi-GeV
ν/ν̄ Ratio

SK Only

Fit Value(σ)
SK+θ13

SK+T2K Model

NormalInverted

NormalInverted

NormalInverted

0.20
1.32

0.23
1.33

0.20
1.32

0.20
1.31

0.70
1.50

0.72
1.51

0.46
−0.27
0.82

0.44
−0.20
0.80

0.44
−0.28
0.82

0.44
−0.34
0.83

0.27
−0.05
0.61

0.16
−0.04
0.61

0.09
−0.10
1.33

0.08
−0.11
1.35

0.09
−0.10
1.32

0.09
−0.12
1.30

0.10
−0.11
1.30

0.07
−0.11
1.30

0.22
−0.38
−0.09
−1.08
0.34
−0.11

0.26
−0.38
−0.12
−1.05
0.43
−0.12

0.24
−0.38
−0.08
−1.09
0.34
−0.11

0.22
−0.39
−0.02
−1.09
0.41
−0.19

0.54
−0.30
−0.10
−0.99
0.32
0.05

0.64
−0.30
−0.09
−1.00
0.31
0.06

−1.33
0.31

−1.32
0.35

−1.33
0.31

−1.29
0.29

−1.64
0.54

−1.64
0.54

−0.33
0.19
−0.23
0.13
−0.10
−0.51

−0.33
0.18
−0.23
0.14
−0.10
−0.44

−0.33
0.19
−0.23
0.14
−0.10
−0.53

−0.33
0.18
−0.23
0.14
−0.10
−0.42

−0.27
0.14
−0.24
0.12
−0.08
−0.77

−0.27
0.14
−0.24
0.12
−0.07
−0.81

−0.44
—
2.05
0.86
0.66
0.85

−0.42
—
2.06
0.83
0.66
0.82

−0.45
—
2.06
0.83
0.66
0.84

−0.45
—
2.05
0.83
0.65
0.85

−0.82
0.59
—
—
—
—

−0.86
0.58
—
—
—
—

Continued on next page

226

Table B.1 – Continued from previous page

Systematic
Uncertaintyνµ /νe Ratio
2p2h Norm.
BeRPA ABeRPA BBeRPA D
BeRPA E
BeRPA U
Single Pion Production
π 0 /π ± Ratioν̄/ν Ratio
MARes.
5
CA
I 21
CC Coherent Norm.
NC Coherent Norm.
NC π ± Contamination
DIS
PDF Difference
World Average Difference
q 2 , W > 2 GeV
q 2 (vec.), W < 2 GeV
q 2 (axi.), W < 2 GeV
q 2 Norm., W < 2 GeV
Hadron Multiplicity
Neutrons
Multiplicity, Transverse p
Generator Comparison
Neutron Tagging
FSI
FSI Max. Set
FSI Min. Set
Other Xsec.
NC/CC Ratio
NC 1γ Norm.
NC Other Norm.
CC ντ Cross Section

SK Only

Fit Value(σ)
SK+θ13

SK+T2K Model

NormalInverted

NormalInverted

NormalInverted

0.74
−0.38
—
—
—
—
—

0.77
−0.40
—
—
—
—
—

0.74
−0.38
—
—
—
—
—

0.73
−0.36
—
—
—
—
—

0.41
−0.97
0.44
0.55
0.62
−0.02
−0.04

0.40
−0.96
0.43
0.53
0.61
−0.02
−0.03

−0.83
0.13
−1.55
0.76
−0.91
−0.20
—
0.16

−0.84
0.17
−1.53
0.76
−0.93
−0.21
—
0.14

−0.84
0.14
−1.55
0.76
−0.91
−0.19
—
0.16

−0.84
0.13
−1.55
0.74
−0.95
−0.20
—
0.15

−0.09
0.69
1.10
2.68
−0.94
−0.23
0.63
0.06

−0.09
0.70
1.09
2.68
−0.95
−0.23
0.64
0.01

1.13
0.41
−0.30
0.75
−1.09
−0.41
0.36

1.13
0.42
−0.29
0.74
−1.11
−0.41
0.37

1.14
0.41
−0.30
0.75
−1.09
−0.41
0.36

1.17
0.42
−0.28
0.73
−1.08
−0.40
0.35

0.82
0.50
−0.38
1.02
−1.12
−0.55
0.23

0.83
0.49
−0.39
1.02
−1.11
−0.55
0.24

−0.31
0.40
−0.22

−0.31
0.40
−0.22

−0.31
0.40
−0.22

−0.31
0.40
−0.24

−0.23
0.45
−0.42

−0.23
0.45
−0.42

−0.06
−0.01

−0.06
0.00

−0.06
−0.01

−0.06
−0.01

−0.10
0.14

−0.10
0.14

1.83
—
—
−0.07

1.81
—
—
0.03

1.83
—
—
−0.09

1.85
—
—
0.15

—
0.03
1.25
−0.18

—
0.03
1.23
−0.21

Continued on next page

227

Table B.1 – Continued from previous page

Systematic
Uncertainty
Oscillation∆m221
sin2 θ12
Matter effectsin2 θ13
T2K
Post-ND280 Flux
νe Appearance Norm.
ν̄e Appearance Norm.

SK Only

Fit Value(σ)
SK+θ13

SK+T2K Model

NormalInverted

NormalInverted

NormalInverted

−0.01
0.02
0.18
—

−0.01
0.01
0.19
—

−0.01
0.02
0.18
0.01

−0.01
0.01
0.08
0.03

0.01
0.03
0.26
0.25

0.01
0.03
0.27
0.22

—
—
—

—
—
—

—
—
—

—
—
—

−0.30
0.45
0.00

−0.33
0.41
−0.01

Table B.1: Flux-, cross section-, and oscillation-related fitted systematic uncertainty parameters (epsilons) at the best-fit point for each of
the three analyses presented in this thesis.

228

Fit Value (σ)

Systematic
Uncertainty

NormalI

II

Fiducial Volume−0.45 −0.13
FC Reduction
−0.05
0.01
PC Reduction
0.05 −0.53
Up-µ Reduction
0.00 −0.11
FC/PC Separation −0.22
0.05
PC Stop/Through SeparationTop
0.39 −0.26
Barrel0.54 −0.35
Bottom−0.58
0.01
Ring Counting
0.54
0.42
Single-Ring PID
−0.13
0.45
Multi-Ring PID
−0.21
0.60
Non-ν Backgrounde-like0.00
0.05µ-like0.53
0.32
Non-νe BackgroundSingle-Ring
0.10 −0.56
Multi-Ring
0.24
0.21
Two-Ring π 0−0.39 −0.02
Energy Scale
Absolute−0.12 −0.38
Up/Down
0.03
0.08
Decay-e Tagging−0.63 −0.22
Multi-Ring BDT
Efficiency
−0.18
1.25
Migration
−0.10
0.16
Up-µ
Path Cut−0.01
0.42
Momentum Cut
0.01
0.04
Separation
Stop/Through
0.00
0.02
Non-showering
1.65 −0.52
Showering

III

Inverted
IV

V

I

II

III

IV

V

0.62 −0.42 −0.28 −0.46 −0.13
0.62 −0.43 −0.28
0.25
0.70
0.20 −0.05
0.01
0.25
0.71
0.20
0.00 −0.75 −0.25
0.05 −0.54
0.00 −0.76 −0.26
0.23
0.12
0.01
0.00 −0.11
0.23
0.12
0.01
−0.10
0.02
0.00 −0.22
0.05 −0.10
0.02
0.00
−0.28
−0.36
−0.03
0.22
−0.74
−0.21

−0.62
0.73
−0.32
−0.40
1.02
−0.93

−0.28
0.39 −0.26 −0.27 −0.62 −0.27−0.60
0.54 −0.35 −0.35
0.74 −0.59−0.44 −0.58
0.01 −0.03 −0.31 −0.44
−0.23
0.55
0.42
0.22 −0.40 −0.23
0.15 −0.13
0.45 −0.74
1.03
0.15
0.20 −0.21
0.60 −0.21 −0.93
0.20

0.22 −0.03 −0.15
0.19 −0.07
0.41

0.00
0.52

0.05
0.30

0.22 −0.03 −0.15
0.17 −0.11
0.39

0.01
0.14 −0.04
0.10 −0.56
0.01
0.13 −0.04−0.32 −0.23
0.03
0.24
0.20 −0.33 −0.23
0.02
0.00
0.57 −0.04 −0.39 −0.02
0.00
0.57 −0.04−0.05 −0.30
0.21 −0.12 −0.37 −0.04 −0.29
0.21
0.58
0.08 −0.27
0.06
0.10
0.59
0.12 −0.26
0.00
0.19
0.87 −0.64 −0.22
0.00
0.21
0.89
−0.09
0.10

0.06
0.39

0.29 −0.18
0.08 −0.10−0.34 −0.60 −0.37 −0.01
0.48 −0.01 −0.10
0.00

1.24 −0.09
0.16
0.10

0.00

1.05

1.66 −0.52

0.35

0.28
0.08

0.42 −0.34 −0.59 −0.37
0.04
0.49
0.00 −0.10

0.08 −0.09 −0.12
0.98

0.05
0.39

0.03

0.08 −0.09 −0.11
1.05

0.98

0.36

Background Subtraction
Stopping
0.75 −0.63 −0.26 −0.56 −0.39
0.74 −0.63 −0.27 −0.56 −0.39
Non-showering −0.22 −0.24
0.05
0.18 −0.07 −0.22 −0.24
0.04
0.18 −0.08
Showering−0.26 −1.87
0.16 −0.16
0.05 −0.26 −1.87
0.16 −0.16
0.05

Table B.2: Detector-related fitted systematic uncertainty parameters(epsilons) at the best-fit point for the atmospheric-only analysis.

229

Fit Value (σ)

Systematic
Uncertainty

NormalI

II

Fiducial Volume−0.45 −0.13
FC Reduction
−0.05
0.01
PC Reduction
0.06 −0.53
Up-µ Reduction−0.01 −0.11
FC/PC Separation −0.22
0.05
PC Stop/Through SeparationTop
0.39 −0.26
Barrel0.54 −0.35
Bottom−0.58
0.01
Ring Counting
0.54
0.42
Single-Ring PID
−0.13
0.45
Multi-Ring PID
−0.21
0.60
Non-ν Backgrounde-like0.00
0.05µ-like0.53
0.32
Non-νe BackgroundSingle-Ring
0.10 −0.56
Multi-Ring
0.24
0.21
Two-Ring π 0−0.39 −0.02
Energy Scale
Absolute−0.12 −0.38
Up/Down
0.03
0.08
Decay-e Tagging−0.64 −0.22
Multi-Ring BDT
Efficiency
−0.18
1.25
Migration
−0.10
0.16
Up-µ
Path Cut−0.01
0.42
Momentum Cut
0.01
0.04
Separation
Stop/Through
0.00
0.02
Non-showering
1.65 −0.52
Showering

III

Inverted
IV

V

I

II

III

IV

V

0.62 −0.42 −0.28 −0.45 −0.13
0.62 −0.42 −0.28
0.25
0.70
0.20 −0.05
0.01
0.25
0.71
0.20
0.00 −0.75 −0.25
0.05 −0.54
0.00 −0.76 −0.26
0.23
0.12
0.01
0.00 −0.11
0.23
0.12
0.01
−0.10
0.02
0.00 −0.23
0.05 −0.10
0.02
0.00
−0.27
−0.36
−0.03
0.22
−0.74
−0.21

−0.62
0.73
−0.32
−0.40
1.01
−0.93

−0.28
0.37 −0.26 −0.29 −0.66 −0.29−0.60
0.54 −0.36 −0.36
0.73 −0.60−0.44 −0.59
0.00 −0.03 −0.32 −0.44
−0.23
0.54
0.42
0.21 −0.41 −0.22
0.15 −0.15
0.43 −0.73
1.00
0.15
0.20 −0.19
0.62 −0.20 −0.87
0.20

0.22 −0.03 −0.15
0.19 −0.07
0.41

0.00
0.54

0.06
0.32

0.22 −0.02 −0.15
0.19 −0.07
0.41

0.01
0.13 −0.04
0.11 −0.55
0.01
0.13 −0.04−0.32 −0.22
0.03
0.24
0.20 −0.33 −0.24
0.03
0.00
0.57 −0.04 −0.39 −0.02
0.00
0.57 −0.04−0.05 −0.30
0.21 −0.11 −0.37 −0.04 −0.29
0.22
0.58
0.08 −0.27
0.04
0.09
0.59
0.09 −0.27
0.00
0.20
0.88 −0.63 −0.22
0.01
0.22
0.89
−0.08
0.10

0.06
0.39

0.29 −0.18
0.08 −0.10−0.34 −0.60 −0.37 −0.01
0.48 −0.01 −0.10
0.01
0.08 −0.09 −0.12 −0.01
1.05

0.97

0.35

1.23 −0.09
0.16
0.10

0.04
0.41

0.27
0.08

0.42 −0.35 −0.60 −0.37
0.04
0.48 −0.01 −0.10
0.02

1.66 −0.52

0.08 −0.09 −0.12
1.06

0.99

0.36

Background Subtraction
Stopping
0.75 −0.63 −0.26 −0.56 −0.39
0.76 −0.62 −0.25 −0.55 −0.38
Non-showering −0.22 −0.24
0.05
0.18 −0.07 −0.22 −0.24
0.05
0.18 −0.08
Showering−0.26 −1.87
0.16 −0.16
0.05 −0.26 −1.87
0.16 −0.16
0.05

Table B.3: Detector-related fitted systematic uncertainty parameters(epsilons) at the best-fit point for the SK analysis with θ13 constrained.

230

Systematic
Uncertainty

Fit Value (σ)
NormalI

II

Fiducial Volume−0.42 −0.08
FC Reduction
−0.04
0.02
PC Reduction
0.03 −0.57
Up-µ Reduction
0.01 −0.11
FC/PC Separation −0.18
0.07
PC Stop/Through SeparationTop
0.41 −0.23
Barrel0.58 −0.31
Bottom−0.55
0.01
Ring Counting
0.69
0.38
Single-Ring PID
−0.09
0.51
Multi-Ring PID
−0.21
0.59
Non-ν Backgrounde-like−0.01
0.06µ-like0.50
0.28
Non-νe BackgroundSingle-Ring
0.13 −0.55
Multi-Ring
0.22
0.25
Two-Ring π 0
−0.19
0.09
Energy Scale
Absolute−0.15 −0.38
Up/Down
0.02
0.06
Decay-e Tagging−0.66 −0.17
Multi-Ring BDT
Efficiency
−0.30
1.10
Migration
−0.10
0.14
Up-µ
Path Cut
0.00
0.42
Momentum Cut −0.01
0.02
Separation
Stop/Through
0.01
0.03
Non-showering
1.59 −0.56
Showering
Background Subtraction
Stopping
0.73 −0.64
Non-showering −0.22 −0.24
Showering−0.24 −1.85
fiTQun (T2K Model)
Decay-e Tagging
Ring Countinge/µ Separatione/π 0 Separationµ/π + Separation

III

Inverted
IV

V

I

II

III

IV

V

0.64 −0.34 −0.25 −0.42 −0.08
0.64 −0.33 −0.25
0.27
0.80
0.22 −0.04
0.02
0.27
0.80
0.22
−0.01 −0.79 −0.26
0.03 −0.56 −0.01 −0.78 −0.26
0.23
0.13
0.01
0.01 −0.11
0.23
0.13
0.01
−0.09
0.02
0.00 −0.18
0.07 −0.09
0.02
0.00
−0.23
−0.34
−0.02
0.23
−0.80
−0.21

−0.57
0.79
−0.27
−0.29
1.05
−0.99

−0.24
0.41 −0.23 −0.23 −0.56 −0.24−0.56
0.59 −0.31 −0.34
0.79 −0.56−0.42 −0.55
0.01 −0.02 −0.27 −0.42
−0.17
0.71
0.38
0.23 −0.29 −0.17
0.09 −0.09
0.51 −0.81
1.03
0.09
0.10 −0.21
0.59 −0.21 −0.99
0.10

0.24 −0.03 −0.08 −0.01
0.13 −0.13
0.37
0.50

0.06
0.28

0.24 −0.03 −0.08
0.13 −0.13
0.37

0.01
0.24 −0.03
0.13 −0.55
0.01
0.23 −0.03−0.35 −0.20
0.01
0.23
0.25 −0.34 −0.19
0.02
0.05
0.89
0.04 −0.19
0.09
0.05
0.89
0.04
−0.06 −0.40
0.20 −0.15 −0.37 −0.06 −0.40
0.20
0.57 −0.04 −0.25
0.03
0.06
0.57 −0.03 −0.25
0.05
0.24
0.94 −0.66 −0.17
0.04
0.25
0.95
−0.19 −0.03
0.10
0.41

0.24 −0.30
0.08 −0.10−0.33 −0.57 −0.35
0.00
0.49
0.06 −0.09 −0.01

1.11 −0.19 −0.03
0.14
0.10
0.40

0.42 −0.33 −0.57 −0.35
0.02
0.49
0.06 −0.08

0.08 −0.06 −0.10

0.01

1.02

1.59 −0.57

0.90

0.33

0.24
0.08

0.03

0.08 −0.06 −0.10
1.02

0.90

0.33

−0.28 −0.58 −0.39
0.73 −0.64 −0.28 −0.57 −0.39
0.04
0.18 −0.07 −0.22 −0.24
0.04
0.18 −0.07
0.17 −0.15
0.06 −0.24 −1.85
0.17 −0.15
0.06
0.00
0.30
0.08
−0.09
0.00

0.00
0.29
0.07
−0.07
0.00

Table B.4: Detector-related fitted systematic uncertainty parameters(epsilons) in the SK+T2K model analysis.

231

List of Journal Abbreviations
AIP Conf. Proc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .AIP Conference Proceedings
APJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . American Journal of Physics
Adv. Space Res. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Advances in Space ResearchAnn. Math. Stat. . . . . . . . . . . . . . . . . . . . . . . . . . . The Annals of Mathematical Statistics
Ann. Phys. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Annals of Physics
Annu. Rev. Nucl. Part. Sci. . . . . . . . Annual Review of Nuclear and Particle Science
Astron. Astrophys. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Astronomy and Astrophysics
Astropart. Phys. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Astroparticle Physics
Astrophys. J. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Astrophysical Journal
Comput. Phys. Commun. . . . . . . . . . . . . . . . . . . . . . Computer Physics Communications
Eur. Phys. J. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .The European Physical Journal
Front. Astron. Space Sci. . . . . . . . . . . . . . . . Frontiers in Astronomy and Space Science
Front. Phys. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frontiers in Physics
IEEE Trans. Nucl. Sci. . . . . . . . . . . . . . . . . . . . . . IEEE Transactions on Nuclear Science
J. Geophys. Res. Space Phys. . . . . . Journal of Geophysical Research: Space Physics
J. Phys. G Nucl. Partc. . . . . . . . . Journal of Physics G: Nuclear and Particle Physics
JHEP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Journal of High Energy Physics
JINST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Journal of Instrumentation
Nucl. Instrum. Meth. . . . . . . Nuclear Instruments and Methods in Physics Research
Nucl. Phys. Proc. Suppl. . . . . . . . . . . . . . . Nuclear Physics - Proceedings Supplements
Nucl. Phys. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nuclear Physics
Nuovo Cimento . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Il Nuovo Cimento232

PTEP . . . . . . . . . . . . . . . . . . . . . . . . . . . Progress of Theoretical and Experimental Physics
Phys. Earth Planet. Inter. . . . . . . . . . . . .Physics of the Earth and Planetary Interiors
Phys. Lett. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physics Letters
Phys. Rept. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physics Reports
Phys. Rev. Lett. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physical Review Letters
Phys. Rev. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physical Review
Phys. Today . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physics Today
Prog. Part. Nucl. Phys. . . . . . . . . . . . . . . . . . . Progress in Particle and Nuclear Physics
Prog. Theor. Phys. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Progress of Theoretical Physics
Rept. Prog. Phys. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reports on Progress in Physics
Rev. Mod. Phys. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reviews of Modern Physics
Sov. J. Nucl. Phys. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Soviet Journal of Nuclear Physics
Sov. Phys. JETP . Soviet Physics Journal of Experimental and Theoretical Physics

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Biography
Thomas Benjamin Wester was born in Chicago, Illinois to Drs. William andBarbara Wester in July, 1995. Thomas’ proximity to Fermi National Accelerator
Laboratory (Fermilab) proved fortuitous. In the summer of 2012, Thomas worked
with the liquid argon R&D group at Fermilab where his interest in computer programming quickly transferred to dark matter detectors. Thomas spent the following
summer at Gran Sasso National Laboratory, where he helped with the commissioningof the DarkSide-50 detector. Thomas continued working with dark matter detectors
as an undergraduate at the University of Chicago with Professor Luca Grandi. During Thomas’ time at UC, the Grandi group deployed and operated a small-scaleliquid xenon detector, which would become the subject of Thomas’ undergraduate
thesis. Thomas spent UC’s summer breaks back at Fermilab, where he worked on
the MicroBooNE neutrino experiment as part of Professor Janet Conrad’s group. After graduating UC, Thomas moved to Boston and continued work with the Conradgroup, now on the ion source demonstrator for the upcoming IsoDAR neutrino experiment. By this point, Thomas had decided to pursue neutrino physics, and happily
accepted the opportunity to enroll as a graduate student at Boston University in 2018.
He joined the Super-Kamiokande collaboration before starting his first semester ofclasses, and the timing could not have been better: The SK tank was opened thatsummer for the first time in 12 years, giving Thomas first-hand experience with thedetector. Throughout his Ph.D., Thomas worked under the supervision of Professors Ed Kearns and Roger Wendell to develop improvements for the SK atmospheric
neutrino oscillation analysis. Upon graduating from BU, Thomas will return to the
University of Chicago, his alma mater, as a McCormick postdoctoral fellow.

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