Doctoral Dissertation
博士論文

Measurement of neutrons produced in
atmospheric neutrino interactions at Super-Kamiokande(スーパーカミオカンデに於ける大気ニュートリノ反応に伴う中性子の測定)

A Dissertation Submitted for the Degree of Doctor of Philosophy
December 2023
令和5年12月博士(理学)申請

Department of Physics, Graduate School of Science,
The University of Tokyo東京大学大学院理学系研究科物理学専攻

Seungho Han
韓 昇澔

Abstract
Reconstructing recoil neutrons from neutrino-nucleus interactions poses a challenge,
yet it holds significant merit due to its relevance to neutrino kinematics, nuclear
effects, and potential for background rejection. However, adding to the difficulty
in detection is the substantial uncertainty in neutron production. Several measurements suggest that the current default simulation models not only overpredict
overall neutron production but also display considerable variability in predictions.
In this study, we measured the average neutron multiplicity and track length
in atmospheric neutrino interactions at Super-Kamiokande as a function of visibleenergy. Using a neural network, we identified delayed neutron capture signals in adataset spanning 4,270 days of livetime, including 561 days in a 0.011w% Gd-loaded
phase featuring improved neutron tagging efficiency. The neutron tagging efficiency
and its uncertainty were calibrated using an Am/Be neutron source, extensively
studied in this thesis. Various hadron intranuclear cascade (INC) models were tested
and compared with observations for the first time.
The observed average neutron multiplicity exhibited a clear linear increase withvisible energy. While the results showed a deficit compared to the predictions of
INC models widely used in the field — consistent with earlier studies — INCL, amodel offering a more sophisticated description of the nucleus, demonstrated better
agreement. This was particularly evident in sub-GeV neutrino events, where thedeficit was more pronounced. This underscores the significance of accurately modeling nuclear effects for predicting nucleon production, especially at lower projectile
energies where semi-classical assumptions of INC models are less applicable.

iAcknowledgements
First and foremost, I want to express my sincere gratitude to the Super-Kamiokande
collaborators. Their unwavering dedication over more than two decades, working day
and night to maintain the detector and gather invaluable data, is the cornerstone of
my work. Without their efforts, none of my achievements would have been possible.
I stand on the shoulders of these giants, and the significant discoveries resulting
from their consistent dedication paved the way for my contributions.
I would like to express my heartfelt gratitude to Okumura-san for introducing
me to the realm of atmospheric neutrinos. His guidance served as a guiding lightthroughout my demanding five-year journey in graduate studies, steering throughthe challenges of a pandemic, and balancing a hectic personal life that often felt likenavigating in the dark towards a mountain top. I am truly thankful for his patienceand mentorship, which have been invaluable to me.
I extend my sincere gratitude to Hayato-san, whose wholehearted support has
touched every aspect of my life, from personal matters to my career and research.
This work owes a great deal to the works of past pioneers, among them Akutsusan, Ka Ming, Pablo, Okajima-san, Tristan, and Haibing.
I want to express special thanks to Yano-san for automating the deploymentof calibration sources and overseeing the collection of calibration data. I learned agreat deal and heavily relied on Yano-san’s simulation work and Gd-neutron capture
models, which were instrumental in my neutron calibration studies.

ii

I want to express my gratitude to the Okayama group, with special thanks to
Koshio-san, Hagiwara-san, Harada-san, Sakai-san, and Hino-san. The detector simulation developed by Koshio-san and Hagiwara-san serves as the foundation of myneutron calibration studies. The countless discussions with Harada-san on improving
Am/Be simulation and NTag was something that kept me moving forward. Sakaisan’s work on various INC models was a critical input to my model comparison
studies. I am also grateful to Hino-san for resolving the Gd-capture ratio issue.
I would like to thank colleagues in the oscillation group, namely Roger, Thomas,
and Miki-san. My work on atmospheric neutrino oscillation sensitivity studies withneutron tagging, as well as cosmic ray calibration, heavily relied on their work.
I would like to express my gratitude to colleagues in ATMPD, including Linyan,
Noguchi-san, Ishitsuka-san, and a special acknowledgment to Ed. His questions
played a crucial role in steering me in the right direction, pushing me forward when
I felt stuck. Additionally, I want to thank Miura-san, Mine-san, and Kameda-san
for their assistance in the simulation production and data reduction processes.
Appreciation is also due to colleagues in the LOWE group: Sekiya-san for overseeing Gd-loading, Izumiyama-san and Kanemura-san for improving NTag, Michaelfor offering insights into neutron vertex reconstruction, and H. Ito-san for measuring
the Am/Be source intensity, critical in enhancing our calibration.
I would like to thank Tairafune-san and Nakajima-san for allowing me to contribute to the neutron tagging analysis in T2K.
Special thanks to those based in Kashiwa: Patrick, Junjie, Guille, Xubin, andRhea, are among the few friends I made in Kashiwa, and they made my life enjoyable. Abe-san provided crucial advice that shaped my research, and Tashiro-san’s
assistance with simulation production and data reduction was invaluable. Mashimasan supported me through every administrative process for five years. My apologies
to Tomiya-san and Yoshida-san for not being the best senpai.
I acknowledge the financial support from the MEXT scholarship, which sustained
my five-year graduate studies in Japan.

iii

I dedicate this thesis to my parents and family,
who have supported me throughout the entire journey.

iv

Contents

1 Introduction1

1.1

Neutrino oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2

Parameter measurements and challenges . . . . . . . . . . . . . . . .

8

1.3

Advantages of neutron-tagging . . . . . . . . . . . . . . . . . . . . . . 15

1.4

SK-Gd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.5

Sources of neutron-related uncertainties . . . . . . . . . . . . . . . . . 23

1.6

Past neutron measurements . . . . . . . . . . . . . . . . . . . . . . . 24

1.7

Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.8

Author’s contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2 Atmospheric neutrino interactions at Super-Kamiokande

31

2.1

Atmospheric neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2

The Super-Kamiokande detector . . . . . . . . . . . . . . . . . . . . . 33
2.2.1

Working principle . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.2.2

Detector components . . . . . . . . . . . . . . . . . . . . . . . 37v

Contents

2.3

2.4

2.5

2.2.3

Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.2.4

Operational phases . . . . . . . . . . . . . . . . . . . . . . . . 57

Event simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.3.1

Atmospheric neutrino flux model . . . . . . . . . . . . . . . . 62

2.3.2

Neutrino-nucleus interaction model . . . . . . . . . . . . . . . 68

2.3.3

Hadronic interaction model . . . . . . . . . . . . . . . . . . . 80

2.3.4

Neutron capture reaction model . . . . . . . . . . . . . . . . . 88

2.3.5

Detector simulation . . . . . . . . . . . . . . . . . . . . . . . . 88

2.3.6

Background modeling . . . . . . . . . . . . . . . . . . . . . . . 90

2.3.7

Event weights . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Event reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
2.4.1

Atmospheric neutrino interaction . . . . . . . . . . . . . . . . 92

2.4.2

Neutron capture vertex . . . . . . . . . . . . . . . . . . . . . . 94

Atmospheric neutrino event selection . . . . . . . . . . . . . . . . . . 98

3 Neutron detection
3.1

Characteristics of signal and background . . . . . . . . . . . . . . . . 101

3.2

Signal identification algorithm . . . . . . . . . . . . . . . . . . . . . . 104

3.3

vi

101

3.2.1

Stage 1: Candidate selection . . . . . . . . . . . . . . . . . . . 105

3.2.2

Stage 2: Candidate classification. . . . . . . . . . . . . . . . 109

Validation with Am/Be neutron source . . . . . . . . . . . . . . . . . 125

CONTENTS

3.4

3.3.1

Am/Be neutron source . . . . . . . . . . . . . . . . . . . . . . 125

3.3.2

Calibration setup . . . . . . . . . . . . . . . . . . . . . . . . . 128

3.3.3

Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

3.3.4

Data reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 134

3.3.5

Neutron tagging efficiency estimation . . . . . . . . . . . . . . 150

3.3.6

Systematic uncertainties in simulation model . . . . . . . . . . 152

3.3.7

Data-MC comparison . . . . . . . . . . . . . . . . . . . . . . . 166

Michel electron separation . . . . . . . . . . . . . . . . . . . . . . . . 170
3.4.1

3.5

Validation with cosmic ray muon data . . . . . . . . . . . . . 171

Effect of PMT afterpulsing on signal detection performance in SK-VI 175

4 Neutron multiplicity measurement
4.1

4.2

4.3

177

Data and simulation configurations . . . . . . . . . . . . . . . . . . . 178
4.1.1

Data run selection . . . . . . . . . . . . . . . . . . . . . . . . 178

4.1.2

Simulation configuration . . . . . . . . . . . . . . . . . . . . . 180

4.1.3

Event reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 181

Data quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
4.2.1

Prompt atmospheric neutrino interaction signals . . . . . . . . 183

4.2.2

Delayed neutron capture signals . . . . . . . . . . . . . . . . . 192

Estimation of signal detection performance . . . . . . . . . . . . . . . 209
4.3.1

Dependence on neutrino event variables . . . . . . . . . . . . . 210

vii

Contents
4.3.2
4.4

Regression on neutrino event variables . . . . . . . . . . . . . 212

Average neutron capture multiplicity . . . . . . . . . . . . . . . . . . 220
4.4.1

Systematic uncertainties . . . . . . . . . . . . . . . . . . . . . 220

4.4.2

Linear regression on visible energy

4.4.3

Various hadronic interaction models . . . . . . . . . . . . . . . 230

4.4.4

Generation of model predictions . . . . . . . . . . . . . . . . . 240

4.4.5

Comparison of model predictions with data . . . . . . . . . . . 245. . . . . . . . . . . . . . . 226

5 Discussion and conclusions

253

5.1

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

5.2

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

5.3

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

Appendix264

A Reconstruction algorithm of neutrino interaction

264

A.1 Vertex Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . 264
A.2 Ring counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
A.3 Particle identification . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
A.4 Momentum reconstruction . . . . . . . . . . . . . . . . . . . . . . . . 270

B Data selection stages

272

B.1 FC1: Initial cuts to reject evident backgrounds . . . . . . . . . . . . . 272

viii

CONTENTS
B.2 FC2: Subsequent cuts to further reject backgrounds . . . . . . . . . . 273
B.3 FC3: Cosmic ray rejection based on fit . . . . . . . . . . . . . . . . . 274
B.4 FC4: Flasher rejection based on likelihood . . . . . . . . . . . . . . . 279
B.5 FC5: Additional cuts to reject remaining backgrounds . . . . . . . . . 282
B.6 FCFV: Events fully-contained in fiducial volume . . . . . . . . . . . . 283

C Comparison of atmospheric neutrino data: SK-IV with SK-VI284

D Statistical uncertainties

288

D.1 Statistical inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
D.2 Signal detection performance spread . . . . . . . . . . . . . . . . . . . 289

E Comparison of neutrino event generators

293

Bibliography

319

ix

Chapter 1
Introduction
Neutrinos were first theorized [1] in 1933 as part of the “weak” interaction theory
to explain the prolonged lifetime of beta decay compared to electromagnetic interaction. The properties of neutrinos are defined by weak interaction characteristics
and conservation laws. Neutrinos must be electrically neutral, possess an extremely
light mass, and have a spin of 1/2. They exhibit an incredibly low interaction crosssection with matter when compared to other known particles, making them one of
the most abundant elementary particles. However, despite their abundance, neutrinos remain mysterious, featuring various puzzling and unique aspects that set themapart from other particles.
One notable aspect is that, so far, only left-handed neutrinos and right-handed
anti-neutrinos have been observed [2]. For decades, physicists had ample reasons
to believe that neutrinos were massless, given the chiral nature [3] of weak interaction, an extremely small neutrino mass limit, and the apparent lack of right-handed
neutrinos. The electroweak theory [4, 5, 6], which unifies weak interaction with electromagnetic interaction and predicts mediator boson masses, was formulated basedon this assumption.

1

The surprising discovery of neutrino oscillation [7] suggests neutrinos have massand hints at the possible existence of right-handed neutrinos, which do not interact
weakly. While this finding sheds light on longstanding mysteries such as observed
neutrino deficits [8], it also raises new questions. What is the origin of their smallmass, and what are the properties of right-handed neutrinos and their relationship toleft-handed neutrinos? Precision measurements of neutrino oscillation parameters,
including mixing angles, differences in neutrino masses, and the correct “order”
of the neutrino masses can provide valuable insights. Additionally, investigating
whether the oscillation violates CP symmetry, as observed in quarks [9, 10], may
offer an extra source of CP symmetry to explain the observed baryon-antibaryon
asymmetry [11].
In this chapter, we briefly discuss the theory of standard three-flavor neutrino
oscillation, outlining the remaining challenges and the current status of oscillation
parameter measurements, as well as the obstacles faced by experiments. We thenfocus on how the detection of “neutrons” from atmospheric neutrino interactionsin Super-Kamiokande can not only enhance future neutrino oscillation parametermeasurements but also contribute to other rare physics searches where atmospheric
neutrinos serve as backgrounds. Finally, we explore the motivation behind measuring the properties of neutrons produced in atmospheric neutrino interactions inwater and conclude with an overview of this study.

2

CHAPTER 1. INTRODUCTION

1.1

Neutrino oscillation

The three known weak flavor eigenstates of neutrinos [12, 13, 14] form a complete
and orthonormal basis for the three lepton generations in the Standard Model. Withnonzero neutrino mass, we can establish another complete and orthonormal basiswith three mass eigenstates, which may not necessarily align with the flavor basis.
Let the flavor eigenstate be a coherent superposition of the mass eigenstates, linked
by a 3 × 3 unitary matrix U :X|νl ⟩ =

k

Ulk∗ |νk ⟩(1.1)

where |νl ⟩ represents the flavor eigenstate of flavor l ∈ {e, µ, τ }, and |νk ⟩ is thek th mass eigenstate where k ∈ {1, 2, 3}.

Let |ν(x)⟩ denote a neutrino state produced via weak interaction at time x0 = 0.

Each mass eigenstate should acquire a distinct phase as it propagates in (vacuum)
spacetime, so that:|ν(x)⟩ =
=

X
k

X
l′ ,k

Ulk∗ |νk ⟩ e−iϕk (x)Ulk∗ Ul′ k |νl′ ⟩ e−iϕk (x)(1.2)

where ϕk (x) := pk · x represents the phase gain of |νk ⟩ after propagation x.
Naturally, the probability that the state |ν⟩ which started with flavor l will be

detected with another flavor l′ is given by:
P (νl → νl′ )(x) =X

2Ulk∗ Ul′ k e−iϕk (x)kX= δll′ + 2Re
Ulk∗ Ul′ k Ulk′ Ul∗′ k′ e−i∆ϕkk′ (x)(1.3)

k′ ̸=kwhere ∆ϕkk′ (x) ≡ ϕk (x) − ϕk′ (x).

3

1.1. Neutrino oscillation
If the neutrino masses are non-degenerate so that the mass eigenstates may fallout of phase, the probability in Equation 1.3 becomes non-zero for l ̸= l′1 . Thisflavor transition, permitted by the non-degeneracy of neutrino masses, is referred
to as “oscillation,” owing to the sinusoidal nature of the transition probability inEquation 1.3. Experimental evidence of neutrino oscillation has been establishedthrough experiments involving various neutrino sources: atmospheric [7], solar [8],reactor [16], and artificial [17].
In a scenario where the Dirac masses are considered, the unitary 3 × 3 neutrino

mixing matrix U can be parameterized with four real free parameters. This considers
the neutrino and lepton spinors of three generations absorbing the relative phasesand eliminating 32 − 1 degrees of freedom from U , so that only the remaining four

degrees of freedom can be considered physically meaningful. The standard approach
to parameterizing the mixing matrix U is:

#
" # "
1
0
0
νe
νµ = 0 cos θ23 sin θ23
0 − sin θ23 cos θ23ντ
cos θ130 sin θ13 e−iδCP


0
1
0
iδCP
− sin θ13 e0cos θ13"#" #cos θ12 sin θ12 0 ν1
− sin θ12 cos θ12 0 ν2
0
0
1 ν3

(1.4)

Here, θij , where i, j ∈ {1, 2, 3}, denotes the mixing angles representing the inter-

nal mixing of mass eigenstates, and δCP signifies the overall complex phase of U . It
is worth noting that the complex phase is not allowed for dimension smaller than 3.
δCP ̸= 0 or π implies P (νl → νl′ ) ̸= P (ν̄l → ν̄l′ ) and thus violation of CP symmetry.

1

One might speculate about the oscillation of charged leptons produced through weak decayof a heavy particle. However, due to the substantially larger masses of charged leptons comparedto neutrinos, verifying this experimentally with O(1) m path length would necessitate an energy
scale on the order of O(1013 ) GeV. Refer to [15] for further details.

4CHAPTER 1. INTRODUCTION
In the ultrarelativistic limit, the expression ϕk (x) in Equation 1.3 simplifies to
ϕk (x) ≈ m2k L/2E, where mk represents the mass of the k th mass eigenstate |νk ⟩,L = |⃗x|2 denotes the distance traveled by the neutrino, and E = p0k = Eν is assumed
to be the energy of the neutrino, consistent across all k. Consequently, the oscillation

probability P (νl → νl′ ) is fully characterized by the traveled distance L, the neutrinoenergy E, and the so-called neutrino oscillation parameters: the three mixing angles
θij , the (Dirac) CP phase δCP , and the two independent mass-squared differences∆m221 and ∆m232 , where ∆m2ij ≡ m2i − m2j .
It is important to note that, at the leading order, the (vacuum) oscillation probability as shown in Equation 1.3 is not sensitive to the relative ordering of neutrino
masses mi , as it depends on the mass-squared differences ∆m2ij . Currently, two−5measured values of mass-squared differences exist, ∆m221 = 7.41+0.21
eV2 and−0.20 × 10−3∆m23{1,2} = 2.507+0.026
eV2 . This information tells us that m1 and m2 are−0.027 × 10

much closer, and m3 is either much larger or smaller than the other two. The un-

known direction of this mass difference is referred to as the mass ordering/hierarchy
problem. In the normal mass ordering (NO) scenario, m3 is larger than m1 and m2 ,while in the inverted mass ordering (IO) scenario, m3 is smaller. This distinction
has implications for the associated neutrino flavors with different mass ranges, as illustrated in Figure 1.1. Neutrino mass ordering is also believed to have implications
for the origin of neutrino mass as well as the Dirac/Majorana nature of neutrinos.
Most of the oscillation parameters, except δCP , are relatively precisely measured
as shown in Table 1.1. Several inferences can be drawn from Table 1.1: firstly,
normal neutrino ordering is slightly favored by 2.5σ if atmospheric neutrino data is
considered.2 Secondly, whether θ23 is in the 0-45◦ or 45-90◦ is not yet known. This
uncertainty arises because θ23 appears in the oscillation probability in the form ofsin 2θ23 , leading to a degeneracy between the two angle regions. Finally, the current
best fit δCP is close to 1.1π with normal mass ordering, allowing for CP conservation(δCP = π) at 1-2σ confidence level. Figure 1.2 visually demonstrates the relatively
larger mixing in neutrinos compared to quarks.

2

The atmospheric neutrino’s sensitivity to mass ordering is discussed in Section 1.2.

5

1.1. Neutrino oscillation

Figure 1.1: Illustration of neutrino mass ordering. (Reprinted from [18])Figure 1.2: Visual comparison of quark (CKM, left) and neutrino (PMNS, right)
mixing matrices. (Credit: S. Stone)

6CHAPTER 1. INTRODUCTION

without SK atmospheric data

NuFIT 5.2 (2022)

sin2 θ12θ12 /◦sin2 θ23
θ23 /◦sin2 θ13θ13 /◦δCP /◦∆m221
10−5 eV2∆m23`
10−3 eV2

with SK atmospheric datasin2 θ12θ12 /◦sin2 θ23
θ23 /◦sin2 θ13θ13 /◦δCP /◦∆m221
10−5 eV2∆m23`
10−3 eV2

Normal Ordering (best fit)bfp ±1σ
3σ range

Inverted Ordering (∆χ2 = 2.3)
bfp ±1σ
3σ range

0.303+0.012−0.0110.270 → 0.341

0.303+0.012−0.0110.270 → 0.341

33.41+0.75−0.72

31.31 → 35.74

33.41+0.75−0.72

31.31 → 35.74

0.572+0.018−0.0230.406 → 0.620

0.578+0.016−0.021

0.412 → 0.623

49.1+1.0−1.3

39.6 → 51.9

49.5+0.9−1.2

39.9 → 52.1

0.02203+0.00056−0.00059

0.02029 → 0.02391

0.02219+0.00060−0.00057

0.02047 → 0.02396

8.54+0.11−0.12

8.19 → 8.89

8.57+0.12−0.11

8.23 → 8.90

197+42−25

108 → 404

286+27−32

192 → 360

7.41+0.21−0.20

6.82 → 8.03

7.41+0.21−0.20

6.82 → 8.03

+2.511+0.028−0.027+2.428 → +2.597−2.498+0.032−0.025−2.581 → −2.408

Normal Ordering (best fit)bfp ±1σ
3σ range

Inverted Ordering (∆χ2 = 6.4)
bfp ±1σ
3σ range

0.303+0.012−0.0120.270 → 0.341

0.303+0.012−0.0110.270 → 0.341

33.41+0.75−0.72

31.31 → 35.74

33.41+0.75−0.72

31.31 → 35.74

0.451+0.019−0.016

0.408 → 0.603

0.569+0.016−0.021

0.412 → 0.613

42.2+1.1−0.9

39.7 → 51.0

49.0+1.0−1.2

39.9 → 51.5

0.02225+0.00056−0.00059

0.02052 → 0.02398

0.02223+0.00058−0.00058

0.02048 → 0.02416

8.58+0.11−0.11

8.23 → 8.91

8.57+0.11−0.11

8.23 → 8.94

232+36−26

144 → 350

276+22−29

194 → 344

7.41+0.21−0.20

6.82 → 8.03

7.41+0.21−0.20

6.82 → 8.03

+2.507+0.026−0.027+2.427 → +2.590−2.486+0.025−0.028−2.570 → −2.406

Table 1.1: Three-flavor oscillation parameters obtained from different global analysesof neutrino data. (Reprinted from [19])71.2. Parameter measurements and challenges

1.2

Parameter measurements and challenges

In Section 1.1, we discussed how, for a given neutrino (antineutrino) flavor, the threeflavor neutrino oscillation probability can be expressed as a function of oscillation
parameters and the ratio between the neutrino energy Eν and the path length L:

P (να → νβ ) ≈ f (θij , ∆m2ij , δCP ; Eν /L)(1.5)

When the neutrino (or antineutrino) α, β, Eν , L, and the flux from the source
are known, we can deduce the oscillation parameters from the observed event ratesof each neutrino flavor. Different neutrino sources, spanning from MeV to GeV andincluding atmospheric, solar, reactor, and accelerator neutrinos, contribute to themeasurements of oscillation parameters, as summarized in Table 1.1.
The dominant interaction channel in oscillation experiments is the neutrinonucleus charged-current quasi-elastic (CCQE) interaction, which produces a single
charged lepton (carrying the neutrino flavor) and scatters a single nucleon from thetarget nucleus, as in νl n → l− p or ν̄lp → l+ n. This channel is “clean,” because
the neutrino energy Eν can be mostly inferred by reconstructing lepton kinematics∗

Z
only. In the ideal 2-body CCQE scenarios, such as νl + A
X → l− + Z−1
A X + p and∗Z−1Zν̄l + AX → l+ + A−1
Y + n, where one outgoing lepton l and one nucleon n/p are

involved, the energy of the incoming neutrino Eν is determined as:

Eν ≈

2
2
eff
(mN )2 − (meffN ) − ml + 2(mN )El2(meffN − El + pl cos θl )(1.6)

Here, mN is the mass of the incoming nucleon, meffN ≈ mN − Eb is the effective

mass of the outgoing nucleon, accounting for the nuclear binding energy Eb , and ml ,El , pl , and θl represent the mass, energy, momentum, and scattering angle of theoutgoing lepton, respectively.

8CHAPTER 1. INTRODUCTION
In the context of δCP , the comparison between P (νµ → νe ) and P (ν̄µ → ν̄e ) withaccelerator neutrinos is considered to be the most sensitive. This sensitivity arises
from the well-characterized neutrino type, flavor, path lengths, and the reasonably
accurate energy reconstruction performance of Equation 1.6, thanks to the wellknown neutrino direction.
For determining the mass ordering, the most sensitive approach involves comparing the rates of νe and ν̄e in atmospheric neutrinos. This sensitivity is attributed
to the Mikheyev-Smirnov-Wolfenstein (MSW) effect, also known as the “matter effect” [20, 21]. In contrast to the vacuum, ordinary matter, filled with bound e− ,introduces a matter effect through interactions such as νe e− → e− νe mediated by
the charged current W + . This alters the effective Hamiltonian experienced by prop-

agating neutrinos, with distinct changes for neutrinos and antineutrinos based on
different mass ordering scenarios, as illustrated in Figure 1.6.
Several challenges limit the precision of these measurements:
Lack of statistics
The smallness of neutrino interaction cross sections, on the order of 10−14 barns,
makes it difficult to accumulate event statistics. Long-baseline accelerator neutrino
experiments targeting δCP face this challenge, with oscillation path lengths typicallyreaching O(100) km. For example, over the last decade, the T2K experiment hascollected around 100 νe and ν̄e events at the far detector (Super-Kamiokande), whichis O(10−2 ) smaller than the events collected with atmospheric neutrinos. The small
cross-section necessitates large, dense detectors with target materials like water,
plastic, oil, or metal, often compromising event reconstruction capabilities.
Hadron reconstruction
Large, non-magnetized detectors face challenges in distinguishing between neutrino and antineutrino interactions, as the sign of the scattered lepton charge islost. Additionally, these detectors struggle to accurately reconstruct recoil nucleon
kinematics, posing a significant challenge to the neutrino reconstruction, especiallyfor atmospheric neutrinos that are not well-characterized.

91.2. Parameter measurements and challenges
Nuclear effects
The use of composite nucleus targets introduces complications. Experiments
relying on the “cleanliness” of CCQE events define their samples based on specificobserved final states, e.g., one charged lepton and one proton, as expected by νl n →l− p. However, events like ν̄l p → l− pπ − with π − absorbed by the target nucleus
cannot be distinguished from CCQE events. Accurate reconstruction of such eventsis crucial, demanding a comprehensive understanding of nuclear effects to estimate
the fraction of non-identifiable backgrounds.
Degeneracy between δCP and mass ordering
In the first order, P (νµ → νe ) with the matter effect can be expressed [22]:sin2 2θ12
sin2 [(A − 1)∆31 ](A − 1)2J0 sin δCPsin ∆31 sin(A∆31 ) sin[(1 − A)∆31 ]∓αA(1 − A)J0 cos δCP+αcos ∆31 sin(A∆31 ) sin[(1 − A)∆31 ]A(1 − A)sin2 2θ12+ α2 cos2 θ23
sin2 (A∆31 )A2

P (νµ → νe ) ≈ sin2 θ23(1.7)

assuming natural units, where:
α = ∆m221 /∆m231∆ij = ∆m2ij /4Eν√A = ±2 2GF ne Eν /∆m231
J0 = sin 2θ12 sin 2θ13 sin 2θ23 cos θ13
The opposite signs are for antineutrinos. The second term in the equation allows for
CP violation for δCP ̸= 0 or π, enhancing/suppressing νe /ν̄e appearance for sin δCP >0. The third term changes sign for different mass orderings, enhancing/suppressing
νe /ν̄e appearance for normal mass ordering, potentially mimicking the δCP effect, as
demonstrated in Figure 1.3.

10

782 Page 26 of 50

CHAPTER 1. INTRODUCTION

O(MeV) energies,
run, the water’s atte
going cosmic-ray m
consistent with dat
slightly longer. This
tor uncertainties shdata periods, and s
confirm this.
Figure 15 show
cosmic-ray muon d
ing the run 9 and rof the distributions
ity of the detector
the refurbishment
Michel momentum
ticle ID parameter
Figure 1.3: Comparison of observed e-like candidates in the T2K far detector (Superother reconstructed
Kamiokande) with νµ and ν̄µ beam modes against predictions based on varying
oscillation parameters. The assumed normal mass ordering mimics changes in tion
the showed similar
observed νe : ν̄e ratios corresponding to δCP = −π/2, while the inverted ordering
tests of the expected
mimics changes in the direction of δCP = π/2. T2K estimates that the δCP -mimickingtrue for other calibr
effect (the relative size of the third term in Equation 1.7) of different mass ordering
trino data, and sma
scenarios is ∼30% compared to the true δCP effect. (Reprinted from [23])within current unce
Good detector st
One proposed solution involves jointly analyzing data samples from accelerator
selection of eventsand atmospheric neutrinos that share the same (far) detector. This approach levertion of event times
ages on the distinct sensitivities of the two data sets, as well as the common detector
is shown in Fig. 1
systematic uncertainties. Figures 1.4 and 1.5 illustrate the expected power to reject
tor activity, labelled
CP symmetry and incorrect mass ordering across various assumed true δCP values.
10 showed a 34.2 n
The additional mass ordering sensitivity in atmospheric neutrino data restoresbunch
the timing (dotterejection power lost due to the degeneracy between the genuine CP effect and fromthe previous runs
matter effect demonstrated in Figure 1.3, particularly for the two scenarios: trueAmongst the 35
ν-mode1Re
18 normal
The numberof ν-mode
1Re1deversus10, 75 were selectesin δCP > Fig.
0 withordering
and true1Re
sin +δCP< 0 with
invertedordering.
events (top, leading sin δCP dependence) and ν-mode 1Re + 1Re1de +new 1Re1de event
ν-mode 1Re events above and below Er ec = 550 MeV (bottom, leadsection. The numbe
ing cos δCP dependence), with the predicted number of events for varin Sect. 8, Table 9.
ious sets of oscillation parameters, as shown by the different coloured
ellipses. The values for the neutrino mass splitting are from the frequentist analysis of data, where ∆m 232 = 2.40 × 10−3 eV2 (∆m 231 =−2.46×10−3 eV2 ) is the best-fit point in the normal (inverted) ordering.
The uncertainties represent the 68% confidence interval for the meanof a Poisson distribution given the observed data point. The underlaid
contours contain the predicted number of events for 68% of simulated
experiments, varying the systematic uncertainty parameters around thebest-fit values from the fit to ND data, and oscillation parameters set tothe best-fit values from a fit to data. The overlaid triangle point shows

8 Oscillation anal
11

This section presen
from the full data se
straints from the N1.2. Parameter measurements and challenges

Figure 1.4: The expected power to reject CP symmetry compared across three different analyses: SK atmospheric neutrino data only with T2K near detector constraints(black), T2K accelerator neutrino data only (blue), and a joint fit of the two (red),
for true normal (NO, left) and inverted (IO, right) mass ordering scenarios. The vertical axis represents the minimum χ2 in the CP-symmetric scenario (δCP ∈ {0, ±π})
subtracted by the global minimum χ2 . (Adapted from [24])Figure 1.5: The expected power to reject incorrect mass ordering hypothesis. The
vertical axis is the minimum χ2 in the wrong mass ordering scenario subtracted by
the minimum χ2 in the correct mass ordering scenario (Adapted from [24])12CHAPTER 1. INTRODUCTION
The future determination of δCP and neutrino mass ordering can benefit significantly from the complementary strengths of accelerator neutrinos, known for
their well-characterized properties, and atmospheric neutrinos, which offer abundantstatistics, diverse path lengths, and a range of energies. While there is considerable
potential for improvement on both fronts, the prospects for advancement appear
particularly promising in the realm of atmospheric neutrinos, especially concerning
the ability to differentiate between neutrinos and antineutrinos, and the reconstruction of neutrino path length and energy (L/Eν ). In the following section, we discuss
how the detection of neutrons, especially with the Super-Kamiokande detector, cancontribute to addressing these challenges.

131.2. Parameter measurements and challenges

Figure 1.6: Oscillation probabilities for atmospheric νµ (top panel) and ν̄µ (bottom
panel) at Super-Kamiokande, as functions of true neutrino energy and zenith angle,
assuming normal mass ordering. Distortions induced by matter effects in the Earthare observable only in the O(1) GeV range for cos θzenith < −0.5 in the case of νµ ,not for ν̄µ . This angular region corresponds to neutrino propagation through the
Earth’s core, mantle, and crust. Conversely, under the assumption of inverted massordering, matter effects manifest solely in the ν̄µ oscillations, not in those of νµ .
(Reprinted from [25])14CHAPTER 1. INTRODUCTION

1.3

Advantages of neutron-tagging

In this section, we discuss the advantages of “neutron-tagging” physical events inSuper-Kamiokande (SK), with a particular focus on atmospheric neutrino eventsand oscillation analysis, though not limited solely to these.
Since 2008, SK has actively detected neutron signals in its atmospheric neutrino
data. The primary objective has been to identify recoil neutrons from antineutrino
CCQE interaction ν̄l p → l+ n, and differentiate them from neutrino CCQE interaction νl n → l− p. Most recoil neutrons undergo 1 H(n, γ), resulting in the emissionof a single 2.2 MeV gamma-ray. The detection efficiency of this gamma-ray at SK
stands at approximately 20%. Figure 1.7 shows the difference in the expected neutron signal multiplicity between true neutrino and antineutrino events. It implies
that requiring no neutron in the event allows the effective classification of antineutrinos from neutrinos. It is important to note that the neutron multiplicity in ν/ν̄
CCQE events is not strictly confined to zero or one. This variability arises due tocontamination from non-CCQE events, as well as the influence of nuclear effects
and subsequent hadronic interactions within the detector medium. These factors
contribute to the smearing of the neutron signal multiplicity, as evident in the tails

Probability

displayed in Figure 1.7.

0.6
0.5
0.4
0.3
0.2
0.1
0.0

MC: Sub-GeV (FCFV)
CCQE
CCQE

0

1 2 3 4 5
Expected (n, ) interactions

6

Figure 1.7: The probability density of expected neutron signal multiplicity for subGeV atmospheric neutrino events (purple) and antineutrino events (olive) with reconstructed visible energy below 1.33 GeV, and vertices within the fiducial volume(FCFV). Refer to Section 2.3 for a description of the simulation configuration.

15

1.3. Advantages of neutron-tagging
By using the neutron-tagging technique for ν/ν̄ classification, we can potentiallyleverage the sensitivity of δCP and mass ordering inherent in the expected νe /ν̄eflux ratio at SK. When considering opposite-signed values of δCP or different massordering scenarios, they yield opposite-signed effects on P (νµ → νe ) and P (ν̄µ →ν̄e ), as indicated in Equation 1.7 and demonstrated in Figures 1.3 and 1.6. To
magnify this effect, we can examine the ratio between observed νe and ν̄e events,
particularly focusing on “up-going” events where neutrinos are expected to have
traversed through the Earth.
Figure 1.8 shows the expected ratios of strictly up-going νe /ν̄e fluxes observableat SK, assuming different values of δCP and mass ordering scenarios. The figure
indicates that νe /ν̄e flux ratios show sensitivity to δCP in the O(0.1) GeV range andto mass ordering scenarios in the O(1) GeV range.

Figure 1.8: The predicted ratios of strictly up-going νe /ν̄e fluxes, assuming different
values of δCP (left) and mass ordering scenarios (right). Neutrinos were assumed tohave traveled a path length of 12,000 km, with a zenith angle of 180◦ . In making
these predictions, the HKKM 2011 flux model, detailed in Section 2.3.1, and thedefault oscillation parameters from Table 4.1 were used. To accommodate imperfectenergy resolution, fast oscillations were averaged out in the figures.

16

CHAPTER 1. INTRODUCTION
Figure 1.9 shows the enhancement in the expected sensitivity to δCP and massordering due to the ν/ν̄ classification through neutron-tagging. Notably, significant
improvement in mass ordering sensitivity is expected regardless of the assumed δCP .
Additionally, there is some extra sensitivity gain in δCP , particularly around theleast probable value of δCP .

Figure 1.9: The expected Asimov (simulation-only) sensitivity of SK atmospheric
neutrino observations to δCP and neutrino mass ordering, assuming an additional
10 years of operation from 2019, both with (thicker line) and without (lighter line)
neutron-tagging. The oscillation parameters were adjusted to best describe fakedata generated with a predefined set of ground truth parameters.
The ground truth assumes δCP = 3π/2 and normal mass ordering. The solid lines
represent the fit results assuming the correct mass ordering, while the dashed lines
show results under the assumption of the incorrect mass ordering. The analysis
incorporates the default systematic uncertainties used in SK atmospheric neutrino
oscillation studies [25].

171.3. Advantages of neutron-tagging
Another advantage of detecting recoil neutrons from neutrino events lies in thepotential to enhance the reconstruction of the incident neutrino kinematics. This
becomes particularly valuable for atmospheric neutrino events, where the direction(indicative of path length) and energy are initially unknown. Currently, proxies suchas the momentum and direction of scattered charged leptons are utilized, but theiraccuracy in representing incident neutrino kinematics is limited.
To achieve a more accurate reconstruction of neutrino energy, we can leverage
the count of neutrons, expected to be proportionate to the energy transferred to thehadron system [26, 27]. Figure 1.10 illustrates the expected linear increase in thefraction of “missing neutrino energy” with respect to the detected neutron signals.

Figure 1.10: The expected fraction of energy reconstruction error ((Eν − Erec )/Erec ),as a function of detected neutron signals. The simulation assumes 0.01w% Gd in
the SK water volume. (Reprinted from [27])For a more accurate reconstruction of the neutrino direction, we can leverage
the fact that the expected neutron track length, from its production to capture,tends to be longer for more energetic neutrons. By reconstructing both the neutrino
interaction vertex and the subsequent neutron capture vertices, we can obtain aset of vectors that positively correlate with the initial kinetic energy. Using these
vectors, we can deduce the recoil neutron’s kinetic energy and, consequently, theoriginal neutrino direction, given the reconstructed lepton kinematics.

18

CHAPTER 1. INTRODUCTION
Figure 1.11 illustrates enhanced performance in the reconstruction of neutrinoenergy and direction [27, 28]. Notably, we observe a significant enhancement inneutrino energy reconstruction and a moderate improvement in neutrino direction
reconstruction. The complete atmospheric neutrino oscillation analysis, incorporating all the mentioned advancements in ν/ν̄ classification and energy/direction
reconstruction, is currently in progress. Importantly, these techniques extend beyond atmospheric neutrinos and can be applied to accelerator neutrino events aswell, offering further improvements in neutrino energy reconstruction and reducing
the wrong sign (ν/ν̄) background contamination.

Figure 1.11: A comparison between true and reconstructed neutrino energy (left,[27]) and direction (right, [28]). Both simulations assume 0.01w% Gd in the SK
water volume. The Sub-GeV sample consists of events with visible energy (leptononly, highlighted in red) smaller than 1.33 GeV. (Reprinted from [27, 28])

There are also advantages beyond the scope of atmospheric neutrino oscillation
analysis, particularly in analyses where signals are anticipated to exhibit a specificneutron count. For instance, in proton decays like p → e+ π 0 or p → ν̄K + , as pre-dicted by beyond-Standard-Model theories, no neutron is produced (unless nuclear
de-excitation following a bound proton decay emits one). In the energy range of
proton decay around ∼ 1 GeV, neutrinos are expected to produce an average of 2-4
neutrons.

191.3. Advantages of neutron-tagging
The diffuse supernova neutrino background (DSNB), expected to be in the rangeof O(10) MeV, represents the postulated isotropic flux of neutrinos originating fromall core-collapse supernova events throughout the universe’s history. The most dominant interaction channel for DSNB is through ν̄e p → e+ n, commonly known as

inverse beta decay, which is expected to yield a single neutron.

Several typical backgrounds can be significantly reduced by requiring a single
neutron in the detection process:• β-radioactivity (0 neutron)• Solar νe -e scattering (0 neutron)• Michel electrons from cosmic ray muons3 (0 neutrons)• Atmospheric neutrino4 inelastic scattering (0-2 neutrons)

Figure 1.12 visually demonstrates the variations in neutron counts among the
aforementioned physics signals observable in SK.

Figure 1.12: Illustration of the expected neutron counts for different physical events
observable in Super-Kamiokande.
3

Capture of a muon by 16 O can result in neutron emission through nuclear de-excitation, andsuch events may contaminate the signal sample.
4
Neutral-current quasi-elastic (NCQE) interactions of atmospheric neutrinos, which knock out
a single neutron, and the residual oxygen nucleus de-excites by emitting a ∼ 6 MeV gamma-ray.
This becomes the dominant background in characterizing the DSNB flux.

20

CHAPTER 1. INTRODUCTION

1.4

SK-Gd

To enhance the efficiency of neutron detection, Super-Kamiokande (SK) has introduced Gadolinium (Gd), recognized as the most strongly neutron-absorbing element,
into its inner water volume. The details of the SK detector and the Gd-loading process can be found in Section 2.2.2.
When neutrons are captured by 155/157 Gd, the resulting compound nucleus 156/158 Gd∗
undergoes de-excitation by emitting multiple gamma-rays with a combined energy of
8.54 and 7.94 MeV, respectively. These gamma-rays are more distinguishable thanthe 2.2 MeV signal from 1 H(n, γ) in the pure water phase. In Figure 1.13, the typicalenergy deposition on the photosensors on the cylindrical detector wall is illustrated
for both H and Gd(n, γ) signals. In contrast to H(n, γ) signals, the Gd(n, γ) vertex
can be estimated with much greater resolution. This capability enables studies ofrecoil neutron kinematics and neutrino direction reconstruction [28].

Figure 1.13: Event displays for 1 H(n, γ) (left) and 157 Gd(n, γ) (right). Yellow dots
indicate the arrival of a single photon recorded on a photomultiplier tube (PMT).
PMTs are not drawn to scale.
In pure water, more than 99.9% of neutron captures occur on Hydrogen (H).
With Gd-loading, the fraction of neutrons captured by Gd increases, leading to amuch faster rate of neutron captures. Figure 1.14 shows both the characteristic
neutron capture time constant and the fraction of neutrons captured by Gd.

21

1.4. SK-Gd

Figure 1.14: The expected neutron capture time constant τ(n,γ) and the Gd(n, γ)
fraction are plotted against the loaded Gd in mass fraction. For the SK-VI phase,
which is within the scope of this thesis, the Gd mass fraction in the target volume
is 0.0110w%, resulting in an expected capture time of around 120 µs and a Gd(n, γ)
fraction of 50%. These predictions were generated using Geant4.9.5 [29, 30] with
ENDF/B-VII.1 evaluated nuclear cross-section data [31].

22CHAPTER 1. INTRODUCTION

1.5

Sources of neutron-related uncertainties

Unfortunately, there are many sources of uncertainties within the prediction of neutron production, and there exists no consensus on the exact magnitude of these
uncertainties. Nevertheless, we can categorize these uncertainties into two maingroups: a) modeling of neutrino-nucleus interactions, handled by neutrino event
generators, and b) detector simulation. The details for both categories are provided
in Sections 2.3.2 and 2.3.5. We outline the dominant sources of uncertainties here.
• Primary neutrino-nucleon interaction (PI)- Fermi motion and binding energy of target nucleon- Nucleon-level interaction cross-sections and kinematics- Hadronization• Hadronic final-state interaction (FSI)- Hadron transport within target nucleus• Nuclear de-excitation• Hadronic secondary interaction (SI)- Hadron transport within detector volume• Neutron capture detection efficiency

Numerous models, particularly those simulating hadronic interactions, contributeto a significant variability in their predictions. Furthermore, there are only a few
measurements of nucleon-16 O cross-sections available in the 0.1-1 GeV range, which
we can use to estimate uncertainties and constrain the models. For a detailed
overview and visualization of the PI, FSI, and SI components of neutron production, refer to Section 2.3.2 and Figure 2.46.

23

1.6. Past neutron measurements

1.6

Past neutron measurements

Over the past decade, efforts have been made to measure recoil and shower neutrons
generated from neutrino interactions, aiming to evaluate the consistency between
observed data and model predictions. We provide a brief overview of these neutron
measurements, their implications, and potential avenues for future investigations.
The very first measurement of neutron multiplicity emerged from SK in 2011[32]. The study presented results showing the mean neutron multiplicity as a function of visible energy (Evis ) in atmospheric neutrino events. The dataset spanned
approximately 2 years, and the neutron detection efficiency stood at around 20%.
Notably, the measurement lacked consideration of systematic uncertainty and was
not compared with any model predictions.
A comparable study was published by the Sudbery Neutrino Observatory (SNO)
in 2019 [33], using a Cherenkov detector similar to SK but with heavy water D2 O
as the target. The dataset spanned about 4 years of atmospheric neutrino data,with an average neutron detection efficiency of 15% for D(n, γ) signals and 45% forCl(n, γ) signals. The dominant uncertainties were from neutron detection efficiency
and neutrino interaction modeling, with no consideration given to uncertainties inhadronic interactions within the detector volume. The SNO results, based on atmospheric neutrino data, showed reasonable agreement with the 2011 SK findings, aswell as simulations from GENIE 2.10.02 and Geant4.10.0, as shown in Figures 1.15
and 1.16. In the sub-GeV visible energy range, the observed neutron multiplicity
was slightly lower than the predictions, but the sizable statistical uncertainty in theobservation makes it challenging to definitively label any model as inaccurate.

24

CHAPTER 1. INTRODUCTION

PHYS. REV. D 99, 112007 (2019)
Phase I

DATA
CCQE
CCnonQE
NC

I

10

DATA
MC (stats. + systs.)

5

Averaged number of
produced neutrons

Figure 1.15: The average neutronMCsignal
multiplicity observed in atmospheric neuTRUTH ALL
MC
TRUTHPrimaries
trino interactions for SK (left, [32]) and SNO (right, [33]), presented as a function of
event visible energy Evis . In the right figure, the SNO H2 O prediction is represented
5
by purple points with error bars, including both statistical and prediction errors.
showing reasonable agreement with the overlaid SK data represented by dark green
points with statistical error bars.
010

utron multiplicity

102

103

Visible Energy [MeV]Phase II
10

10

Averaged number of
produced neutronsII

MC (stats. + systs.)
MC TRUTH ALL
MC TRUTH Primaries

5

0

20

DATA

102

103

Visible Energy [MeV]

utron multiplicity

Figure1.16: A comparison
betweenpredicted(red) andobserved(black, with statiscted neutrons per neutrino
interaction
FIG. 14. Averaged
numberof produced
neutrons
vs visible
and Phase II (bottom).
energy
for
Phase
I
(top)
and
Phase
II
(bottom).
The
pointsrepresenttical errors) average neutron signal multiplicities in SNO is presented. The predicdatausingwith statisticaluncertainties.
The reconstructed
MC is showntion was generated
neutrino
event simulation
with GENIE
2.12.02 and detector
with red boxes with the size corresponding to the systematicGeant4.10.0.
Thelineoverallpredictionof 25%
hase I and 10.8=6 simulation
for Phase II,with uncertainties.
The green
represents
the averageerror
total numberof includes a 13%
correlations and correspondto p-due neutrons
uncertainty
to PI/FSI
GENIE
a 16%
uncertainty
givenwithinby the MCtruth,andand theblue line
correspondsin
to signal efficiency.
9, respectively. We(Reprinted
performed afromthe
average number of primary neutrons given by the MC truth.
[33])

omparing the efficiency-corrected
C (red band) with the true neutronThis shows an excellent agreement,
efficiency correction is properly
arates out the number of primaryo show how the production isy neutrons at higher energies, as
he measured neutron production
between both phases, despite theon efficiencies.
tibility between phases, we per2

are calculated with respect to this observable, as shownin Fig. 16.
We compared the total number of produced neutronsobtained by this work with the SK results [8]. Since ourmeasurement of neutron production is a combination of lightand heavy water, we estimated the neutron production in a
SNO detector filled with light water, in order to compare to
the SK results. We calculate the expected neutron production
difference between light water and heavy water by generating neutrino interactions in two SNO configurations: one25

1.6. Past neutron measurements
T2K [34] conducted a measurement of neutron multiplicity in νµ /ν̄µ -CCQE-likeinteractions at its far detector (Super-Kamiokande), utilizing accelerator neutrinos.
The measurement explored the dependence on the transverse muon momentum pTµ ,which is believed to be positively correlated with the four-momentum transfer andthe energy transferred to the hadronic system. This represents the first neutron
production measurement employing well-controlled accelerator neutrino events.
The uncertainty in neutron detection efficiency in the far detector (SK) during
the pure water phase was extensively examined. The observed neutron multiplicity
was notably smaller than predictions, particularly for neutrino interactions withlow energy transfer. Furthermore, the study revealed significant variability in theneutron multiplicity predictions among different neutrino event simulators. The
results are shown in Figure 1.17.
Since this measurement utilized events observed at SK, the results currently serve
as a basis for estimation of neutron production uncertainty in SK’s atmospheric
neutrino oscillation analysis. It is important to note that the result is dominated bystatistical errors, and the uncertainty estimate is substantial, reaching up to 60%
for events with hightransverse
transfer.OF MEAN NEUTRONCHAPTER
8. momentum
MEASUREMENTMULTIPLICITY

FigureMeanneutronmultiplicity
as a function
muon transFigure 8.42:
1.17: The
averageneutronsignal multiplicity
forofνµreconstructed(left) and ν̄µ (right)
beam
verse
momentumincomparison
totheequivalent
expectationsoftheNEUT-,
modes, observed at the far detector (SK), as a function of the reconstructedmuon
NuWro-,and GENIE-based
Thestatisticalleft and errors
right figures
showandthesystematic
FHC andpTµ . The error
bars in the dataMCs.
includes(dominant)
RHC
respectively.
µ samples,errors1Rν
relatedto signal
efficiency. The shaded regions are predictions from three

distinct neutrino event generators, incorporating errors in signal efficiency.

26

CHAPTER 1. INTRODUCTION
In 2019, the MINERvA collaboration [35] presented the first measurement of
neutron kinematic variables, such as 1/β (β = vn /c). with a ν̄µ beam with an average energy of 3 GeV and a fine-grained polystyrene (C8 H8 )n scintillator tracker
as the target. Neutrons scatter protons and leaves a visible signal a small distanceapart from the production vertex. Neutron kinematics can be deduced from thetrack length and time-of-flight of the signal. While the production of energetic neutrons aligned with predictions, there was a noticeable deficit of low-energy neutrons,
surpassing the comprehensive uncertainty estimates.
In 2023, MINERvA published another measurement [36]. This time, they investigated the ν̄µ multi-neutron production cross-section as a function of pTµ , using ahigher average beam energy of 6 GeV. Once again, a deficit of low-energy neutrons
was observed. The measured differential cross-section was compared with various
PI models that considered the knock-out of multi-nucleon pairs. The measurement
revealed a significantly lower cross-section compared to the expected values fromthe models. Figure 1.18 summarizes the observed deficit in neutron production in
MINERvA.

Figure 1.18: A comparison between predicted and observed distributions for thesignal energy deposit (a proxy for neutron kinetic energy, left figure) and the multineutron production cross-section (as a function of pTµ , right figure) in the MINERvA
hydrocarbon target. (Reprinted from [36])Table 1.2 summarizes the past neutron measurements and this study.

27

1.6. Past neutron measurements

SK

SNO

T2K

2011 [32]2019 [33]2019 [34]ν source

Atm.

Atm.

Acc.
(νµ /ν̄µ )E range [GeV]

0.1-200.1-5

0-5

Target

H2 O

D2 O

H2 O

ν events∼ 7000512

311

n signal

H(n, γ)D(n, γ)Cl(n, γ)H(n, γ)

Recoil p

H(n, γ)Gd(n, γ)Detected n∼ 2000476

77

23,492n efficiency

∼ 20%

∼ 15%
∼ 44%

∼ 104
∼ 2 × 105

∼ 20%

∼ 40-60%

Year

Observable (y)Considered
uncertainty (δy)
Variable (x)PI/FSI
modeling
SI modelingn distribution
Multi-n xsec

Mean n multiplicity
Stat only

PI/FSI+eff.

Evis
NEUT

GENIE

MINERvA2019 [35]2023 [36]Acc.
(ν̄µ )
2-6
2-12
C8 H8
16,129∼3×105

SK
2023
Atm.
0.1-10H2 O
39,376∼25%∼50%
Mean nmultiplicity

PI/FSI+SI+eff.

pTµ

1/β
pTµ

Evis

NEUT
GENIE
NuWro

GENIE

NEUT
GENIE

Bertini

Bertini
INCL

Table 1.2: A simplified summary of previous neutron measurements and this study(rightmost column). The numbers marked with the ∼ symbol signify approximations
or rough estimation by the author.

28

CHAPTER 1. INTRODUCTION

1.7

Thesis outline

This thesis focuses on measuring the average neutron capture multiplicity of atmospheric neutrino interactions observed in the Super-Kamiokande (SK) detector,
as a function of neutrino event’s visible energy. With the largest dataset of bothneutrinos and neutron events in water target, this measurement also presents thefirst measurement of “track length” for neutrons in atmospheric neutrino interactions, made possible by recent Gd-loading. The results were compared with various
models, some not included in any previous neutron production measurements.
Key motivations include:• Quantifying systematic uncertainty in total neutron production
in atmospheric neutrino interactions in water• Comparing model predictions with data and providing constraints

Chapter 2 explains the Super-Kamiokande detector and the modeling of atmospheric neutrino interactions, as well as subsequent hadron and neutron interactions.
Chapters 3 and 4 outline the core contributions of this thesis. Chapter 3 covers
the establishment and calibration of the neutron detection algorithm for use in SK,
addressing both pure water and Gd-loaded phases. Various sources of uncertainty
affecting neutron detection efficiency in a Gd-loaded water Cherenkov detector are
studied in detail.
Chapter 4 explores data selection and quality for both atmospheric neutrino andneutron signals. It then discusses modeling neutron detection performance, considering variations with neutrino event properties, and contributing to the reduction of
uncertainty. The chapter also details neutron multiplicity and uncertainty estimation, comparing results with various simulation models. Finally, Chapter 5 explores
the implications and future prospects of the research.

29

1.8. Author’s contributions

1.8

Author’s contributions

While the work presented in this thesis involved collaboration with members of theSuper-Kamiokande collaboration, the primary content in Chapters 3, 4, and 5 is theresult of the author’s efforts. The key contributions are summarized below.
In Chapter 3, the author overhauled the neutron detection software, building
upon the original algorithm for pure water phase data. Although the core algorithm
remained mostly unchanged, a significant redesign of the neural network, including a streamlined feature set, followed modern machine learning standards. The
introduction of a new algorithm (RECO vertex mode, detailed in Section 3.2) ensured consistent neutron detection performance across various capture distances,contributing to the software’s robustness.
In the collaborative Am/Be neutron source calibration, the author made three
key contributions. First, a comprehensive Am/Be simulation (explained in Section
3.3.3) was developed, including continuous source activity and fine-tuned scintillator
optical parameters. Second, the author performed a complete extraction of signal
efficiency from calibration data across SK-IV, SK-V, and SK-VI phases, comparing
results and deriving scaling ratios for data/simulation efficiency (refer to Table 4.6).
Third, systematic uncertainties were evaluated and an overprediction of the Gd neutron capture ratio in the Geant4 simulation was identified. These efforts significantly
contributed to resolving the previously unexplained data-simulation discrepancy inestimated signal efficiencies.
Additionally, the author estimated potential contamination of Michel electronsand PMT afterpulses, as described in Sections 3.4 and 3.5.
The core content in Chapters 4 and 5 represents original work by the author, including the estimation of neutron detection efficiency through non-linear regression
on multiple event variables, measurement of neutron multiplicity, systematic uncertainty evaluation, prediction of neutron multiplicity estimates for various hadronicinteraction model combinations available in the detector simulator, linearity measurements, and model comparisons with observations.

30

Chapter 2
Atmospheric neutrino interactionsat Super-Kamiokande
2.1

Atmospheric neutrinos

Atmospheric neutrinos originate from interactions between cosmic rays, mainly protons making up about 90%, and the Earth’s atmosphere. The source of these cosmic
rays is currently under investigation, but most are believed to come from outside
the Solar System, from sources such as supernova remnants. When these cosmic
rays collide with atoms in the Earth’s atmosphere, predominantly nitrogen and oxygen nuclei, they create a cascade of secondary particles, primarily pions and kaons.
These particles then undergo decay processes, resulting in lighter leptons, including
muon neutrinos and electron neutrinos. The schematic is illustrated in Figure 2.1.
Muon neutrinos result from both meson decays and subsequent muon decays, whileelectron neutrinos are dominantly produced in muon decays. As a result, the expected count of muon neutrinos is anticipated to be roughly twice that of electron
neutrinos.

31

2.1. Atmospheric neutrinos

15

E2 Φ [GeV cm-2 sec-1 sr-1]

Figure 2.1: Schematic of atmospheric neutrino production. (reprinted from [37])

10-1
10-2
10-3
10-4
10-5
10-6
10-7
10-8
10-9
-1

Super-Kamiokande I-IV ν µFrejus ν µ
IceCube ν µ unfolding
IceCube ν µ forward folding
AMANDA-II ν µ unfolding
AMANDA-II ν µ forward folding
ANTARES ν µ
HKKM11 ν µ +ν µ (w/ osc.)
Super-Kamiokande I-IV ν eFrejus ν e
IceCube/DeepCore 2013 ν e
IceCube 2014 ν e
HKKM11 ν e +ν e (w/ osc.)

0

1

2

3

4

5

Log (Eν /GeV)
10(color measured
online) The measured
energy spectraof the atmospheric⌫ and ⌫ fluxes by SK,
withνmeasurements
Figure FIG.
2.2:14. Theenergyspectraof the atmosphericνeshownandµ fluxes by SK,
by other experiments, Frejus [39], AMANDA-II [40, 41], IceCube [42–45], and ANTARES [75]. The phrase ”forward folding”
by IceCube
and AMANDA-II isbysynonymouswithforward-fitting. The HKKM11
flux model predictions
for the Kamiokashown usedwithmeasurements
otherexperiments.
(Reprinted
from
[38])
site are also shown in solid (with oscillation) and dashed (without oscillation) lines. The error bars on the SK measurementeinclude all statistical and systematic uncertainties.

32µ

CHAPTER 2. ATMOSPHERIC NEUTRINO INTERACTIONS AT
SUPER-KAMIOKANDE

2.2

The Super-Kamiokande detector

The Super-Kamiokande (SK) detector is an underground water Cherenkov detector.
Figure 2.3 illustrates schematic drawings of the setup. The detector consists of acylindrical tank divided into two coaxial volumes: the inner detector (ID) and theouter detector (OD), both filled with water. The ID serves as the active target forphysics analyses, detecting Cherenkov radiation through inward-facing photosensors
mounted on the tank wall. Meanwhile, the OD, equipped with outward-facing photosensors, acts as a veto for cosmic rays and captures a portion of the energy fromphysical events escaping the ID. Our primary signal involves atmospheric neutrinos
interacting within the ID volume, which typically show minimal activity in the OD.
The distinct Cherenkov radiation signatures from scattered charged leptons (electrons or muons) with energies in the range of O(0.1-1) GeV are observable within
the ID. In cases where neutrons are produced during atmospheric neutrino interactions in water, faint neutron capture signals with energies around O(1) MeV become
visible in the detector within approximately O(100) µs from the interaction. The
following sections describe the operational principles of the detector, as well as description of hardware and software components used for signal detection, calibration
procedures, and a recent Gd-loading campaign aimed at enhancing neutron capture
detection efficiency.

Figure 2.3: A schematic representation of the SK detector along with its relevant
infrastructure (left [39]), and a sectional view of the detector system (right [40]).

33

2.2. The Super-Kamiokande detector

2.2.1

Working principle

As a charged particle traverses a dielectric (polarizable) medium, it undergoes electromagnetic interactions with the surrounding molecules. This interaction excites
the molecules, inducing radiation. Notably, if the particle’s speed surpasses thephase velocity of the electromagnetic wave in the medium, radiation emitted simultaneously from different positions can combine on the wavefront, generating a kindof shockwave characterized by an angle of cos−1 (β/n). In this context, β denotes
the ratio of the charged particle’s speed to the speed of light in a vacuum, while n
represents the medium refractive index which may vary with the wavelength of theradiated photons. These emitted photons, peaking at around 375 nm, are detectableusing a photosensor.

Figure 2.4: Illustration of Cherenkov radiation dynamics (left) and the resulting
ring-shaped image on a photosensor plane (right). (Adapted from [41])When a particle slows down below the Cherenkov threshold speed (βmin = 1/n),
the particle ceases to radiate, resulting in a distinctive ring-shaped pattern, as illustrated in Figure 2.4. By setting up a photodetector plane (as shown in Figure2.4) to measure the ring radius and estimate the number of emitted photons, we
can technically reconstruct both the position and energy of the incoming charged
particle. Figure 2.5 presents a typical Cherenkov ring generated by a muon-neutrino
charged-current interaction observed in the detector.

34

CHAPTER 2. ATMOSPHERIC NEUTRINO INTERACTIONS AT
SUPER-KAMIOKANDE

Figure 2.5: A sample display of 0.6 GeV muon-neutrino charged-current interaction.

The energy radiated per unit track length is described by the Frank-Tamm formula (in cgs units) [42]:
dE
(ze)2
= 2
dx
c

Z



1
ω 1− 2β ϵ(ω)
ϵ(ω)>1/β 2



dω

(2.1)

Here, ze represents the charge of the moving particle, ϵ is the permittivity ofthe medium, and ω is the frequency of the radiation. (We assume a magnetic permeability µ(ω) = 1) For an ultrarelativistic particle with β ≈ 1, both the energy

loss per path length and the Cherenkov angle remain nearly constant, given theparticle’s charge and the dielectric properties of the medium. The energy loss is in
the range of O(1-2) keV/cm or O(200-1000) visible photons/cm, which is exceptionally small compared to the energy lost through ionization, typically on the order ofO(1) MeV/cm. Nevertheless, it proves to be a practical option for reconstructing acharged particle.

35

2.2. The Super-Kamiokande detector
The water Cherenkov technique offers a distinct advantage in efficiently distinguishing electrons and muons. Electrons are more prone to induce electromagnetic
interactions, such as pair production and bremsstrahlung1 . As a result, Cherenkov
rings produced by electrons often appear “fuzzier” compared to their muon counterparts. This “fuzziness” can be quantified and utilized as a particle identification(PID) discriminator, as illustrated in Figure 2.6. Further details on this discrimination technique are provided in Section 2.4.1.

Figure 2.6: Illustration of the νe /νµ identification among beam (T2K) neutrino datacollected in SK, based on the clarity of Cherenkov rings. (Reprinted from [43])However, a drawback to this technique is that low-energy heavy particles are
not visible, due to the energy threshold of Cherenkov radiation increasing withparticle mass. The kinetic energy thresholds are 0.26 MeV for electrons, 54 MeV
for muons, 71 MeV for charged pions, 255 MeV for charged kaons, and 481 MeV
for protons. This is an inherent trade-off in the technique, prioritizing the neutrino
event statistics.
1

The critical energy for electrons in water is approximately 70 MeV, while for muons, it isaround 1 TeV.

36

CHAPTER 2. ATMOSPHERIC NEUTRINO INTERACTIONS AT
SUPER-KAMIOKANDE

2.2.2

Detector components

Detector location
The detector is positioned at coordinates 36◦ 25’32.6”N, 137◦ 18’37.1”E, located about
1,000 meters below ground in the Kamioka Mine in Gifu, Japan, with an averageoverburden of 2,700 meters water equivalent (m.w.e.). This depth effectively diminishes the cosmic ray flux by a factor of roughly 10−5 in comparison to sea-level
intensity. At this depth, the cosmic muon flux at SK reaches approximately 10−7
cm2 ·s−1 ·sr−1 , resulting in an event rate of around 2 Hz at the detector.
Detector geometry and design
The detector is a cylindrical tank divided into two distinct, optically isolated volumes. Surrounding the tank are 26 Helmholtz coils to reduce the geomagentic field(∼450 mG) down to 50 mG, maximizing PMT photoelectron collection efficiencies.
The ID volume, measuring 33.8 meters in diameter and 36.2 meters in height,
contains 32 kilotons of water. This space is currently equipped with 11,129 inwardfacing 50 cm PMTs (Hamamatsu R3600) covering the entire inner surface. To minimize reflections, the inner wall (excluding the PMTs) is coated with black polyethylene terephthalate sheets. The effective photocoverage is approximately 40%. Due topotential background radioactivity near the tank wall, our analyses focus on events
sufficiently separated from the tank wall. In recent atmospheric neutrino analyses,
the Fiducial Volume (FV) of the ID is defined as a cylindrical space located 1 meter
inward from the ID wall, with a total mass of 27.2 kilotons.
The OD wall is sparsely equipped with 1,885 outward-facing 20 cm PMTs (Hamamatsu R1408), serving as an active veto counter against incoming particles andproviding a passive shield against neutrons and gamma-rays from the surroundingrocks. The OD region, with a width of about 2.5 meters, contains a total water massof approximately 18 kilotons. To enhance reflection and photon detection efficiency,
the OD wall is covered with a white Tyvek® sheet.

37

2.2. The Super-Kamiokande detector
Water purification
To maintain optimal water purity in the SK detector, a continuous circulation andpurification process is imperative. Unwanted radioactivity, particularly Radon andRadium, naturally present in air and water, increases noise hits and mimics lowenergy PMT hits. The both the tank water and water sourced from the mine
undergoes constant circulation through a purification system at a flow rate of 30 to
60 tons per hour, achieving full recirculation approximately every month.
The purification system, consisting of multiple stages, focuses on three key objectives: filtering particulates, controlling bacterial growth, and reducing Radonthrough degasification. Water is cooled to 13◦ C, aiding in maintaining uniform
PMT dark noise levels and preventing bacterial growth. Water purity is assessedby measured resistivity, increasing from around 11 MΩ·cm pre-purification to 18.2
MΩ·cm post-purification, reaching the chemical limit for ultra-pure water. Particulates smaller than 0.2 µm entering the ultra-filter decrease from an estimated 1000
cm−3 to 6 cm−3 . The Radon concentration in water returning to the SK tank after
filtration is estimated at 0.4 mBq·m−3 . The additional features of the water purification system with Gd-loading is explained in Section 2.2.4.

Air purification
The mine’s air, richer in radon gas due to radon-containing rocks, sees concentrationsof about 1,000 to 2,000 Bq/m3 in summer and 100 to 300 Bq/m3 in winter due tovarying air-flow patterns. An air purification system is employed to reduce radonlevels, resulting in concentrations below 3 mBq/m3 , supplied between the water
surface and the top of the SK tank.

38

CHAPTER 2. ATMOSPHERIC NEUTRINO INTERACTIONS AT
SUPER-KAMIOKANDE
Inner detector photomultiplier tubes

The ID utilizes 50 cm Hamamatsu R3600 photomultiplier tubes (PMTs), and theirdesign is illustrated in Figure 2.7. The photocathode of the ID PMT is coated withbialkali (Sb–K–Cs) to match the wavelength of Cherenkov light, with a sensitivity
range from 280 to 660 nm. At its peak (λ ∼380 nm), the maximum quantum

efficiency (QE) is approximately 21%. The average collection efficiency at the firstdynode is around 70%, maintaining uniformity within ±7% across the entire PMT

surface. The dynode structure follows the Venetian Blind style, and the gain isO(107 ) at a high voltage of 2,000 V. The relative transit time spread for the single
photoelectron signal is about 2.2 ns (1σ). An individual signal pulse with a charge
deposit equivalent to or exceeding 0.25 photoelectrons is digitized as a hit. The
average dark-hit rate at the 0.25-photoelectron threshold is approximately 3 kHz.

All ID PMTs are coated with an acrylic layer with over 96% transparency fora wavelength of 350 nm, and they feature a fiber-reinforced plastic (FRP) case toshield against pressure and prevent potential damage from spreading to other PMTs.

Figure 2.7: ID PMT design. (Reprinted from [40])392.2. The Super-Kamiokande detector

Figure 2.8: Quantum efficiency of the ID PMT photocathode plotted against incident photon wavelength (left) and the distribution of single photoelectron pulse
heights observed with the ID PMT (right). (Reprinted from [40])Frontend electronics, readout, online data processing
The signal from each ID PMT is directed to a module called “QTC-based Electronics with Ethernet” (QBEE). Each QTC (charge-to-time converter) has three input
channels, and there are eight QTCs on each QBEE board, allowing it to manage
input analog signals from 24 PMTs. The schematic of the QBEE module is shownin Figure 2.9.

Figure 2.9: The actual QBEE board (left) and a schematic illustrating the data flow
within the module (right). (Reprinted from [44])40CHAPTER 2. ATMOSPHERIC NEUTRINO INTERACTIONS AT
SUPER-KAMIOKANDEA QTC takes an analog input signal and turns it into a square gate pulse, where
the time width of the pulse corresponds to the integrated charge. If the input
signal is above a certain threshold (for SK, -0.69 mV, which is equivalent to 0.25
photoelectrons), the signal’s leading edge initiates the timing process. The charging
and discharging gates, along with a VETO gate, remain open for 400 ns, 350 ns,
and 150 ns, respectively. All signal currents are ignored while one of the gatesis open. Consequently, the total time needed to process a single input current isapproximately 900 ns. The time it takes for the capacitor to charge and discharge
hit a certain voltage level is measured, which is proportional to the input charge.
The linearity of this time width with respect to the input charge is within ±1%.

This process is illustrated in Figure 2.10.

Figure 2.10: The schematic illustrating the gates and output issued following a PMT
pulse above the discriminator threshold in the QTC. (Reprinted from [45])The QTC incorporated in the QBEE modules features three gain stages: Small,
Medium, and Large, with gain ratios set at 1 : 1/7 : 1/49. The QTC applicationspecific integrated circuit (ASIC) performance is outlined in Table 2.1. Notably, theQTC exhibits a high dynamic range, ensuring the accurate reconstruction of lightyield from GeV-scale events.

41

2.2. The Super-Kamiokande detector

Input channels
Processing speed
Gain ratios
Discriminator threshold
Dynamic range
Charge resolution
Output linearity
Timing resolution

3∼900 ns / cycle
1 : 1/7 : 1/49
-0.3 ∼ -14 mV (Small)
0.2 ∼ 51 pC (Small)∼ 357 pC (Medium)∼ 2500 pC (Large)∼0.1 pC (Small)<1%
0.3 ns (2 pC input)< 0.2 ns (>10 pC input)

Table 2.1: Performance of the QTC ASIC. (Adpated from [44], [46])The QTC output pulse duration is digitized by the time-to-digital converter(TDC), which temporarily stores hit time and charge information within its internalbuffer. A single 60 MHz master clock issues a trigger every 1,024 clocks (approximately 17 µs) to the field programmable gate arrays (FPGA) on each QBEE board.
Care was taken to ensure that signal transfer cables for this master clock trigger
have nearly equal lengths. Following the master clock trigger, each FPGA reads
and compresses the hit information stored in the TDC buffer simultaneously during
the 17 µs period, saving them in a first-in-first-out (FIFO) memory.
The FIFO memory content is read by a network interface card, and the information is transferred to online PCs via a Fast Ethernet connection, achieving athroughput of 11.8 MB/s per QBEE. All recorded PMT hits are sorted in time andsegmented into 22-ms length blocks by a parallel processing approach involving 20
frontend PCs. The segmented data are then merged, and triggers are applied toconstruct physical events. The triggered events are reformatted into CERN ZEBRA
and ROOT formats in real-time, serving various physics analyses. The data flow
schematic within the online system is illustrated in Figure 2.11. Notably, this system can handle data flow from individual PMT hit rates of up to 10 kHz without
loss of information.

42

CHAPTER 2. ATMOSPHERIC NEUTRINO INTERACTIONS AT
SUPER-KAMIOKANDE

Figure 2.11: The schematic of the online data processing. (Reprinted from [47])Software trigger
The Merger PCs, illustrated in Figure 2.11, use a “software” trigger to construct anevent from time-sorted PMT hits. If the count of PMT hits within a 200-ns sliding
window exceeds a predetermined threshold, the PMT hits within the corresponding
event window (usually [-5, +35] µs from the trigger) are combined into a single event.
The thresholds and event window details for each event trigger type can be foundin Table 2.2. Originally configured to mimic trigger conditions and performances
before the 2008 electronics upgrade introducing QBEEs, Fast Ethernet, and software
triggers, these trigger definitions are continuously updated during data collection in
response to changes in PMT dark rates and failed PMTs.

Trigger TypeSuper High Energy (SHE)High Energy (HE)Low Energy (LE)Super Low Energy (SLE)
Outer Detector (OD)

Threshold [hits/200 ns]
70 → 5850
47
34 → 3122 (in OD)

Event window [µs][-5, +35][-5, +35][-5, +35][-0.5, +1.0][-5, +35]

Table 2.2: Overview of software trigger types. Thresholds, indicated with arrows,
may vary within the specified range.

432.2. The Super-Kamiokande detector
For SHE triggers exclusively, a special attachment flag named AFT (after-trigger)is raised to capture neutron capture reactions occurring with a time scale of O(100)µs. When this flag is attached, an extra 500-µs record of PMT hits is appended to
the leading SHE-triggered event. If another event trigger is issued within 500 µs
from the SHE event, the AFT attachment is omitted.
Another unique trigger type is the random (also known as “dummy”) trigger.
A “dummy” event is triggered by a clock with a frequency of O(1) Hz. Initiallydesigned to synchronize with the accelerator beam spill frequency for the T2K experiment, this trigger serves to extract background PMT hits when the beam isinactive. Dummy-triggered events have an event window of [-500, +500] µs, and therecorded PMT hits in these events are appended to our simulation to model both
PMT dark noise hits and background radioactivity.

2.2.3

Calibration

This section discusses the calibration processes for both hardware and simulation.
For additional information, refer to [46]. We focus specifically on the calibration
related to the ID, as we do not incorporate OD information in our physics analyses,
but rather use the OD solely for the purpose of cosmic ray rejection.

ID PMT and electronics calibration
To interpret the digitized output from the detector system in terms of physicalquantities, it is crucial to calibrate the ID PMT and electronics. We focus on
key parameters such as photon counts and the arrival time of each digitized PMT
hit. The calibration process follows a specific sequence: we first calibrate the PMT
charge output, and then, we fine-tune the timing response, given that the latter isdependent on the former.

44

CHAPTER 2. ATMOSPHERIC NEUTRINO INTERACTIONS AT
SUPER-KAMIOKANDE
High-voltage determination
Initially, 420 “reference PMTs” were pre-calibrated within a µ-metal shield, usinga Xe lamp and scintillator ball combination. This pre-calibration aimed to maintainthe variability of the output ADC counts for the reference PMTs below 1.3% RMS,
given the same light input. These reference PMTs were symmetrically installed in
the ID wall, as illustrated in Figure 2.12. Once all ID PMTs were in place within thetank, the scintillator ball used for pre-calibration was positioned at the tank center.
This source then was used to adjust the high-voltage for each individual PMT,aligning it with the output charge of the reference PMTs in the same “group,” i.e.,
in azimuthally symmetric positions as each tested PMT.
This adjustment successfully achieved an output charge variability below 1.3%,
consistent with the earlier pre-calibration results. This adjustment took place only
once in 2006, following an incident where approximately half of the PMTs weredamaged. The scintillator ball used for the pre-calibration remains permanently
positioned for real-time and long-term monitoring of the detector response.

Figure 2.12: The positions of the reference PMTs (left) and a schematic illustrating
the grouping of individual PMTs with reference PMTs based on their symmetricalgeometrical relationship with the calibration source (right).

45

2.2. The Super-Kamiokande detector
Gain measurement
The “gain” refers to the charge amplification rate of the PMT dynode. The
process of measuring individual PMT gain involves two steps. Firstly, we determine
the relative gain differences using a variable-intensity light source. Secondly, using asingle-photoelectron level light source, we establish the average absolute gain scale
applied to all ID PMTs, correcting for the earlier determined relative gain differences.
To evaluate relative gain differences, we utilize a scintillator ball coupled with an
N-laser-driven dye laser capable of adjusting light flash intensities. By comparing
the charge output during high-intensity flashes with the number of hits at a singlephotoelectron level, we obtain a factor proportional to the gain of each PMT. The
RMS of the distribution of this factor is 5.9%.
erage gain
gain ofthe entire detector.
A nickelsource, whichemitscombination
9 MeV
Absolute
measurement
involvesa “Nickelball,”isotropicallya sphericalat theof the detector.housing
The photons
the nickel
sourceof 6.5 kgphotons,of NiOis placed
and 3.5
kg centerof polyethylene
a Cffrom
sourceat thecenter.
58 a single
suﬃcientlyby
low-energy
sucharethatthermalized
virtually any hit
be attributed
Neutronsare
produced
Cf fission
andcancaptured
by toNi, emitting

photoelectron
emission.
The nickel-induced
single-photoelectron
each these
9 MeV energy
throughgamma-rays.
Over 99%of PMT hitsdistribution
resulting offrom
PMT,
corrected
relative gain, isarrival.
then added
intoabsolutea single distribution,
signals are
expected
tobybeeach
duePMT’s
to single-photon
Thegain is derived
shown ofin theFigure2·8. A fit to thiswhole-detector
single-photoelectron
distributionacrossfrom the peakgain-corrected
singlephotoelectron
charge peak, averaged
establishes
a single picocoulombs-to-photoelectrons
conversion factor.
all ID PMTs,
as illustrated
in Figure 2.13.

Figure 2.13: An
example of Sum
combinedgain-corrected charge distributions from all
Figure 2·8:of relative gain-corrected ID PMT singleID PMTs, displaying
a
peak
at
2.5
pC.
(Reprinted[48]) charge, inphotoelectron distributions. The
peak showsfrom
the averagepicocoulombs, corresponding to one photoelectron.
PMT signal times are calibrated to account for diﬀerences in the photoelectron

46

transit 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,CHAPTER 2. ATMOSPHERIC NEUTRINO INTERACTIONS AT
SUPER-KAMIOKANDE
Relative QE measurement
The “QE” of a PMT is defined as the product of its quantum efficiency (theconversion ratio of photons to photoelectrons at the photocathode) and the collection
efficiency (the ratio of photoelectrons collected at the first dynode). Essentially, thisquantifies the probability of a PMT recording a single photon arrival as a hit.
Prior to measurement, the water underwent convection, ensuring temperature
variations below 0.01◦ C to guarantee uniform water transparency throughout the
ID volume. Subsequently, the QE was measured using the same Nickel ball used
for absolute gain measurement. This measurement is based on the number of hits
recorded by individual PMTs, with corrections applied for photon attenuation andthe angular acceptance of individual PMTs obtained from MC simulation, as illustrated in Figure 2.14. As the exact number of photon arrivals at each PMT is notknown, the results provide relative differences in QE, which are then tabulated for
integration into the MC simulation of the detector.

Figure 2.14: The normalized hit probabilities of PMTs, derived from single-photon
level light emitted by the Nickel source, in both expected (blue) and observed (red)distributions. These distributions are shown in relation to the vertical positions of
PMTs on the tank barrel (top figure) and the radial positions of PMTs on the tank
top and bottom walls (bottom two figures). (Adapted from [46])47

2.2. The Super-Kamiokande detector
Charge linearity calibration
The ID PMTs are expected to show approximately 5% non-linearity for signals
surpassing 250 photoelectrons. To assess the charge linearity of each PMT’s response, a N-laser-driven dye laser connected to a MgO light diffuser ball was used.
This setup was inserted into the ID volume at various positions and light intensities. Figure 2.15 shows the calibration configuration as well as the ratio between theobserved and expected charge outputs as a function of the expected values. These
results are then incorporated into the MC simulation of the detector response toaddress the apparent non-linearity observed in the PMT charge output response.

Figure 2.15: Setup featuring the N-dye laser diffuser ball for PMT charge linearity
and timing response calibration (left), along with the observed charge linearity plotted against the expected charge output (right).
Timing
The configuration shown in Figure 2.15 is also used for calibrating the PMT
timing response, which is dependent on factors of O(1) ns, such as transit time,
lengths of PMT signal cables, readout electronics, and the pulse height (“timewalk” effect). The timing of the laser signal is accurately known to the sub-ns level.

48

CHAPTER 2. ATMOSPHERIC NEUTRINO INTERACTIONS AT
SUPER-KAMIOKANDE

Figure 2.16: A 2D histogram representing the charge and relative timing of hits
recorded by a PMT in the laser calibration setup (left), with a specific charge binslice projected onto the time axis (right). A smaller value of T indicates later timing.
The tail on the later timing side is attributed to light reflection and scattering.

Figure 2.16 displays a 2D histogram representing the charge and relative timingof hits recorded by a PMT in response to laser pulses with varying intensity andpositions. Notably, the relative timing shows a dependence on signal charge, where alarger charge corresponds to faster timing, as expected due to the time-walk effect.
This relationship is modeled using a segmented polynomial function with 15 freeparameters, as detailed in [46]. Termed the TQ-map, the fitted function for eachindividual PMT is used to correct the time response of its respective readout channel.
The corrected time distribution for each PMT can be fitted with an asymmetric
Gaussian function, as shown in Figure 2.16, to estimate the PMT’s timing resolution.
An N-laser light constantly flashes at a frequency of 0.03 Hz, resulting in a daily
timing difference for each readout channel with a statistical accuracy of 0.05 ns. The
timing response demonstrates a stability within 0.1 ns.

Photon tracking calibration
Photons undergo either scattering or absorption during their propagation in water,
with the extent varying based on wavelength. The light scattering and absorption inwater is studied using a monochromatic laser beam of different wavelengths, injectedfrom the top of the tank to the bottom, as illustrated in Figure 2.17.

49

2.2. The Super-Kamiokande detector

Figure 2.17: A schematic illustrating the photon tracking calibration throughmonochromatic laser injection (left) and the observed PMT hit time distributions
divided into various tank regions (right). In the right figure, the milder peak on
the left is attributed to scattered photons, while the sharper peak on the right is a
result of photons reflected at the tank bottom. (Adapted from [46])
SK has its own empirical model for photon propagation in water. The model
begins with an assumption of exponential light attenuation represented by e−l/L(λ) ,where l is the photon track length, and L is the characteristic attenuation length,
also called water transparency. L is influenced by photon absorption or scattering,which is dependent on the photon wavelength λ. In the SK detector simulation,
L(λ) is defined as:

L(λ) =

1αabs (λ) + αsym (λ) + αasy (λ)(2.2)

Here, αabs , αsym , and αasy represent amplitudes (in m−1 ) of absorption, symmetric scattering, and asymmetric scattering, respectively. To align with the laser
data for each wavelength, we determine the amplitudes through simulation comparisons. The relationship between these amplitudes and the photon wavelength λ isparametrized, as described in [46].

50CHAPTER 2. ATMOSPHERIC NEUTRINO INTERACTIONS AT
SUPER-KAMIOKANDE

Figure 2.18: Illustration of fitted absorption, symmetric scattering, and asymmetricscattering amplitudes parametrized by photon wavelength (left), along with thetemporal variation of these amplitudes (right). (Reprinted from [46])51

2.2. The Super-Kamiokande detector
These parameters, characterizing photon scattering and absorption rates in water, are referred to as “water parameters.” Figure 2.18 shows α and L−1 as functionsof λ with the continuously monitored and fitted water parameters. The fitted amplitudes in Figure 2.18 correspond to a water transparency L of approximately 120
m (for λ = 440nm).
Additionally, photons can be reflected when encountering a boundary between
materials with different refractive indices. In SK simulation, four materials are
considered: water, glass (PMT cover), bialkali (PMT photocathode), and vacuum(air). Reflection can occur on the PMT surface or the black sheet covering the tankwall. The complex refractive index of bialkali (nreal + i · nimag ) is specifically tuned
for various laser wavelengths, aligning with the steeper photon reflection peak in thelaser calibration data in Figure 2.17.
Finally, a dedicated measurement of the reflectivity of the black sheet covering
the wall was conducted. A laser light injector and a sample of the black sheet were
submerged in the tank, and the light reflected off the black sheet sample was directedto the ID PMTs. The reflectivity was adjusted based on the ratio between the direct
and reflected charge output with and without the black sheet sample.

Absolute QE measurement
Following the calibration of water parameters in each SK phase, the absolute QE
scale (COREPMT), which normalizes the overall charge output response of the PMTs,
is determined by comparing simulation results with calibration data. Distinct calibration data sets are used for the two analysis groups within SK.
The LOWE group, focusing on solar and astrophysical neutrinos with signal
energies in the range of O(1-10) MeV, utilizes a linear accelerator (LINAC). This
LINAC generates an electron beam with energies ranging from 5 to 18 MeV. The
electron beam, originating from the klystron-based LINAC, is directed straight into
the tank through three bending magnets and a beam pipe from the tank top. This
setup is depicted in Figure 2.19.

52

CHAPTER 2. ATMOSPHERIC NEUTRINO INTERACTIONS AT
SUPER-KAMIOKANDE

Figure 2.19: Illustration of the LINAC setup (left) and the calibration arrangement(right). (Reprinted from [49])The ATMPD group, which focuses on GeV-level events like atmospheric neutrino
interactions or proton decays, has used through-going cosmic ray muons. Thesemuons traverse both the top and bottom of the detector with relatively consistentpath lengths. The absolute QE value that best fits the observed through-going muon
charge distribution is selected, as illustrated in Figure 2.20.

Figure 2.20: An illustration of how the absolute QE is obtained from the observed
and expected charge per track length distributions of through-going cosmic muons.
The left figure shows the simulated distribution with absolute QE of unity, while
the right figure shows the distribution with the fitted value. (Adapted from [50])The values obtained from two distinct calibration datasets show an agreement
within 1%, owing to the high dynamic range and charge linearity of the ID PMTs.

53

2.2. The Super-Kamiokande detector
Absolute energy scale calibration
Using a fully tuned simulation, we can reconstruct the momentum of charged particles based on their Cherenkov ring. The detailed reconstruction process itself isoutlined in Section 2.4.1. The precision of the PMT charge response simulation significantly impacts the accuracy of the reconstructed momentum. Estimated errors
in momentum reconstruction, spanning from O(101 -104 ) MeV/c, are determined using natural calibration sources specific to each SK phase. For the lower energy range
around O(10) MeV/c, Michel electrons resulting from cosmic-ray muons stopping
inside the detector are used. In the 135 MeV range, neutral pion decays originating from atmospheric neutrino neutral-current interactions are used. For thehigh-energy range, up to 10 GeV/c, fully contained stopping muon events are used.
Results up to SK-V demonstrate a reasonable agreement between simulation anddata within 2%, as illustrated in Figure 2.21. As of the current writing, calibrationfor SK-VI has not been completed.

54

CHAPTER 2. ATMOSPHERIC NEUTRINO INTERACTIONS AT
SUPER-KAMIOKANDE
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

Momentum [MeV/c]

103

104

Figure5·12: energy
Absolute
energyscale measurements
for all SK
phasesFigure 2.21:
Absolute
scale
measurementsfrom SK-I throughV. “Convenand refers
separately
measuredin themore
conventionallines)theandadditional FV”
to events
with vertices
than 2 m (solidaway from
tank
wall,
tional(dashedlines)
fiducial
volumeregions.
The
measurements
spanwhile “Expanded FV” refers to events with vertices between 1 to 2 m from the wall.
a rangeof energies
relevant
to atmospheric
neutrinos,
and arefound
Both setsof events
are utilized
for analysis
in this thesis.
(Reprintedfrom[48])tobe within a few percent for all SK phases & regions.
on the eﬃciency
is 1.5 %offordetector
SK I-III modelingand 0.8 % parameters
for SK IV-V, and corresponds to aLong-termmonitoring
few-percent change in the decay-electron-selected samples for a 1σ variation.
The following parameters are monitored on a run-by-run basis, with each normal
NeutronTagging:
Thea fulluncertainty
on the neutron tagging algorithm’s eﬃciency isruntypically
spanningday:
estimated from a combination of two smaller systematic eﬀects: the dependence ofWater transparency
photonattenuation
length)the• algorithm’s
eﬃciency(i.e.,on theneutron’s
distancefrom the primary event vertex,
Cosmicray muons,primarily
minimum-ionizing
particleswitha 2neural
MeV/cmand onchanging
detector
conditions.
An overall data
versusMC
networkenergy loss in water, serve as a natural calibration source. Their track lengths
eﬃciency uncertainty is also included. The estimated overall uncertainty is 8 % toare directly proportional to the radiated energy, allowing us to discern the
16 %,relativeand is separately
estimatedfor sub-GeV
andmulti-GeV
e-likeandµ-like singleamount of energy
dissipatedthroughphotonscattering
andabsorption
in water. A
Through-goingreasonably
fixed path
are usedof thering samples.
1σ variation muonsin thiswith
uncertainty
increases
the lengths
normalization
for monitoringwater neutrons,
transparency.
sampleswith no tagged
and decreases the normalization of the samples

148
55

2.2. The Super-Kamiokande detector• Dark rate
The average hit rate per PMT is monitored using off-timing data, primarily
consisting of dark noise and minimal background radioactivity.
• PMT relative gain
The single photoelectron peaks of individual PMTs in off-timing data areutilized to determine the relative gains of each PMT.
• Top-Bottom Asymmetry
The hit probability shows a vertical asymmetry, quantified by a parameter
defined as (⟨NTop ⟩ − ⟨NBottom ⟩)/⟨NBarrel ⟩, where NTop , NBottom , and NBarrel

are the averages of the hit probabilities of top, bottom, and barrel PMTs,
respectively. This rate is monitored using the Xe-lamp-driven scintillator ballthat is permanently fixed at the tank center.

Figure 2.22 shows an example of the time variation of these parameters in SK-V.

Figure 2.22: Temporal fluctuations (SK-V) in detector modeling parameters, continuously monitored on a run-by-run basis.

56

CHAPTER 2. ATMOSPHERIC NEUTRINO INTERACTIONS AT
SUPER-KAMIOKANDE

2.2.4

Operational phases

SK initiated its operations in 1996 and has been in continuous operation for over 26
years. The first phase, SK-I, experienced an interruption in 2001 due to an accident
that damaged 60% of the ID PMTs. Following this setback, SK-II operated withapproximately half of the ID PMTs starting in 2002. The successful re-installationof PMTs in 2005 marked the beginning of the third phase, SK-III, in 2006.
A significant electronics upgrade in 2008 marked the start of the fourth phase,
SK-IV. This upgrade not only increased data processing speed but also facilitateda transition from hardware to software triggers. As a result, the recorded event
length was expanded to 1 ms, allowing the tagging of neutron capture signals witha characteristic time constant of O(100) µs following a neutrino interaction.
After a decade of operation in the fourth phase, the detector underwent comprehensive maintenance in 2018, including draining, cleaning, and overhaul, in preparation for Gd-loading. The fifth phase, SK-V, began in 2019, representing the final
phase with an ultra-pure water target.
In July 2020, about 13 tons of Gd-sulfate were loaded into the ID volume toenhance neutron detection efficiency, marking the sixth phase, SK-VI. The Gd concentration of the ID water reached 0.011w% by August 2020.
The seventh phase, SK-VII, started in June 2022 with additional Gd-loading,reaching a Gd concentration of 0.033w% by July 2022. This phase is currently
ongoing and actively collecting data.
A summary of the operational phases and their corresponding livetimes for atmospheric neutrino data is provided in Table 2.3. The data analysis in this thesis
relies on fully-contained atmospheric neutrino events collected during the SK-IV,
SK-V, and SK-VI phases, covering approximately 12 years of detector livetime. SKI, SK-II, and SK-III data were excluded due to insufficient event lengths for neutron
signal searches, while SK-VII was not considered in the thesis scope as it is stillongoing and not yet completed.

57

2.2. The Super-Kamiokande detector
Phase
Dates
Livetime [days]
SK-I-III
1996-2008
2805.9SK-IV
2008-20183244.4
SK-V
2019-2020
461.0
SK-VI
2020-2022
564.4
SK-VII 2022-present-

Gd concentration [w%]0.0110 ± 0.0002 [51]0.0332 ± 0.0001

Table 2.3: A summary of SK operational phases. SK-IV, SK-V, and SK-VI data
were used in this analysis.
0.01w% Gd-loading for SK-VI
From July 14th to August 17th , 2020, 13 tons of Gd2 (SO4 )3 · 8 H2 O was introduced
into the tank using a dedicated dissolving system (refer to Figure 2.24). This quan-tity corresponds to a Gd-sulfate concentration of 0.026% by weight, dissolved in 50
ktons of pure water within the detector system. The Gd mass fraction is 0.0110w%.
Figure 2.23 illustrates the dissolved Gd-sulfate powder and the loaded amounts oneach date.

Figure 2.23: Gd-sulfate powder (left), along with the corresponding loading amounts
into the detector by date (right). (Adapted from [52]).

58

CHAPTER 2. ATMOSPHERIC NEUTRINO INTERACTIONS AT
SUPER-KAMIOKANDE

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 2.24: A schematic diagram of the SK-VI water circulation system with the
Gd-dissolving system.Figure(Reprinted
from [51])
1: Schematic diagram of SK-Gd water system.
To remove positively charged impurities, and radium ions in particular, astrongly acidic cation exchange resin, AMBERJET™1020 [6] (“C-Ex Resin”
in Fig. 1) is used. For the SK-Gd water system, this resin has been modified
to contain gadolinium as the ion exchange group such that the resin’s cationexchange action never results in a loss of dissolved Gd content; Gd2 (SO4 )3 ·7

59

2.2. The Super-Kamiokande detector
The Gd2 (SO4 )3 · 8 H2 O powder was dissolved in a small amount of water to createa solution. Before introducing the solution into the water circulation system, it wassent to the “pre-treatment” system to eliminate impurities and prevent bacterial
growth. To maximize water purity, both the pre-treatment and water re-circulation
systems incorporated cation and anion exchange resins featuring Gd and sulfate
ions as ion exchange groups. This effectively removed all ionic impurities in thewater, except for Gd3+ and SO2−
4 . The solution was further diluted with pure waterto achieve the desired Gd concentration. The resulting solution was then directedto the SK tank via the “re-circulation” section of the system, which continued tocirculate SK water even after the loading was completed. To facilitate a gradualbottom-up filling of the tank, the temperature of the supplied Gd water was set
approximately 0.3◦ C lower than the tank water temperature just before the loading,

0.026%
Gd2(SO4)3∙8H2O

Day 1

Day 9

pure water
0.026%
Gd2(SO4)3∙8H2O

Day 180.026%
Gd2(SO4)3∙8H2O

41.4m

pure water

20.7m

pure water

10.4m 0.02 %0.026 %

0.0 %as illustrated in Figure 2.25.

Day 35Figure 2.25: IllustrationFigureof the 3:Gd-solution
ascending from the bottom toGadoliniumgraduallyloading scheme.
the top. (Adapted from [51]).

13.90 C at Temperature Control Unit B in Fig. 1. Water was recirculated
As thesethe Gd-loadedwater
introduced
the bottom,
concentration
graunderconditions
forwasabout
45 days.from
Afterwards
thethesupplytemperaturewasto 13.55
C to begin
the Gd Absorption
loading. This
created an
additional
dientloweredby height
was assessedusing Atomic
Spectroscopy(AAS).
This
densitydi↵erence
theof Gd-loaded
water andthe absorbed
pure waterin thetechniquemeasures between
the amountspecific wavelengthsof light
by vaportank beyond that caused by the compound itself. In this way the spatialized Gd. The results, illustrated in Figure 2.27, demonstrate consistent uniformity
profile of the Gd-loaded water could be monitored by measuring the water
across sampled positions. Simultaneously, water conductivity was measured by extemperature
at di↵erent positions in the detector.
tracting samples from various heights in the Gd-loaded tank, as demonstrated in

3.2.
Water
Injection
and Extraction
Figure4.1. The
water attenuation
length, measured through cosmic ray muon data,
Fig. 4 toshows
schematicof thepiping returnedand approximate
flowlevel
ratedecreased
75 m aduringGd-loading
andwatersubsequently
to the baselineateach locationin theinner
detector(ID) andobservedin the purewater
phases(approximately
90outerm), asdetector
shown in (OD)
Figureregions
2.26.
of the tank. This piping was newly installed during a major in-tank refurbishment of the SK detector in 2018 and 2019 conducted in preparation for
Gd loading. There are 12 inlets at both the top and bottom parts of the ID,
60 in the annulus of the OD near the top and bottom, and four in the top
eight
and bottom OD endcaps. The end of each pipe is made of 50A polyvinyl
chloride (PVC) tubing. In order to suppress convection due to the verticalflow of injected vertical water at the bottom, di↵user caps made of 14 cm
diameter stainless-steel plates were installed in February 2020 on each of theCHAPTER 2. ATMOSPHERIC NEUTRINO INTERACTIONS AT
SUPER-KAMIOKANDE

Figure 2.26: Attenuation length of water measured using through-going cosmic
ray muons observed towards the end of SK-V and the first six months of SK-VI.
(Adapted from [51]).

Figure Gd
10: Latest
AAS-measured Gd2 (SO4 )3 · 8H2 O concentration vs. Z position in the
Figure 2.27:
2 (SO4 )3 · 8 H2 O concentration (260 ppm corresponding to 0.0011w%
ID and OD from samples taken on March 25, 2021, showing the complete homogenization
Gd concentration)
in water sampled from various vertical positions, measured withwithin the detector.
AAS. (Reprinted from [51])156

Gd + ’s [8.5 MeV in total], n + 157 Gd ! 158 Gd + ’s [7.9 MeV in total].
The data were taken by deploying the Am/Be source into the SK inner detector through a calibration port near the center in the X-Y plane
3.9 m,
Y= of
0.7backgroundm). Three positions
along
the fiducial
Z-coordinate
wereisse-expected
The (X=
primarysourceneutronsin thevolumelected for periodic monitoring: Z=0 m, Z=+12 m and Z= 12 m. Whento be the spontaneous fission of 238 U impurities in the loaded Gd-sulfate. Withsearching for neutron capture events in SK, specialized data acquisition trigeffectivegers
removalof radioactive
impurities,
the background neutronrate
is expected
were applied:
the so-called
SHE (super-high-energy)
and AFT(after)triggers
[18].
An
SHE
trigger
wasgenerated
whenever
more
than
60
ID
photo be below 600 neutrons per day [53]. This corresponds to an negligibly small
tomultipliers tubes (PMTs) detected at least one photon within a 200 ns
O(10−6 )time
backgroundneutronperinatmospheric
neutrino
window. These
SHEcontribution
triggers resulted
all photons detectedbyevent.
PMTs
during the 35 µs following the SHE trigger being recorded, while those in a
subsequent 500 µs window were also recorded by a sequentially issued AFT
trigger. Gd(n, )Gd event candidates were then extracted from this recorded
data by looking for greater than 30 active PMTs in a 200 ns time window
61
and applying event vertex reconstruction [19].
The time distribution of neutron capture event candidates are shown in
Fig. 12. Event selections were applied using the following event reconstruction parameters: the reconstruction timing goodness gt had to be greaterthan 0.4, the hit pattern goodness gp smaller than 0.4, and the event vertex

2.3. Event simulation

2.3

Event simulation

In this section, we discuss the default models used in SK for simulating atmospheric
neutrino interactions and the subsequent hadronic interactions leading to neutron
capture signals. We also explore the simulation of detector response to these signals
and the methodology used to model backgrounds. Finally, we briefly touch on how
we handle the simulation of neutrino oscillation.

2.3.1

Atmospheric neutrino flux model

As briefly outlined in Section 2.1, the primary production mechanism for atmospheric neutrinos is the collision of cosmic rays with nuclei in the atmosphere. Modeling the atmospheric neutrino flux is thus based on the measurement of the primarycosmic ray flux and the secondary hadronic interaction models predicting the outgoing hadron content from collisions between cosmic rays and air nuclei.
Additionally, the geomagnetic field deflects incoming charged particles, potentially preventing low-energy cosmic rays from reaching the atmosphere, as illustrated
in Figures 2.28 and 2.29. This geomagnetic cutoff is a crucial consideration in modeling the cosmic ray flux.

Figure 2.28: Illustration of the geomagnetic cutoff (left, credit: NMDB) and theEarth’s magnetosphere blocking solar wind. (right, credit: NASA).

62

ontours that would have been
ically quiet period during thein 1859. We also calculateds for selected ‘‘equatorward’’sighted during the period of
ompare those values with theudes appropriate for EpochFig. 2. Vertical2.
cutoﬀ
rigidity contours calculated
for Epoch INTERACTIONS
1850. TheCHAPTERATMOSPHERIC
NEUTRINO
AT
contours are in 1 GV intervals. BGS designates the model developed bySUPER-KAMIOKANDE
the British Geological Survey (see Barraclough, 1978).

y calculations

d grid of vertical geomagnetic
e International Geomagnetic
2000 (Macmillan et al., 2003)
978) geomagnetic field modelsr 5) for 1850. We utilized eachodels with our cosmic ray tragram and determined the verdity parameters for a world
d 15! in longitude. Details ofs for cosmic rays in a model
etermination of cutoﬀ rigidity

ussion of the temporal Figurechanges in2.29:
theFig. 3. Vertical cutoﬀ rigidity contours calculated for Epoch 2000. The

The
geomagnetic
imposesana energy
threshold,foralso known as
contours
are in 1 GV fieldintervals.
IGRF isgeneric abbreviation
cutoff rigidity (expressedin GeV perReference
unit charge),
International Geomagnetic
Field. which is higher at the geomagnetic
equator and lower at the poles. Due to the gradual change in the geomagnetic field,
the rigidity values exhibit seasonal variation. (Adapted from [54])For this analysis, we used the Honda-Kajita-Kasahara-Midorikawa (HKKM)
2011 model [55] as the default flux model. In this model, both the energy andcomposition of primary cosmic rays are sampled from distributions based on satellite measurements (AMS [56, 57]) and high-altitude balloon measurements (BESS[58, 59]).
The composition and density of Earth’s atmosphere nuclei are modeled using
the US-standard ’76 model [60], and the geomagnetic field is represented by the International Geomagnetic Reference Field (IGRF) 2005 model [61]. The uncertainty
arising from potential time variations in these models is expected to be negligible
when compared to other sources of uncertainty.
Hadronic interactions are handled by the JAM model, a component of PHITS(Particle and Heavy-Ion Transport System, [62]), for E < 32 GeV, and a modified
DPMJET-III [63] for E > 32 GeV. Notably, uncertainties in pion and kaon production from these models, particularly in the sub-GeV and above 10 GeV ranges, areconsidered to carry a substantial (10-15%) uncertainty, as illustrated in Figure 2.30.

63

2.3. Event simulation

CALCULATION OF ATMOSPHERIC NEUTRINO FLUX . . .

PHYSICAL REVIEW

0.30

Relative Uncertainty

difference is a little larger, but simila
left panel of Fig. 10. Note, the maxi
0.25
the modified DPMJET-III is seen influx model in most of the cases.
0.20
For $& , we assumed the difference
Fig.
10 of Paper I. Since the uncertai
0.15
cross section works with opposing ef
muons and neutrinos, the error of
0.10
section introduces an error in the calimodel with the atmospheric muon fl
0.05
hand, as we use the observed atmosp
the calibration is not affected by th
0
0
1
2
3
10
10
10
10
pheric model. We use !"% only in F
E ν (GeV)$air . All these uncertainties, $! !$# ", $$tot , are summarized in Fig. 11. Not
FIG. 11 (color online). The uncertainty of each error source for
Figure 2.30: Uncertainty
estimation in the HKKM 2007 model [64]. The
sources andof the maximum unc
conservative,
atmospheric neutrino flux and their sum with Eq. (8). Note,
all
kind
of
neutrinos
and zenith anglerror include uncertainties
in validity
pion flux
flux
meanπ ), kaon
K ),error
Eq. (9) loses its
in the(δshadedregion.
The(δtotal
for free path (crossWe
note,
Eq.
(9)
is
valid only for *& 1 GeV(δisσ ),estimateddifferently from Eq.
(8), as profilestated in thesection) uncertainties
and atmosphericdensity
uncertainties (δair ).
estimate $! without using the atmosp
text. Notethe statisticalsystematic
errorupdates
are not shown
the hadronic(Adapted from [64])In theHKKMand2011
model,
in intheinteracat ground level. In Fig. 12, we show t
figure.
tion simulation are anticipated to reduce the uncertainty in the sub-GeV
flux atrange
balloonby
altitudes at Fort Sumnapproximately 5%.
DPMJET-III reproduces the muon fl
almost only from the ! decay, we use the sum of the%1 GeV=c, and p# =p% ratio for theexperimental error and the residual of the reconstruction
parent !’s remains %3 even at the lo
as the !"# in Eq. (9) (see Fig. 15 of Paper I). Then we
the small energy loss of muons
replace $! in Eq. (8) with !!"# ="# ""% , where "% is the
However, the distance of the produc
sum of ! and K contributions for a conservative estimaplaces are longer than the muons obs
tion. The estimated uncertainty is depicted by the solid line
The muon decay in this distance mak
above 1 GeV in Fig. 11.
for & 1 GeV. We conservatively est
For the $K , we used the modified calculation schemes
pion productions responsible to the atstudied in Sec. III. We assumed the maximum neutrino flux
%0:3 GeV.
difference from the modified DPMJET-III among them as
Note, the uncertainty studied abov$K . The maximum difference for all kinds of neutrino forneutrinos, and for all zenith angles.
vertical direction is depicted by the dashed line in Fig. 11,
neutrino and the zenith angle, we msince that variation is the largest of all zenith angles. Eachmation for the uncertainty. Especiall

δ tot
δπ
δK
δσ
δ air

µ+

pµ / p ν< flux / depth >cal< flux / depth >obs

1.8
−

µ

4.0

Balloon Altitude(Fort Sumner)

1

Figure 2.31: Energy distribution of cosmic rays (left) and their composition (right).

0.6

64

1.0

Ground level

10

0

10
p µ (GeV/c)

1

0.4

10

0

10

1

FIG. 12 (color online). The muon fluxes at balloon altitudes at Fort Sumner (Sept. 2001). Left: comparison
muon fluxes. The dashed lines show $10% deviation from the calculated values. Right: p# =p% ratio for the sam!’s in average. The dashed lines are the same quantities but for the muons at the ground level.

CHAPTER 2. ATMOSPHERIC NEUTRINO INTERACTIONS AT
SUPER-KAMIOKANDE
Flux calculation starts by tracking cosmic rays. Nuclei and flux are sampledfrom data-based distributions (see Figure 2.31), positioned randomly on a sphericalsurface 100 km above Earth. If they pass the geomagnetic cutoff and interact withthe atmosphere, trajectories, and subsequent shower particles (pions, kaons, muons,and neutrinos) are tracked in a modeled atmosphere and geomagnetic field in three
dimensions. Monitoring continues until particles reach the escape sphere (sphericalsurface with 10 times Earth’s radius) or the Earth’s surface. Neutrinos are traced
within the Earth to check interactions from below. Overburden and terrain aroundthe detector are considered.
The HKKM 2011 model provides flux predictions for various detectors, including
SK. This is accomplished by counting neutrinos (Eν > 100 MeV) arriving on the
Earth’s surface within an azimuthal radius of 10◦ from the detector. The surface forthe Kamioka site (SK) is shown in Figure 2.32. The use of such an extensive surface(approximately a 1,000 km radius circle) is essential for efficiently counting incoming
neutrinos, as the actual size of the detector is significantly smaller compared to the
Earth’s dimensions. Any bias introduced by employing such a large counting surface
and averaging the flux over a wide area is accounted for. For instance, larger weights
are applied to horizontally arriving neutrinos than vertically arriving ones. The
anticipated bias is at the 1% level. The uncertainty in the prediction is estimatedto be roughly between 10-20%, as shown in Figures 2.33 and 2.34.

Figure 2.32: The surface designated for neutrino detection, also referred to as the“virtual detector,” for Kamioka site (SK). The boundary of the surface is shown as
the outermost white circle, with the radius indicated by the red arrow.

65

5.5 results and discussions

103

E2Φ [GeV/cm2/sec/sr]

2.3. Event simulation

10-1
10-2

νµ

10-3
10-4

νe

1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2

νe

Ratio

Ratio

10-5

104

1.8
νµ
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
-1
-0.5 of 0the energy
0.5
1spectra1.5 of2 the 2.5
3
3.5
measurement
neutrinoflux

4

Log (E / GeV)
10

The42:
measuredenergyagreesusing
withalloscillated
HKKM11FigureThe measured
fluxspectrum
energy spectra,
SK-I to SK-IV
data,
Figure 2.33: Atmospheric
neutrino
flux
prediction
at Kamioka
by
HKKM 2011 (solid
for thene (blue)and uncertainties.
nµ (red). The flux
are shown
flux, within
estimated
The values
unoscillated
flux ismultialso
2 , following a common convention. Error bars inpliedbyE
lines: oscillated, dashedlines:
comparedνe (blue)and νµ (red)plotted, where
the unoscillated)
deficit of nµ flux due
to neutrinowith
oscillation
beclude allbelow
statisticalsystematic uncertainties. The curves show
comesapparent
100andGeV.
flux measurements
at SK[38]. (Adaptedfrom [38]).

Ratio

Ratio

5.5.2

the HKKM11 [19] model’s prediction with (solid) and without(dashed) neutrino oscillation (same color scheme). The lower part
Comparison
to flux
modelsof the figureshows
the ratio of data / HKKM11 prediction.

1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2

νe

1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
-1

νµ

HKKM11

BartolHKKM07

Fluka

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Log (E / GeV)
10

Figure 43: The measured energy spectra compared with several flux model
predictions (dotted lines), HKKM11 [19], HKKM07 [18], Bartol [21], and Fluka [20]. Vertical axis is the ratio to the HKKM11µ
model. Error bars include all statistical and systematic uncertainties.

Figure 2.34: Comparison of different flux model predictions at Kamioka (oscillated)with νe (blue) and ν (red) flux measurements at SK [38]. The vertical axis represents the ratio to the HKKM 2011 model. (Adapted from [38]).

66

The observed fluxes are compared graphically to several flux models, including the HKKM11 [19], HKKM07 [18], Fluka [20], and Bartol [21] models in Figure 43.
To perform a quantitative comparison, we perform a c2 test whichtakes into account the error correlation matrix, defined as
N N

c2 = Â Â ( F iij

FMC,i ) T Cij 1 (F j

FMC,j )

(62)

where Fi is the observed flux and FMC,i is the expectation of the flux
model being tested, at the i-th energy bin, and Cij is the error covariance matrix (calculated as described in Sec. 5.4 and shown in Fig. 40).

CHAPTER 2. ATMOSPHERIC NEUTRINO INTERACTIONS AT
SUPER-KAMIOKANDE
It is important to recognize that the cosmic ray flux is influenced by solar activity,
exhibiting an 11-year periodicity. During periods of maximum solar activity, thesolar wind can disperse low-energy cosmic rays, diminishing their flux. The HKKM
flux model offers atmospheric neutrino flux as a function of neutrino energy acrossthree solar activity levels: minimum, maximum, and intermediate. The generationof simulated neutrino events assumes the cosmic ray flux at the intermediate solar
activity (later adjusted by the livetime-weighted fractions spent in minimum andmaximum solar activity, as explained in Section 2.3.7). These fractions are computed
based on variations in sunspot numbers and neutron monitor data, which tracks
the count of cosmic neutrons on the ground as a proxy for cosmic ray flux. The
fluctuations in sunspot numbers and neutron monitor counts are shown in Figure
2.35.

Neutron Monitor Counts

II

III

IV

V

700

100

600

90

500

80

400

70

300

60

200

50

Sunspot Number

I

110

100

40
1997

2000

2003

2006

2009

2012

2015

2018

0

FigureFigure
2.35: Neutron
monitormonitor
counts counts[65] during
thetheSKSK
I-VI-V
data-taking
phases.
4·2: Neutron
duringdata-taking
Each SKphase
isrepresented
byashadedgrey
region.
(Adaptedfrom[48])phases. The neutron monitor data is from [92]. Each SK phase isindicated by a filled grey region. The solar minimum and maximum
for each
SK phaseandare maximum
calculated solar
as theactivity
neutronarecount
ratio for
Thefractions
proportionsof minimum
calculated
during periods of solar minimum and maximum activity. The calculated
each SK
operational
phase. Theof solar activity on the overall atmosphericfractions
are tabulated
in impact
Table 4.1.
neutrino flux is expected to be negligible (at the ∼5% level, as indicated by cosmic

neutronin Figure2.35). These fractions are calculated using counts from aandsolarcounts
maximum
conditions.

neutron[91], flux
which
countsthe number
cosmic neutrons as a proxy for the
The monitor
HKKM 2011
model
providesνe , ν̄e , νofµ , and ν̄µ fluxes within the 100 MeV
to 1 TeV
which
arerays.
considered
reliablyare
reconstructablewith thedetector. thenumberof range,
primary
cosmicThe fractions
formed by linearly
extrapolating

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 67
andminimum fractions used for the flux re-weighting.

4.2

Neutrino Interaction Models

2.3. Event simulation

2.3.2

Neutrino-nucleus interaction model

The interactions of neutrinos with the target nucleus in the detector medium are
simulated using “neutrino event generators.” These generators use theoretical models with parameters adjusted to match external measurements, allowing them tocalculate cross-sections for various interaction channels. The simulated event rateis determined by sampling the fractions of these interaction channels based on thecalculated cross-section ratios. Neutrino event generators also sample the kinematics of the outgoing final state based on the theoretical models.
In the energy range of 0.1-1 GeV, the dominant interaction is the quasi-elastic(QE) interaction (refer to Figures 2.37 and 2.38), involving the scattering of a boundnucleon. Since fully solving the neutrino-nucleus quantum many-body problem is
intractable, event generator assumes that the neutrino interacts with one target
nucleon, which is bound in the target nucleus. This approach, known as Impulse
Approximation (IA), assumes that the interaction occurs on one nucleon while theother nucleons act as spectators. Subsequent nuclear effects, such as hadronic finalstate interaction (FSI), are factorized out (illustrated in Figure 2.36).

Figure 2.36: Illustration of the factorization process in modeling neutrino-nucleus
interactions. (Credit: C. Andreopolous)68⌫ µ 0.8 < cos ✓µ < 0.9
CHAPTER 2. ATMOSPHERIC NEUTRINO INTERACTIONS AT
SUPER-KAMIOKANDE 11
Chapter 1. Introduction
Universe 2020, 6, 80

4 of 24

Coherent

Nucleon response•••

Nucleus response

respectively. A CC interaction produces a charged lepton, electron or muon, whose flavor corresponds
to that of a neutrino, ne or nµ . Therefore, the original neutrino flavor is identified by distinguishing
QE
the flavor of the related charged lepton. However, an NC interaction does not indicate the neutrino
flavor since the outgoing lepton is a neutrino. The following neutrino interactions are dominant in theN⇤
DIS
atmospheric neutrino energy region,
GR
Charged-Current quasi-elastic scattering : n + N ! l +2N
N’Charged-Current pion production : n + N ! l + N’ + p
Deep Inelastic ScatteringElastic(DIS) : n + N ! l + N’ + hadrons

Nucleuswhere N and N’ are nucleons (proton or neutron) andN l⇤ is a charged
DIS lepton (CC) or neutrino (NC).
Here, pion production is realized via D resonance excitation. To generate neutrino interactions, there
Nucleonare several pieces of simulation software. Figure 2 shows the total cross-section of nµ in total and!
each interactions predicted by NEUT [24] version 5.3.6, which was used in the latest atmospheric
neutrino
analysisSuper-Kamiokande.
In thismodel,
charged-current
interactions
Figure2.37: inIllustrationof the nucleon
andnucleusresponses to quasi-elastic
a leptonic probe.
Figure1.8:
Schematic
nucleonandnucleusresponses
toaleptonic
probe.
are simulated
using
the[66])
Llewellyn-Smith formalism [25] with nucleons distributed according to the(Reprinted
from
Smith-Moniz relativistic Fermi gas [26] assuming an axial mass M A = 1.21 GeV/c2 and form factors
on correlated
pairs ofnucleons
have been included
following
thetomodel
QEfrompeak[27].
for Interactions
nuclei comparedto the elastic
peak
for lepton-nucleon
scattering
is duethe ofNieves[28]. Pion
production
Fermi
motionof thenucleons.processes are simulated using the Rein-Sehgal model [29] with Graczyk
form factors [30]. The cross-section in this model are consistent with several experimental results.

Serpukhov

IHEP-ITEP 79

σ/E (10-38cm2/GeV)

σ/E (10-38cm2/GeV)

Between the coherent and the QE peak lies the giant resonance (GR) region. These excitations occur1.4when the neutrino interacts with a single0.7nucleon, but due to long-range nuclear
CC single
pion isCC
Total
CCQE
CC Totalcorrelations, CCQE
the energy
transfersharedamong the nuclear
constituentsand the nucleus
ANL
ANL 82CCFR 90
GGM 77
CCFR 90
0.6
1.2
GGM 77
BNL 86CDHSW 87
GGM 79CDHSW 87
undergoes a collective
excitation.
GGM 79IHEP-JIRN 96
Serpukhov
IHEP-JIRN 96
IHEP-ITEP 79

CCFRR arises
84
84
1
A second typeof multinucleon e↵ect
when 0.5
the neutrino CCFRR
interactswith a pair of
BNL 82
correlated nucleons in the target, causing both of them to be knocked out of the nucleus,
0.8
0.4
resulting in a two-particle two-hole (2p2h) final state. These two-nucleon (2N ) knockout
interactions
0.6 are one of the main topics of this work.
0.3 Adopting the same conventions as
before, we denote the 2N knockout reactions by

0.4
0.2

0.2

l + A æ l + (A ≠ 2) + Na + Nb ,
Õ

ú

0.1

where Na andNb are the two emitted nucleons. Whenboth emitted nucleons are detected
0
0
-1
2
-1
10 the lepton
10 is called 1an exclusive
102
1 in the final
10
10 2N knockout
together withstate, 10the process
E
(GeV)
E
(GeV)
cross section, denoted as A(l, lÕ Na Nb ). If only one of the two of the emitted nucleons isÕdetected,
we callthisa(left)semi-exclusive
2N
knockoutenergy
eventforA(l,Na ).andThecase when
only
Figure2.2.38:Total
by neutrinonµ l(left)n¯µ (right)
to nucleon
Figureνµcross-sectionand ν̄dividedµ (right) cross-sections for CC interactions (per nucleon)
the finalnormalizedstate lepton
detectedisagainst
an inclusive
2Nversionknockout
section.
Throughout
thischarged-current
interactions
by NEUT
5.3.6crossoverlaid
with several
experiments.
by is
and
plottedcalculated
neutrino
energy,alongside
variousmeasurements
work, wediscusswhich
nucleonstoappearpairs:
short-range
correlations
Data
pointstwo
are e↵ects
taken fromthe cause
following
experiments:
ANLin[31],GGM77[32], GGM79(left) [33]with error bars. The predictions are produced using the NEUT 5.3.6 generator with an(right) [34], Serpukhov
ANL82[36], BNL86[37], CCFR90[38], CDHSW87[39],
IHEP-JINR96
and meson-exchange
We=stress
that andthesee↵ectsinfluence
the [40],
1N
assumedaxial currents.
mass [35],
of M
1.21
GeV
thetwo-bodyglobalFermi
gascanmodel
for CCQE.
A
IHEP-ITEP79[41],
CCFRR84
[42]
and
BNL82
[43].
knockout cross section as well.
(Reprinted from [67])Experiments
analyze CC neutrino interaction data by only considering the lepton in
1.4. Neutrinooften
Oscillation
the final state, meaning that inclusive cross sections are studied. The MiniBooNE experiNeutrino oscillation occurs if flavor eigenstates are mixed with mass eigenstates, and a difference
ment, for example, used a Cherenkov-type detector, and they only used the final state muon
in mass exists. The mixing between flavor eigenstates (na ) and mass eigenstates (ni ) can be written as
for CC events.
⇤|na i = 2N
In a situation like that, it is hard to distinguishevents from genuine QE events (1)Â U-knockoutai | ni i .
69
discussed above, where a single nucleon is emitted from the nucleus. Thus, the measured
cross section will consists of a sum of 1N and 2N (and eventually 3N etc.) knockout crosssections. This led to some confusion between the experimental and theoretical community,

2.3. Event simulation
Nuclear ground state model
The Fermi gas model is used to describe nucleons within a target nucleus. In thismodel, the nucleus is regarded as a collection of non-interacting nucleons (fermions),
each interacting with a general potential field. This approach allows for the straightforward summation of cross-section contributions from individual nucleons. The potential field can be represented as constant across the entire nuclear volume (global
Fermi gas model), or it can be modeled based on the nucleon density that varies withposition (local Fermi gas model). Neutrons and protons, being distinct particles, experience different forces (protons undergo additional Coulomb repulsion), leading toseparate potential wells with distinct energy levels for neutrons and protons. The
maximum energy that a nucleon can occupy is determined by the nucleon density
and the size of the nucleus, referred to as the Fermi energy.
In the global Fermi gas model, the constant Fermi momentum is set to 225
MeV/c for 16 O, while in the local Fermi gas model, the position-dependent Fermi
momentum is determined by the nuclear density profile ρ(r) that best explains elastic
electron-nucleus scattering data [68].
Figure 2.39 provides an illustration of the Fermi gas model and Fermi momentum
as a function of relative position inside the nucleus.

Figure 2.39: Illustration of the Fermi gas model for a nucleus (left) and Fermi
p/nmomentum (right). EF represents the Fermi energy of protons/neutrons, and EB
denotes the minimum binding energy. (Adapted from [69])70CHAPTER 2. ATMOSPHERIC NEUTRINO INTERACTIONS AT
SUPER-KAMIOKANDE
Momentum of a nucleon undergoing free “Fermi” motion can be sampled fromthe Fermi momentum distribution. Interactions that result in a final momentum ofa struck nucleon smaller than the Fermi momentum are prohibited (Pauli blocking).
The ejection of a nucleon is further restricted by the nuclear binding energy, thedifference of which from the Fermi energy per nucleon averages around 7-8 MeV.
The key feature of the global Fermi gas model lies in the one-to-one correspondence
between individual nucleon momentum and the energy required to remove the nucleon, owing to the constant potential depth. This is not the case for the local Fermi
gas model, making it more suitable for describing low-energy interactions where nuclear effects are significant.
In NEUT 5.4.0, the default event generator for SK atmospheric neutrino simulation, CCQE simulation follows the local Fermi gas model.

Interaction channels
The weak interaction between neutrinos and quarks occurs through the exchange ofa charged W ± boson or a neutral Z 0 boson. The former is termed Charged-Current(CC) interaction, while the latter is termed Neutral-Current (NC) interaction. CC
interaction has an energy threshold equivalent to the lepton mass due to the production of a charged lepton, whereas NC interaction has no energy threshold and merely
scatters a nucleon. CC interactions serve as target signals for neutrino oscillation
experiments because the produced charged leptons carry the neutrino flavor.
In the energy range of interest (O(0.1-1) GeV) for neutrino event generators inaccelerator/atmospheric neutrino oscillation experiments, the dominant interaction
channels include quasi-elastic (QE or 1p1h), two-nucleon-correlated quasi-elastic (2Nor 2p2h), ∆-resonance (RES), coherent pion production (COH), and deep inelastic
scattering (DIS). The corresponding energy scales are illustrated in Figures 2.37
and 2.38. Figure 2.40 displays the Feynman diagrams for these major interaction
channels. Both CC and NC interactions can undergo these types of interactions.

71

2.3. Event simulation
In NEUT, these interactions can be categorized based on the number of pions(mesons) produced:• Quasi-elastic: 1p1h, 2p2h
2.1. PRIMARY INTERACTION

21• Single-pion production: RES, COH

inelastic scatterings with a quark, called deep inelastic scattering (DIS), take place. Fur• Multi-pion/nucleon
production:
thermore,in the energy region
between QEDISand RES, known as the dip region, the 2p2hinteraction, which is the scattering with two correlated nucleons, is considered to exist.
νl

l−

ν

p

p/n

(a) CCQE

νl

p/n

νl

π+

p

A
A

N

(d) CCCOH

π+
∆++

p

(c) CCRES

l−
W+

l−
W+

(b) NCQE

l−
W+

νl

Z0

W+
n

ν

νl

l−
W+

n
hadrons p/n(e) CCDISpp/n(f) CC 2p2h

2.2: diagrams
Feynman diagramsof major
neutrino interactions.
Figure 2.40: Figure
Feynmanof the primary
neutrino-nucleon/nucleus
interaction
channels in the O(0.1-1) GeV range. (Reprinted from [70])The eﬀective Lagrangian density of neutrino-nucleon interaction is written by" of each interaction chanHere, we provide a brief overview
characteristics
GF ! † of the′ λ′
Lef f = √j (k, k )J (p, p ) + h.c. ,

(2.1)

λ
nel, along with details about the2 default
theoretical model employed in NEUT 5.4.0

for predicting cross sections and final state kinematics of these interactions.

where GF is the Fermi constant, jλ is leptonic current, Jλ is hadronic current, k (k ′ )represents the four-momentum of the initial (final) state lepton, and p (p′ ) represents thefour-momentum of the initial (final) state nucleon. The diﬀerential cross section can be
written in terms of the leptonic tensor Lµν and hadronic tensor Hµν , which are calculated
from the leptonic and hadronic current,dσ1
1=
G2 c2 Lµν H µν ,22dQ
32π M Eν2 F EW

(2.2)

where M is the nucleon mass, Eν is the neutrino energy, and q 2 = (k − k ′ )2 ≡ −Q2 issquared of four-momentum transfer. cEW is a constant: cos θc (θc : Cabibbo angle) for
CC interaction, and 1/4 for NC interaction. The leptonic tensor is written by72

Lµν = kµ kν′ + kµ′ kν − gµν (k, k ′ ) ± iϵµνρσ k ′ρ k σ ,(2.3)

where gµν is Lorentz metric tensor and ϵµνρσ is Levi-Civita symbol. The hadronic tensor
H µν is usually complicated and depends on the models.
The following sections contain the formalism of the various neutrino interaction channels. Fig. 2.3 shows the neutrino and antineutrino cross section on carbon obtained using
NuWro [27]. Below 1 GeV, where the dominant region of atmospheric neutrino flux, theCHAPTER 2. ATMOSPHERIC NEUTRINO INTERACTIONS AT
SUPER-KAMIOKANDE
Quasi-elastic (QE)
The differential cross section for neutrino quasi-elastic scattering on a “free” nucleon,
per four-momentum transfer q 2 , is expressed as [71]:2dσ ν(ν̄)
G2F M 2 (f CC/N C )22
2 s−u
2 (s − u)=A(q ) ± B(q )+ C(q )
d|q 2 |
8πEν2
M2
M4

(2.3)

Here, M denotes the nucleon mass, GF is the Fermi coupling constant, f CC =cos θc , where θc is the Cabibbo angle, and f N C = 1. The variables s and u represent
Mandelstam variables, and Eν is the neutrino energy. The + sign is for neutrinos,
and the − sign is for antineutrinos.
The functions A(q 2 ), B(q 2 ), and C(q 2 ) are expressed in terms of vector and axial
nucleon form factors. Vector form factors are related to nucleon electromagnetic
form factors constrained by electron scattering experiments, while the axial form
factor can only be studied through weak interactions with neutrinos, making it
challenging to constrain. The axial form factor FA is approximated in dipole form
as:

−2
q2
FA (q ) = gA 1 −
MA
2

(2.4)

Here, gA = FA (0) is the axial coupling constant, well-constrained by β-decay
measurements, and MA is the “axial mass,” determining the q 2 -dependence of theform factor and carrying substantial uncertainty, as illustrated in Figure 2.41. The
choice of vector form factor parametrization and the value of axial mass significantly
influence the overall q 2 -dependence of the inclusive QE cross sections. For instance,larger values of MA are expected to increase both the total and differential crosssections with larger q 2 (Figure 2.41).

73

2.3. Event simulation
For CCQE interactions, NEUT 5.4.0 utilizes the Local Fermi Gas (LFG) modelfor the nuclear ground state supplemented by random phase approximation (RPA)
correction, accounting for the long-range weak charge screening effect from othernucleons (not applicable to free nucleons). By default, NEUT 5.4.0 follows the socalled “Valencia model” [72] (provided by the group at IFIC, University of Valencia),
naturally extending to describe neutrino interaction on strongly-correlated nucleon
pairs, potentially knocking two nucleons (2p2h) instead of one (1p1h). The axial
mass MA is set to 1.05 GeV, and the BBBA05 [73] vector form factor parametrization
is employed.

Figure 2.41: The measured axial mass values as a function of mean neutrino energy
used in each measurement (left, reprinted from [74]) and the predicted differential
cross section per target neutron for 1 GeV νµ -12 C CCQE interaction with various
values of the axial mass MA (right, reprinted from [70]). There is a slight tension
between the axial mass measured with deuterium targets and measurements withcomposite nucleus targets.

74

CHAPTER 2. ATMOSPHERIC NEUTRINO INTERACTIONS AT
SUPER-KAMIOKANDE
Single-pion production (RES, COH)

l−

νl
W+

p

d
u
u

u
u
u
∆++

u
d̄

π+

d
u
u

p

Figure 4·5:diagram
A Feynman
diagram for atheresonantpro- producing a
Figure 2.42: Feynman
illustrating
CC pionRESproduction
interaction
cess. A neutrino+++ with lepton flavor l interacts with a nucleus, changing
single π through the ∆ resonance (Reprinted
from [48]).
++
the flavor of one of the quarks. A ∆baryon is temporarily createdin an excited state, which then radiates a gluon, producing a protonand pion.

An additional energy transfer of 300 MeV from the QE energy peak represents
the resonanceenergyforeachnucleons
excitedthat
to produces∆-baryonswith afinalmass
1.2 GeV,contributions
frombaryon resonancethe desired
state.ofThewhich is largest
the lowest
masstoresonanceabove
the pion
production
threshold.
contribution
these resonancesat few-GeV
energies
is the ∆(1232),
which is Figure
2.42 provides
an example
Feynmandiagramillustrating
the However,∆++ resonance.
the lowest-mass
resonance
above thepion production
threshold.
there are NEUT
suchcrossresonances
below
neut simulates
resonant
single pion
processes
calculates18 thesectionfor2 GeV.
producinga baryon
resonance,
takingintoinaccounttwo steps.
In multiple
the first step,
neut calculates
the statescross section
a baryoninterferenceamongpossibleresonance
up toforWproducing< 2 GeV.
Next, NEUT
for formalism
interference between
the bymultipleresonance states
5.4.0, by resonance,
default, accounting
follows theprovided[75],possible
with corrections
from [76].
up tocalculates
q < 2 GeV. theIn thesecond step,crossneut sections
computes as
decaysof the excited
baryonThis model
differential
a productof theamplitude
into pions, considering
branchingfractionsand the kinematic
dependence
on angularof each resonanceproduction
and theprobabilityof the baryon
resonance
decay tomomentumof the crossparent section
baryon state.
pions. The
differential
is given by:neut uses the Rein-Sehgal model [100] for resonant pion production. As in the

d2 σ
1
1 X
1
Γ
case of =quasi-elastic crosssections,
the resonantcon·|T (νN→ lNj∗pion)|2 production cross sections
2
2
dq dW
32πM Eν 2 j,spin
2π (W − Mj ) + Γ2 /4(2.5)

tains an axial form factor,

where M"−2
q2
(4.19)
2MAResWis the mass of the target nucleon,is the invariant mass of the!
CA5 (q 2 ) = CA5 (0) 1 −hadronicInsystem,
the themassof thepion
baryonresonance
Nj∗ ,MΓResis, andthethedecay
j is both
EquationM
4.19,
resonant
production
axial mass,coeﬃ-width ofA

the resonance. The first term in the summation computes amplitudes of intermediatebaryon resonances, and the second term computes
decays of the excited baryon into
103
pions, considering branching fractions and the kinematics.
NEUT considers 28 baryon resonances in the W < 2 GeV range and extends themethod to calculate single K, η, γ, Λ production cross sections as well.

75

2.3. Event simulation
Additionally, NEUT 5.4.0 adopts the Graczyk-Sobczyk formulation [77, 78] of form
factors by default, with a dipole axial form factor C5A (q 2 ) = C5A (0)(1 − q 2 /MA2 )−2 .

The parameters are set to C5A (0) = 1.01 and axial mass MA = 0.95 GeV, tuned

based on νµ p → µ− pπ + cross-section measurements with bubble chamber data [79].
Another interaction channel that can produce a single pion is through coherent
interaction with the entire nucleus, represented by CC and NC coherent pion production. For CC:
νl + A → l − + π + + A(2.6)

ν̄l + A → l+ + π − + A(2.7)

ν(ν̄) + A → ν(ν̄) + π 0 + A(2.8)

and for NC:

Coherent pion production occurs with low q 2 , and little momentum is transferredto the nucleus, resulting in forward-scattered outgoing leptons and pions. Coherent
pion production is expected to be very small compared to resonant pion production.
In NEUT, the Berger-Sehgal model [80] is used. The energy ranges were determined to qualitatively explain the recent measurement of coherent pion production
[81].

76

CHAPTER 2. ATMOSPHERIC NEUTRINO INTERACTIONS AT
SUPER-KAMIOKANDE
Multi-pion/nucleon production (DIS)
DIS is the dominant interaction process for multi-GeV neutrinos. DIS interactions
are characterized by a neutrino interacting at the quark scale, breaking apart thenucleon, and resulting in the production of multiple hadrons in the final state.
The calculation of the CC DIS cross section for the range 1.3 < W GeV involves
integrating the following double-differential cross section (based on the formalism
introduced in [82]):






y
G2F M Eν
y2
d2 σ
=
1−y+
+ C1 F2 (x) ± y 1 − + C2 xF3 (x)
dxdy
π
2
2
2
2
m (y − 2) M xy
m
C1 =
−
−
4M Eν x
2Eν
4Eν2
2
m
C2 = −
4M Eν x(2.9)
(2.10)(2.11)

Here, x = −q 2 /(2M ∆E) and y = ∆E/Eν are Bjorken parameters where ∆E

represents the difference between the incoming neutrino energy Eν and the outgoing
charged lepton energy El , M is the nucleon mass, and m is the lepton mass.
Three structure functions that are not shown are incorporated into F2 andF3 with modified structure function correlations based on Callan-Gross [83] andAlbright-Jarlskog [84] relations. The calculation of F2 and xF3 relies on the GRV98 LO[85] parton distribution function (PDF) with Bodek-Yang correction [86, 87], which
offers improved accuracy in the low-q 2 region of lepton inelastic scattering data.
The computed cross-section is normalized by the fraction of multi-pion events(two or more pions). This normalization is applied to exclude the contribution fromresonant single-pion production, as explained earlier.
Using the sampled W , the number of produced hadrons (either a nucleon or apion) is determined from a hadron multiplicity model. This model is a linear functionof log(W 2 ) and is tuned to deuterium bubble chamber data [88] for accuracy.

77

2.3. Event simulation
For W > 2 GeV, PYTHIA v5.72 [89] is used. Similar to low-W DIS simulation,
the overlap with single-pion production is excluded. The full event is simulatedwith PYTHIA based on the sampled W , and, unlike the low-W case, the productionof heavier mesons such as K and η is permitted.
Table 2.4 summarizes the default models and options used for major channels of
nucleon-level neutrino (CC) interactions in NEUT 5.4.0. Figures 2.44 and 2.45 show
the fraction of simulated interaction channels in the atmospheric neutrino event
simulation, plotted against the reconstructed event visible energy.

ChannelCross section

CCQE

Valencia [72]RES

Berger-Sehgal [76]DIS

Kretzer-Reno [82]OptionMA
Vector form factor
Nuclear model
Form factor
MA
C5A (0)Max. W limit
PDF
Low-W model
High-W model

Value
1.05 GeVBBBA05
Local Fermi Gas
Graczyk-Sobczyk (dipole) [78]0.95 GeV
1.01
2.0 GeVGRV98 + Bodek-Yang [85, 87]
Custom model [88]PYTHIA v5.72 [89]Table 2.4: Overview of the default models and settings applied to the calculation ofcross-sections for major neutrino-nucleon CC interaction channels in NEUT 5.4.0.

78

CHAPTER 2. ATMOSPHERIC NEUTRINO INTERACTIONS AT
SUPER-KAMIOKANDE

Figure 2.43: NEUT 5.4.0 default ν-12 C cross section averaged per nucleon (i.e,.
{Zσ(νp) + (A − Z)σ(νn)}/A.) (Reprinted from [90])Figure 2.44: MC-simulated atmospheric neutrino interaction type with NEUT 5.4.0.

Figure 2.45: Distribution of MC-simulated atmospheric neutrino interaction types
using NEUT 5.4.0, categorized by the number of reconstructed Cherenkov rings.

792.3. Event simulation

2.3.3

Hadronic interaction modelThrough the neutrino-nucleon interaction described in the preceding subsection,
various types of hadrons are generated. The most crucial ones for neutrino experiments to reconstruct are the recoil nucleons from CCQE scattering. Additionally,
long-lived mesons, particularly charged pions, are abundantly produced in multiGeV events. The energy of these hadrons can range from a few MeV to several
GeV, depending on the energy of the incoming neutrinos. The transport of thesehadrons within the target nucleus needs to be accounted for separately. The numberof outgoing hadrons from the target nucleus, as well as their charge and kinematics,
may differ from the expectations based on nucleon-level model predictions due tothese nuclear effects. In the current suite of neutrino event generators, these effects
are factored out and referred to as “final-state interaction” (FSI).
It is also essential to consider hadronic interactions within the detector target
volume, which, in our case, is water. Particularly significant are inelastic hadroninteractions on 16 O that can alter the hadron content. This type of hadronic interaction within the detector is often termed “secondary interaction” (SI). The schematic
representation of the FSI and SI effects on the detectable hadron content is illustrated in Figure 2.46.

Figure 2.46: Illustration of the production of hadrons via neutrino-nucleon level
scattering and the potential change of detectable hadron content due to FSI (green)
within the target nucleus and SI (blue) within the detector volume.

80

CHAPTER 2. ATMOSPHERIC NEUTRINO INTERACTIONS AT
SUPER-KAMIOKANDE
As outlined in Section 1.7, these effects impact the reconstruction of assumed
nucleon-level neutrino interactions. For instance, events reconstructed as QE-like,
may actually include pion-producing events with pions absorbed through FSI/SI
processes, introducing bias to our reconstruction. Similarly, neutrons intended for
detection in SK to enhance neutrino reconstruction may experience “smearing” effects due to these re-scatterings. Therefore, it is crucial to simulate these effects
with high accuracy.

Intranuclear cascade (INC) model
Most neutrino event generators and detector simulators used in neutrino oscillation
experiments rely on“intranuclear cascade” (INC) models to simulate these hadronic
re-scatterings. In the INC model of hadron-nucleus scattering, the nucleus is represented using the local Fermi gas model as an assembly of non-interacting freenucleons, with a continuous nuclear density function ρ(r). The overall outcome ofthe hadron-nucleus scattering is approximated by the convolution of a “cascade” involving binary (two-body) collisions between a projectile (either the initial incoming
hadron or subsequent recoil hadrons) and a nucleon. The schematic is illustrated
in Figure 2.47. Theoretically, it has been shown that summing over the INC calculations corresponds to a solution to a Boltzmann transport equation [91]. In theenergy range of O(0.1-1) GeV, which aligns with the size of nucleons, this model is
expected to perform effectively, similar to the factorization of nucleon-level interactions in considering neutrino-nucleus interaction.
p

p

n
ﬁ≠
n

p

ﬁ+

ﬁ≠
ﬁ0

n

ﬁ0
n

p
ﬁ

≠

n

n

ﬁ

p

+

Figure 2.16: Possible scenario of final state interactions.

Figure 2.47: Schematic representation of the (space-like) INC model MC computaon the final proton momentum is negligible (see Fig. 2.13b). However, if one looks also
tion for hadron-nucleus
interaction. (Reprinted from [69])at correlated nucleon (in the case of SF), the total momentum distribution of all final
state nucleons is shifted to the higher energies.
Usually, in MC generators Fermi gas is used. It is well known from electron scattering
data that cross section predictions obtained using spectral function are much closer tothe data (see e.g. Ref. [90]). For many purposes FG is a good approximation. However,one must be aware that it aÄects significantly the cross section prediction (see Fig. 2.14).
For the neutrino energy E‹ ≥ 1 GeV the diÄerence is about 10%. The disagreement
between global and local FG models is caused by the Pauli blocking - the eÄect of PB islower for LFG. Besides the normalization, the shape of the diÄerential cross section is
also aÄected around the pick (see Fig. 2.15), but in this region IA is doubtful anyway.
There is also a disagreement at high Q2 .

2.4

Final state interactions

Final state interactions describe the propagation of particles created in the primary81

2.3. Event simulation
In INC models, particles are treated classically, moving in straight lines within
a nucleus described by the Fermi gas model, as illustrated in Figure 2.47. Two
approaches are used to MC-simulate the cascades of binary collisions:• Space-like approach
The discrete collision points of each projectile are sampled from the probability
density function P (λ) = e−λ/⟨λ⟩ , with the mean free path ⟨λ⟩ estimated as:
⟨λ⟩ =

1
1
A
=
ρ⟨σ⟩
ρ Zσp + (A − Z)σn(2.12)

Here, ⟨σ⟩ is the averaged nucleon scattering cross section based on the atomic

and mass numbers Z and A, and projectile-to-proton/neutron cross sections
are denoted as σp/n . Cross sections are often either directly extracted fromavailable external data or parametrized as a function of incident particle speed
to fit external data. The iterations of collision steps continue until all hadrons
exit the nucleus, determined by a pre-determined radius.
• Time-like approach
In this approach, both the incoming projectile and each nucleon in the nucleus
are assigned a position and momentum. Initial nucleon positions are sampledfrom the nuclear density ρ(r). All particles, both projectiles and target nucleons, are propagated until any two of them get closer than the characteristicpdistance dint = ⟨σ⟩/π. The iterations of collision steps stop at a specifictime, regardless of particle positions.

82

CHAPTER 2. ATMOSPHERIC NEUTRINO INTERACTIONS AT
SUPER-KAMIOKANDE
Final-state interaction (FSI) modeling
The modeling is separated based on whether the incident particle is a pion or a
nucleon. For pions, four channels — scattering, absorption, production, and charge
exchange — are considered. For low-momentum (p < 500 MeV/c) pions, crosssection calculations are based on [92], with outgoing pion kinematics modeled by [93,94]. For nucleons and higher momentum pions, NEUT uses Bertini’s implementation[95] of space-like INC model. The nuclear density profile is described by a WoodsSaxon type distribution for radial displacement r from the nucleus center:
ρ(r) =ρ00
1 + exp r−Ra(2.13)

Here, R0 represents the nucleus radius, and a is a parameter determining the“diffuseness” of the nuclear surface. For 16 O in NEUT, specific values [96] are assigned:
R0 = 2.69 fm, a = 0.40961 fm, and ρ0 ∼ 10−3 .
Higher momentum pion cross-sections are extracted from π ± -p scattering data,and pion multiplicities are tuned using bubble chamber data [97]. Nucleon multiplicity after internal pion absorption is adjusted using external π + measurements[98] under the assumption of isospin symmetry. Kinematics for two-body nucleon
ejection are determined based on [99], while multi-body ejection kinematics are randomly sampled within the allowed phase space. Internal parameters governing pion
interaction strengths are tuned based on pion scattering measurements [100], asshown in Figure 2.48.
For nucleon re-scattering within the target nucleus, NEUT considers elastic scattering, single and double pion production processes. The kinematics of the products
are determined by the INC model, whose input nucleon-nucleon cross sections are
based on [101]. Limited nucleon-16 O cross-section measurements, especially in theenergy range of O(0.1-1) GeV, make it challenging to precisely characterize the nucleon scattering model. The uncertainty of the nucleon-nucleon cross section [101]for the O(0.1-1) GeV range is estimated to be 30% [34] based on NEUT-predictedcross sections for p-12 C compared with data [102].

83

2.3. Event simulation

USING WORLD π " -NUCLEUS SCATTERING DATA…

PHYS. REV. D 99, 052007 (2019)

FIG. 25. Ratios of the +π þ -12C, 16O, 27Al data and the GEANT4 (black), GENIE (red), NuWro (blue), and GiBUU (magenta)
Figureto2.48:
π best-AfitcrosssectionratiostheTheNEUT
5.4.0
default
andpredictions
the NEUT
for the fiveinteraction
channelsbetween
used in the fit.
green band
represents
the "1σband.exter-

nal data and other models. σREAC is the total reaction cross section, σQE is thequasi-elastic cross section, σABS is the absorption cross section, σCX is the charge
exchagnge cross section. (Reprinted from [100])
052007-1984

CHAPTER 2. ATMOSPHERIC NEUTRINO INTERACTIONS AT
SUPER-KAMIOKANDE
Figure 2.49 displays the sub-GeV proton interaction probability and scattering
angles within a 16 O nucleus in NEUT.

Figure 2.49: Probability of nucleon-nucleon interaction as a function of incoming nucleon momentum (left) and proton deflection angle (right) within 16 O. The incident
proton momentum is uniformly sampled within the 0.5-1 GeV/c range. (Adaptedfrom [103])
When the mean free path is smaller than the wavelength of the scattering particle,
the generation of hadrons may be suppressed over a specific distance/time, known
as the formation length/time [104]. This effectively decreases the hadron production
cross-section. In NEUT, the formation length of pion production is parameterized to
be proportional to the incident hadron momentum, with the coefficient determinedby a neutrino-bubble chamber experiment [105]. For hadrons with a momentum of
1 GeV/c, the formation length is approximately 2.5 fm, which closely matches thesize of the 16 O nucleus.

852.3. Event simulation
Nuclear de-excitation modeling
NEUT incorporates a straightforward nuclear de-excitation scheme for 16 O. Originally motivated by studies of proton decays in 16 O, this scheme gained further significance in the investigation of atmospheric neutrino NCQE interactions. NCQE
interactions serve as a major background for inverse beta decay signals of diffuse
supernova neutrinos. Figure 2.50 illustrates a neutrino knocking out a nucleon occupying the p3/2 state of the 16 O nucleus.
In NEUT, the de-excitation of the residual nucleus after nucleon knockout can
result in the emission of another nucleon or a gamma-ray. Deeper holes (such asones in the s1/2 state) in the residual nucleus with higher excitation energy are
more likely to eject a nucleon or gamma-ray during the de-excitation process. The
occupation probabilities of nucleon energy states are derived from [106], and thebranching ratios for the knock-out of each state are based on [107].

PRL 108, 052505 (2012)PHYSICAL REVIEW LETTERS

nucleon-emission th
governed only by entron holes yield pho
ences being as smal
The calculation
16
8 O ! # þ X$ þ Napproach discussed
charged-current (CCing ratios BrðX$Refs. [12,20].
Following Refs.
section in the form
3/2
Figure 2.50:
Illustrationofaneutrino
knocking
outaproton
from
the
p
state
in
FIG. 1 (color online). Schematic representation of neutralXd"#A
16
a simple shellmodelof the scattering
O nucleus.
from [106])
current
neutrino
off (Reprinted
oxygen.
¼
d!dE0# N¼p
In our approach, the cross section of !-ray production
following a NC QE interaction, "! , is written in the form
X
O
!
#
þ
!
þ
Y
þ
NÞ
¼
"ð# þ 16"! ! "ð# þ 16
8
8 O
$

86

! # þ X$ þ NÞBrðX$ ! ! þ YÞ;(1)

where N is the knocked out nucleon, X$ denotes theresidual nucleus in the state $, and Y is the system resulting from the electromagnetic decay of X , e.g., 15 O, 15 N,

pﬃﬃﬃﬃﬃﬃﬃﬃ
where EN ¼ M2d"#N =d!dE0# denocross section, andthe probability of rfrom the target leav
E þ E0 ' M, E0 be
In the nuclear sparticle states %$ w
a consequence, kno

CHAPTER 2. ATMOSPHERIC NEUTRINO INTERACTIONS AT
SUPER-KAMIOKANDE
Secondary interaction (SI) modelingIn the SK detector simulator, detailed in Section 2.3.5, the modeling of hadron interactions is designed to closely follow the approach used in NEUT. For low-momentum
charged pions (p < 500 MeV), NEUT is directly called. Hadronic interactions involving higher momentum pions and nucleons are simulated using GCALOR 1.04/08.
GCALOR provides interfaces to various MC simulation packages covering a range of
hadron energies; for the relevant energy range, HETC [108], based on Bertini’s INC
model [95], handles hadronic scatterings, using mostly the same input cross sections implemented for NEUT nucleon interactions. Neutron interactions with kinetic
energy below 20 MeV, including thermalization through elastic scatterings and neutron capture reactions, are modeled with MICAP [109]. The cross sections for MICAP
are sourced from ENDF/B-V (1979). In SK simulation, MICAP with ENDF/B-V has
demonstrated accurate reproduction of 1 H(n, γ)2 H signals, capturing both gammaray energy and the characteristic time constant (∼ 200 µs) [34, 110]. These settings
are used for simulating secondary interactions in pure water, corresponding to SKIV and SK-V atmospheric neutrino MC simulation.
For the simulation of low-energy neutron interactions in Gd-loaded water, specifically targeting neutron captures on Gd, MICAP is replaced with the NeutronHP
package [111] in Geant4.10.5.p01. This newer package relies on the updated evaluated nuclear cross sections of ENDF/B-VII.1 (2011, [31]). The validity of using
Geant4 and ENDF/B-VII.1, especially regarding the simulation of the Gd(n, γ) reaction, will be discussed further in Section 3.3.6.

87

2.3. Event simulation

2.3.4

Neutron capture reaction model

For the 1 H(n, γ) reaction, a single 2.2 MeV gamma-ray is emitted isotropically.
Regarding neutron captures on Gd, our detector simulator offers several models,
including GLG4Sim [112] and GGARNET [113]. The default model is the ANNRIGd [113] model, which specifically considers the two most strongly neutron-absorbing
Gd isotopes, 155/157 Gd. In this model, the gamma-ray energy and multiplicity aresampled from a theoretical model fitted to data observed with the ANNRI Germanium spectrometer surrounding a natural Gd target irradiated with a neutron beam.
The neutron energy was constrained to the thermal range by neutron time-of-flight.
The uncertainties associated with using this model are further explored in Section
3.3.6.

2.3.5

Detector simulation

The outgoing particle kinematics from NEUT 5.4.0, the neutrino event generator with
the HKKM 2011 atmospheric neutrino flux, were input into the SK detector simulator skdetsim, based on Geant 3.21, for the simulation of the detector response toatmospheric neutrino events. Table 2.5 provides a list of all the processes consideredin particle tracking. The PMT charge/timing response, as well as optical photon
propagation in water, are tuned with the detector calibration detailed in Section
2.2.3.

88

CHAPTER 2. ATMOSPHERIC NEUTRINO INTERACTIONS AT
SUPER-KAMIOKANDE

γ

e

µ

Hadrons

Optical photons

Pair production
Compton scattering
Photoelectric effectMultiple scattering
Ionization / δ-ray production
Bremsstrahlung
Annihilation
Cherenkov radiation
Decay in flight
Multiple scattering
Ionization / δ-ray production
Bremsstrahlung
Pair production
Nuclear interaction / µ− capture
Cherenkov radiation
Decay in flight
Multiple scattering
Ionization / δ-ray production
Hadronic interactions / π − capture
Cherenkov radiation
Rayleigh scattering
Mie scattering
Absorption / Reflection

Table 2.5: Processes simulated in skdetsim, for each particle type.

89

2.3. Event simulation

2.3.6

Background modeling

The primary backgrounds for the neutron capture signals, typically ranging from
1 MeV with 1-10 PMT hits, are accidental coincidences attributed to PMT darknoise. Although a significant portion of dark noise is inherently random, there
exist other types of background noise and radioactivity with characteristics that are
not qualitatively characterized. To accurately simulate the background noise andminimize data-simulation bias, we extracted PMT hits recorded with random event
triggers (also known as “dummy trigger” or “dark trigger,” see “Software trigger”
in Section 2.2.2). These dark events were pre-shuffled, and the PMT hits extracted
from each of the dark events were then appended to the signal events simulatedwith the skdetsim detector simulator. The appended background hits did not
overlap with the neutrino interaction signal following the event trigger, preventing
interference with event reconstruction. At the same time, it was ensured that thebackground hits covered the entire neutron capture signal search time range, which
is a few µs delayed from the event trigger.

2.3.7

Event weights

Once the full set of MC-simulated atmospheric neutrino events in the detector was
generated, each event received a weight based on the three-flavor oscillation probability, calculated with a given true neutrino flavor and zenith angle. The neutrino
production height was sampled from a height probability, a function of flavor andzenith angle provided by the flux model. The zenith angle and the production height
fully characterizes the neutrino path length. For each neutrino event with a true
lepton flavor l ∈ {e, µ}, neutrino path length, and energy, oscillation probabilities

P (νe → νl ) and P (νµ → νl ) (the same applies for antineutrinos) were calculatedusing Prob3++ [114] under the three-flavor standard oscillation scenario, including
the Earth matter effect. The Earth matter density was modeled using the Prelim-

inary Reference Earth Model (PREM, [115]). The three-flavor standard oscillation
parameters used in this thesis are outlined in Table 4.1 [116]. The “oscillation”
weight for each event νl (l ∈ {e, µ}) was calculated as:

90

CHAPTER 2. ATMOSPHERIC NEUTRINO INTERACTIONS AT
SUPER-KAMIOKANDE

wosc =ϕe (Eν , θzenith )P (νe → νl ) + ϕµ (Eν , θzenith )P (νµ → νl )ϕl (Eν , θzenith )(2.14)

where ϕl is the νl flux prediction with neutrino energy Eν and zenith angle θzenith(the same applies for antineutrinos). This weight considers contributions from both
flavors, e and µ, oscillating to the observed flavor l.
In addition to oscillation, solar activity, as discussed in Section 2.3.1, was considered by normalizing the total event rate according to the average solar activity ineach SK phase. (See Figure 2.35) If the livetime-averaged neutron monitor count isc, then the fraction spent in maximum solar activity for each SK phase is determinedby rmax ≡ (cmax − cavg )/(cmax − cmin ), where c{min/max} is the neutron monitor countat maximum and minimum solar activity. With rmax , for each event, the “solar
activity” weight is calculated as:

wsol =

rmax ϕmax (Eν , θzenith ) + (1 − rmax )ϕmin (Eν , θzenith )ϕmid (Eν , θzenith )(2.15)

where ϕ{max/mid} is the flux prediction of the given neutrino type and flavor at
maximum and intermediate solar activities, provided by the flux model.
In this analysis, the fractions determined for SK-IV (44% rmax ) were used for allphases for convenience, as most of our analyzed data is SK-IV (∼ 10 years), which
is much longer than the other two phases SK-V and SK-VI (∼ 3 years combined).
The final event weight assigned to each event is the product of both oscillation andsolar activity weights, i.e., wosc wsol .

91

2.4. Event reconstruction

2.4

Event reconstruction

2.4.1

Atmospheric neutrino interaction

In this analysis, our goal is to characterize the properties of neutrons produced in
atmospheric neutrino interactions within water, with a focus on understanding theirdependence on neutrino interaction type and kinematics. For instance, we aim toinvestigate the increase in neutron production as a function of momentum transfer,and we seek to compare these observations with predictions from various models.
Additionally, we want to examine the differences in neutron production between
inelastic and elastic interactions and compare our findings with model predictions.
Also, there are inherent correlations between the variables in neutrino interactions and neutron detection efficiency. Neutrons produced in high-energy (on theorder of O(1-10) GeV) neutrino interactions may travel farther from the interaction
vertex, making them less likely to be detected compared to those produced in subGeV interactions. Moreover, neutrons generated close to the tank wall are difficult
to distinguish from background radioactivity occurring around the wall.
While accurate reconstruction of the neutrino interaction is crucial, it remains
challenging to precisely determine the interaction type or neutrino energy with atmospheric neutrinos given our current detector setup. Therefore, we rely on proxyvariables that best capture the relevant information:• Visible energy
Visible energy is the sum of the reconstructed momenta from all rings in an
event, assuming the ring was produced by an electron. This quantity provides
a calorimetric measure of energy dissipated by the neutrino interaction, and,on average, exhibits a linear relationship with four-momentum transfer andthe energy transferred to the hadronic system.

92

CHAPTER 2. ATMOSPHERIC NEUTRINO INTERACTIONS AT
SUPER-KAMIOKANDE

Average value

GENIE 3.4.0
Momentum transfer Q 2 [GeV2 ]Hadronic invariant mass W [GeV]

6
4
2
0

2000

4000
6000
8000
Visible energy [MeV]

10000

Figure 2.51: Average values of squared momentum transfer Q2 (blue) and hadronic
invariant mass W (orange) as a function of visible energy of SK atmospheric neutrino
events simulated with GENIE 3.4.0.
• Number of Cherenkov rings
The number of Cherenkov rings can indicate whether the interaction was quasielastic or inelastic, as implied in Figure 2.45. While the most energetic ring islikely associated with the charged lepton from the CC interaction, the presenceof multiple rings may suggest the production of hadrons, particularly pions,indicating the inelasticity of the interaction and a larger energy transfer to thehadronic system, as shown in Figure 2.51.
• Lepton typeLeveraging the excellent charged lepton PID performance of the detector, we
can compare neutron properties between νe and νµ interactions.
• Interaction vertex
The knowledge of the interaction vertex is essential for separating neutrino
interactions from backgrounds such as cosmic rays or PMT flashers, which
typically have vertices reconstructed on the tank wall.

93

2.4. Event reconstruction
SK uses APFit [117] to reconstruct observed Cherenkov rings in atmospheric
neutrino events. We provide a brief overview of the Cherenkov reconstruction algorithm, covering aspects such as ring vertex determination, multiplicity, PID, andmomentum in Appendix A. Further details can be found in [117, 118].

2.4.2

Neutron capture vertex

As discussed in Section 3.1, during the pure water phases of SK, 99.9% of neutron capture signals originated from neutron captures on free protons in water(1 H(n, γ)2 H), emitting a single 2.2 MeV gamma-ray. With 0.01w% Gd-loading in
SK-VI or the current 0.03w% Gd in SK-VII, a significant portion, about half ormore, of the neutron captures occur on Gd, primarily 155/157 Gd(n, γ ′ s)156/158 Gd. Insuch cases, multiple gamma-rays with a total energy of 7.94 MeV (for 157 Gd captures) or 8.54 MeV (for 155 Gd captures) are emitted.
The 2 MeV 1 H(n, γ) signal results in approximately 6-7 PMT hits on averagein our detector of size O(10) m with approximately 12,000 ID PMTs. As a consequence, accurately estimating its vertex becomes extremely challenging. Therefore,
we rely on the a priori knowledge of the neutron production vertex and assume thatneutrons are captured in the vicinity of the production vertex.
On the other hand, with Gd-loading, the 8 MeV Gd(n, γ) signals typically deposit
more than 20 PMT hits. In this energy range, we already have a vertex fitter thathas been applied to reconstruct e-like rings of O(1 − 10) MeV energy from solar andastrophysical neutrino interactions. Consequently, the neutron capture vertex canbe independently reconstructed without relying on any a priori vertex information.
The primary motivation for neutron capture vertex reconstruction is to use thedistance between neutron production and capture vertices as a proxy for the outgoingneutron momentum. This provides a handle on the outgoing neutron kinematics,
which has been entirely lost during the 10 years of pure water phase data. In thisthesis, we aim to investigate whether this new handle can offer additional constraints
on the existing variability in hadronic interaction model predictions.

94CHAPTER 2. ATMOSPHERIC NEUTRINO INTERACTIONS AT
SUPER-KAMIOKANDE
For neutron capture vertex reconstruction, we utilize the vertex fitter (BONSAI[119]) widely used in low-energy neutrino analyses (for example, [120]). This vertex
fitter assumes that all light is emitted from a single vertex ⃗x at an instant time at
T0 , given that O(1) MeV electron tracks are O(1) cm. It attempts to find them byminimizing the log likelihood L(⃗x, T0 ) of PMT hits, given by:L(⃗x, T0 ) =

NX
PMT
i

log P (ti − tToF(⃗x) − T0 )i

(2.16)

where ti is the hit time of the ith PMT, and tToF(⃗x) is the photon time-ofi
flight (ToF) between the assumed vertex ⃗x and the ith PMT. PMT charge is notconsidered, as most hits are due to single photon arrivals in the O(1) MeV energy
range. The probability P is determined by the distribution shown in Figure 2.52,
empirically obtained from the O(1) MeV electron beam data collected during LINAC
calibration, as explained in Section 2.2.3.

Figure 2.52: PMT hit probability density function as a function of hit time corrected by photon time-of-flight and the electron injection time T0 . The probability
is highest at 0 as expected, while the peak width determines the timing resolution.
The tails are due to reflected and scattered light, and the smaller peaks are due to
PMT afterpulses whose time is characterized by the internal design of the PMT.
(Reprinted from [121])95

2.4. Event reconstruction
One challenge in optimization tasks involving maximization or minimization isthe computational time and the risk of the computation getting trapped in local
extrema. The BONSAI algorithm employs a distinctive strategy to efficiently explore
the global extremum. Here, we provide a qualitative overview of the algorithm.
To begin with, the algorithm identifies “good PMT hits” based on “causality
constraints.” If two PMTs register hits from photons originating at a single vertex,
the time difference between the two hits must be smaller than the distance between
the two PMTs (assuming the speed of light, c = 1). Essentially, if two PMTs exhibit
significantly different hit times, it is unlikely that they are detecting photons fromthe same source. We select the maximal set of “good PMT hits” satisfying thecausality constraint between any two PMT hits. PMTs located more than 12.5 m
or 35 ns apart from any other PMTs are rejected to prevent contamination fromrandom PMT noise hits.
The goal is to determine the values of ⃗x and T0 , representing three vertex coordinates and a single signal time. If we have four equations related to these values,
we can find a unique solution. From the “good PMT hits,” we can randomly selectfour hits and solve four equations, stipulating that the distance between each PMT
and the solution vertex must align with the expected photon time-of-flight.
Multiple combinations of four good PMT hits can yield a corresponding setof “candidate” vertices on a one-to-one basis. However, to avoid computational
inefficiency (especially when having a large number of good PMT hits), we limit
the number of possible combinations. Additionally, when candidate vertices are
sufficiently close (less than 1.5 m apart), they are averaged.
Next, a list of vertices with significant log likelihoods is iteratively searched forand retained. Utilizing the selected candidate vertices and time, a dodecahedron(initial side length of 7.8 m) is drawn from each candidate vertex. Log likelihoods
are computed for the 12 vertices of the dodecahedron, and only a fraction of vertices
with the highest log likelihoods are preserved, while the rest are pruned. This process
is repeated with a smaller (×0.38) dodecahedron for each remaining vertex until thesize becomes sufficiently small (10 cm).

96

CHAPTER 2. ATMOSPHERIC NEUTRINO INTERACTIONS AT
SUPER-KAMIOKANDEThis method enables the exploration of the entire tank without being trappedin local extrema, and it efficiently selects likely vertices without the need to iterateover the entire grid points in the tank. The vertex resolution of this algorithm forMichel electrons, Gd(n, γ) signals, and p(n, γ) signals is shown in Figure 2.53. The
vertex resolution for the target Gd(n, γ) signals is approximately 1 m, which falls
between that for Michel electrons (∼0.5 m) and 1 H(n, γ) signals (> 3 m).

BONSAI vertex resolution

0.025

Michel-e
Gd(n, )
p(n, )

a.u.

0.020
0.015
0.010
0.005
0.000

0

100
200
300
|xTrue xReco| [cm]400

Figure 2.53: Comparison of vertex reconstruction performance for Michel electron,
Gd(n, γ), and H(n, γ) signals. A smaller value indicates better performance.

972.5. Atmospheric neutrino event selection

2.5

Atmospheric neutrino event selection

Atmospheric neutrino events observed in the SK detector are classified into three
distinct categories, as illustrated in Figure 2.54:• Fully contained (FC)In FC events, there is no entry or exit of particles visible in the OD. Consequently, the majority of energy radiated in visible form is expected to bedeposited within the ID.
• Partially contained (PC)In PC events, a portion of the radiated energy is deposited in the OD after theinteraction, suggesting an exit of charged particles, mainly muons. However,there is no entry of particles visible in the OD, indicating that the interaction
itself should have occurred within the ID.
• Upward-going muon (UPMU)
UPMU events involve the entry of a charged muon visible in the OD but fromthe bottom of the tank. Cosmic rays are unlikely to come from the bottom,
given their inability to penetrate the entire Earth radius. Therefore, these
events are considered atmospheric muon neutrinos passing through the Earthand interacting within the rock surrounding the detector. Electron neutrinos
are less likely as they are not as penetrative. UPMU events can be further
classified into stopping muon events and through-going muon events.

The expected mean energies for atmospheric neutrinos are approximately ∼1

GeV for FC, ∼10 GeV for PC and stopping UPMU, and ∼100 GeV for throughgoing UPMU, as depicted in Figure 2.55. In this analysis, our focus is exclusively on

FC events. This choice is to measure neutron properties as a function of the energy
transferred by the neutrino interaction. FC events provide the complete calorimetricenergy from the interaction deposited within the detector system, allowing us tominimize energy loss escaping the detector.

98

is deposited inside the ID, the event is classified as FC. If high energy muon exits the ID anddeposits its energy in the OD, the event is classified as PC.
UPMU events are high energy muons produced by neutrino interactions with the rock surrounding the detector. AsCHAPTER
these events
cannot be distinguishedcosmic raysATwhich travel
2. ATMOSPHERIC
NEUTRINO from
INTERACTIONSSUPER-KAMIOKANDE
in downward direction, only muons traveling in the upward direction through the detector are
selected. UPMU events are divided into two types: for events which come to rest in the detector,
they are categorized as upward stopping muons, and those which traverse through the entire
detector volume, they are categorized as upward through-going muons.

5.1 shows the schematic view of the above three event types. The neutre ⇠1 GeV for FC, ⇠10 GeV for PC and stopping UPMU, ⇠100 GeV thro
Figure 2.54:5.2.
Illustration of SK atmospheric neutrino data event categories.
shown in Figure
Fully contained

Partially contained

Upward stoppingmuon

Upward through-going
muon(Reprinted[122]) view of observed atmospheric neutrino in SK.
Figure5.1:from
Schematic
1000
FC νe
FC νµ
PC

800

70

Events / 1000 days

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

Upward stopping µ
Upward through-going µ

1

10
10
Eν (GeV)

2

10

3

10

4

Figure 2.55: Expected true neutrino energy spectra of each event category.
(Reprinted from [122])

Figure 5.2: Expected neutrino spectra of di↵erent event categories.

ent selection process consists of reduction and reconstruction processes.
tion is to select neutrino events by removing background events99 in the colle
several background observed in SK:

mic ray through-going muons which pass through the detector

2.5. Atmospheric neutrino event selection
Approximately 106 events are recorded daily in SK, predominantly consisting of
background events such as cosmic rays and background radioactivity. To enhance
the purity of atmospheric neutrino events, several data reduction cuts are applied.
The primary backgrounds include:• Cosmic ray muons and subsequent Michel electrons• Low-energy gamma rays originating from radioactivity• PMT ”flashers,” characterized by PMTs emitting light through internal discharge

We reduce the background level down to O(0.1)%, using the following comprehensive five-stage data selection process:(FC1) Initial cuts to reject evident backgrounds(FC2) Subsequent cuts for further reject backgrounds(FC3) Cosmic ray rejection based on muon reconstruction(FC4) Flasher rejection based on PMT hit pattern likelihood(FC5) Additional cuts to reject remaining backgrounds
We provide a description of the data selection criteria applied to event observables at each stage of data selection in Appendix B. For a detailed comparison
between the data and simulation after passing these selection stages, refer to [118].

100

Chapter 3
Neutron detection
In this chapter, we begin by examining the characteristics of neutron capture signal
and common background within the detector. Following this, we discuss the signal
identification algorithm and evaluate its performance using calibration data obtained
from a neutron point source, and compare the results with MC simulations. Sourcesof uncertainty in the evaluated performance are thoroughly examined.

3.1

Characteristics of signal and backgroundIn pure water, signals primarily result from the 1 H(n, γ) reaction, constituting over
99.9% of total neutron captures, emitting a single 2.2 MeV gamma-ray. For Gd(n, γ),
which represents roughly half of neutron captures in 0.0110 w% Gd-loaded water,
multiple gamma-rays are emitted, with a combined energy of either 7.94 MeV (for157

Gd) or 8.54 MeV (for 155 Gd), as shown in Figures 3.1 and 3.2. The backgrounds

primarily stem from coincidences of PMT noise, which includes dark noise, afterpulse, and scintillation within the PMT glass window. Another source of backgroundare radioactive impurities found in water, PMT glass window, and the surrounding
rock, notably 222 Rn, which produces O(1) MeV electrons during its decay chain.

101

1.0

H
157Gd
155Gd

PDF

0.8
0.6
0.4
0.2
0.0

0

2

4
6
Gamma multiplicity

810

Number of occurrences in 3000 events

3.1. Characteristics of signal and background

1400

H
157Gd
155Gd

1200
1000
800
600
400
200
0

0

2

4
6
8
Total gamma-ray energy [MeV]10

Figure 3.1: The gamma-ray multiplicity in neutron captures on H and Gd (left),
and the energy sum of all emitted gamma-rays (right) obtained from simulation
with GCALOR [123] and ANNRI-Gd model [113]. (area-normalized, reprinted from[124])

Figure 3.2: Sample MC event displays for a 1 H(n, γ) (left) and a 157 Gd(n, γ) event(right). The top figures are the 3D drawings of the inner tank dimensions withall neutron signal hits (red dots) and gamma-rays (red arrows) in each event. The
bottom histograms are the raw PMT hit time distributions of each event with bin
width being 1 µs and 10 ns, respectively. (reprinted from [124])

102

CHAPTER 3. NEUTRON DETECTION
Following are the features that distinguish between signal and background:• Number of PMT hits
The likelihood of random coincidental PMT noise hits within a small fixed time
window decreases exponentially with the total number of coincident PMTs.
Conversely, neutron captures emit a consistent amount of energy (2-8 MeV per
isotope) which typically triggers a specific number of PMTs. The deposited
charge is less relevant, as most of the signal PMT hits result from a single
Cherenkov photon.
• Angular distribution of PMT hits
Gamma-rays from neutron captures scatter off bound electrons, generating
Cherenkov light if their kinetic energy exceeds the Cherenkov threshold of 0.3
MeV. As a result, signals manifest as one or more faint electron-like Cherenkov
rings with a distinctive angular distribution of roughly 42 degrees around the
direction of the gamma-ray, and a total energy of 2-8 MeV. Conversely, coincidental random noise hits exhibit little angular correlation with each other.
Radioactive decays near the tank wall typically have a highly localized angular
distribution close to the tank wall.
• Positional correlation of hit PMTs to signal vertex
Given that most neutrons in water can travel at most a few meters in a 36
m-high ID, the neutron capture vertex itself should be relatively close to theestimated neutron production vertex. Real neutron captures usually activate
fewer PMTs that are farther from the capture vertex due to light attenuation inwater. Fake signals caused by coincidental PMT noise display little correlation
with the vertex. Furthermore, the reconstructed vertex for signals tends tobe within the fiducial volume of the ID, while the reconstructed vertex for
backgrounds is often on the tank wall.
• Correlation to noisy PMTs
Certain PMTs that are relatively noisier are more prone to generating a false
signal, whereas actual signal photons can trigger any PMT regardless of thenoise rate of each individual PMT.

103

3.2. Signal identification algorithm

3.2

Signal identification algorithm

Here, we describe the algorithm implemented in the signal search software NTag[125]. Figure 3.3 shows a typical PMT hit time distribution.

Number of hits / 200 ns

Sample event PMT hit time distribution (MC)Event PMT hits
True (n, ) time
N200 trigger = 25

103
102
101
0

100

200
300
PMT hit time from trigger [µs]

400

500

Figure 3.3: A sample hit time distribution for a simulated neutrino event. Events
typically have a large number of triggering PMT hits and a small number of PMT
hits due to neutron captures buried in background noise spanning over 500 µs.
Before initiating the signal search, we perform a preliminary cleaning of unnecessary PMT hits. We first eliminate PMT hits with charge deposits equivalent to over
10 photoelectrons, as the majority of signal PMT hits are expected to be caused bya single photon. Next, we discard the latter hit among two consecutive hits on thesame PMT within a 1,000 ns interval to ensure uniform deadtime across all PMTs.
Then, the signal detection algorithm operates in two main stages:
Stage 1: Candidate Selection
We establish a threshold for the number of PMT hits to identify potential signal“candidates.” A group of hits within a sliding time window is considered a candidate
if the number of hits surpasses a predefined threshold.
Stage 2: Candidate Classification
Each candidate is then categorized as either signal or background based on its characteristics. In this particular analysis, we trained a neural network on simulated
candidate features and then applied it to candidates observed in previously unseen
data.

104

CHAPTER 3. NEUTRON DETECTION

3.2.1

Stage 1: Candidate selection

For each event (which records PMT hits over a fixed period of time), we execute thefollowing steps:
1. We begin by correcting for the photon time-of-flight (TOF) from the estimatedsignal vertex to each hit PMT. This correction reduces the variance of signal
PMT hit times from the order of O(100) to O(10) ns, significantly enhancing the signal-to-noise ratio in a smaller time window as in Figure 3.4. The
vertex information should be provided a priori. For instance, in the context
of searching for neutrons produced in a neutrino interaction, a reconstructed
neutrino vertex can be given. Alternatively, in the case of seeking neutrons

Number of hits / 1 ns

originating from a neutron source, the position of the source can be used.

Raw
ToF-corrected
Dark noise (avg.)

100

10 1

10 2

0

50 100 150 200 250 300
PMT hit time from signal generation [ns]Figure 3.4: Gd(n, γ) signal PMT hit time distributions.
True neutron vertex was used as the input for photon TOF
correction. Dark noise represents the SK-VI average. The
tails are mainly due to light reflection on the tank wall.
2. Next, we arrange all PMT hits chronologically and scan for sets of PMTs witha hit count exceeding a threshold of around 5-20 hits within a O(10) ns sliding
time window. In cases where there are adjacent sets (i.e., two distinct sets of
PMTs separated by less than a predetermined time width), we discard the set
with the smaller hit count and consider the set with the larger number of hits
as a potential signal candidate, to prevent double counting of a signal.

105

3.2. Signal identification algorithm

Figure 3.5: Number of PMT hits within 14 ns after photon
TOF correction based on the true particle-gun simulation
vertex, for PMT dark noise (gray), H and Gd neutron capture signals (pink), and Michel electron signals (olive).

Vertex options for photon TOF correction
The photon TOF correction’s effectiveness in reducing hit time variance, as shownin Figures 3.4 and 3.5, depends on the proximity of the actual neutron capture
vertex to the a priori vertex used for TOF correction. In most cases, the given
vertex is the estimated position of neutron production, determined from prompt
signals generated upon neutron production. This method is the most efficient way
to approximate the neutron capture vertex, given that neutron capture gamma-rays,
especially from 1 H(n, γ), have low energy levels for accurate vertex reconstruction.
The majority of neutron captures occur close to their production vertex.
However, this presents a challenge as signal detection performance starts to relyon the distance between the neutron capture and the provided input vertex. Thisis particularly relevant when detecting neutron captures far from the production
vertex, indicating the production of highly energetic neutrons. To address this,we may independently reconstruct the neutron capture vertex and use it for TOF
correction and subsequent feature extraction. This approach is feasible for 8 MeV
Gd(n, γ) signals, with a vertex resolution of approximately 1 m. In the Gd-loaded
phase, we can choose one of two strategies for signal detection:106

CHAPTER 3. NEUTRON DETECTION• INIT vertex mode (default mode used in this study)
No additional vertex reconstruction is performed; the selected candidates areused as they are. Features, such as signal vertex distance to the tank wall andhit timing distributions, are computed based on the initial provided vertex
used for TOF correction in the candidate selection stage. This yields higher
overall detection performance, though the performance does depend on the
distance between the initial provided vertex and the actual neutron capture
vertex. This approach is the only applicable method in the pure water phase.
• RECO vertex mode (used for vertex reconstruction only)
The neutron capture vertex and time are independently reconstructed using
only PMT hits contained in each candidate. Using the algorithm explained in
Section 2.4.2, we find a vertex that maximizes the likelihood of hit PMTs in[-0.5, 1] µs range around each candidate. The TOF for each PMT is adjusted
based on the distinctly reconstructed vertex for each candidate. A set of PMT
hits that constitute a candidate are rearranged as the TOF changes. Due tothe relatively lower vertex resolution of neutron capture signals compared tomuch stronger prompt signals, a larger sliding time window is necessary toinclude more signal hits. Features, such as signal vertex distance to the tankwall and hit timing distributions, are computed based on the reconstructed
neutron capture vertex. A candidate is rejected if the reconstructed signal time
deviates significantly (200 ns) from the central time of the original candidate.

Table 3.1 list the parameters used in this study. Figure 3.6 compares the performance of the INIT and RECO vertex modes.

1073.2. Signal identification algorithm

Parameter

Description

TWIDTHTCANWIDTHNHITSTHNHITSMX
TMINPEAKSEPTMATCHWINDOW

Sliding time window width [ns]Hit selection time width for each candidate [ns]Minimum PMT hits to allow as a candidate
Maximum PMT hits to allow as a candidate
Minimum separation of two adjacent candidates [ns]Match window used for candidate labeling in MC [ns]Vertex mode
INIT RECO
14
14
14
30
5
7
400
400
50
200
50
200

Table 3.1: Parameters used in the vertex modes for candidate selection.
The param√
eter TWIDTH was adjusted to maximize the signal significance S/ S + B for 1 H(n, γ)
signals, where S represents the number of signal PMT hits and B represents thenumber of background PMT hits in a candidate. Other parameters were determinedheuristically. For more details, refer to [124].

Figure 3.6: Comparison of overall signal detection efficiencies in SK-VI simulated
atmospheric neutrino events, between the two vertex modes — INIT vertex mode
(blue) and RECO vertex mode (orange), as a function of the distance from the reconstructed neutrino vertex to true neutron capture vertex. While the detection
performance of the RECO vertex mode is generally not as robust as the INIT vertex
mode due to the poorer vertex resolution, it is independent of the distance from theinitially provided vertex to the neutron capture vertex.

108

CHAPTER 3. NEUTRON DETECTION

3.2.2

Stage 2: Candidate classification

Extracting features
For each signal candidate selected in the steps described in 3.2.1, the following 14
features based on the discussion in 3.1 are extracted according to each vertex mode:• NHits
This represents the number of PMT hits within each candidate’s time width(see TCANWIDTH in Table 3.1). Because the likelihood of accidental noise coincidences decreases exponentially with an increasing number of coincident PMT
hits, this feature stands out as one of the most potent discriminators between
signal and background.
• NResHits
This denotes the count of ”residual” hits, which is the number of PMT hits
within a 200-ns time window ([-100, 100] ns, with 0 as the midpoint of thecandidate time width) minus NHits. Signal hits typically have very small
variance in time after TOF correction, resulting in a low count of residual
hits. Conversely, background hits tend to have a larger time spread and ahigher count of residual hits. This feature also aids in the rejection of PMT
afterpulses that can lead to an increase of residual hits.
• TRMS [ns]This represents the root-mean-squared (RMS) value of the TOF-corrected
PMT hit times. Signal hits tend to yield smaller values.
• FitGoodness
This metric quantifies the goodness of hit timing based on the formula [119]:g(ttest , ⃗xtest ) =

N
X
i=1

wi e

− ti −|x⃗i −⃗
xtest |2
−ttestc/(2σ52 )

(3.1)

109

3.2. Signal identification algorithm
Here, ⃗xtest represents the tested vertex position, N is the number of hit PMTs
in a candidate, and ti , ⃗xi are the recorded hit time and position of the ith PMT
respectively. c is the speed of light, σ5 is the assumed timing resolution (5 ns)of the PMTs, and wi , given by the equation below, is the Gaussian weightwith a larger timing resolution σ60 (60 ns) to filter out outlier hits.

wi = P− ti −e|x⃗i −⃗
xtest |2
−tf itcN− ti −
i=1 e



2 )
/(2σ60|x⃗i −⃗
xtest |2
−ttestc

2 )/(2σ60(3.2)

For the INIT vertex mode, ttest is defined as the central time of the candidate
time window, and ⃗xtest is given by the initially provided vertex. For the RECO
vertex mode, ttest is defined as the most likely signal generation time returned
by BONSAI, and ⃗xtest is given by the BONSAI-reconstructed vertex. This metricis larger for signals and smaller for background.
• DWall [cm]This is the distance from the assumed neutron capture vertex to the nearest
tank wall. In the RECO vertex mode, some background events originating fromradioactivity near the tank wall or in the PMT glass window may have their
vertices reconstructed very close to the tank wall.
• DWallMeanDir [cm]This represents the closest distance from the assumed neutron capture vertex
to the tank wall in the average hit direction, or the mean of the unit vectors
connecting the assumed neutron capture vertex to each hit PMT. Signal hits
tend to have smaller values, as Cherenkov photons are more likely to hit PMTs
closer to their point of origin due to light attenuation in water.
• BurstRatio
This is the ratio of the number of “burst” PMTs to the total number of hit
PMTs in a candidate, where a “burst” PMT is defined as a PMT hit witha preceding hit within 10 µs. When radioactive impurities trigger a PMT,
scintillation within the PMT glass window occurs with a time constant of a110

CHAPTER 3. NEUTRON DETECTION
few µs, as shown in Figure 3.7. This feature captures the correlation between
two consecutive noise hits.

Figure 3.7: Time difference of two consecutive hits in asame PMT, for signal (cyan) and noise (gray) hits.

• DarkLikelihood
This metric is related to the likelihood of the coincident noise scenario of hit
PMTs, considering their individual average dark rates which are measured foreach run in SK. The value is defined as:


N
Y

ri
σ log
⟨r⟩All
i=1



(3.3)

where σ represents the sigmoid function σ(z) = 1/(1 + e−z ), N is the numberof hit PMTs in a candidate, ri is the measured average dark rate of the ith
PMT, and ⟨r⟩All is the average dark rate of all PMTs. The dark rates may varyfrom one run to another. When ri is similar to ⟨r⟩All , the value approaches

0.5. As ri approaches ∞, the value approaches 1, and as ri approaches 0, thevalue approaches 0. Signals tend to have smaller values compared to accidental
background, which tends to include more noisy PMTs.
• OpeningAngleStdev [◦ ]This is a measure of the variation in opening angles across all possible 3hit-PMT combinations in the candidate. Opening angles are determined by111

N10n =13

0.012

N10n =120.01

N10n =110.008

N10n =100.006

N10n = 90.004

N10n = 8

3.2. Signal identification algorithm
CHAPTER 6. NEUTRON TAGGING

0.002

N10n<= 7

0

10

20

30

40

50

60

70

80

90

0

3 hits combination Cherenkov angle (deg)0.02

N10n>=160.018

N10n =15

0.016

N10n =140.014

N10n =13

0.012

N10n =120.01

N10n =110.008

N10n =100.006

N10n = 90.004

N10n = 80.002

N10n<= 7

0

10

20

30

40

50

60

70

80

90

0

0.018

N10n =15

0.016

N10n =140.014

N10n =13

0.012

N10n =120.01

N10n =110.008

N10n =100.006

N10n = 90.004

N10n = 80.002

N10n<= 7

0

10

20

30

40

50

60

70

80

90

0

3 hits combination Cherenkov angle (deg)3 hits combination Cherenkov angle (deg)

0.02Figure 6.12: Array of one-dimensional PDF of 3 hits Cherenkov angle distribution for(b) Background hits(a) Signal hits

N10n>=16

0.02

N10n>=16

the signal (left) and background (right) events. For a given N 10nvx , the corresponding
0.018

N10n =15

horizontal slice is used to calculate llrCAng. In the left and right plots, N 10n
0.016
denotes N 10nvx .

N10n =14 3.8: Opening angle distributions of signal and background hits. N10n in verFigure
0.014
N10n =13
ticalaxes represents the number of PMT
candidate time window.
0.012 hits within 10-ns 104
N10n =12(Reprinted from [34])0.01
N10n =110.008

N10n =10

0.006

creating a cone using the base circle
0.004 formed by each 3-hit-PMT combination,

N10n = 9N10n = 8

with the apex placed at the initial0.002
provided vertex or the reconstructed neutron

N10n<= 7

0

10

20

30

40

50

60

70

80

90

0

capturevertex,Cherenkov
depending
3 hits combination
angle (deg)on the vertex mode. The half-angle of the cone isthen taken as the opening angle. While the opening angle distribution of signal

Figure 6.12: Array of one-dimensional PDF of 3 hits Cherenkov angle distribution forthe signal (left) and background (right) events. For a given N 10nvx , the corresponding
horizontal slice is used to calculate llrCAng. In the left and◦ right plots, N 10n
denotes N 10nvx .

hits generally peaks at 42 , for background hits the spread can range from verysmall to large as illustrated in Figure 3.8b, particularly for a portion of hits
104that exhibit anisotropic behavior due to background radioactivity.
• Beta1-5
The parameter Beta(l) is defined by the equation [126]:
βl =

X
2
Pl (cos θij )NHits(NHits − 1) i,j

(3.4)

i̸=j

where Pl represents the Legendre polynomial of degree l (with 1 ≤ l ≤ 5), andθij is the angle between vectors extending from the given vertex to the ith andj th hit PMTs. These parameters characterize the angular distribution of the
PMT hits and have demonstrated significant discriminatory power between
signal and background hits in the SNO experiment [126]. In general, lower
values of the Beta(l) parameters indicate a higher degree of angular isotropy
in hits. While βl values for different l are correlated, including βl for all five l112

CHAPTER 3. NEUTRON DETECTION
values in the classification process has demonstrated a slightly improved signal
efficiency compared to using βl for only two different l values.

For candidates selected in MC-simulated events, we assigned the label signal
if the central time of the candidate fell within O(10-100) ns time window (specified by the parameter TMATCHWINDOW in NTag, refer to Table 3.1) of a simulated
physics event (such as neutron capture, Michel electron, and non-capture gammaray with energy over 1 MeV). Candidates not meeting this criterion were labeled as
background. Candidates labeled as signal were further classified into H-capturePP( Eγ ≤ 6 MeV), Gd-capture ( Eγ > 6 MeV), Michel-e, and Gamma, based on
the coincident type of simulated physics event. We can use the terms ‘true’ back-ground, 1 H(n, γ), and Gd(n, γ) candidates interchangeably with the correspondinglabels in this context. Figure 3.9 shows the expected feature distributions.

Dark noisep(n,γ)
Gd(n,γ)Figure 3.9: Comparison of 14 extracted features in the INIT vertex mode, forbackground (gray), H-capture (cyan), and Gd-capture (pink) candidates, whichwere selected from events of simulated PMT hits from SK-VI thermal neutron MC,
along with background PMT hits extracted using random event triggers. (All histograms are normalized by area.)113

3.2. Signal identification algorithm
Neural network for feature classification
HiddenHiddenHiddenInput

OutputFigure 3.10: An illustration of a standard neural network architecture featuringmultiple input nodes, three hidden dense layers, and a single output node.
A neural network is a computational model, similar to a multivariate function,that takes specific inputs and generates one or more outputs. In a feedforward, fully
connected network (see Figure 3.10), nodes process incoming data from all nodes in
the preceding layer using a function represented as:
All inputs

Node output = Activation

Xi(Inputi × Weighti ) + Bias

!

(3.5)

Each node is associated with a set of weights, which scale the inputs, and abias term, which adjusts the output. Typically, activation functions like hyperbolic
tangent tanh, sigmoid f (x) = 1/(1 + e−x ), or Rectified Linear Unit (ReLU) given by
f (x) = max(0, x) are chosen to introduce non-linearity and increase the complexityof the input-output relationship.
One of the most remarkable aspects of neural networks is that given a set of input
data X and a loss function L(X, W, B), we can find a set of weights W and biases
B (the parameters of the neural network) that minimize the given loss function.

114

CHAPTER 3. NEUTRON DETECTION
This process, often referred to as ‘learning’ or ‘training’, is carried out iteratively
using the gradient descent algorithm for each batch of input features. This involves
computing the gradients ∇W L and ∇B L and updating the weights and biases with
−α∇W L and −α∇B L respectively. Here, the learning rate α is a hyperparameterthat influences the learning process. The non-linearity introduced by the activation
functions enables the network to learn complex relationships between the input andoutput. For more in-depth information, refer to [127].
For the task of signal and background classification in this analysis, we designed
a feedforward, fully connected neural network with an input layer comprising 14
features, three hidden layers with 128 ReLU-activated nodes, a 50% dropout rate pereach hidden node, and a single sigmoid output node. This network was implemented
using Keras 2.6.0 [128]. The architecture which was optimized heuristically during
training is summarized in Table 3.2.
OptionLayer architecture
Hidden layer activation
Dropout rate
Output layer activation

Settings
14-128-128-128-1
ReLU
50%
Sigmoid

Table 3.2: Neural network configuration settings.
For the network training, we utilized signal features generated from thermal
neutron MC simulations using skdetsim v14 for the pure-water phases (SK-IV, SKV) and v15 for the Gd-loaded phase (SK-VI). Detector parameters, including PMT
gain as well as light attenuation and scattering parameters, were fine-tuned using
cosmic ray muon and laser data specific to each phase. Background hits were derived
from randomly triggered data events, which were shuffled in advance to averageout the impact of varying dark rates over time. Additionally, background eventswith more than 60 ID PMT hits or 20 OD PMT hits within a 200-ns window werediscarded to prevent the network from misidentifying a physical event as background.
The background hits extracted were added to each simulated signal event within therange of [-1, 501] µs from T0 , the start of the simulation, without any duplicates.
The training simulation configurations are outlined in Table 3.3.

115

3.2. Signal identification algorithm
OptionTraining data size
Signal search duration from T0
Neutron vertex
Neutron kinetic energy
Neutron direction
Background PMT hits
(SK-VI only) Gd concentration(SK-VI only) Gd(n, γ) reaction model

Settings
100,000 neutron particle-gun events[0, 500] µs
Random in ID
0.025 eV
Random in 4π sr
Extracted with random event triggers
0.011w% [51]
ANNRI-Gd [113]Table 3.3: Training simulation configuration settings.
Candidates were selected and features were extracted according to the steps
outlined in Sections 3.2.1 and 3.2.2. True simulated neutron vertex was used for the
initial TOF correction. Extracted features were standardized before being fed into
the input layer.
For the loss function, we used binary cross entropy, a widely used metric forbinary classification tasks. It is defined as:

L(yi , yitrue ) = −N
1 X true
yi log yi + 1 − yitrue log (1 − yi )
N i=1

(3.6)

Here, yi represents the sigmoid output (ranging from 0 to 1) of the neural network, which is a function of input features ⃗xi , weights W , and biases B, while yitrue
is the true label, either 0 or 1 for the two classes. In our analysis, yitrue = 0 signifies
a candidate labeled as background, and yitrue = 1 denotes a candidate labeled as
H-capture or Gd-capture. It’s important to note that as yi approaches yitrue , thevalue of L tends towards 0, regardless of whether yitrue is 0 or 1. To prevent theweights from becoming overly complex, a small L2 regularization term was added,
which is the sum of the squares of all weights with a coefficient of 0.00001.

116

CHAPTER 3. NEUTRON DETECTION
Weights and biases were initialized using the He initialization technique [129],
which helps mitigate the issues of vanishing or exploding gradients. The widelyused stochastic gradient descent algorithm, Adam [130], was employed to update
the weights using a minibatch of size 2048 per iteration. The initial learning rate
was set to 0.0001. 80% of the training data was utilized for updating the weights,
while the remaining 20% was reserved for validation. The hyperparameters, including the minibatch size, initial learning rate, and regularization term, were determinedheuristically while closely monitoring the training process. The process of updating
weights using the training data, continued until the signal efficiency in the validation set ceased to improve for 5 consecutive epochs. The training conditions are
summarized in Table 3.4, and Figure 3.11 shows an example of loss versus epochs.
Four neural networks, all with the same architecture and training conditions,were trained on simulations tuned to each SK phase and vertex mode: SK-IV, SKV, SK-VI with INIT vertex mode and SK-VI with RECO vertex mode.

OptionTrain : validation split
Minibatch size
Input scaling
Weight initialization
Loss
Regularization
Loss optimizer
Initial learning rate

Settings
80% : 20%
2048
Standardized
He [129]
Binary cross entropy
L2: 0.00001
Adam [130]0.0001

Table 3.4: Neural network training configuration settings.

117

3.2. Signal identification algorithm

Algorithm evaluation
We evaluated the signal efficiency and false positives per unit time, using a distinct
test set of thermal neutron MC-simulated events for each SK phase (SK-IV, SKV, and SK-VI). These simulations were generated with unique random seeds andunseen background PMT hits. The remaining configurations were aligned with thetraining data conditions outlined in Table 3.3. Moving forward, we will implicitly
refer to the INIT vertex mode unless stated otherwise. The notation ‘SK-VI (R)’denotes the results for SK-VI in the RECO vertex mode.

Stage 1 candidate selection performance
In the INIT vertex mode, the efficiency in selecting signals as candidates ranged
from 71% to 80% for 1 H(n, γ) signals, and while it reached 98% for Gd(n, γ) signals. The accidental background rate averaged 178, 454, and 486 background candidates per 500 µs interval (the typical search time range per neutrino event) for
SK-IV, SK-V, and SK-VI, respectively. Notably, SK-V and SK-VI exhibited a higher
background rate and 1 H(n, γ) efficiency compared to SK-IV. This discrepancy is attributed to a larger average PMT dark rate (8 kHz) in SK-V and SK-VI in contrast
to SK-IV (6 kHz).
In contrast, in the RECO vertex mode of SK-VI, a substantially lower count of 21
background candidates were selected per 500 µs. This reduction can be attributed
to a higher threshold and an additional constraint on the reconstructed signal time.
This mode also demonstrated a lower efficiency of 32% for 1 H(n, γ) signals, whilestill maintaining a robust efficiency of 94% for Gd(n, γ) signals. Table 3.5 provides
a summary of the candidate selection performance.

118

CHAPTER 3. NEUTRON DETECTION

Loss

SK4
Train
Validation

0.150
0.125
0.100
0.075
0.050
0

50

100
150
Epochs200

Figure 3.11: The evaluated loss against training epochs for both the training andvalidation sets (SK-IV INIT-VERTEX mode). Beyond 100 epochs, the loss stabilizes,
showing minimal differences between the training and validation sets.

Figure 3.12: Distributions of neural network outputs for true signal and backgroundcandidates from the SK-IV, SK-V, and SK-VI thermal neutron MC simulations.
SK-VI (R) shows the result in the RECO vertex mode.

119

3.2. Signal identification algorithm
Figure 3.12 shows the output of the trained neural networks for each SK phase. It
is clear that the output for signals peaks at 1, while backgrounds peak at 0. Notably,
the Gd(n, γ) signals show a much sharper peak at 1 compared to the 1 H(n, γ) signal,
owing to the significantly better separability in NHits as opposed to the background.
SK-V shows a slightly diminished signal peak compared to SK-IV. This is attributed
to the larger dark rate, which causes signal features other than NHits to resemble
those of the background, and vice versa. Furthermore, in the RECO vertex mode of
SK-VI, there is a noticeable reduction in signal peaks compared to the INIT vertex
mode. This is attributed to the degradation in vertex resolution and the resulting
hit time resolution.
We can establish a cut threshold on the network output to distinguish signals
from backgrounds. By decreasing the cut threshold from 1, we can enhance signal
efficiency but at the expense of an increased number of false positives. This correlation is clearly illustrated in Figure 3.13. Therefore, selecting an appropriate cut
threshold necessitates striking a balance between these two metrics.
To maintain the false positive rate per neutrino event at around O(10−2 ), we
have opted for a uniform cut threshold of 0.7 across all SK phases.

Signal efficiency

1.0
0.8
0.6
SK-IV
SK-V
SK-VI
SK-VI (R)

0.4
0.2
0.0

10 2

10 1
100
False positives / 500 µs

101

Figure 3.13: Performance of the overall algorithm plotted in terms of signal efficiency
and false positives per 500 µs, with varying neural network output cut thresholds.
The data points correspond to the performance achieved with a threshold of 0.7.

120

CHAPTER 3. NEUTRON DETECTION

Background candidates / 500 µs
1
H(n, γ) efficiency
Gd(n, γ) efficiency
Overall signal efficiency

SK-IV
177.7
70.52%
70.52%

SK-V
453.8
73.90%
73.90%

SK-VI
485.6
80.13%
98.40%
89.52%

SK-VI (R)
21.2
31.58%
94.27%
63.80%

Table 3.5: Summary of Stage-1-only candidate selection performance.

The performance of the entire algorithm, including both the candidate selection
and the neural network classification stages with a 0.7 cut threshold, is summarizedin Table 3.6. Specifically, the INIT vertex mode demonstrated an efficiency of 2030% for 1 H(n, γ) signals and 94% for Gd(n, γ) signals.
Overall signal detection performance on thermal neutron MC
Stage 1 and 2 combined
SK-IV
SK-V
SK-VI
False positives / 500 µs 0.0267 0.0307 0.0156
1
H(n, γ) efficiency
32.47% 28.24% 24.27%
Gd(n, γ) efficiency
94.41%Overall signal efficiency 32.47% 28.24% 60.33%

SK-VI (R)0.0082
3.07%
69.50%
37.22%

Table 3.6: Summary of signal detection performance of Stage 1 and 2 combined.

1213.2. Signal identification algorithm
One might wonder which features contribute how much, on average, to the final network output. A widely-used metric in interpreting machine learning model
predictions is the Shapley value [131], a concept from cooperative game theory thatprovides a solution for fairly distributing payoffs to players in a coalition.
The Shapley value ϕi is determined by the average marginal contribution of theith player to the total payoff, across all possible coalitions of N players:ϕi (v) =XS⊆N \{i}|S|!(|N | − |S| − 1)!
[v(S ∪ {i}) − v(S)]
|N |!

(3.7)

Here, v represents the total expected sum of payoffs within a coalition S. ThisP
value possesses unique properties, such as efficiency ( i ϕi (v) = v(N )) and symme-

try (ϕi (v) = ϕj (v) if and only if v(S ∪ {i}) = v(S ∪ {j}) for all S that excludes bothi and j), which ensure equitable distribution of payoffs.

Recently, this value has gained attention for fairly estimating feature contributions in individual model predictions. This value can be extended to models withmultiple features, treating features as players, and v as the prediction for feature
values in set S, marginalized over features not included in set S.
SHAP (SHapley Additive exPlanations) [132], one of the “additive” feature attribution methods, employs this approach and implements the computation of v as wellas ϕ, which is referred to as a SHAP value. SHAP also defines “feature importance”
as the mean of absolute SHAP values for all data instances. This can be interpreted
as the average impact of each feature on output magnitude.
Figure 3.14 compares SHAP feature importance values calculated for randomly
selected 200 candidates in each SK phase test data, normalized by the sum of thevalues. This allows us to compare the fraction of average impact a feature has on
the network output among different SK phases. We can infer that NHits, TRMS,
and DWallMeanDir are the top three features that contribute most to the network
output in the INIT vertex mode, while DWall is equally important as NHits in the
RECO vertex mode for SK-VI.

122

SK-IV
SK-V
SK-VI
SK-VI (R)0.6
0.4

DarkLikelihood

FitGoodness

BurstRatio

DWallMeanDir

DWall

OpeningAngleStdev

Beta5

Beta4

Beta3

Beta2

Beta1

TRMS

0.0

NResHits

0.2
NHits

Normalized
SHAP feature importance

CHAPTER 3. NEUTRON DETECTION

Figure 3.14: Comparison of normalized SHAP feature importance values across different SK phases.
We can also explore something similar to partial dependence plots. Figure 3.15
illustrates the partial dependence of the average neural network true positive rates(representing signal classification efficiencies) in relation to each feature. These plots
indirectly reveal the features favored by the neural network, such as a high NHits
count or a low TRMS value, for instance.
Most of the neural network preferences align with our expectations. Additionally,
it is evident that neural networks tend to favor candidates with a larger fraction of
signal hits (SignalRatio), even though they have not been explicitly trained on
this. They also do not exhibit a preference for signal timing relative to the event
trigger (FitT), a feature they have not been trained on either.
However, for certain features like NResHits, the neural network preference isentirely opposite between the pure water phases and the Gd-loaded phase. This
discrepancy could be attributed to differences in gamma-ray multiplicity and totalenergy between 1 H(n, γ) and Gd(n, γ) reactions.
From this point forward, “signal” candidates refer to candidates with a neuralnetwork output greater than 0.7.

123

3.2. Signal identification algorithm

1.0

1.0

1.0

1.0

0.8

0.8

0.8

0.8

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

10

20

30
NHits

40

0.0

50

0

10

20
30
NResHits

40

50

0.0

0

2

4

TRMS

6

8

10

0.0
0.0

1.0

1.0

1.0

1.0

0.8

0.8

0.8

0.8

0.6

0.6

0.6

0.6

0.4

0.4

0.4

0.4

0.2

0.2

0.2

0.2

0.2

0.4
0.6
FitGoodness

0.8

1.0

0.0
0.0
0.0
0.0
0.50 0.25 0.00 0.25 0.50 0.75 1.00 0.50 0.25 0.00 0.25 0.50 0.75 1.00 0.50 0.25 0.00 0.25 0.50 0.75 1.00 0.50 0.25 0.00 0.25 0.50 0.75 1.00
Beta1
Beta2
Beta3
Beta4
1.0
1.0
1.0
1.0
0.8

0.8

0.8

0.8

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
0.50 0.25 0.00 0.25 0.50 0.75 1.00
0
Beta5
1.0
1.0

10

20
30
40
OpeningAngleStdev

50

0.0

0

500

1000
DWall

0.0

1500

1.0

1.0

0.8

0.8

0.8

0.8

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

0.1

0.2
0.3
BurstRatio

0.4

0.5

0.0
0.0

0.2

0.4
0.6
DarkLikelihood

0.8

1.0

0.0
0.5

0.6

0.7
0.8
SignalRatio

0.9

1.0

0.0

0

1000

2000 3000
DWallMeanDir

4000

5000

0

100

200

400

500

FitT

300

Figure 3.15: Partial dependence of neural network true positive rates (signal classification efficiency, vertical axes) for each feature, shown for SK-IV (blue), SK-V
(cyan), SK-VI (pink), and SK-VI (R) (red). SignalRatio represents the ratio of
signal PMT hits in each candidate, while FitT denotes the candidate time relative
to the event trigger, both of which were not included in the training data.

124

CHAPTER 3. NEUTRON DETECTION

3.3

Validation with Am/Be neutron source

3.3.1

Am/Be neutron source

The Am/Be (Americium/Beryllium) neutron source is a mixture of AmO2 and Be
powders enclosed in a thin stainless steel container, commonly used as a fast neutron
point source. The radioactive isotope 241 Am undergoes alpha decay, a process witha half-life of 432 years. The resulting alpha particle, with a maximum kinetic energyof 5.5 MeV, collides with 9 Be, leading to the formation of a compound nucleus, 13 C∗ .
This state quickly decomposes into 12 C∗ and emits a neutron. Approximately half[133] of these emitted neutrons are accompanied by a 4.438 MeV gamma-ray. Thisdetectable gamma-ray is produced during the de-excitation of the 12 C∗ nucleus in its
first excited state, enabling us to identify neutrons associated with this specific state.
However, the remaining neutrons linked to other states (such as the ground stateor higher excited states) either do not or rarely accompany detectable gamma-rays.
This makes them an unwanted source of continuous pile-up. Figure 3.16 illustrates
the expected kinetic energy distributions for neutrons in each excited state of 12 C∗ ,ranging from approximately 1 to 8 MeV for gamma-ray associated neutrons.

Figure 3.16: Kinetic energy distribution of neutrons from Am/Be source for each excited state of 12 C∗ , g.s. (red), 1st e.s. (green), and 2nd or higher e.s. (black) (Adaptedfrom De Guarrini and Malaroda [134] and ISO8529-1 [135], reprinted from [136])1253.3. Validation with Am/Be neutron source
6.2. Americium-Beryllium Calibration

85

Acrylic case

Am/Be source
Acrylic case

Am/Be

Am/Be

BGO

BGO

BGO scintillator(a) Am/Be capsule(a) Appearance and Schematic of the 1BGO (b) Appearance and Schematic of the 8BGO(b) Am/Be capsulegeometry.
geometry.surrounded by

BGO scintillator crystals

Figure 6.10: Appearance and schematic of 1BGO geometry (left)
8BGOusedgeometry(right).
Figure 3.17: Am/Be neutronandsourcefor thecalibration (left) and capsule sur-rounded by BGO scintillator crystals (right), which is the actual setup placed inside
theDatatank taking
for calibration (reprinted from [136]).
In January 2021, data from nine distinct AmBe points were taken. Figure 6.11 visualizes
the measurement
points. At oftheourcentralpoint,
8BGO data
was gathered
over inTo assessthe actual capability
signaldetection
algorithm,
as outlined
1 hour, while 1BGO data spanned 2 hours. Measurements at other oﬀ-center points
Section 3.2, and to confirm any differences between observations and our signal andwere conducted over a 30-minute interval, primarily to evaluate position dependence
detector
simulation,
we employed
Am/Beneutrondeveloped
of eﬃciency.
Note that
1BGO dataan
was
exclusively
takensourceat the center
point.in the early

1990s, containing 241 Am with a strength of 97 µCi (3.6 MBq). This source wasz

encapsulated within a stainless steel cylinder with an outer diameter and height of+12 neutronm
0 m total
12 m The12.5 mm, as illustrated in Figure 3.17a.
emission rate as well asx

the emission rates of neutrons corresponding to each excited state of 12 C∗ (includingmthe ground state, 1st+12
excitedstate, and 2nd or higher excited states) were thoroughly

examined, as detailed in [137]:0m• Total neutron12emission
rate: 236.8 ± 5.0 n/smThis measurement was conducted at National Institute of Advanced Industrial
Science and Technology (AIST) in Japan; the source was positioned at thecenter of a standard graphite pile, and the fully thermalized neutron fluence
rate wasmeasureda 3 Heproportionalpoints.
counterpositioned
Figure6.11: using
Illustrationof measurement
Orangesquares 90 cm away
exhibit measurement points, and the darker blue circle denotes cali-from thebration
center.holes,
Thewheremeasuredvalue
with the referencex = 1237,
35.3,was1080then
cm ⇠ compared
12, 0, +12 m.
value obtained with the standard Am/Be neutron source provided by AIST.
6.2.2

AmBe MC simulation

To compare the neutron tagging eﬃciency between the measured data and MC, theAmBe source geometry was reproduced by Geant4 and introduced to the SKG4 detector simulation.

126

CHAPTER 3. NEUTRON DETECTION• 4.438 MeV γ / 1st e.s. n emission rate: 110.1 ± 15.5 γ(n)/s
The source was placed at distances of 10 mm and 50 mm from a high-purity
Germanium (HPGe) detector. Energy spectra were measured and the gammaray peak was compared with Geant4 simulation. The 1st e.s. neutron emission
rate was set to be equal to the measured 4.438 MeV gamma-ray emission rate.
• Ground-state n emission rate: 74.9 ± 11.9 n/s
Deposited energy spectra in the NaI(Tl) crystal caused by the source activity
as shown in Figure 3.18a were recorded using a PMT. The source was placedat various distances from the crystal, and the ratio Rtail/peak of the high-energy
tail (visible energy > 5 MeV) to the 4.438 MeV gamma-ray peak was comparedwith Geant4 simulation to fit the neutron-to-gamma ratio. This ratio varies
with distance L as shown in Figure 3.18b, due to lower likelihood of gammaneutron coincidences which form a part of the high-energy tail. Since thecontribution from neutrons from the 2nd or higher e.s. with kinetic energy
below 3 MeV is negligible, the neutron-to-gamma ratio determines the ratioof the ground-state neutrons to the 1st e.s. neutrons of 12 C∗ .

(a) Observed energy spectrum(b) Rtail/peak vs. L

Figure 3.18: The energy spectrum of Am/Be source activity (left) andthe tail-to-peak ratio Rtail/peak as a function of source distance L (right).
(reprinted from [137])• 2nd or higher e.s. n emission rate: 51.8 ± 17.2 n/s
This value was estimated by subtracting the ground-state and 1st e.s. neutron
emission rates from the total emission rate.

127

3.3. Validation with Am/Be neutron source

3.3.2

Calibration setup

The Am/Be neutron source was surrounded by Bismuth Germanate (Bi4 Ge3 O12 ,BGO) scintillator crystals, as illustrated in Figure 3.17b. This configuration was
designed to initiate scintillation responses induced by 4.438 MeV gamma-rays andto identify neutrons in the 1st excited state of 12 C∗ originating from the Am/Be
neutron source. BGO is non-hygroscopic, allowing it to be safely immersed in water.
The scintillation properties of BGO are outlined in Table 3.8. When a 4.438 MeV
gamma-ray deposits all its energy within the scintillator, the ID PMTs register
scintillation light equivalent to 1,000 - 2,000 photoelectrons in total.
The setup encased in Polymethyl methacrylate (PMMA, acrylic glass) was inserted into the tank through calibration holes located on the top. It was secured
in a fixed position within the ID for an extended period, typically ranging from 30
minutes to 1 hour, during which PMT hits were recorded per each event triggers.
While the triggering conditions have varied slightly over the course of more than 10
years of operation, PMT hits within 535 µs (or 835 µs for SK-IV) from the event
trigger were logged for SHE-triggered events. This allowed for subsequent searches
for neutron signals within the time frame.
It is important to consider that the calibration setup, including the use of
BGO scintillators and the constant pile-up background neutrons stemming fromthe source, may introduce non-negligible effects on the signal efficiency estimate.
For instance, BGO (Bi4 Ge3 O12 ) has an effective neutron capture cross section of
1.47 barns [138]. The total excited energy resulting from neutron captures on 76 Ge
or 74 Ge is 6.072 MeV and 6.505 MeV, respectively. These captures inside the scintillator, which can mimic the 4.438 MeV gamma-ray scintillation and consequently
lead to false signal events, may account for a few percent of genuine signal events,
considering the background source intensity and the relative capture rates on Ge, as
illustrated in Figure 3.19. Furthermore, multiple gamma-rays from Gd(n, γ) reactions of piled-up neutrons in Gd-loaded water, in case roughly half of the energy isaccidentally deposited in the scintillator, can also mimic the signal event trigger. We
assessed these potential nuisance effects using MC simulations conducted in Geant4.

128

a prompt event is calculated. The 2.2 MeV gamma ray from hydrogen capture should
not influence the prompt event selection, given that the QISMSK is likely half thatof the 4.4 MeV gamma ray. Instead, we examined the captures from other isotopes,with Ge(n, s)Ge being the primary candidate,as illustrated in Figure 6.23.
CHAPTER 3. NEUTRON DETECTION

10−1
10−2
10−3
10−4
10−5

e-

S34

deuteron

S33

O19

O18

O17

Ni65

Ni63

Ni61

Ni62

Ni59

Mo99

Mo98

Mo97

Mo96

Mo95

Mo93

Ge77

Mo101

Ge75

Ge74

Ge71

Ge73

Gd161

Gd159

Gd158

Gd157

Gd156

Gd155

Fe59

Gd153

Fe58

Fe57

Cr55

Fe55

Cr54

Cr53

C14

Cr51

C13

Bi210

10−6

Figure 3.19: The thermal neutron capture probabilities on nuclides (shown are
the final states) existent in the SK-VI Am/Be calibration setup, simulated with
10−1
Geant4.10.5.p01[139, 140] and ENDF/B-VII.1 [31]. (Reprinted from [136])
10−2
10−3

3.3.3

Simulation

10−4
10−5

e-

deuteron

S35

S33

O19

O18

O17

Ni65

Ni63

Ni62

Ni61

Ni59

Mo99

Mo98

Mo97

Mo96

Mo95

Mo93

Mo101

Ge77

Ge75

Ge74

Ge73

Ge71

Gd161

Gd159

Gd158

Gd157

Gd156

Gd155

Gd153

Fe59

Fe58

Fe57

Fe55

Cr55

Cr54

Cr53

Cr51

C14

C13

Bi210

Neutronparticle-gun simulation with skdetsim in Geant3
10−6
To evaluate the neutron signal simulation capabilities of the detector simulator
skdetsim (Geant3.21 [141]) used in atmospheric neutrino event simulations, theFigure 6.23: The probability of thermal neutron capture in the SK
with AmBe source geometry for 8BGO (top) and 1BGO (bottom).
v15p2.
to practicalchallenges
in implementing
TheseUnfortunately,probabilitiesdue
were
calculated
based onthe result ofthe
thecalibration
five milsetup
intheno-longer-maintained
Geant3
in
FORTRAN,thesimulation
conlion neutron MC, which uses ENDF.VII.1 as a nuclear datawas
library[47].
ducted in a particle-gun setup. The configuration
settings, including the detectorbase MC simulation for comparison with Am/Be data was generated using skdetsim

parameters tuned for each SK phase, as well as the inclusion of background PMT
hits extracted from randomly triggered data events, were similar to the thermal neu-

To ascertain
the probability,
these 3.2.2
steps
tron simulation
described in Section
andarethefollowed:
ones outlined in Table 3.3 whichwere used
for training
and evaluating
1. Generate
5 million
neutron
MC. the signal detection algorithm.
The notablecapture
differencesdata,
includedneutron kinetic
energy,which followedthe
2. Store neutron
likethevertex,
capturenucleus,and energyof eachstandard(thegreen
distribution
inFigure3.16)
corresponding
togamma-associatedcapture gamma-ray.
neutrons from Am/Be, and the fixed vertex position set to the source position of

3. Based
this data,
millionsof gamma-ray
MC appliedfor Gd(n,eachoncalibration
run. generate
Additionally,no background
event cuts werewhen s)Gd and
30,000MC events
for hits
Ge(n,appendingbackgroundto thes)Ge.
simulated signal events.
4. Determine the likelihood by applying identical cuts as the prompt event selection.
129

The outcomes indicate that the Gd(n, s)Gd probability is 0.285%, and for Ge(n, s)G
it was 10.2%. MC calculations yield neutron capture fractions of 48.1% for Gd and0.7% for Ge. Thus, the final estimated probability stands at 0.2% (Table 6.8). Using3.3. Validation with Am/Be neutron source
For this simulation, no consideration was given to factors such as the scintillator
geometry, constant pile-up from the source, or resulting trigger dead time effects.
The input Gd concentration for SK-VI was set to 0.0110 w%.

Full simulation of Am/Be calibration setup with SKG4 in Geant4
We used SKG4 v2.4.3, based on Geant4.10.5.p01 [139, 140], for the comprehensive
simulation of the Am/Be calibration setup. This included modeling the stainless steel capsule, BGO scintillator crystals, and plastic casing, while considering
the complete kinetic energy distributions of both gamma-ray associated and nonassociated neutrons, as illustrated in Figure 3.16. To accurately replicate the BGO
scintillation response to 4.438 MeV gamma-rays, we fine-tuned the input opticalparameters for BGO by comparing the simulation results against Am/Be data.
For the tuning of BGO optical parameters, we used Am/Be data collected withthe source at the center of the tank during the SK-V phase. Refractive indicesand photon wavelength spectrum were sourced from [142] and [138], respectively.
We generated MC simulations using various light yield parameters and estimatedthe peak value of the integrated charge spectrum. For each simulated event, theintegration range was set to [-0.5, 5] µs from the simulation’s T0 to fully encompass
all scintillation light with a decay constant of 300 ns. Then, we applied linear
regression to establish the relationship between light yield and corresponding peakvalues. The regression was used to estimate the light yield corresponding to thepeak value observed in the Am/Be data. The light yield value was further refined bymodeling the peak as a combination of Gaussian and linear background components.
Finally, we fine-tuned the Fano factor to match the observed peak width. This
process is illustrated in Figure 3.20. The tuned light yield was 5,935 photons per
MeV γ, which was significantly lower than the reference value of 8,400 photons [138].
The scintillation decay time was adjusted based on the pulse shape of 4.438
MeV gamma-ray scintillation, averaged among events with integrated charge within
1,250 to 2,000 photoelectrons. This involved applying TOF correction to the PMT
hit time distribution, which was then weighted by deposited charge per event.

130

CHAPTER 3. NEUTRON DETECTION

Figure 3.20: Correlation between the input BGO scintillation photon yield and thesimulated peak position in the integrated charge spectrum (left) and a comparison
between the integrated charge spectrum obtained from Am/Be data and the corresponding spectrum from the fine-tuned MC simulation (right). The minor variation
observed in the lower tail is attributed to event pile-ups, a factor not yet incorporated in the MC simulation at this stage.
Following the approach outlined by Gundacker et al. [143], we fitted the averagepulse shape with the following function:f (t|θ) = Θ(t − θ)Xi∈{fast, slow}

e−(t−θ)/τd,i − e−(t−θ)/τr,i· ρi
τd,i − τr,i

(3.8)

Here, t represents the PMT hit time, Θ is the Heaviside step function, θ is thestart time of scintillation, τd is the decay time constant, τr is the rise time constant,and ρi is the fraction of the fast/slow component. The timing response of the PMTs
on the order of O(1) ns was neglected, as the pulse shape time scale was on theorder of O(1) µs.
The best-fit values were τd,slow = 433 ns and τd,fast = 110 ns, which aligned withthe reference [144] for a water temperature of 13.5◦ C. The fitted fast component
ratio was 12%, also in agreement with the reference value of 10% [145]. The fitted
rise time values were on the order of O(1) ns, which was negligibly small, so they
were assumed to be 0 in the actual simulation input.

131

3.3. Validation with Am/Be neutron source

Figure 3.21: The average BGO scintillation pulse shape observed in the Am/Be
data, along with the corresponding fitted pulse spectrum (left), and a comparison
between the average scintillation pulse shapes from the data and the fine-tuned MC
simulation (right).

SCINTILLATIONYIELDRESOLUTIONSCALEFASTTIMECONSTANTSLOWTIMECONSTANTYIELDRATIO

5,935 photons / MeV γ
0.86
109.92 ns
432.60 ns
0.1191

Table 3.7: Tuned BGO optical parameter inputs for Geant4.

Light yield
Emission peak
Rise time
Decay time
Radiation length
Temperature response
Afterglow (after 20 ms)

8,400 photons / MeV γ
480 nm< 2 ns
300 ns1.1 cm
-1.2%/◦ C
150 ppm

Table 3.8: Reference values for BGO properties (Room temperature, [138])

The fit results and the data-MC comparison of the average BGO scintillation
pulse shapes are displayed in Figure 3.21. Table 3.7 provides a summary of theadjusted BGO input optical parameters for Geant4.

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CHAPTER 3. NEUTRON DETECTION
At this point, the simulator can only simulate ‘discrete’ events of signals, suchas a single instance of n + γ (4.438 MeV) corresponding to the 1st excited state of
12

C or just n corresponding to the other states of 12 C. In reality, these events occur

continuously from the source, and a single triggered event may contain multiple
pile-ups of gamma-rays and neutrons. Some of these continous emissions, throughGe(n, γ) or Gd(n, γ) reactions of 6-8 MeV for example, may eventually trigger an
event that is indistinguishable from a genuine signal event — specifically, a 4.438
MeV gamma-ray scintillation-triggered event with a single neutron.
To accurately reproduce these effects, we developed a dedicated pile-up simulator, which reorganizes the ‘discrete’ simulated events of each neutron emission modeof Am/Be on a single global ‘continuous’ time axis, based on the emission rate ofeach mode as described in Section 3.3.1. The simulated PMT hits and randomly
selected background PMT hits are rearranged on the global time axis, and event
triggers are applied to generate events. This approach enables us to faithfully replicate both the constant source activity and the actual event triggering scheme used inour data. We performed a comprehensive simulation of the Am/Be calibration setup
using a combination of SKG4 v2.4.3 with finely-tuned BGO scintillation parameters,
along with the pile-up simulator for each source position. The simulation was used
to evaluate any potential adverse impacts of the calibration setup on the estimationof signal efficiency, thereby complementing the comparison of Am/Be data with theskdetsim neutron particle-gun simulation, which does not include such features.
As an additional point, the input Gd concentration was adjusted from the actual
estimated value of 0.0110 w% to 0.0104 w% to align the simulated neutron capture
time with the observations in the Am/Be data. This adjustment was made byanother analysis group to compensate for the overestimation of Gd(n, γ) reactionsin Geant4.10.5.p01 which takes neutron capture cross sections from ENDF/B-VII.1.

133

3.3. Validation with Am/Be neutron source
In the case of SK-IV/V, candidates were selected within the time range of [18,534] µs from the event trigger, effectively eliminating the possibility of PMT afterpulse or Michel electron contamination. For SK-VI, which had a shorter capture
time, a broader window of [3, 534] µs was used. The assessment of PMT afterpulse
and Michel electron contamination for SK-VI is necessary and is detailed in Section
3.4 through 3.5. The rationale for the different search ranges arises from a technicalissue (which could be addressed in the future), rather than a consideration related
to physics. The initial TOF correction was determined based on the source position.
Table 3.9 summarizes the signal search time range used for each SK phase.
TargetPure waterGd-water

Phase
SK-IV
SK-V

Vertex mode

SK-VI

INIT

Trigger

Signal search time range
Start
End
18 µs

0 µs

RECO

534 µs
3 µs

Table 3.9: Summary of the signal search time range.

3.3.4

Data reduction

Data reduction involved three main steps: (1) selecting 4.438 MeV gamma-ray scintillation, (2) reducing cosmic rays, and (3) minimizing pile-ups. The following section explains the relevant 7 event cuts applied to the Am/Be data. As an example,
figures comparing the cut variables of SK-VI data with the source positioned at thetank center for 1.1 hours (Run 85622, 4,002 seconds) and the corresponding MC
simulation for a livetime of 1.7 hours (6,000 seconds) are provided.
In each figure, the legend label ‘(all)’ denotes histograms before applying any
cut, while ‘(cut)’ indicates histograms after applying all previously mentioned cuts,
excluding the one being discussed. The shaded areas in gray represents the data
events that were rejected due to the discussed cut.

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CHAPTER 3. NEUTRON DETECTION
Cut #1: Selecting SHE and AFT event triggers
The full 4.438 MeV gamma-ray scintillation is expected to activate the SHE
trigger which has the highest threshold in SK. Due to the deadtime resulting fromelectronic issues and pile-ups, SHE-triggered events are accompanied by AFT triggers — an additional 500-µs record of PMT hits — roughly half of the time. Eventswith both SHE and AFT triggers were selected, as shown in the figure below. This
selection allows us to search for neutron capture signals with a characteristic decay
time constant of O(100) µs. Figure 3.22 shows the trigger type distribution expected
and observed with Am/Be neutron source deployed near the tank center.

Events / sec

104

SK-VI Run 85622: (35.3, -70.7, 0.0) [cm]Data (all)MC (all)

Data (cut)MC (cut)

SHE SHE+AFT LE
Event trigger type

HE

102
100
10 2

Other

Figure 3.22: Event trigger type with Am/Be neutron source in SK tank.

1353.3. Validation with Am/Be neutron source
Cut #2: Rejecting event triggers due to cosmic-rays
To identify the entry of a cosmic ray or the exit of an outgoing particle, we
employ a clustering algorithm for the OD PMT hits. These hits are binned by theazimuthal and polar angles. For each bin, we calculate the gradient of total charge
with respect to each neighboring grid. The bin is then paired with its neighboring
bin that shows the highest charge gradient. Events with more than 10 OD PMThits in the largest cluster (NHITAC) were rejected, as they may suggest the entry ofa cosmic ray muon. The distribution of NHITAC is shown in Figure 3.23.

SK-VI Run 85622: (35.3, -70.7, 0.0) [cm]Data (all)MC (all)

Events / sec

103

Data (cut)MC (cut)

101
10 1
10 3
10 5

0

20
40
60
80
Number of clustered OD PMT hits

Figure 3.23: Number of clustered OD PMT hits.

136

100

CHAPTER 3. NEUTRON DETECTION
Cut #3: Selecting event trigger from 4.438 MeV γ scintillation
The total deposited charge within the time window of [-0.5, 1.0] µs from the event
trigger was integrated for each event to estimate the energy content of the triggering
signal. The resulting distribution exhibits a distinct peak attributed to the 4.438
MeV gamma-ray scintillation, along with tails arising from neutron captures andinelastic scattering within the scintillator, as illustrated in Figure 3.25. The peakcharge, measured in the number of photoelectrons, is determined from a histogram
within the range of [500, 1500] photoelectrons, divided into 100 bins. Events fallingwithin ±200 photoelectrons around the estimated peak were subsequently selected.

Figure 3.24 shows the distribution.

103
Events / sec

102

SK-VI Run 85622: (35.3, -70.7, 0.0) [cm]Data (all)MC (all)

Data (cut)MC (cut)

101
100
10 1
10 2

0
500
1000
1500
2000
Integrated ID charge around event trigger [p.e.]

Figure 3.24: Integrated ID charge with Am/Be neutron source in SK tank.

137

3.3. Validation with Am/Be neutron source

Figure 3.25: Integrated charge (QISMSK) distribution for SK-VI Am/Be data (black)and corresponding MC simulation (red) for Run 85622, prior to applying any cuts,
along with a breakdown of triggering events. While the fraction of Am/Be γ triggers (green) is ∼ 95%, ∼ 5% of the remaining events are expected to triggered bybackgrounds, such as neutron capture gamma-rays (blue) or neutrons (orange) interacting within the scintillator.

138

CHAPTER 3. NEUTRON DETECTION
Cut #4: Rejecting event triggers too close to the previous trigger
For two events in close proximity, there is a chance of late neutron captures fromthe prior trigger interfering with the subsequent event. To mitigate this, events
triggered less than 1 ms after the previous trigger were rejected. Figure 3.26 shows
the distribution of the time difference between two consecutive events.

Figure 3.26: Time difference between two consecutive event triggers.

1393.3. Validation with Am/Be neutron source
Cut #5: Rejecting cosmic rays within the signal search range
Cosmic rays can penetrate the detector even during events triggered by Am/Be
gamma-rays and affect the signal efficiency estimate. Events with more than 14 OD
PMT hits within a 200-ns window in the signal search range were rejected. Figure
3.27 shows the distribution of the expected and observed maximum OD PMT hits.

SK-VI Run 85622: (35.3, -70.7, 0.0) [cm]
Events / sec

103

Data (all)MC (all)

Data (cut)MC (cut)

101
10 1
10 3
10 5

0
20
40
60
80
100
Maximum number of OD PMT hits within 200 ns

Figure 3.27: Maximum number of OD PMT hits within a 200-ns window in thesignal search range. Events with a larger value implies a cosmic ray entry within
the signal search range.

140

CHAPTER 3. NEUTRON DETECTION
Cut #6: Rejecting events with pile-up scintillation
Scintillation resulting from pile-up gamma-rays or neutron captures can impact
the signal efficiency estimate. For each event, the maximum number of ID PMThits within a 200-ns window in the signal search range was determined, as shownin Figure 3.28. To find the most probable value, a histogram was generated in
the range of [0, 500] PMT hits with 100 bins, and the peak bin value was derived.
Minor peaks near 500 hits and 1,000 hits indicate scintillation caused by the 2.2
MeV gamma-ray from the 1 H(n, γ) reaction and the 4.438 MeV gamma-ray fromthe Am/Be neutron source, respectively. Events with more than 50 hits beyond theestimated peak were rejected to mitigate such contamination.

Figure 3.28: Maximum number of ID PMT hits within a 3-µs window in the signal
search range. Events with a larger value implies a pile-up causing scintillation within
the signal search range. Both figures show the same histograms, but with different
horizontal axis ranges.

1413.3. Validation with Am/Be neutron source
Cut #7: Selecting events with 535-µs duration
Although rare, some data events, even after the event trigger type selection, may
have an event duration shorter than 535 µs, potentially due to electronics issues. The
total number of PMT hits in the signal search range follows a normal distribution,with its mean determined by the average dark rate at the time of measurement. It
is important to note that data from a specific run and the MC with backgroundhits spanning an entire SK phase may exhibit different distributions. Outlier eventswith an unusually low number of hits may indicate abnormal events with shorter
durations. The peak position was estimated in the histogram within the range of [0,100,000] PMT hits, divided into 1,000 bins. Events falling within ±3,000 hits fromthe peak value were selected to ensure an event length of 535 µs.

SK-VI Run 85622: (35.3, -70.7, 0.0) [cm]Data (all)MC (all)

Events / sec

103

Data (cut)MC (cut)

101
10 1
10 3
10 5

20000

30000
40000
50000
Number of all PMT hits

60000

Figure 3.29: The distribution of all PMT hits within the signal search range, primarily influenced by the PMT dark rate. The disparity between data and MC
results stems from the random selection of background PMT hits throughout each
SK phase in the MC simulations. In contrast, the data specifically reflects the dark
rate measured during the 0.5-1 hour timeframe. Outlier events resulting from dataacquisition problems or background contamination were excluded from the analysis.

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CHAPTER 3. NEUTRON DETECTION
The event rates before and after applying each cut are outlined in Figure 3.30.
The event rates in general show agreement between the data and the MC, while small
difference in Cut #3 reduction rates suggests there is potential for improvement insimulating both the Am/Be source activity and the scintillator response.

SK-VI Run 85622: (35.3, -70.7, 0.0) [cm]
MC
Data

200
175
Events / sec

150

Cut #1: Select SHE+AFT trigger
Cut #2: Reject cosmic-ray triggers
Cut #3: Select 4.4 MeV scintillation
Cut #4: Reject leakage from previous trigger
Cut #5: Reject comsic rays in search range
Cut #6: Reject pile-up scintillation

125
100
75
50

0

1

2

3

4

5

6

7

Cut #1

Cut #2

Cut #3

Cut #4

Cut #5

Cut #6

Cut #7

0
1.00
0.95
0.90

No cutData / MC25
8

Figure 3.30: Am/Be event rates compared between data and MC.

143

3.3. Validation with Am/Be neutron source

SK-VI Run 85622: (35.3, -70.7, 0.0) [cm]
Events / sec

0.2
0.1
0.0

300

400
500
600
Number of candidates

700

Candidates / event / 5 µs

Data
MC

0.3

5.0

SK-VI Run 85622: (35.3, -70.7, 0.0) [cm]
Data
MC

4.8
4.6
4.4
4.2
4.0

0

100
200
300
400
Time from event trigger [µs]500

Figure 3.31: The distribution of total number of candidates in each event (left) andthe candidate time distributions (right) of SK-VI Am/Be data and simulation.
Figure 3.31 illustrates the number of candidates and their corresponding time
distributions in the selected Am/Be data and MC events. Specifically, for Run
85622, the overall number of candidates in the Am/Be data is 5% smaller comparedto the MC, as the MC incorporates background hits from various times across theSK phase, each with differing dark rates. The candidate time distributions are in
general flat over time. The 5% uptick in the number of candidates within 15 µs fromthe event trigger is attributed to PMT afterpulses, as also demonstrated in MC.
Figures 3.32 through 3.36 display the distributions of candidate features andnetwork outputs from Am/Be data and MC simulations for the source positioned
at the tank center, across various SK phases. Notably, the distributions from both
the data and MC show excellent shape agreement.

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CHAPTER 3. NEUTRON DETECTION

SK-IV Am/Be (x, y, z) = (0,-1,0) [m] (APFit)
101

Data
MC

MC: p(n, )SK-V Am/Be (x, y, z) = (0,-1,0) [m] (APFit)MC:MC: Noise

101

10 3
10 5
10 7

10 5
10 7

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Neural network output (signal likelihood)

Data
MC

MC: Gd(n, )MC: p(n, )

101

Data
MC

MC: Gd(n, )MC: p(n, )MC:
MC: Noise

10 1
a.u.

a.u.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Neural network output (signal likelihood)

SK-VI Am/Be (x, y, z) = (0,-1,0) [m] (BONSAI)MC:MC: Noise

10 1
10 3
10 5
10 7

MC:
MC: Noise

10 3

SK-VI Am/Be (x, y, z) = (0,-1,0) [m] (APFit)
101

MC: p(n, )

10 1
a.u.

a.u.

10 1

Data
MC

10 3
10 5

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Neural network output (signal likelihood)

10 7

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Neural network output (signal likelihood)

Figure 3.32: The neural network responses for candidates selected in both Am/Be
data and full Am/Be MC simulation with SKG4, for SK-IV (top left), SK-V (topright), SK-VI (bottom center), and SK-VI (R) (bottom right). Both data and MC
histograms are normalized by area. The colors in the legend indicate the fractionof labels derived from the full Am/Be calibration simulation. The label “MC: γ”
corresponds to the 4.4-MeV gamma-ray emitted by the source.

145

3.3. Validation with Am/Be neutron source

Data

MC

Dark noisep(n,γ)
Am/Be γ

Data

MC

Dark noisep(n,γ)
Am/Be γFigure 3.33: The feature distributions of signal candidates before (top) and after(bottom) the neural network selection for both the Am/Be data (Run 75502) andMC simulations, taken with the Am/Be neutron source at the tank center in SK-IV.
Both data and MC histograms are normalized by area.

146

CHAPTER 3. NEUTRON DETECTION

Data

MC

Dark noisep(n,γ)
Am/Be γ

Data

MC

Dark noisep(n,γ)
Am/Be γFigure 3.34: The feature distributions of signal candidates before (top) and after(bottom) the neural network selection for both the Am/Be data (Run 82406) andMC simulations, taken with the Am/Be neutron source at the tank center in SK-V.
Both data and MC histograms are normalized by area.

147

3.3. Validation with Am/Be neutron source

Data

MC

Dark noisep(n,γ)
Gd(n,γ)
γ

Data

MC

Dark noisep(n,γ)
Gd(n,γ)Am/Be γFigure 3.35: The feature distributions of signal candidates before (top) and after(bottom) the neural network selection for both the Am/Be data (Run 85622) andMC simulations, taken with the Am/Be neutron source at the tank center in SK-VI.
Both data and MC histograms are normalized by area.

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CHAPTER 3. NEUTRON DETECTION

Data

MC

Dark noisep(n,γ)
Gd(n,γ)
γ

Data

MC

Dark noisep(n,γ)
Gd(n,γ)Am/Be γFigure 3.36: The feature distributions of signal candidates before (top) and after(bottom) the neural network selection for both the Am/Be data (Run 85622) andMC simulations, taken with the Am/Be neutron source at the tank center in SK-VI(R). Both data and MC histograms are normalized by area.

149

3.3. Validation with Am/Be neutron source

3.3.5

Neutron tagging efficiency estimation

When comparing Am/Be data and MC, we utilize the neutron tagging efficiency,
which is calculated based on the total number of neutrons rather than the numberof neutron captures. This approach simplifies the analysis by avoiding additional
complexities. The neutron tagging efficiency ϵ represents the ratio of identified
neutron capture signals to the overall number of neutrons. In the case of Am/Be
data events, it can be approximated using the following formula:ϵ≈Candidates classified as signals − Estimated backgrounds
Selected event triggers(3.9)

For the denominator, we assume that each event trigger corresponds to the production of one neutron, accompanied by a 4.438 MeV gamma-ray scintillation.
To estimate the backgrounds in the numerator, we fit the time distribution ofcandidates classified as signals with an exponential curve given by:

f (t) = A(1 − e−t/τthermal )e−t/τcapture + B(3.10)

Here, A, τthermal , τcapture , and B are the fitted parameters. A represents thescaling coefficient, τthermal denotes the neutron thermalization time constant, τcapture
denotes the neutron capture time constant, and B is the constant offset. This offset
can be interpreted as the constant background, resulting from pile-ups and dark
background hits mis-identified as signals. The number of backgrounds is estimatedby multiplying the number of bins by B and the number of selected event triggers.
Figure 3.37 provides an illustrative example of such a fit applied to the time
distribution of signal candidates observed in the Am/Be data. For the pure waterphases, we set τthermal to 0, since most neutrons are already thermalized within the
search time window of [18, 535] µs from their production. Besides, the neutron
capture probability in pure water is largely independent of neutron kinetic energy,
due to the 1/v-dependence of the 1 H(n, γ) cross section.

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CHAPTER 3. NEUTRON DETECTION
It is worth noting that pile-up backgrounds actually have time dependence in theform of (1 − e−t/τcapture ) as shown in Figure 3.38. This is because neutrons passing
through the scintillator may initiate an event trigger, preventing their captures fromleaking into the subsequent event. This leads to overestimation of B, which cannot
be reproduced using a ‘discrete’ event generator alone.

100

Signal n-capture
BG n-capture
4 MeV Am/Be
Other

10 1
10 2
10 3
10 4
0

100
200
300
400
Time from event trigger [µs]500

BG n-captures / event / 50 µs

Signal candidates / event / 5 µs

Figure 3.37: Fitted signal candidate time distribution for SK-VI Am/Be data, withsource positioned at the tank center.

0.004
0.003
0.002
0.001
0.000

0

100
200
300
400
Time from event trigger [µs]500

Figure 3.38: Comparison of time distributions for signal candidates between Am/Be
data and MC (left), and the time distributions showing only the MC backgroundneutron captures from pile-ups (right). The MC signal neutron captures correspondto neutrons accompanying a 4.438 MeV gamma-ray.

151

3.3. Validation with Am/Be neutron source

3.3.6

Systematic uncertainties in simulation model

We compared the estimated neutron tagging efficiency derived from both Am/Be
data and MC simulation. For data estimation, only statistical uncertainty wasconsidered. All major systematic uncertainties were propagated to our simulation
model prediction. The propagated uncertainties were evaluated by contrasting thetrue neutron tagging efficiencies obtained from a nominal model, which is the sameas neutron particle-gun setup in skdetsim described in Section 3.3.3, with thosefrom a varied model. Below, we provide a description of each source of systematic
uncertainty and its influence on the signal efficiency estimate.

Neutron kinetic energy
The Am/Be neutron source emits fast neutrons with kinetic energy on the order ofO(1) MeV. In this energy range, the inelastic alpha emission process, specifically16O(n, α)13 C, becomes non-negligible as shown in Figure 3.39. This alpha emission

is not visible in the detector and competes with the process of elastic collisions withfree protons, which serves to moderate neutrons.

Figure 3.39: Cross sections of 16 O(n, α)13 C and 1 H(n, γ)2 H reactions as a functionof incident neutron kinetic energy, in the range of 0-20 MeV. The data is taken from
ENDF/B-VII.1 [31] nuclear data.

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CHAPTER 3. NEUTRON DETECTION
We assume a ±0.5 MeV uncertainty in the standard kinetic energy distributionof Am/Be neutrons that accompany a 4.438 MeV gamma-ray. This is based on acomparison between the standard ISO distribution [135] and the recently measured
spectrum by H. Ito et al. [137], as shown in Figure 3.40. This 0.5 MeV uncertainty
in the neutron kinetic energy distribution propagates to the fraction of neutrons
undergoing alpha emission and consequently to the neutron tagging efficiency estimate, as outlined in Table 3.10.

Figure 3.40: Comparison of Am/Be neutron kinetic energy distributions associatedwith the 4.438 MeV gamma-ray: Standard ISO distribution (black) and H. Ito etal. (red, [137]) (reprinted from [136])

MC neutron KE
Nominal(ISO-8529)
Shifted(-0.5 MeV)
Fractional change

SK-IV

SK-V

SK-VI

SK-VI (R)29.80 ± 0.16

26.09 ± 0.15

57.90 ± 0.17

39.55 ± 0.17

30.09 ± 0.16

26.47 ± 0.15

58.66 ± 0.17

40.11 ± 0.17

1.0%

1.5%

1.3%

1.4%

Table 3.10: The true neutron tagging efficinecies in percentage for nominal MC and‘shifted’ MC, where the neutron kinetic energy distribution is shifted by 0.5 MeV.

153

3.3. Validation with Am/Be neutron source
Detector modeling parameters
The detector modeling parameters including the relative gain and dark rates of individual PMTs, water transparency, and the top-bottom asymmetry (TBA) parameterreflecting the asymmetry in the mean PMT hit rate between the top and bottom of
the tank, undergo gradual yet continuous changes over time. These parameters areautomatically monitored on a run-by-run basis using randomly triggered data andthe isotropic Xe light source fixed at the tank center. MC simulations were generatedusing these measured parameters, allowing us to observe how efficiency is affectedby the varying parameters in comparison to the nominal MC. It is worth noting that
not all runs were simulatable, as background PMT hits were only recorded during
specific periods, typically correlated with the T2K accelerator neutrino schedule.
Figure 3.41 displays the true neutron tagging efficiency values obtained from runby-run MC simulations as a function of the Run number. In Figure 3.42, histogramsof the efficiency values are shown with varying detector modeling parameters measured in each run, categorized by each SK phase. The standard deviation of theefficiency distributions was 2.0% for SK-IV, 2.8% for SK-V, 1.0% for SK-VI, and1.3% for SK-VI (R), relative to the average of each phase. These values were considered as the 1σ uncertainty resulting from the varying detector conditions.

Figure 3.41: The true neutron tagging efficiencies obtained from run-by-run MC(blue, showing INIT vertex mode only), compared with the average for each SK
phase (cyan) and the efficiency derived from the nominal MC (red).

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CHAPTER 3. NEUTRON DETECTION

Figure 3.42: The distributions of true neutron tagging efficiencies derived from runby-run MC simulations (blue histogram), categorized by each SK phase. The verticalaxes represent the number of runs used in the simulation, with 100 runs in total foreach phase. The cyan line represents the average of the blue histogram, while thered line represents the efficiency derived from the nominal MC.
Overall PMT response scaling factor (COREPMT)
The overall PMT response scaling factor is the final scaling factor to the PMT
response that is introduced to match the simulated PMT charge with the observed.
It is not measured run-by-run, but fitted to minimize the data-MC discrepancy inthe charge per track length distributions of penetrating cosmic ray muons for each
SK phase. The 1σ uncertainty of this factor was assumed to be 1% based on [34].
Table 3.11 summarizes the fractional changes due to 1σ shift in this factor, which areassumed as the propagated uncertainties to the neutron tagging efficiency estimate.

COREPMT+1σ
Nominal-1σ
Fractional change

MC neutron tagging efficiency [%]
SK-IV
SK-V
SK-VI
SK-VI (R)
31.13 ± 0.16 27.73 ± 0.16 58.30 ± 0.17 39.94 ± 0.17
30.89 ± 0.16 27.06 ± 0.16 57.90 ± 0.17 39.55 ± 0.17
30.60 ± 0.16 26.73 ± 0.16 57.52 ± 0.17 39.17 ± 0.17
0.9%
1.8%
0.7%
1.0%

Table 3.11: Fractional changes in efficiency with shifted scaling factor.

155

3.3. Validation with Am/Be neutron source
Bias due to calibration setup
As discussed in Section 3.3.2, the calibration setup, including the pile-up backgrounds from the Am/Be neutron source and the BGO scintillator surrounding it,
can impact the estimate of neutron tagging efficiency. Conducting the analysis
without this setup is challenging, so we rely on simulations to estimate its effect.
Additionally, generating multiple sets of simulations with different models — varied
source neutron emission rates or different cross section datasets — using a complete
Am/Be MC simulation with both SKG4 and the pile-up simulator is an extremely
time-consuming process. Therefore, we conservatively estimate the uncertainty bycomparing only the existing nominal MC results with varying source positions.
We define the efficiency “bias factor” due to calibration setup as:
Bias factor =Estimated signal efficiency in full Am/Be MC
True signal efficiency in neutron particle-gun MC(3.11)For all source positions in each SK phase, the uncertainty of the bias factor was
estimated as the quadratic sum of the mean of the MC statistical errors and thestandard deviation of the bias factors for different source positions.
The two sets of simulations were all conducted in SKG4 (Geant4.10.5.p01) withthe same setup, including hardware and water modeling parameters, except forthe calibration setup. The estimated bias factor (93.3%) for SK-VI was slightly
larger than the value for the pure water phases (96.8% for SK-IV, 95.6% for SKV), as gamma-rays from 8 MeV Gd(n, γ) reactions introduced an additional sourceof fake event triggers that mimic scintillation due to 4.438 MeV Am/Be gammarays. The RECO vertex mode for SK-VI showed even larger bias factor, possibly
due to scintillator interference with the signals, affecting the performance of signal
vertex reconstruction. These results are summarized in Figure 3.43. No significant
vertex dependence was expected by the simulations, as shown in Figure 3.44. The
estimated uncertainties of the bias factors for each SK phase were 6.9% for SK-IV,
4.6% for SK-V, 1.1% for SK-VI, and 1.2% in the SK-VI RECO vertex mode.

156

CHAPTER 3. NEUTRON DETECTION

Am/Be MC vs. n-gun MC (Geant4.10.5.p01)
SK-IV
SK-V

Bias factor

1.05

SK-VI
SK-VI (R)

1.00
0.95
0.90
Calibration run with distinct source position

Estimated Am/Be bias factor

Figure 3.43: MC-predicted bias factors (points with MC-statistical errors) and theconservatively evaluated uncertainties (colored shade), resulting from the Am/Be
calibration setup. The dashed lines represent the mean of the predicted bias factors
for each SK phase.

SK-VI Am/Be MC (Geant4.10.5.p01)

1.00

Vertex mode
INIT-NEUTRON
RECO-CAPTURE

0.98
0.96
0.94
0.92
0.90
0.88

0

200

400 600 800 1000 1200 1400 1600
Am/Be source distance to ID wall [cm]Figure 3.44: The MC-predicted bias factors plotted against the distance from thesource to the ID tank wall, categorized by the two different vertex modes. The
dashed lines correspond to the linear functions that best fit the simulations in eachmode, while the shaded areas indicate the 1σ prediction intervals. The determinedslopes aligned closely with zero, with a deviation of within 1.1σ.

157

3.3. Validation with Am/Be neutron source
Neutron capture ratios on H and Gd
The detection efficiencies for 1 H(n, γ) and Gd(n, γ) signals show significant differences. Specifically, the efficiency for 1 H(n, γ) signals is approximately 20%, while
the Gd(n, γ) signal efficiency is around 95%. This makes the H/Gd-capture ratio a
crucial factor in determining the overall signal detection efficiency. In the nominal
MC simulation using skdetsim with 0.0110 w% input Gd concentration and neutron
capture cross sections from ENDF/B-VII.1 [31], the thermal neutron capture ratios
on Gd and H were found to be 52 : 48. To evaluate the uncertainty in this ratio,
we compared the MC-simulated ratio with analytical predictions based on the input
Gd concentration and cross sections. We also considered the ratios estimated basedon the scales of 1 H(n, γ) and Gd(n, γ) signal peaks in the signal energy distributions
observed in Am/Be data and MC.
First, we estimated the neutron capture ratio on H analytically using the following equation:nH gwH (T )σH,2200 m/sPrH ≈ii̸=H ni gw (T )σi,2200 m/s(3.12)

where ni is the number density of the ith isotope, σi is the neutron capture crosssection of the ith isotope, v is the neutron speed, ρ is the neutron density, gw is
the Westcott g-factor (the ratio between the Maxwellian average cross section andthe cross section evaluated at thermal neutron energy of 0.0253 eV), and T is thetemperature. The Gd(n, γ) ratio rGd can be approximated as 1 − rH .
In the actual calculation, we only considered three isotopes:
157

1

H, 155 Gd, andGd, as the fraction of neutron captures on the other isotopes such as 16 O and 32 S

was negligible (less than 0.1%). The number densities were estimated based on theactual loaded weight of Gd2 (SO4 )3 · 8(H2 O) powder for SK-VI [51]. The estimatednumber densities, along with the relevant cross section and the g-factors for thesethree isotopes, are summarized in Table 3.12.

158

CHAPTER 3. NEUTRON DETECTION

1

H
Gd
157
Gd
155

Number density [/cm3 ]6.68×1022
6.22×1016
6.57×1016

Cross section [barns]
0.332015
60740.1
252928Westcott g-factor
1.001
0.84390.8528

Table 3.12: Values used for estimating neutron capture ratios in SK-VI. Cross sections were obtained from ENDF/B-VII.1 [31] for a neutron kinetic energy of 0.0253
eV, and the Westcott g-factors (T = 293.6 K) were sourced from [146].
As for uncertainties, we assumed a 0.5% uncertainty for the number density of1

H and 0.8% for those of the gadolinium isotopes. For cross section uncertainty, we

adopted the standard values assigned in the ENDF/B-VII.1 [31] evaluation: 2.553%
for 1 H(n, γ), 3% for 155 Gd(n, γ), and 4% for 157 Gd(n, γ) reactions. Given recent
measurements with smaller cross sections [147, 148] that conflict with the evaluatedcross sections in ENDF/B-VII.1 [31], we may need to consider larger uncertainties,
potentially up to around 10% for Gd isotopes. In addition, we accounted for a 1%
uncertainty in the g-factor, which includes considerations of variations in nuclear
data evaluation and the SK water temperature. The resulting estimated neutron
capture ratio on 1 H with SK-VI Gd concentration was (56.2 ± 1.5)%.
The 1 H(n, γ) ratio rH can be experimentally determined using Am/Be data,by analyzing the distribution of the number of PMT hits (NHits), as illustrated in
PnData ≥5 Data MC
Figure 3.45. The chi-squared statistic χ2 (rH ) = i i(ni −ni (rH ))2 /nMC
i (rH )
— nData
represents the observed signal counts in the ith data NHits bin, and nMCii (rH )

represents the expected signal counts in the ith NHits bin scaled by rH — wassummed for all source positions. The χ2 -minimizing value of rH was (56 ± 3)%(Figure 3.46). This result is consistent with the analytically predicted rH of (56.2± 1.5)%. When fitting only the H(n, γ) peak (NHits∈ {6, 7, 8, 9}), which is lessinfluenced by the shape uncertainty of the Gd(n, γ) NHits distribution, a similar
result of (59 ± 5)% was obtained.

159

3.3. Validation with Am/Be neutron source

Figure 3.45: Comparison of NHits distributions for different source positions using
Am/Be data (black) and MC simulations (red) with rH set to 50% (left) and 56%(right). In the plots, colors correspond to various components: true Gd(n, γ) signals(pink), H(n, γ) signals (cyan), 4.438 MeV gamma-rays (green), and background
PMT hits (gray). The histograms are normalized based on the number of events.

Figure 3.46: The distribution of ∆χ2 as a function of rH . The shaded gray area
represents the 1σ uncertainty of the fitted rH .

160

CHAPTER 3. NEUTRON DETECTION
The MC-predicted rH is 48%, showing a significant discrepancy with both theanalytically predicted and measured rH as shown in Figure 3.47. It seems like the MC
simulation is giving an overestimated Gd-capture ratio, likely because the thermal
motion of water molecules has not been accurately modeled1 . Assuming an rH valueof 48% with uncertainties ranging from +8.2% (including both data-MC difference
and the cross section uncertainties) to -1.5% (cross section uncertainties only), theresulting relative change in SK-VI neutron tagging efficiency is +2.1
−9.6 % (INIT) and+3.1−14.7 % (RECO).

Figure 3.47: Comparison between the MC-predicted rH (red) and the analyticalpredictions (blue) as a function of input Gd concentration. The values of Gd concentration and rH estimated based on Am/Be data are indicated by dashed graylines, with the shades representing the corresponding uncertainties.

1

In Geant4, it seems that hydrogen is treated as a free proton without considering the molecularmass of water. This results in hydrogen moving 18 times faster in Geant4, decreasing the probabilityof neutron captures on hydrogen. Disabling the simulation of thermal motion in Geant4 changes
the MC predictions to align with both the analytical prediction and observed results. As of thetime of writing, this solution has not been officially adopted or integrated into the current analysis.

161

3.3. Validation with Am/Be neutron source
Gamma-ray emission model for Gd(n, γ) reaction
The ANNRI-Gd model which is used in the nominal MC demonstrates the closest
agreement with our data, as illustrated in Figure 3.49, although it is not without
its imperfections. To estimate the impact of the gamma-ray emission model to theneutron tagging efficiency, we compared the the nominal model with an alternative,
GGARNET [113]. GGARNET uses a different photon strength function model usedin simulating gamma-rays corresponding to de-excitation within quasi-continuum
levels of 156/158 Gd∗ .
Specifically, while ANNRI-Gd considers only the two most dominant electric
dipole transition peaks in the de-excitation of 156/158 Gd∗ isotopes using the standard Lorentzian (SLO) model — a sum of two Lorentzian peaks, the simplest
model —GGARNET employs an extended model known as the enhanced generalized Lorentzian (EGLO) model [149]. GGARNET incorporates magnetic and higher
order transitions, but it also introduces more empirical parameters that require dedicated tuning. The two models exhibit distinct gamma-ray energy spectra, while the
ANNRI-Gd model aligns slightly better with the Ge detector measurements [113], asshown in Figure 3.48. Both models sample gamma-rays from discrete de-excitationsin the same way, based on the observed multiplicities and discrete energy peaks.
K. Hagiwara et al.
Data / MC [1/200 keV]Counts [1/10 keV]

PTEP 2019, 023D01

106
105
104
103

Data
SLO model

102

EGLO model

10
0

4.5

SLO model

4.0

EGLO model

3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0

0

1000 2000 3000 4000 5000 6000 7000 8000
Energy [keV]

Fig. C.1. Comparison of the measured M1H1 energy spectrum from thermal 157 Gd(n, γ ), i.e., data for periphcrystalIndividual
3 (black) andgamma-ray
the energy spectra
generatedby the(left)
SLO and
EGLO models (left),
and theFigureeral3.48:
energyspectraof GGARNET(EGLO,
corresponding ratios of DATA/MC (right).

blue) and ANNRI-Gd (SLO, orange) models, and the comparison with Ge detector
array measurements of single gamma-ray events from 157 Gd(n, γ) reactions (right,
Here,fromk0 is an[113]).
empirical enhancement factor and Eγ 0 is a reference energy. A value for the temperature
reprinted

162

Ti (Tf ), which can be a constant or a function of the initial (final) state excitation energy, is usuallyobtained from the NLD model [46]. The term proportional to 1/Ei3 in Eq. (C.1) ensures "i (Ei , 0) = "i .
For a comprehensive summary of PSF models not only for E1 but also for M1 and E2 transitions,which are not explicitly included in our spectrum model, the reader is referred to Ref. [46].
We generated the energy spectrum for Gd(n, γ ) using the EGLO model, using T = 0.4 MeV andk0 = 2. The energy spectra generated using the EGLO model are compared with the data and thespectrum by the SLO model in Fig. C.1 (left). The data/MC ratio for both the models is shown on
the right. Both appear consistent with the data.
The contribution of the prominent discrete γ rays to the total γ -ray energy spectra is less than 7%
in the thermal 157 Gd(n, γ ) reaction, implying that the majority of γ rays are produced via cascade

Downloaded from https://academic.oup.com/ptep/article/2019/2/023

1000 2000 3000 4000 5000 6000 7000 8000
Energy [keV]

5.0

CHAPTER 3. NEUTRON DETECTION

Signals / event

Am/Be at tank center
SKCNTL-GDMODEL
0: Geant4 default
1: GLG4Sim
2: GGARNET
3: ANNRI-Gd
4: Geant4 Photoevaporation

0.03
0.02
0.01
0.00

0

20
40
60
80
Number of PMT hits in 14 nsec

100

Figure 3.49: The NHits distribution of signal candidates taken from Am/Be data
(black) with the source positioned at tank center compared with corresponding
Am/Be MC with various Gd(n, γ) gamma-ray emission models. The neutron capture
ratio on Gd was tuned to 44%.
As GGARNET tends to produce fewer energetic gamma-rays (>2 MeV, see
Figure 3.48) that are more likely to scatter electrons over the Cherenkov threshold,the signal NHits distribution shifts to the left in comparison to the ANNRI-Gd
model, as illustrated in Figure 3.49. This leads to a reduced true neutron tagging
efficiency in SK-VI with GGARNET: (58.62 ± 0.17)% in INIT vertex mode and(36.75 ± 0.17)% in RECO vertex mode. In contrast, with ANNRI-Gd, the efficiencies

were (60.17 ± 0.17)% (INIT) and (41.10 ± 0.17)% (RECO). We assigned fractionalchanges of 2.6% for the INIT vertex mode and 11.8% for the RECO vertex mode
as uncertainties. The greater uncertainty associated with the RECO vertex modesignifies the fact that vertex reconstruction performance is heavily dependent on
the gamma-ray emission model.

Neutron cross sections
The impact of cross section uncertainty appeared to be minimal. When comparing
different cross section datasets, such as ENDF/B-VII.1 [31] with JEFF-3.3 [150], we
observed a fractional difference on the order of 0.1% in the neutron tagging efficiency.
As a result, it was not factored into the overall uncertainty estimate.

163

3.3. Validation with Am/Be neutron source
Neutron capture time constant
The neutron tagging efficiency estimate in Equation 3.10 relies on the width of thecandidate selection time window ([18, 534] µs for SK-IV/V, [3, 534] µs for SKVI, starting from the event trigger). The neutron capture time constant τcapture in
Equation 3.10, which depends on neutron capture cross sections, can influence theefficiency estimate. The observed τcapture in Am/Be data was compared with ananalytical prediction given by:1

τcapture ≈ P(3.13)

i
i ni gw (T )σi,2200 m/s (2200 m/s)

The parameters used were obtained from Table 3.12. The weighted mean of theobserved time constants was 200.35 ± 3.74 µs for the pure water phases and 116.81± 0.27 µs for the Gd-loaded phase. The SK-VI result, obtained with the RECO vertex

mode, was 116.85 ± 0.32 µs, consistent with the INIT vertex mode. These valuesaligned well with the predicted thermal neutron capture time constants 204.69 ±5.33 µs for pure water and 114.87 ± 2.47 µs for 0.011 w% Gd-loaded water.

While the skdetsim simulation showed a consistent capture time for pure waterphase, it suggested a slightly smaller capture time constant of 112.4 µs for SK-VI.
However, the 4 µs discrepancy in neutron capture time constant led to a change inthe efficiency estimate of less than 0.1%, and therefore was neglected.

260
240

Data
Data weighted mean
Prediction (ENDF/B-VII.1)

220
200
180
Am/Be data Run

150
Capture time constant [µs]Capture time constant [µs]SK-IV and SK-V

140
130

SK-VI
Data
Data weighted mean
Prediction (ENDF/B-VII.1)

120
110
100

Am/Be data Run

Figure 3.50: Neutron capture time constants obtained from the time distributions ofsignals processed with the INIT vertex mode, in SK-IV and SK-V (left), and SK-VI(right).

164

CHAPTER 3. NEUTRON DETECTION
The major sources of uncertainty in estimating neutron tagging efficiency using
the Am/Be neutron source discussed so far are outlined in Table 3.13. For the purewater phases, the most significant source of uncertainty arises from calibration setup
bias, whereas in the Gd-loaded phase, the largest uncertainty stem from neutron
capture ratios and the Gd(n, γ) gamma-ray emission model.
SourceAm/Be neutron kinetic energy
Detector modeling parameters
Overall PMT response scale
Bias due to calibration setup
Neutron capture ratios
Gd(n, γ) gamma-ray emission model

SK-IV
0.5%
2.0%
0.9%
6.9%
-

SK-V
0.9%
2.8%
1.8%
4.6%
-

SK-VI
0.5%
1.0%
0.7%
1.1%

SK-VI (R)
0.6%
1.3%
1.0%
1.2%

+2.1%
−9.6%

2.6%

+3.1%
−14.7%

11.8%

Total

7.3%

5.7%

+3.8%
−10.1%+12.4%−18.9%

Table 3.13: Evaluated major uncertainties to the neutron tagging efficiency calibration with the Am/Be neutron source.

165

3.3. Validation with Am/Be neutron source

3.3.7

Data-MC comparison

To validate the neutron capture signal detection efficiency on real data, we compared
its estimates on Am/Be neutron source data with those obtained from MC simulations generated using skdetsim in a particle-gun setup. Since skdetsim operates inGeant3 and does not incorporate calibration setup (as implemented in Geant4), we
did not apply Cuts #3 and #4 to skdetsim MC, as these cuts rely on calibration
setup simulation. Instead, for each source position, we multiplied the estimated“bias factor”, derived by comparing simple neutron particle-gun simulations withfull Am/Be simulations using SKG4 (as detailed in Section 3.3.6), with the true neutron tagging efficiency in skdetsim. This approach allowed us to account for thebias effect stemming from calibration setup and make a meaningful comparison between skdetsim and Am/Be data.

Figure 3.51 illustrates the estimated neutron tagging efficiency of INIT vertex
mode for both Am/Be data and neutron simulation with the neutron source atthe tank center. The data points are plotted against corresponding measurement
dates. The observed efficiencies were consistent with the expected efficiencies within
the total uncertainties for each SK phase, differing by approximately 10% at most.
Notably, the evaluated efficiencies in data display variations over time due to fluctuations in detector modeling parameters.
We can also take an average of multiple measurements at the same position.
Figure 3.52 (in the INIT vertex mode) and Figure 3.53 (in the RECO vertex mode)compare the weighted average of the efficiency estimates for each source position withthe expected values from the simulation. SK-IV and SK-V demonstrate agreement
between data and MC results, while for SK-VI in both vertex modes, a significant
difference is observed across all source positions. This discrepancy is likely attributed
to an approximately 10% larger Gd-capture ratio in our simulation. The average
Data/MC ratios can be found in Table 3.14.

166

CHAPTER 3. NEUTRON DETECTION

Figure 3.51: Neutron tagging efficiencies estimated for Am/Be neutron source data(in black) with the source positioned near the tank center, plotted against the measurement date. The expected efficiencies obtained from skdetsim neutron particlegun MC simulations, adjusted by the corresponding calibration bias factors estimated with SKG4, are shown in red. The red shaded areas for the MC simulations
incorporate major systematic uncertainties evaluated in Section 3.3.6.

167

SK-IV

0.60
0.55
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15

SK-VI

7

8

9

10

11

12

13

14

15
16 mean17 18
Weighted

19Standard
20 error21 (stat
22 only)23(0,-1,-13)(0,-1,0)(0,-1,15)(0,-1,13)(-4,-1,12)(-4,-1,0)(-4,-1,-12)(-12,-1,-12)(-12,-1,0)(11,-1,-12)(11,-1,0)(11,-1,12)(0,-1,0)(0,-1,12)(-4,-1,0)

6(-4,-1,-12)

5(-12,-1,12)

4(0,-1,-12)

3(0,-12,12)2(0,-12,0)1(0,-12,-12)

0(0,-1,15)

Am/Be data average (stat)MC estimation (stat+syst)(0,-1,0)

1.2
1.1
1.0
0.9

SK-V

(0,-12,0)

Data / MC

Estimated efficiency

3.3. Validation with Am/Be neutron source

Source position [m]

SK-VI

0.60
0.55
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15
5

6

7

8

9 Weighted
10mean 11(-4,-1,-12)(-12,-1,-12)(-12,-1,0)(11,-1,-12)(11,-1,0)(11,-1,12)(0,-1,0)(0,-12,-12)

Source position [m]1213 (stat only)
14Standard
error(-12,-1,12)

4(0,-1,-12)

3(0,-1,12)2(0,-12,12)1(0,-12,0)0(-4,-1,0)

1.1
1.0
0.9
0.8

Am/Be data average (stat)MC estimation (stat+syst)(-4,-1,12)

Data / MC

Estimated efficiency

Figure 3.52: The estimated neutron tagging efficiencies of the INIT vertex mode for
Am/Be data (black) and skdetsim neutron MC with SKG4 calibration bias factors
applied, plotted against the neutron source position.

Figure 3.53: The estimated neutron tagging efficiencies of the RECO vertex mode for
SK-VI Am/Be data (black) and skdetsim neutron MC with SKG4 calibration bias
factors applied, plotted against the neutron source position.

168

CHAPTER 3. NEUTRON DETECTION
In our primary analysis, we utilize the obtained average data/MC efficiency ratios
to account for the apparent disparity between calibration data and MC-estimatedefficiencies. Especially for SK-VI, applying this correction allows us to disregard thesubstantial uncertainty in the Gd-capture ratio arising from simulation errors.

SK phase
SK-IV
SK-V
SK-VI
SK-VI (R)

Data / MC efficiency ratio
1.10000.97340.90860.8770

Uncertainty
Statistical Systematic
0.06840.08030.04430.05550.00690.03450.00700.1087Total
0.10550.07100.03520.1089

Table 3.14: Weighted mean and its statistical error for the data/MC efficiency
ratios. These ratios and their uncertainties are utilized to adjust the signal efficiency,
initially derived from the MC simulation, in our neutron multiplicity measurement(refer to Chapter 4). When incorporating systematic uncertainty into the analysis,
we exclude the Gd-capture ratio uncertainty related to potential inaccuracies inmodeling molecular thermal motion. Instead, we include uncertainties associatedwith neutron capture cross-section.

1693.4. Michel electron separation

3.4

Michel electron separation

Michel electrons, the decay products of stopped muons, have an energy range of
0-50 MeV and occur within a few µs after muon production. In SK-VI, where theneutron capture time is faster, this time range overlaps with the neutron capture
signal search window, which begins 3 µs after the event trigger. Michel electrons are
more likely to be classified as signals because they tend to produce a much higher
NHits count, typically in the hundreds, compared to typical background events.
To distinguish Michel electrons in SK-VI, a straightforward rectangular cut was
employed based on the number of PMT hits and the time elapsed since the event
trigger. Among the candidates with a neural network output greater than 0.7,
those satisfying NHits > 50 and a time interval of less than 20 µs from the event
trigger were identified as Michel electrons. The remaining candidates were classified
as neutron captures. Remarkably, this straightforward cut proved to be highly
effective in cleanly distinguishing Michel electrons from neutron capture signals andvice versa, as illustrated in Figure 3.54.

Figure 3.54: The neural network output distributions (left) for all candidates selectedin MC-simulated atmospheric neutrino events, and the 2D scatter plot of the signal
candidates in the plane of time and energy. The red arrows represent cuts forseparating neutron captures from Michel electrons.

170

CHAPTER 3. NEUTRON DETECTION

3.4.1

Validation with cosmic ray muon data

Cosmic ray muons continuously penetrate the overburden and reach the detector.
Some of these muons come to a stop inside the ID and generate Michel electrons.
This cosmic ray muon data serves as valuable calibration data for testing the Michelelectron separation algorithm.
Stopping muon data was collected throughout SK-VI. The major reduction steps
included requiring a large charge deposit in the ID to reject low-energy radioactivity,
checking for OD PMT hits at the entrance and ensuring no other OD PMT hit
clusters due to exiting particles to reject penetrating muons.
For each stopping muon event, the muon’s entry point, momentum, and direction
were reconstructed using an algorithm described in [28]. The muon range was determined by interpolating from tabulated data of muon Continuous Slowing Down
Approximation (CSDA) ranges in water [151]. The stopping point of each muon
was estimated using this information, and the estimated 1σ vertex resolution forthis estimation was approximately 100 cm. The muon’s estimated stopping vertex
for each event was utilized for the initial TOF correction before the candidate selection stage. Candidates were then selected and classified according to the procedure
outlined in Sections 3.2 and 3.4. MC simulations using skdetsim were generated
based on the observed distribution of reconstructed muon momenta.
To validate the Michel electron separation cuts described in Section 3.4, eventswith an estimated muon stopping point farther than 3 m from the ID tank wall wereselected. This ensured that Michel electrons were at least partially contained within
the ID and prevented corner-clipping muons —penetrating muons that typicallyhave a reconstructed stopping point close to the top corners of the tank wall — fromcontaminating the sample. Figure 3.55 displays the time distribution of candidates
classified as Michel electrons, with the characteristic muon decay time accuratelyaligning with the expected value of 2.027 µs in water.

1713.4. Michel electron separation

Figure 3.55: The observed muon decay time, fitted using a function of the form
Ae−t/τ + B, where τ represents the muon decay time constant.
Figure 3.56 compares the features of Michel electron signals observed in stopping
muon data with those from MC simulations. By examining the composition ofMC-simulated signals within the data histograms, we can estimate the purity ofMichel electrons in the selected signals. To find the purity in the selected candidates
classified as Michel electrons, we minimized the following metric [152]:− ln L(ae ) ≈

X 

bins
nMC >0nData
nMC (ae ) − nData + nData ln
nMC (ae )(3.14)

Here, ae is the Michel electron scaling parameter to be estimated, and L(ae ) is
the Poisson likelihood of the observation as a function of ae . nMC (ae ) is the scaledeMC bin value calculated as ae neMC + (1 − ae )nBG
MC , where nMC is the MC true Michelelectron count and nBG
MC is the MC true background count in each bin. nData is theobserved count of candidates classified as Michel electrons in each bin.

The distribution of the metric as a function of ae is shown in Figure 3.57. The
minimizing ae was multiplied by the MC true Michel electron purity to estimate thesignal purity in the stopping muon data. The Michel electron detection efficiency
was estimated by assuming the same Michel electron production ratio per event as
in the MC simulation. The estimated Michel electron signal efficiency and purity
for both stopping muon data and MC simulation are summarized in Table 3.15.

172

CHAPTER 3. NEUTRON DETECTION

Figure 3.56: Feature distributions for candidates classified as Michel electrons, observed in the SK-VI stopping muon data (blue) and its corresponding MC simulation:
MC true Michel electron signals (olive) and backgrounds (gray). Both the data andMC histograms are normalized by area.

173

3.4. Michel electron separation

Figure 3.57: The change in the metric L(ae ), defined in Equation 3.14, with respect
to the Michel electron scaling parameter ae . The shaded regions indicate the 1σ
uncertainty in the fitted ae .

Efficiency
PurityData [%]
98.4 ± 1.3
98.7 ± 0.5

MC [%]
98.8 ± 0.8
99.3 ± 0.7

Table 3.15: Estimated Michel electron signal efficiency and purity in both the stopping muon data and its corresponding MC simulation, with statistical uncertainties.

The estimated signal efficiency and purity for Michel electrons in both the stopping muon data and its MC simulation were exceptionally high, exceeding 98%,
and showed good agreement. This result demonstrates the effectiveness of the twodimensional cut on signal energy and time in distinguishing Michel electrons fromneutron captures. Consequently, we can confidently employ this technique to countneutron captures in neutrino events that involve both Michel electrons and neutron
captures, with minimal concern for Michel electron contamination.

174

CHAPTER 3. NEUTRON DETECTION

3.5

Effect of PMT afterpulsing on signal detection performance in SK-VI

Each hit PMT has a chance of experiencing an afterpulse, typically occurring approximately 10-20 µs after the initial PMT hit (main pulse), with a likelihood ofaround 0.1%. Due to the low probability, it is usually of minimal concern. However,in cases of highly energetic signals where multiple PMTs are triggered with significant charge deposits, this concern becomes more pronounced. The time frame ofthese afterpulses coincides with the widened signal search window in SK-VI, whichstarts 3 µs after the event trigger. We conducted a qualitative assessment of theirimpact on the signal detection efficiency and the false positive rate.
We compared the performance of signal detection on Multi-GeV (electron-equivalentvisible energy larger than 1.33 GeV) atmospheric neutrino events, both fully contained and within the fiducial volume, with their corresponding MC simulation. We
simulated afterpulses for each hit PMT, using a uniform afterpulse hit probabilityof 0.56%, along with a Gaussian-shaped timing probability with a mean of 14 µs
after the initial hit and a standard deviation of 2.1 µs. Multi-GeV neutrino events
typically has O(103 ) PMT hits, or more than O(104 ) photoelectrons in total. If afterpulsing does indeed impact signal detection performance, this event sample would
be the most significantly affected.
Figure 3.58 displays the time distributions of all candidates along with thoseclassified as neutron captures and Michel electrons, comparing the data to the MC.
The afterpulse peak is clearly visible in the time distribution of all candidates, which
are mostly due to accidental coincidences of background PMT hits including afterpulses. However, this peak disappears after the neural network’s signal classification
stage. This indicates that the neural network effectively filters out accidental candidates associated with PMT afterpulsing. Since the contamination from Michelelectrons or PMT afterpulsing seems to be negligible, we can confidently start thesignal search from 3 µs after the event trigger, as opposed to the 18 µs used inprevious studies on neutron detection [34, 26, 28, 110, 153].

175

3.5. Effect of PMT afterpulsing on signal detection performance in SK-VI

Multi-GeV (FCFV)

Density (area-normalized)

0.025

Noise
Michel e

0.020

Gd(n, )
H(n, )

0.015
0.010
0.005
0.000

0

5

10
15
20
Time from event trigger [µs]

25

30

25

30

Multi-GeV (FCFV)NoiseMichel e

Density (area-normalized)

0.06
0.05

Gd(n, )
H(n, )

0.04
0.03
0.02
0.01
0

5

10
15
20
Time from event trigger [µs]Density (area-normalized)

Multi-GeV (FCFV)
0.4

Noise
Michel e

0.3

Gd(n, )
H(n, )

0.2
0.1
0.0

0

5

10
15
20
Time from event trigger [µs]2530

Figure 3.58: The time distributions of all selected candidates (top) and those classified as neutron captures (center) and Michel electrons (bottom), comparing theMulti-GeV atmospheric neutrino data (black) and the corresponding MC simulation(red). Both the data and the MC are normalized by area.

176

Chapter 4
Neutron multiplicity measurement
In this chapter, we delve into the specifics of data reduction, MC simulation, andneutron detection schemes tailored to this analysis. Next, we evaluate data quality by examining signals from both atmospheric neutrino interactions and delayedcoincident neutron captures. We then review the method for estimating neutron
capture detection efficiency, which is crucial for accurately evaluating the total produced neutron captures. Based on this method, we present measurements related
to outgoing neutron multiplicity and kinematics in response to energy transferredduring atmospheric neutrino interactions.
Specifically, we studied the average neutron capture multiplicity in atmospheric
neutrino interactions by examining its dependence on the visible energy of the interaction. Neutron capture and visible energy serve as proxies for neutrons and transferred energy, respectively, as neither is directly observable in atmospheric neutrino
interactions in SK. To assess the impact of uncertainties in hadronic re-interactions
on our predictions, we compared the observed neutron capture multiplicity withdifferent interaction models. This chapter presents a comprehensive overview of ourfindings, including an analysis of systematic uncertainties.

1774.1. Data and simulation configurations

4.1

Data and simulation configurations

The data selected for analysis were obtained from Fully Contained (FC) events thatpassed all five FC data reduction stages, as described in Section 2.5.

4.1.1

Data run selection

For the pure water phases (SK-IV and SK-V), data from all “good” runs were used,
excluding calibration or “bad” runs with unusual background rates in the reduction
stages. In the Gd-loaded phase (SK-VI), data from the early phase, when Gd
concentration may not have been uniform throughout the tank volume, were notincluded in the analysis, as such non-uniformity is not reliably simulatable.
The exact cut-off date was determined based on water conductivity and themeasured neutron capture time constants with an Am/Be neutron source positioned
at different locations within the tank. Water conductivity increases with Gd-loading
due to an increase in the amount of free ions in the water. This serves as a reliable
metric to assess ion concentration uniformity throughout the tank volume. Water
conductivity at various vertical positions within the tank was measured throughout
the Gd-loading process. Figure 4.1 displays the measured conductivity at different
vertical positions and on different dates. Conductivity became maximal and uniform
throughout the tank volume only after early September 2020, approximately 1.5
months after the start of Gd-loading (or the start of SK-VI on July 14, 2020).
Figure 4.2 illustrates the measured neutron capture time constants at three different vertical positions. A faster neutron capture time indicates a larger concentrationof Gd. The position dependence is observed at the beginning of the Gd-loading. The
capture time near the top of the tank becomes lower than the other two positions
only after the mid-September measurement (September 15, 2020). Hence, to ensure
Gd concentration uniformity in the SK tank, we used SK-VI atmospheric neutrino
data taken after (including) September 15, 2020, excluding 2 months of data fromthe beginning of the Gd-loading in SK-VI.

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CHAPTER 4. NEUTRON MULTIPLICITY MEASUREMENT
The total detector livetime was 3244.4 days for SK-IV, 461.0 days for SK-V, and564.4 days for SK-VI.

Figure 4.1: Water conductivity measurements taken during the Gd-loading process
in SK-VI, plotted against vertical displacements of the measurement positions within
the ID tank volume. (Reprinted from [51])K. Abe, C. Bronner, Y. Hayato et al.

Nuclear Inst. and Methods in Physics Research, A 1027 (2022) 166248

Fig. 14. The history of the neutron capture time constant in SK in the latter half of 2020, obtained from the analysis of Am/Be source data. Three positions along the Z-coordinate
are shown here: Z=0 m (red circles), Z=+ 12 m (blue squares) and Z=*12 m (magenta triangles). The shaded area at the top of the plot indicates the neutron capture time
constant in pure water [21]. Though the July 29 and August 5 data points at Z=+ 12 m are out of range (too high), they are nevertheless consistent with the pure water shaded
region within about one sigma.

Figure 4.2: Neutron capture time constant measurements taken during and after the
Gd-loading process in SK-VI, plotted against the measurement dates. (Reprinted
from [51])Fig. B.16 shows typical temperature profiles along the Z (vertical)
coordinate of the detector during the SK-IV (2009–2018) and SK-V
(2019–2020) phases. The region at a roughly constant temperature(Z<Ì *10 m for SK-IV) indicates a convection zone, while the regionswith constant slope as a function of z indicate a steady vertical flow.
During the SK-V period there is almost no convection region inside theID, and the steady flow seems to be established at179
Z > *5 m.
The picture described above is supported by evaluating low energy
background rates – events with a few MeV – in the detector. The
dominant source of the low energy events at Super-Kamiokande isradon (Rn) emanated from the detector components. This Rn dissolves
into the SK water and is then carried to various locations by water flow.
If there is convection, Rn from the PMTs and their associated hardware

4.1. Data and simulation configurations

4.1.2

Simulation configuration

The MC simulation of atmospheric neutrino events in the ID volume, with a detector
livetime of 500 years, was generated for each phase of SK, following the procedure
outlined in Section 2.3. In terms of used software, NEUT 5.4.0 [154] was used for
simulation of neutrino interaction in water based on the HKKM 2011 atmospheric
neutrino flux model [55], without considering the oscillation. skdetsim v13p90 and
v14 were used for detector simulation in SK-IV/VI and SK-V, respectively. The
events were weighted based on the three-flavor oscillation probability following thedetails described in Section 2.3.7. The oscillation probability was calculated using
the best-fit oscillation parameters obtained in the previous oscillation analysis [116],with constraints on θ13 with the value evaluated from reactor neutrino experiments[155]. The values used to calculate oscillation weights on an event-by-event basis in
this analysis are outlined in Table 4.1.

Parametersin2 θ12
sin2 θ13
sin2 θ23|∆m231,32 |2|∆m223 |2
δCP

Value
0.307
0.021
0.425
−5
7.53×10 eV2
2.53×10−3 eV2
3.14

Table 4.1: Oscillation parameters [116] used in this analysis to calculate oscillation
weights on an event-by-event basis. Normal mass ordering was assumed.

The parameters used to model the detector, such as the overall PMT response
scaling (COREPMT) and the light attenuation factor in water, were fine-tuned for SKIV and V primarily using cosmic-ray muon data, as explained in Section 2.2.3. As of
the current writing, the official MC for SK-VI is not yet available. The simulation for
SK-VI used the same detector modeling parameters as SK-IV. The effect of 0.011w%
weight of Gd in water quality or atmospheric neutrino reconstruction capability is
expected to be negligible, as discussed in Section 4.2.1 and Appendix A. Background
PMT hits were properly sampled from randomly triggered events recorded in SK-VI.

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CHAPTER 4. NEUTRON MULTIPLICITY MEASUREMENT

4.1.3

Event reduction

We applied standard cuts to both the data and the MC simulation to identify fullycontained events with vertices in the fiducial volume (FCFV events). The selection
stages are detailed in Appendix B. The efficiency of selecting true fully-contained
neutrino events within the fiducial volume, with visible energy exceeding 30 MeV,
during the data reduction process is estimated through simulation to be 96.2% (SKVI MC) to 97.8% (SK-IV, [118]). The fraction of background events in the final
data sample was maintained at the O(0.1)% level or lower, and this was monitored
through visual inspection.
For this analysis, additional cuts were applied. The visible energy was restricted
to below 10 GeV, as higher energy ranges are not expected to be well-suited forthe “intranuclear cascade models” we would like to test. Additionally, an eventwith more than 15 OD PMT hits within a 200-ns time window in the signal searchtime range were excluded from the analysis, as such OD signatures may indicate thepresence of cosmic-ray interference. This cut is similar to Cut #5 for Am/Be data
reduction described in Section 3.3.4.
In addition, we excluded events that did not have a length of 535 µs. Specifically,
events lacking AFT trigger flags (which ensure extended data recording) and eventswith a last candidate time earlier than 405 µs from the event trigger were excluded.
The last cut was essential for SK-IV data, where roughly 2% of events with AFT
trigger flags had a length less than 400 µs, possibly indicating an electronics issue.
SK-V and SK-VI data did not have this problem.
The final data samples contained 29,942 events for SK-IV, 4,231 events for SKV, and 5,203 events for SK-VI. These figures translate to livetime-normalized event
rates of 9.23 ± 0.05 (SK-IV), 9.18 ± 0.14 (SK-V), and 9.22 ± 0.13 (SK-VI) atmo-

spheric neutrino events per day, which are consistent across all phases. The dataset
is summarized in Table 4.2.

181

4.2. Data quality
SK-IV

Gd
Year
Livetime [days]
ν events
ν event / dayVertex mode
Search range
Detected nn / ν event
Analysis

SK-V

SK-VI
0.011w%
2008-20182019-2020
2020-2022
3,244.4
461.0
564.4
29,9424,231
5,203
9.23 ± 0.05
9.18 ± 0.14
9.22 ± 0.13
INIT
RECO (R)[18, 534] µs from trigger[3, 534] µs from trigger
15,705
2,035
5,752
4,359
0.525 ± 0.004 0.481 ± 0.011 1.106 ± 0.015 0.843 ± 0.013
Track length
Multiplicity (used in combined data analysis)
Multiplicity∗* We utilized SK-VI (R) in RECO vertex mode for comparisons with other datasets. However, it
was not included in the final multiplicity data, which combines results from SK-IV through SKVI. This exclusion is to avoid double-counting with INIT mode results.

Table 4.2: Summary of dataset statistics.
The same neutron capture signal detection scheme was applied to both data andsimulated events. The INIT vertex mode was used across all SK-IV, V, and VI
samples, while the RECO vertex mode was additionally applied to the SK-VI samples
only. The signal search time range for each neutrino event was set to [18, 535] µs fromthe event trigger for the pure water phase (SK-IV and SK-V) samples and [3, 535]
µs for the Gd-loaded phase (SK-VI) samples. For the neutron multiplicity analysis,
we exclusively combined and compared results from the INIT vertex mode with thesimulation. Results with the RECO mode were used for algorithmic comparison ofmultiplicity results and for measuring neutron track length.

4.2

Data quality

The final data sample was compared to the simulation to verify that there are no
significant unexplained discrepancies. Figure 4.3 illustrates the simulated PMT hit
time distribution for a typical neutrino event with a delayed coincident neutron capture signal. Both the prompt signals from atmospheric neutrino interactions (mainly
involving e, µ, and π) and the delayed neutron capture signals were examined.

182

CHAPTER 4. NEUTRON MULTIPLICITY MEASUREMENT

SK-VI MC: ν̄μ CCQE, Eν = 0.63 GeV

μMichel e

Gd(n, γ)Figure 4.3: A PMT hit time distribution for a typical ν̄µ charged-current quasielastic (CCQE) event, starting from the event trigger. The event displays feature
the “prompt” muon signal (red) and the two types of “delayed” coincident signals— Michel electrons from muon decay (olive) and neutron captures on Gd (pink).
This event was simulated using skdetsim in the SK-VI configuration.

4.2.1

Prompt atmospheric neutrino interaction signals

We focus on key factors, such as interaction types (CCQE, non-QE, NC), and transferred energy, which affect outgoing neutron multiplicity and kinematics. The number of reconstructed Cherenkov rings proves to be the optimal variable for distinguishing CCQE events (single-ring) from the others (multi-ring) (see Figure 4.9).
The event rates and reconstructed energies in each sample are compared with thesimulation. Additionally, we verify the validity of the simulation’s oscillation weights
by comparing the oscillated distributions of reconstructed lepton zenith angles withthe corresponding observed data.
We examined reconstructed event variables affecting neutron capture detection
efficiency. The precision of the initial photon TOF, crucial for signal detection,
heavily depends on the base vertex quality. We assessed the reconstructed neutrino
interaction vertex and other observables influencing vertex reconstruction performance, including particle type, vertex, momentum, and multiplicity.

183

4.2. Data quality
Lastly, we show that the presence of Gd has minimal effect on event reconstruction. This is illustrated by comparing data from the pure water phase (SK-IV) with
Gd-loaded SK-VI data, alongside SK-IV MC simulation. Specifically, we present
distributions of ring counting likelihood and particle identification (PID) likelihood
which affect event categorization. Further checks on additional distributions are
outlined in Appendix C.
Unless specified otherwise, “data” refers to the combined final data sample (SKIV + SK-V + SK-VI), and “MC” refers to combined MC events weighted by the
livetime of each SK phase. Errors shown in this section are purely statistical.

Atmospheric neutrino event rate

SK-IV-VI combined
Data
MC

Events / day

8
6
4

0

1

2

3

4

5

6

Sub-GeV 1R -like

Sub-GeV MR

Multi-GeV 1R e-like

Multi-GeV 1R -like

Multi-GeV MR

Total

0
1.2

Sub-GeV 1R e-like

Data / MC2

1.0

Figure 4.4: Comparison of the daily event rates in the final sample with the expectedvalues. Events with visible energy below/above 1,330 MeV are categorized as subGeV/multi-GeV. Sub-GeV and multi-GeV samples were classified into single-ring(1R) and multi-ring (MR), with 1R further divided into e-like and µ-like.

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CHAPTER 4. NEUTRON MULTIPLICITY MEASUREMENT
Figure 4.4 presents a comparison between event rates in the data and simulation.
The overall observed event rate is larger than the expected rate by approximately
5%, with the majority of this difference originating from the sub-GeV samples.
Discrepancies in sub-GeV event rates fall within the uncertainty range of around
25% in the absolute sub-GeV atmospheric neutrino flux. Overall, the event ratesalign well with simulation expectations, though there appears to be a slightly higher
multi-ring fraction in the sub-GeV sample than predicted. From this point, alldisplayed histograms are normalized based on the number of events.

Reconstructed neutrino interaction vertex
SK-IV/V/VI combined: All
0.025

0.020

0.015

0.015

a.u.

0.020

0.010

0.010

0.005

0.005

Data / MC

0.000
1.1 0.0

0.5

1.0

1.0

pvr2

1.5

2.0

2.5
1e6

0.9
0.0

0.5

1.0

1.5
R 2 [cm2 ]

2.0

2.5
1e6

Data
MC

0.025

0.000
1.1

Data / MCa.u.

SK-IV/V/VI combined: All
Data
MC1500 1000

500

1500 1000

500

1.0

0
pvz

500

1000 1500

0500

1000 1500

0.9
z [cm]

Figure 4.5: Distributions of the reconstructed atmospheric neutrino interaction vertex radial position R2 (left) and vertical position z (right).
The distribution of reconstructed atmospheric neutrino interaction vertices appears mostly uniform across the entire detector fiducial volume. However, the event
rate near the tank wall is slightly smaller, possibly due to the data reduction algorithm which tends to avoid vertices close to the tank wall in order to mitigate
background radioactivity and cosmic rays. This characteristic is also manifest in
the atmospheric neutrino simulation. The distributions weighted by detected neutron counts are shown in Figure 4.6, showing reasonable agreement with expectation.

185

4.2. Data quality

Figure 4.6: Distributions of the reconstructed atmospheric neutrino interaction vertex radial position R2 (left figures) and vertical position z (right figures) for SK-IV(top figures) and SK-VI (bottom figures) weighted by the number of detected neutron counts.

186

CHAPTER 4. NEUTRON MULTIPLICITY MEASUREMENT
Reconstructed Cherenkov ring multiplicity
The Cherenkov ring reconstruction algorithm is outlined in Section 2.4.1. Figure
4.7 shows the ring counting likelihood calculated for the first (most energetic) ringreconstruction, along with the reconstructed ring multiplicity distributions of bothdata and simulation. The ring counting likelihood, which determines whether an
event is classified as single-ring or multi-ring, shows good agreement between dataand simulation. Additionally, Gd-loaded SK-VI data shows no significant deviationfrom SK-IV data and simulation. The overall data indicates a few % larger multiring fraction, consistent with the observed sub-GeV multi-ring event rates. It isnotable that single-ring events are expected to be predominantly CCQE events,
while multi-ring events are mostly CC non-QE events or NC events.

0.08
0.07
0.06
0.05
0.04

0.6
0.5
0.4

0.03

0.3

0.02

0.2

0.01

0.1
7.5

5.0

2.5

0.0

2.5

5.0

7.5

10.0

Data / MC

Data / MC

0.00
1.5 10.0
1.0
0.5
10.0

7.5

5.0 2.5 0.0 2.5 5.0
Ring counting likelihood

7.5

10.0

Data
MC
MC: CCQE
MC: CC-non QE
MC: NC

0.7

a.u.

a.u.

SK-IV/V/VI combined: All

SK-IV data
SK-VI data
MC (SK-IV)
MC: CCQE
MC: CC-non QE
MC: NC

0.0
1.2

0

1

0

1

2

3

NRing

4

5

6

7

6

7

1.0
2
3
4
5
Reconstructed Cherenkov rings

Figure 4.7: The distributions of the ring counting likelihood for the most energetic
ring candidate (left) and the final reconstructed Cherenkov ring multiplicity (right).

187

4.2. Data quality
Reconstructed visible energy
The total visible energy of an event is determined by summing the charge depositsof each reconstructed Cherenkov ring, converted into electron-equivalent energy.
Figure 4.8 present the predicted and observed visible energy distributions. Shapedisagreement below 200 MeV is associated with a bug in nuclear binding energy
calculation, which is fixed in the newer versions of NEUT.
SK-IV/V/VI combined: All

SK-IV/V/VI combined: All

DataMC
MC: CCQE
MC: CC-non QE
MC: NC

0.20
0.15

Data
MC
MC: CCQE
MC: CC-non QE
MC: NC

10 1

a.u.

a.u.

10 2

0.10
10 3
0.05
0

2000

1.0

4000

EVis

6000

8000

10000

0.5
0

2000

4000
6000
Visible energy [MeV]

8000

10 4
1.5

Data / MC

Data / MC

0.00
1.5

0

2000

4000

6000

8000

10000

0

2000

4000
6000
Visible energy [MeV]

8000

10000

1.0

EVis

0.5

10000

SK-IV/V/VI combined: Multi-ring

0.200

Data
MC
MC: CCQE
MC: CC-non QE
MC: NC

0.175
0.150
a.u.

0.125
0.100
0.075
0.050
0.025
Data / MC

0.000
1.5 0

2000

4000

6000

8000

10000

2000

4000
6000
Visible energy [MeV]

8000

10000

1.0

EVis

0.5
0

Figure 4.8: The reconstructed visible energy distribution of the entire final sample,
with the vertical axis in linear scale (left) and in logarithmic scale (right). The
reconstructed visible energy distributions of the single-ring (left) and multi-ring(right) samples.

188

CHAPTER 4. NEUTRON MULTIPLICITY MEASUREMENT
Cherenkov ring particle identification
Figure 4.9 displays the distributions of particle identification (PID) likelihood as
detailed in Section 2.4.1 for both the final data sample (SK-IV and SK-VI) andthe simulation. The SK-VI distribution aligns well with SK-IV, indicating minimal
impact of Gd on the ring PID.

0.12

SK-IV data
SK-VI data
MC (SK-IV)
MC: e
MC:

0.10

a.u.

0.08
0.06
0.04
0.02

Data / MC

0.00
1.5

20

15

10

20

15

10

5

0

5

10

15

20

5
0
5
10
Particle ID parameter

15

20

1.0
0.5

Figure 4.9: The PID likelihood distribution for all events in the final data sample.
A Cherenkov ring is identified as “e-like” if the likelihood is less than 0, and as“µ-like” otherwise.

189

4.2. Data quality
Reconstructed lepton momentum
The reconstructed lepton momentum distribution for single-ring events are shownin Figure 4.10. The overall shape agreement between data and simulation is goodfor both single-ring e-like and µ-like samples.
SK-IV/V/VI combined: Single-ring (Sub-GeV e-like)

SK-IV/V/VI combined: Single-ring (Sub-GeV -like)

DataMC
MC: CCQE
MC: CC-non QE
MC: NC

0.08
0.07
0.06

Data
MC
MC: CCQE
MC: CC-non QE
MC: NC

0.08
0.06
a.u.

a.u.

0.05
0.04

0.04

0.03
0.02

0.02

0.00
1.25

0

200

400

1.00

600 800 1000 1200 1400
FirstRingMom

0.75
0Data / MC

Data / MC

0.01
0.00
1.25

400

600 800 1000 1200 1400
FirstRingMom

0200 400 600 800 1000 1200 1400
Reconstructed lepton momentum [MeV]

0.75

SK-IV/V/VI combined: Single-ring (Multi-GeV e-like)

SK-IV/V/VI combined: Single-ring (Multi-GeV -like)

DataMC
MC: CCQE
MC: CC-non QE
MC: NCDataMC
MC: CCQE
MC: CC-non QE
MC: NC

100
10 1
a.u.

a.u.

200

1.00

200 400 600 800 1000 1200 1400
Reconstructed lepton momentum [MeV]

10 1

0

10 2
10 3

1.25

10 4
2000

1.00

4000

6000
FirstRingMom

8000

10000

0.75
2000

4000
6000
8000
Reconstructed lepton momentum [MeV]

10000

Data / MC

Data / MC

10 2

1.25

2000

1.00

4000

6000
FirstRingMom

8000

10000

0.75
2000

4000
6000
8000
Reconstructed lepton momentum [MeV]

10000

Figure 4.10: The reconstructed lepton momentum distributions for single-ring SubGeV e-like (top left), Sub-GeV µ-like (top right), Multi-GeV e-like (bottom left),and Multi-GeV µ-like (bottom right) samples. The disagreement in e-like events
below 200 MeV/c is associated with a bug in nuclear binding energy calculation.

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CHAPTER 4. NEUTRON MULTIPLICITY MEASUREMENT
Reconstructed lepton zenith angle
Figure 4.11 displays the reconstructed distributions of lepton zenith angles.

0.08

SK-IV/V/VI combined: Single-ring (Sub-GeV e-like)

SK-IV/V/VI combined: Single-ring (Sub-GeV -like)DataMC
MC (no oscillation)

0.07

0.08

0.06
a.u.

a.u.

0.05
0.04
0.02

0.02

0.01
0.00
1.25 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00
cosz
1.00
0.75
1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00

0.00
1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00
2
cosz

Data / MC

Data / MC

0.06
0.04

0.03

1
1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00

cos zenith

cos zenith

SK-IV/V/VI combined: Single-ring (Multi-GeV e-like)
0.08

0.08
a.u.

0.06

Data
MC
MC (no oscillation)

0.10

0.06

0.04

0.04

0.02

0.02

0.00
1.25 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00
cosz
1.00
0.75
1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00

0.00
1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00
2
cosz

cos zenith

Data / MCa.u.

SK-IV/V/VI combined: Single-ring (Multi-GeV -like)DataMC
MC (no oscillation)

0.10

Data / MCDataMCMC (no oscillation)

0.10

1
1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00
cos zenith

Figure 4.11: Cosine zenith angle distributions for single-ring, Sub-GeV e-like events(top left), µ-like events (top right), Multi-GeV e-like events (bottom left), and µ-likeevents (bottom right). cos θzenith close to -1 indicates up-going neutrinos reaching
the detector from below, indicating a longer path length. Both the data (black) andthe MC simulation (red) are normalized based on the number of events. The reddashed line represents the simulation without applying oscillation weights, illustrating changes in the fraction of event rates due to neutrino oscillation.

191

4.2. Data quality
The direction of the outgoing lepton serves as a proxy for the incoming neutrino
direction, particularly for energetic neutrinos with Eν > 500 MeV, as leptons tend
to be forward-scattered. The zenith angle not only determines the neutrino path
length but also the oscillation probability together with Eν . As illustrated in the
figure, a noticeable decline in the expected event rates is evident for muon neutrinos,
primarily due to the dominance of νµ → ντ transitions in atmospheric neutrinos, aconsequence of the substantial θ23 angle. The oscillation parameters shown in Table
4.1 explains the observed shape of the muon zenith angle distributions very well.

4.2.2

Delayed neutron capture signals

SK-IV data
SK-IV MCSK-V data
SK-V MCSK-VI data
SK-VI MC

10 3

a.u.

10 4
10 5
10 6
10 7
20000 25000 30000 35000 40000 45000 50000 55000
Total PMT hits in event, within signal search range
Figure 4.12: Distributions of the total number of PMT hits in the signal searchrange for each event, across different SK phases.
Before delving into the properties of the neutron capture signal, it is important
to note the challenges in modeling background PMT hits over the entire data acquisition period. Figure 4.12 shows the distributions of the total number of PMT
hits within the signal search time range for both data and simulation in SK-IV, V,
and VI. It is evident that the modeling of background hits, which constitute mostof the PMT hits within the given time range, does not fully describe data.

192

CHAPTER 4. NEUTRON MULTIPLICITY MEASUREMENT
Table 4.3 provides a summary of the candidate rates selected in detection algorithm Stage 1 (PMT hit trigger). Following this initial stage, the candidates
undergo classification into BG-like and signal-like categories in detection algorithm
Stage 2, facilitated by a pre-trained neural network. In the case of SK-VI, where thesignal search time range has been expanded closer to the neutrino event trigger toaccommodate faster neutron capture time with Gd, we further categorize the signallike candidates into e-like and n-like based on signal time and energy, taking into
account the contamination from Michel electrons (refer to Section 3.2 for detailed
information).
There is a noticeable difference between the expected and observed values. In
the case of BG-like candidates, this difference is approximately 10-20%, and it is attributed to the imperfect sampling of background PMT hits included in our simulation, as illustrated in Figure 4.12. The influence of the time variation in background
PMT hit rates is considered through the Am/Be neutron source measurements taken
at different times and simulation with varying PMT dark rates.
The differences in the rates of expected and observed n-like candidates are approximately 20-30%. Evidently, the fluctuation across different time periods andalgorithms is considerably larger than the statistical uncertainty, necessitating theconsideration of signal efficiency variation across operational phases and signal detection algorithms. Naively, a phase-dependence of around 4% is expected whencomparing SK-IV and SK-V, and an algorithm-dependence of around 7% whencomparing SK-VI and SK-VI (R).
The observed e-like candidate ratio aligns with expectations from our simulation
models. The RECO vertex mode demonstrates slightly higher efficiency in detecting
Michel electrons, as muons tend to propagate much further than neutrons. The
incorporation of signal vertex reconstruction aids in identifying Michel electrons
produced by energetic muons.

193

4.2. Data quality

Stage 1 candidate rate per atmospheric neutrino event
SK-IV
BG-likee-liken-likeData
186.52 ± 0.08
0.5245 ± 0.0042

MC
152.29 ± 0.02
0.6529 ± 0.0014

Data/MC
1.2247 ± 0.0005
0.8033 ± 0.0067

SK-V
BG-likee-liken-likeData414.70 ± 0.31
0.4810 ± 0.0107

MC
449.32 ± 0.03
0.5745 ± 0.0013

Data / MC
0.9232 ± 0.0007
0.8372 ± 0.0187

SK-VI
BG-likee-liken-likeData492.31 ± 0.31
0.1240 ± 0.0049
1.1055 ± 0.0146

MC
452.53 ± 0.04
0.1238 ± 0.0006
1.4342 ± 0.0021

Data / MC
1.0879 ± 0.0007
1.0014 ± 0.0398
0.7708 ± 0.0102

SK-VI (R)
BG-likee-liken-likeData
21.803 ± 0.065
0.1323 ± 0.0051
0.8433 ± 0.0128

MC
20.359 ± 0.008
0.1330 ± 0.0006
1.1692 ± 0.0019

Data / MC
1.0709 ± 0.003
0.9939 ± 0.0383
0.7213 ± 0.0110

Table 4.3: A breakdown of how candidates triggered at Stage 1 were classified into
BG-like, e-like, and µ-like categories through the neural network and Michel-electron
rejection cuts. (Statistical errors only)194CHAPTER 4. NEUTRON MULTIPLICITY MEASUREMENT
Figure 4.13 shows the detected neutron capture signal multiplicity per neutrino
event. In Figure 4.14, the average signal multiplicity per event is plotted against thedate. Clearly, the data shows a larger fraction of events with no detected neutron
capture signals and a smaller fraction with multiple detected signals. This indicatesa smaller average signal multiplicity in the data compared to the simulation.

SK-IV

0.8

Data
MC: CCQE
MC: CCnonQE
MC: NC

0.6
Fraction

Fraction

0.6
0.4
0.2
0.0

0

1
2
3
4
Detected (n, ) signals

0.0

5

SK-VI

0

1
2
3
4
Detected (n, ) signals

5

SK6 FCeFV (BONSAI)

0.8
0.6
Fraction

0.6
Fraction

0.4
0.2

0.8

0.4
0.2
0.0

SK-V

0.8

0.4
0.2

0

1
2
3
4
Detected (n, ) signals

5

0.0

0

1
2
3
4
Detected (n, ) signals

5

Figure 4.13: Fraction of detected neutron capture signal multiplicity per atmospheric
neutrino event for each SK phase.

1954.2. Data quality

Figure 4.14: Neutron capture signal detection rate per event by date, for pure waterphases (SK-IV, V: 2008-2020) and Gd-loaded phases (SK-VI, VII: 2020-present).
Colored lines indicate the weighted mean and the statistical uncertainty of the mean.

Figure 4.15: Neural network output distributions for all selected candidates in the
final data sample and their corresponding simulations, for SK-IV (top left), SK-V
(top right), SK-VI (bottom left), and SK-VI (R) (bottom right). All histograms arenormalized by the total number of candidates.

196

CHAPTER 4. NEUTRON MULTIPLICITY MEASUREMENTTo determine whether this discrepancy arises from reduced neutron production
or signal detection efficiency, a precise estimation of signal detection efficiency andits uncertainty is essential. Section 4.3 provides a detailed account of this estimation.
Figure 4.15 shows the comparison of the neural network output of candidates
selected in the actual data with those selected in simulation. The overall shapeagreement is satisfactory, although there is a significant difference in the number ofbackground candidates, particularly in SK-IV, where the background hit rates indata are noticeably different from those in simulation (refer to Figure 4.12).
In Figure 4.16, the false positives per event are displayed for different subsamplesof simulated atmospheric neutrino events. The false positive rates per event observedin simulation were low, specifically 0.0249 for SK-IV, 0.0287 for SK-V, 0.0201 for
SK-VI, and 0.0134 for SK-VI (R). Most false positives in the pure water phase were
attributed to background PMT hits, resulting from accidental coincidences of PMT
dark noise. No correlation between false positive rate and neutrino flavor or energy
was observed. Since false positives due to background PMT hits are unrelated toneutrino interaction, their contamination can be estimated by fitting the signal time
distribution with an exponential function plus a constant background term, as shownin Equation 3.10: A(1 − e−t/τthermal )e−t/τcapture + B.
Figures 4.17 to 4.20 illustrate the feature distributions for all candidates andthose successfully passing the neural network selection. These figures show a reasonable shape agreement between the data and simulation.

197

4.2. Data quality

Figure 4.16: False positive rates per event, for each subsample. The stacked bars
show the false positives per event in simulation, while the crosses are data estimation
based on the constant background term B in Equation 3.10.

198

CHAPTER 4. NEUTRON MULTIPLICITY MEASUREMENT

Data

MC

Dark noisep(n,γ)Data

MC

Dark noisep(n,γ)

Figure 4.17: The feature distributions of signal candidates before (top) and after(bottom) the neural network selection for SK-IV atmospheric neutrino data and MC
simulations. Both data and MC histograms are normalized by area.

199

4.2. Data quality

Data

MC

Dark noisep(n,γ)Data

MC

Dark noisep(n,γ)

Figure 4.18: The feature distributions of signal candidates before (top) and after(bottom) the neural network selection for SK-V atmospheric neutrino data and MC
simulations. Both data and MC histograms are normalized by area.

200

CHAPTER 4. NEUTRON MULTIPLICITY MEASUREMENT

Data

MC

Dark noisep(n,γ)
Gd(n,γ)Michel e

Data

MC

Dark noisep(n,γ)
Gd(n,γ)Michel e

Figure 4.19: The feature distributions of signal candidates before (top) and after(bottom) the neural network selection for SK-VI atmospheric neutrino data and MC
simulations. Both data and MC histograms are normalized by area.

201

4.2. Data quality

Data

MC

Dark noisep(n,γ)
Gd(n,γ)Michel e

Data

MC

Dark noisep(n,γ)
Gd(n,γ)Michel e

Figure 4.20: The feature distributions of signal candidates before (top) and after(bottom) the neural network selection for SK-VI (R) atmospheric neutrino data andMC simulations. Both data and MC histograms are normalized by area.

202

CHAPTER 4. NEUTRON MULTIPLICITY MEASUREMENT
Figure 4.21 shows the time distributions of the selected neutron signal candidates.
The distributions was fitted with a function of the form A(1 − eτthermal )e−t/τcapture + B(Equation 3.10). Table 4.4 compares the fitted parameters in data and MC, showing
reasonable agreement in both time constants τthermal and τcapture and backgroundterm B in all phases, while the normalization term A is in disagreement by 20-30%.

SK-IV

0.04
0.03

0.01

= 206.25 ± 25.44 µs= 201.51 ± 1.06 µs

0.02

0.00
1.25 0

100

200

1.00

300
FitT

400

500

0.75
0

100

200
300
400
Time from event trigger [µs]Data / MC

Data / MC

0.04

0.08

= 211.57 ± 9.41 µs= 201.70 ± 0.99 µs

0.02

0.06

Data
MC
MC: Gd( , )MC: H( , )MC:
MC: Michel
MC: BG

0.10

a.u.

Data
MC
MC: Gd( , )MC: H( , )MC:
MC: Michel
MC: BG

0.05

a.u.

SK-V

0.00
1.25 0

100

0

100

0.04

Data / MC

0.05
0.04

0.01

0.01

1.00

200

300
FitT

400

500

0.75
0

100

200
300
400
Time from event trigger [µs]500

= 116.99 ± 3.98 µs= 114.19 ± 0.24 µs

0.03
0.02

100

Data
MC
MC: Gd( , )MC: H( , )MC:
MC: Michel
MC: BG

0.06

0.02
0.00
1.25 0500

0.07= 117.15 ± 3.39 µs= 113.45 ± 0.17 µs

0.03

200
300
400
Time from event trigger [µs]0.08Data / MC

a.u.

0.05

500

SK-VI (R)

a.u.

0.06

400

0.75

500

Data
MC
MC: Gd( , )MC: H( , )MC:
MC: Michel
MC: BG

0.07

300
FitT

1.00

SK-VI
0.08

200

0.00
1.25 0

100

1.00

200

300
FitT

400

500

200
300
400
Time from event trigger [µs]

500

0.75
0

100

Figure 4.21: Signal candidate time distributions for each SK phase. In this figure,both data and MC distributions are normalized by the detected neutron counts.

2034.2. Data quality

Neutron capture time t: fit parametersA(1 − e−t/τcapture )e1−τthermal + B
SK-IV
A ×Nbins
τthermal [µs]
τcapture [µs]B ×Nbins

Data
1.4539 ± 0.0212
211.84 ± 9.42
0.0023 ± 0.0179

MC
1.9153 ± 0.0034
201.68 ± 0.99
0.0265 ± 0.0024

Data / MC
0.7591 ± 0.0111
1.0504 ± 0.0470
0.8570 ± 0.6808

SK-V
A ×Nbins
τthermal [µs]
τcapture [µs]B ×Nbins

Data
1.3798 ± 0.0587
206.28 ± 25.46
0.0110 ± 0.0450

MC
1.6551 ± 0.0029
201.60 ± 1.00
0.0309 ± 0.0021

Data / MC
0.8336 ± 0.0355
1.0232 ± 0.1264
0.3559 ± 1.4575

SK-VI
A ×Nbins
τthermal [µs]
τcapture [µs]B ×Nbins

Data
5.2064 ± 0.1356
5.3121 ± 0.7442
117.16 ± 3.39
0.0128 ± 0.0146

MC
7.6744 ± 0.0095
4.4432 ± 0.0379
113.47 ± 0.16
0.0128 ± 0.0009

Data / MC
0.7360 ± 0.0192
1.1956 ± 0.1778
1.0324 ± 0.0299
1.0038 ± 1.1448

SK-VI (R)A ×Nbins
τthermal [µs]
τcapture [µs]B ×Nbins

Data
3.9756 ± 0.1218
5.6777 ± 0.8795
118.16 ± 3.99
0.0016 ± 0.0128

MC
5.8000 ± 0.1089
5.4440 ± 0.0531
114.22 ± 0.23
0.0060 ± 0.0010

Data / MC
0.6854 ± 0.0210
1.0429 ± 0.1619
1.0345 ± 0.0350
0.2679 ± 2.1213

Table 4.4: Best-fit parameters and their 1σ uncertainties obtained from fitting thedetected neutron capture time distribution using Equation 3.10.

204

CHAPTER 4. NEUTRON MULTIPLICITY MEASUREMENTIn SK-VI with 8 MeV Gd(n, γ) signals, we can independently reconstruct theneutron capture vertex using the likelihood fitter BONSAI, as detailed in Section
2.4.2. The radial and vertical vertex distributions of candidates selected with the
RECO vertex mode are shown in Figure 4.22. While the signals are expected to beuniformly distributed throughout the ID fiducial volume, the detected signals withthe reconstructed neutron capture vertex close to the tank wall are significantly
fewer. This tendency is consistent with the expectations from SK-VI simulation.
0.09
0.08

SK-VI (R)

0.08
0.06

0.05

0.05

a.u.

0.06
0.04

MC: Gd( , )MC: H( , )

0.04

0.03

0.03

0.02

0.02

0.01

0.01

0.00
1.25 0.0
1.00
0.75
0.0

SK-VI (R)
Data
MC

0.07

0.5

1.0

1.5
fvr2

2.0

2.5

3.0
1e6

0.5

1.0

1.5
2.0
[cm ]

2.5

3.0
1e6

Data / MC

a.u.

0.07

Data / MC

0.09

MC: Gd( , )MC: H( , )

Data
MC

0.00
1.25
1.00
0.75
2000 1500 1000 500

fvz

0500 1000 1500 2000
[cm]

Figure 4.22: Radial (R2 , left) and vertical (z, right) vertex distributions of signal
candidates selected in SK-VI with the RECO vertex mode. The gray shades indicate
regions outside of the ID, and the dashed lines each indicate 1, 2, 3, 4, 5 m awayfrom the ID wall. The fiducial volume for neutrino event selection is defined as 1 m
away from the ID wall. There is no limits with regards to the reconstructed neutron
capture vertex. Signals with reconstructed vertices outside the ID tank indicate thereconstruction algorithm failed to find a good vertex within the ID.
We examined the distance between the reconstructed neutron capture vertex
and the reconstructed neutrino interaction vertex as a proxy for outgoing neutron
kinematics and vertex reconstruction performances. The distributions of both dataand simulation are compared in Figure 4.23, showing good agreement in shape.
Candidates classified as signals but with a reconstructed distance larger than 10-20
m are mostly attributed to pure background PMT hits. The majority of neutrons
were captured within 5 m from the reconstructed neutrino interaction vertex.

205

4.2. Data quality

SK-VI (R)Data
MC
MC: Gd( , )MC: H( , )MC:
MC: Michel
MC: BG

0.10

a.u.

0.08
0.06
0.04

Data / MC

0.02
0.00
1.25
1.00
0.75

0

200

400
600
DPrompt

800

1000

0
200
400
600
800
1000
Reconstructed distance from neutrino vertex to ( , ) vertex [cm]

SK-VI (R)Data
MC
MC: Gd( , )MC: H( , )MC:
MC: Michel
MC: BG

10 1

a.u.

10 2
10 3
10 4

Data / MC

10 5
1.25
1.00
0.75

0

1000

2000
3000
DPrompt

4000

5000

0
1000
2000
3000
4000
5000
Reconstructed distance from neutrino vertex to ( , ) vertex [cm]Figure 4.23: The reconstructed distance between neutron captures to neutrino interaction vertex, in linear scale (top) and in logarithmic scale (bottom).

206

CHAPTER 4. NEUTRON MULTIPLICITY MEASUREMENT
Lastly, the angular correlation between neutron capture and lepton signals isinvestigated. In Figure 4.24, the cosine of angles between the reconstructed lepton direction and the reconstructed neutron direction was examined for single-ring
events. Here, the reconstructed neutron direction is represented by a vector connecting the reconstructed neutrino interaction vertex and the reconstructed neutron
capture vertex. The distribution appears mostly flat, with the data indicating aslight prevalence of forward-scattered neutrons, primarily attributed to neutrons
from muon captures on 16 O.

All
0.14
0.12

a.u.

0.10
0.08

Data
MC
MC: Gd(n, )MC: H(n, )MC:
MC: Michel e
MC: BG

0.06
0.04
0.02
Data / MC

0.00
1.5 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00
costh
1.0
0.5
1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00
cos lnFigure 4.24: The distribution of cosine of angles between the reconstructed lepton
direction and the reconstructed neutron direction, as observed across all single-ring
events in SK-VI. The “forward-scattered” neutrons are indeed mostly emitted from“forward-going” muon captures on 16 O.

Figure 4.25 shows the angular correlation between charged lepton and neutron
captures, divided into four single-ring subsamples: Sub-GeV/Multi-GeV e-like/µlike. In the Sub-GeV e-like category, where neither muons nor pions are abundantly
produced, we observe a diminished forward-going peak (near cos θln = 1) comparedto the other three subsamples, where muons or pions are more prevalent.

2074.2. Data quality
The data reveals a slightly higher fraction of neutrons captured in the forward
direction (aligned with the lepton) compared to our simulation. This discrepancy
may arise from various uncertainties in our simulation models, such as pion production, muon capture, and neutron scattering angle uncertainties.
Sub-GeV e-like
DataMC
MC: Gd(n, )MC: H(n, )MC:
MC: Michel e
MC: BG

0.25

a.u.

0.20
0.15

0.25
0.20
0.15
a.u.

0.30

0.10

0.10

0.00
1.5 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00
costh
1.0

0.00
1.5 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00
costh
1.0

0.5

Data / MC

0.05

Data / MC

0.05

0.5

1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00

1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00

cos ln

0.30
0.25
0.20

Multi-GeV e-like
DataMC
MC: Gd(n, )MC: H(n, )MC:
MC: Michel e
MC: BG

0.25
0.20

0.15

0.15

Data
MC
MC: Gd(n, )MC: H(n, )MC:
MC: Michel e
MC: BG

0.10

0.10

0.05

0.05
0.00
1.5 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00
costh
1.0

Data / MCMulti-GeV µ-like

a.u.

0.35

cos ln0.5
1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00
cos ln0.00
1.5 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00
costh
1.0

Data / MC

0.40

a.u.

Sub-GeV µ-like
DataMC
MC: Gd(n, )MC: H(n, )MC:
MC: Michel e
MC: BG

0.5
1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00
cos lnFigure 4.25: The distribution of cosine of angles between the reconstructed lepton
direction and the reconstructed neutron direction in single-ring sub-GeV e-like (topleft), µ-like (top right), multi-GeV e-like (bottom left), and µ-like events (bottom
right) in SK-VI. The “forward-scattered” neutrons are indeed mostly emitted from“forward-going” muon captures on 16 O. Muons are either from muon neutrino CC
interaction or from π/K decays.

208

CHAPTER 4. NEUTRON MULTIPLICITY MEASUREMENT

4.3

Estimation of signal detection performance

It is crucial to accurately estimate the performance of neutron capture detection
and its associated uncertainty when measuring neutron properties. For instance,
the average neutron capture multiplicity per atmospheric neutrino event can becomputed as follows:

⟨n⟩ =



Nidetected − NiBG
ϵi



(4.1)

Here, ⟨n⟩ represents the average neutron capture multiplicity per neutrino event,

Nidetected

is the count of detected neutron captures in the ith event, NiBG is theestimated number of backgrounds or false positives in the ith event, and ϵi is theestimated signal detection efficiency for the ith event. NiBG and ϵi are the two signal
detection performance metrics that need to be accurately estimated.
We initially rely on MC simulation to estimate performance on an event-by-event
basis. In terms of the MC-estimated signal efficiency, we have already observed nonnegligible discrepancies between SK-VI Am/Be calibration data and its simulation(as discussed in Section 3.3.7), although these variations fall within our uncertainty
estimates. To address this issue, we adjust the overall signal detection efficiency
estimated through simulation by applying a scaling factor that best accounts for theefficiency values obtained from calibration data. In the context of neutron capture
multiplicity, the uncertainty associated with this scaling factor directly influences
the uncertainty in the final results.
Regarding the false positive rate per event, we base our estimates on values
obtained from simulation without applying data-based corrections. This approachis justified because we utilize actual background PMT hits recorded by our detector
to model backgrounds. The uncertainty in the false positive rate mostly arises fromvariations in PMT dark rates over time. Additionally, for SK-VI, uncertainties in thequantities of Michel electrons and gamma-rays resulting from nuclear de-excitation
may also influence the false positive rate.

209

4.3. Estimation of signal detection performance

4.3.1

Dependence on neutrino event variables

Signal efficiency notably varies with neutrino energy due to deviations between thereconstructed and actual neutron capture vertices, especially for higher-energy neutrino events. The uncertainty in the reconstructed neutrino interaction vertex increases with higher energy, influenced by interference among multiple Cherenkov
rings during reconstruction. Additionally, as neutrons gain more kinetic energy,
their capture points shift farther from their production locations. Figure 4.26 illustrates the variation in signal efficiency between sub-GeV and multi-GeV events.

Figure 4.26: Distribution of signal efficiencies estimated for each SK-IV MCsimulated event. Details of the estimation method are provided in Section 4.3.2.
Assigning the average signal efficiency for all events in estimating the average
signal multiplicity (Equation 4.1) could introduce opposite sign biases for sub-GeV
and multi-GeV events. Therefore, minimizing systematic uncertainty requires acomprehensive understanding of factors influencing signal detection performanceand modeling them to the best extent possible.
To understand factors influencing signal detection, we binned MC-simulated
events based on five reconstructed observables in each neutrino event: visible energy, the number of Cherenkov rings, neutrino flavor (flavor of the most energeticring), and the radial (R2 ) and vertical (z) position of the reconstructed neutrino
interaction vertex. This resulted in 19,404 bins across five dimensions, covering
approximately 2 million MC-simulated atmospheric neutrino events meeting event
selection criteria. Table 4.5 summarizes the binning strategy.

210

CHAPTER 4. NEUTRON MULTIPLICITY MEASUREMENT

Reconstructed variable
Visible energy [MeV]
Number of Cherenkov rings
Neutrino flavor
Vertex R2 [cm2 ]Vertex z [cm]Type
Continuous
Discrete
Discrete
Continuous
Continuous

Range[10−5 , 105 ]{Single, Multi}{e, µ}[0, 16902 ][-2200, 2200]

Bins
11
2
2
21
21

Spacing
Log
LinearLinearLinearLinearTable 4.5: Binning strategy used in MC simulation to explore how signal detection
performance varies with reconstructed neutrino event variables.
We then estimated average signal efficiency and false positive rate per event foreach bin, including associated MC statistical errors.
Projected onto a single feature axis (e.g., visible energy, radial and vertical coordinates of the reconstructed neutrino interaction vertex), binned performance metrics are illustrated in Figures 4.28 through 4.29. Signal efficiency notably depends onneutrino event visible energy across all SK phases and on the reconstructed neutrino
interaction vertex, particularly its proximity to the tank wall. A substantial drop in
signal efficiency is observed when the interaction vertex is too close to the tank wall,
possibly due to difficulty distinguishing signals near the tank wall from backgroundradioactivity in PMT glass or surrounding rock. In contrast, the false positive rate
is mostly independent of neutrino visible energy but increases for a vertex close to
the tank wall. Compared to visible energy and the interaction vertex, the numberof rings and ring particle type have a smaller impact, though signal efficiency isslightly lower for multi-ring and e-like events, likely due to worse vertex resolution
compared to single-ring and µ-like events.
Utilizing the estimated average performance metrics for each bin, we may evaluate signal detection performance for a given data event by extracting the fivereconstructed observables and referencing the corresponding bin for average signal efficiency and false positive rate. However, this approach prompts questions
about handling empty bins, events with no corresponding bin, determining an optimal binning scheme considering MC statistics, and addressing bin-to-bin correlation
alongside statistical uncertainty.

211

4.3. Estimation of signal detection performance

4.3.2

Regression on neutrino event variables

To address these issues, we employed a non-linear multivariate regression model
called the Generalized Additive Model (GAM) to obtain reliable performance estimates for each unseen data event, irrespective of the binning scheme or MC statistics.
The simplest linear GAM assumes that the expected output E[y(x)] for given features x is related by:E[y] = b +

N
X
i=0

N X
M
X
fi (xi ) = b +aji Bij (xi )(4.2)

i=1 j=1

Here, fi (xi ) represents the contribution from the ith feature xi , aji are weights
for each j th basis spline Bij (xi ) for xi , b is a constant bias term, N is the numberof features, and M is the number of basis splines per feature. For simplicity, thefeatures xi are considered independent. Once the form and number of splines are
determined, the parameters aji and b can be fitted to best describe (x, y) data points
using a straightforward least-square method, as illustrated in Figure 4.27.

Figure 4.27: Illustration of non-linear regression on a single feature x using GAM.
The weights assigned to basis splines are adjusted so that the sum of the splines
best fit the gray data points, minimizing the sum of squared residuals.

212

CHAPTER 4. NEUTRON MULTIPLICITY MEASUREMENTThe GAM with basis splines can capture any non-linear relationships between
input and output using a simple least-square fitting method. The contributions ofeach output, fi (xi ), are straightforwardly added (hence the term “generalized additive”) to predict the final output. This simplicity provides greater interpretability.
Using this technique, we can construct regression models that model the signal detection performance as a function of neutrino event’s reconstructed variables.
We model two signal detection performance metrics, the efficiency and false
positive (background) rate. So we have two regression models as in Equations 4.4,
for SK-IV, SK-V, SK-VI, and SK-VI (R), resulting in total 8 GAM models.
Expected signal efficiencySK-x = fSK-x (Evis , Nring , Flavor, R2 , z)

(4.3)

Expected false positive rateSK-x = gSK-x (Evis , Nring , Flavor, R2 , z)

(4.4)

Here x ∈ {IV, V, VI, VI(R)}. The distinct models fSK−x and gSK−x are the GAM

models that are five-dimensional extension of the function described in Equation 4.2,
and the parameters such as the weights and biases need to be fitted to some input

using the least-square method as mentioned earlier. The input we can use to fit theparameters are the binned average performance metrics from our simulation.
We used LinearGAM implemented in pyGAM 0.9.0 [156] to construct each of the 8
GAMs. To handle continuous variables like visible energy and vertex coordinates,
we applied 20 cubic splines per feature. Regression on binary variables, such as thenumber of rings (single-ring and multi-ring) or particle type (e-like and µ-like), weremodeled with two Heaviside step functions.
Smoothing was applied by imposing a penalty on the second derivative of thefitted function to prevent overfitting. The hyperparameter λi , determining the sizeof this penalty for each feature xi , was heuristically determined by monitoring thefit with different sets of λ. A default value of λ = 0.001 was used, except for thecase of fitting false positive rate against continuous features. For example, λ =1,000
was applied for visible energy, and λ = 0.1 for vertex coordinates, as smaller valuesof λ led to overfitting — unphysical cubic spikes showed up in the fitted curve.

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4.3. Estimation of signal detection performance
Central bin values were used as feature values x during the fitting process. The
average signal efficiency and the logarithm (to ensure the fit result is always positive)of the average false positive rate in each bin served as the response variable y.
The inverse of the squared ratio between the average performance metric and its
statistical error in each bin was used as the weighting factor for each data point. It
is important to note that only data points with non-zero signal efficiency and false
positive rate were included in the fitting procedure.
Figures 4.28 and 4.29 showcase the GAM fit — a 1-dimensional projection of
the best-fit results, along with the overlaid 1σ prediction interval onto the binned
performance metrics. The partial dependence on feature xi (fi (xi ) in Equation 4.2)
is distinctive for each GAM fitted in different SK phases but is shared across all
subsamples within the same phase without considering feature correlations. Despite
not incorporating feature correlations, our GAMs effectively capture trends in binned
performance metrics across various subsamples.
Figure 4.30 illustrates how well the fitted GAM estimates the average signal
multiplicity in each SK phase using MC simulation data, spanning the visible energy
range from 30 MeV to 100 GeV. The GAM-derived signal multiplicities match thetrue values well, deviating less than 3% for visible energy greater than 100 MeV.
However, for lower energy events, the GAM tends to underestimate signal efficiency
by 5-10%, leading to an overestimation of signal multiplicity by the same margin.

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CHAPTER 4. NEUTRON MULTIPLICITY MEASUREMENT

SK-IV
1R µ-likeSK-IV
MR e-like

SK-VI
MR e-like

Figure 4.28: Partial dependence plots illustrating the average signal efficiency in theMC-simulated SK-IV single-ring µ-like (top), SK-IV multi-ring e-like (center), and
SK-VI multi-ring e-like sample (bottom). The plots vary with visible energy (left),radial (R2 , center), and vertical position (z, right) of the reconstructed neutrino
interaction vertex. Each black dot on the plots represents the average signal efficiency of MC events in its corresponding bin, with error bars indicating statisticaluncertainty. In each plot, variables other than the one on the horizontal axis areheld constant at the values specified above. The red dashed line represents the GAM
that best fits the binned signal efficiencies, and the red shaded area depicts the 1σ
prediction interval of the fitted GAM. Gray dots denote binned values not used in
the fit due to either efficiency or false positive rate being zero.

215

4.3. Estimation of signal detection performance

Figure 4.29: Partial dependence plots for the average MC false positive rate per
event in the MC-simulated SK-VI multi-ring e-like sample.

Figure 4.30: Comparison of true and estimated average signal multiplicity as afunction of visible energy, for each SK phase simulation.

216

CHAPTER 4. NEUTRON MULTIPLICITY MEASUREMENTOur GAM does not account for feature correlation, potentially leading to lessprecise capture of trends between response variables and features across subsamples.
To assess GAM-based measurements, we applied this method to estimate the averagetrue neutron capture multiplicity in an unseen MC simulation generated with anolder version of NEUT 5.1.3. It predicts about 20% less neutron production than
the nominal simulation against which the GAM was fitted. Reconstructed event
variables from the unseen simulated events were used to regress performance metrics,
plugged into Equation 4.1 for estimating signal detection performance —efficiency,
false positive rate, and their 1σ uncertainty — on an event-by-event basis.
Figure 4.31 compares the performance of three different methods in estimating
the average neutron capture multiplicity as a function of neutrino event visibleenergy. This evaluation was conducted on the aforementioned unseen MC simulation
with ∼ 20% smaller neutron multiplicity and varied detector modeling compared to

our nominal MC. As anticipated (see Figure 4.26), relying on a single value forall events significantly skews the curve. Utilizing binned performance metrics foreach event yields improved accuracy compared to using a single value. However, inbins with limited MC statistics, results become less reliable and more erratic due
to statistical errors. The fitted GAM, offering a smooth prediction of performance
metrics, shows the most reliable multiplicity estimation performance. The slightdeviation between true and estimated multiplicities for each visible energy bin should
be considered a systematic uncertainty.

217

4.3. Estimation of signal detection performance

Figure 4.31: Comparison of three methods for estimating signal detection performance in an unseen SK-IV MC simulation with distinct neutron multiplicity anddetector modeling from our nominal MC. The average signal multiplicity is plotted against visible energy, with signal efficiency ϵ and expected false positives N BG
from Equation 4.1 estimated on an event-by-event basis. The blue curve assumes
a single average signal efficiency and false positive rate for all events. The green
curve involves looking up average signal detection performance metrics binned byreconstructed event variables. The red curve uses non-linear regression on binned
average performance metrics with a GAM. For the red and green curves, binned
performance metrics were extracted from the nominal SK-IV MC simulation.

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CHAPTER 4. NEUTRON MULTIPLICITY MEASUREMENT
Calibration-based correction
One final step remains before incorporating the estimated performances into ouranalysis. The MC signal detection efficiency, as estimated by the fitted GAM, isadjusted using the overall scaling factor shown in Table 3.14. This adjustment is
implemented to address the discrepancy in the observed and expected signal efficiencies in the Am/Be neutron source calibration.

219

4.4. Average neutron capture multiplicity

4.4

Average neutron capture multiplicity

We measured the average neutron capture multiplicity per atmospheric neutrino
event using 39,376 selected atmospheric neutrino events observed over 4269.8 days ofdetector livetime from 2008 to 2022. The number of detected neutron capture signals
was adjusted by accounting for the expected false positives and signal detection
efficiency, as described by Equation 4.1. Neutrino events were binned by theirvisible energy, and the average signal multiplicity was calculated for each bin tocharacterize neutron production as a function of neutrino kinematics.
Statistical uncertainties are discussed in Appendix D.

4.4.1

Systematic uncertainties

Systematic uncertainties are classified into three types:(1) Signal efficiency mean
This uncertainty is related to the mean (or scale) of the estimated signal
efficiency. Notably, this uncertainty is independent of event statistics, The uncertainty due to signal efficiency “spread”, which is averaged down to O(0.1)%
due to large statistics, is discussed in Appendix D.2.
(2) Signal efficiency regression with GAM
This uncertainty arises from systematic imperfections in the performance regression using GAM. Factors contributing to this uncertainty include assumptions of no correlation of features, deviations between the fitted GAM and thebinned performance metrics.
(3) Neutrino event reconstruction
This uncertainty arises from the visible energy reconstruction, representing thehorizontal axis of our measurement. As we conduct measurements for both
single-ring and multi-ring subsamples, we also account for the uncertainty
arising from the classification of single-ring vs. multi-ring events.

220

CHAPTER 4. NEUTRON MULTIPLICITY MEASUREMENT(1) Signal efficiency mean
We adjust the signal efficiency from simulation using a scaling factor, calculated as
the ratio between estimated efficiencies from Am/Be data and simulation (details in
Table 3.14). The uncertainties associated with these correction factors are treatedas systematic uncertainty in the signal efficiency mean. Notably, these uncertainties
consider differences in the background PMT hit rate between data and simulation.
Comparing signal multiplicities across SK phases, the top figure in Figure 4.32
displays average signal multiplicity distributions with GAM-estimated efficiency correction applied. The bottom left figure shows the result of additionally applying efficiency scaling factors from Am/Be calibration (Table 3.14). While the agreement
between SK-V and SK-VI improves, there is still room for better agreement between
SK-IV and the others1 .
To address this discrepancy, we found a set of correction factors that ensures thebest consistency among different phase datasets, minimizing the χ2 between any two
datasets. The results of SK-VI (R) were fixed and used as a reference. The obtained
scaling factors are listed in Table 4.6. The result of mutual χ2 minimization is shownin the bottom right figure of Figure 4.32.
Differences between calibration-based and consistency-based scaling factors are
considered as an additional source of uncertainty. These fractional differences wereadded to the calibration systematic uncertainties in Table 3.14 in quadrature. The
total fractional uncertainty for each dataset was set to 15.5% for SK-IV, 9.6% for
SK-V, 5.9% for SK-VI, and 12.4% for SK-VI (R).

1This issue for SK-IV is speculated to be coming from using different versions of the simulator
between Am/Be simulation and atmospheric neutrino simulation, which is currently under investigation.

221

4.4. Average neutron capture multiplicity

Efficiency scaling
No scaling
Calibration-based
Consistency-based

SK-IV
1
1.10000.9534SK-V
1
0.97841.0387SK-VI
10.9086
0.9452

SK-VI (R)1
0.87700.8770 (fixed)

Reduced χ20.88431.37470.7610

Table 4.6: The signal efficiency scaling factors used to plot Figure 4.32, and theresulting reduced sum of mutual χ2 .

Figure 4.32: Comparison of average neutron capture multiplicity across datasets,using GAM correction only (top), with additional calibration-based scaling (bottom
left), and consistency-based scaling (bottom right).

222

CHAPTER 4. NEUTRON MULTIPLICITY MEASUREMENT(2) GAM regression
We assigned uncertainty based on the difference between the actual signal multiplicity and the signal multiplicity estimated through the fitted GAM, shown in Figure
4.31. This assignment was carried out on a bin-by-bin basis, as bins at lower andhigher energy ranges exhibit a larger bias compared to mid-range bins.

(3) Neutrino event reconstruction

Figure 4.33: Assigned event reconstruction uncertainties in each subsample.
The cosmic ray calibration results up to SK-V indicate an energy scale error ofless than 2% [48]. To account for this uncertainty, we shifted the visible energy ofall simulated neutrino events by ±2%, and the resulting fractional changes in binvalues were designated as the uncertainty in each bin due to the energy scale.

Additionally, single vs. multi-ring classification uncertainty was estimated. Assuming a 5% uncertainty in the multi-GeV e-like event rate, derived from a pastoscillation analysis [116], we propagated the corresponding ±0.5 uncertainty in thering counting likelihood near the cutoff at zero. The fractional changes in bin values
were assigned on a bin-by-bin basis.
Uncertainties arising from other reconstructed variables, such as ring PID likelihood and non-neutrino background events, were estimated to be less than 0.1% andthus were omitted in this analysis.

223

4.4. Average neutron capture multiplicity

Fractional uncertainty

0.4

Total
(1) Signal efficiency mean
(2) GAM regression(3) Neutrino event reconstruction
Statistical

0.3
0.2
0.1
0.0

102

103
Visible energy [MeV]104

Figure 4.34: Bin-by-bin assigned systematic uncertainties to all final sample events.
With the exception of extremely low-energy events, the most prominent source of
uncertainty is the uncertainty in the signal efficiency mean, approximately ∼ 15%.
Figure 4.35 displays the mean neutron capture multiplicity distributions of combined data with systematic errors. A clear linear relationship between the mean
neutron multiplicity and the event visible energy is evident.

Figure 4.35: Mean neutron capture multiplicity distributions from combined data,plotted against event visible energy. The distributions are shown with both linear(left) and logarithmic (right) horizontal axis scales.

224

CHAPTER 4. NEUTRON MULTIPLICITY MEASUREMENT
Figure 4.36 presents a comparison between the combined data, the default MCsimulation, and the pure-water estimation from the SNO collaboration in 2019 [33].
While the agreement with the SNO estimation is reasonably good, a noticeable
deficit is observed across the 0.1-10 GeV range when compared to the default MC.

20.0
17.5
Mean (n, ) multiplicity

15.0

SK-IV-VI combined data (stat+syst)
SNO H2 O estimation [2019]
SK-IV MC

12.5
10.0
7.5
5.0
2.5
0.0
102

103
Visible energy [MeV]104

Figure 4.36: Comparison of SK data with default MC and SNO results [33].
Table 4.7 provides an summary of the average neutron capture multiplicity observed in atmospheric neutrino events with visible energy deposits ranging from30 MeV to 10 GeV in the SK detector, including systematic and statistical uncertainties. The results from SK-V, VI, and VI (R) exhibit consistency, with a slighttension observed with SK-IV and MC results by approximately 1σ.

SK-IV
SK-V
SK-VI
SK-VI (R)SK-IV + V + VI
MC

Mean
2.03852.33312.35972.3016
2.11342.6117

Syst
0.34210.27140.20800.32290.2967-

Stat
0.01370.04620.02230.02870.0119-

Table 4.7: Observed mean (n, γ) multiplicity and uncertainties in each dataset.

225

4.4. Average neutron capture multiplicity

4.4.2

Linear regression on visible energy

We applied the same linear fit to the full dataset and the two subsamples, single-ringand multi-ring datasets. The fitted functions are shown in Figures 4.37 through 4.39,
and the fitted slopes and y-intercepts are listed in Table 4.8 and illustrated in Figure
4.40. The fitted slopes and intercepts among different SK phases and the detection
algorithms (INIT and RECO vertex modes for SK-VI) showed good consistency.
Across all subsamples, we observed reduced neutron production compared to our
predictions. Comparing single-ring and multi-ring subsamples, neutron production
in multi-ring events was nearly twice as much as in single-ring events, which is due
to multi-ring events are dominantly inelastic events with a significant amount ofenergy transferred to hadron production, as predicted by our simulation (refer toFigure 2.44).

SK-IV-VI (APFit): All fully-contained events
Nominal MC (true) + Prediction range
Data (est)Linear fit

Data / MC

Average produced (n, ) / event

20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
1.2
1.0
0.8
0.6

102

103
Visible energy [MeV]

104

102

103
Visible energy [MeV]104

Figure 4.37: Average neutron capture multiplicity (black) distribution with a fitted
linear function (blue).

226

CHAPTER 4. NEUTRON MULTIPLICITY MEASUREMENTSK-IV-VI (APFit): Single-ring events
Nominal MC (true) + Prediction range
Data (est)Linear fit

Data / MC

Average produced (n, ) / event

20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
1.2
1.0
0.8
0.6

102

103
Visible energy [MeV]

104

102

103
Visible energy [MeV]104

Figure 4.38: Average neutron capture multiplicity (black) distribution in single-ring
events with a fitted linear function (blue).

SK-IV-VI (APFit): Multi-ring events
Nominal MC (true) + Prediction range
Data (est)Linear fit

Data / MC

Average produced (n, ) / event

20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
1.2
1.0
0.8
0.6

102

103
Visible energy [MeV]

104

102

103
Visible energy [MeV]104

Figure 4.39: Average neutron capture multiplicity (black) distribution in multi-ring
events with a fitted linear function (blue).

227

4.4. Average neutron capture multiplicity

Figure 4.40: A visual summary of fitted slopes and intercepts. (stat + syst)

All events
Data
MC
Data/MC

Slope [(n, γ) / GeV]
1.2359 ± 0.0574
1.7581 ± 0.0389
0.7029 ± 0.0362

Intercept [(n, γ)]
0.8642 ± 0.0373
1.1587 ± 0.0192
0.7458 ± 0.0345

Reduced χ2
0.51
1.32
-

Single-ring
Data
MC
Data/MC

Slope [(n, γ) / GeV]
0.5546 ± 0.0447
0.8591 ± 0.0388
0.6465 ± 0.0597

Intercept [(n, γ)]0.7708 ± 0.0373
1.1270 ± 0.0218
0.6839 ± 0.0356

Reduced χ2
0.78
0.80
-

Multi-ring
Data
MC
Data/MC

Slope [(n, γ) / GeV]
1.7752 ± 0.0829
2.5209 ± 0.0672
0.7042 ± 0.0380

Intercept [(n, γ)]
1.5973 ± 0.0701
2.0401 ± 0.0483
0.7830 ± 0.0391

Reduced χ2
0.44
1.18
-

Table 4.8: Fitted slopes and intercepts of combined data and MC. (stat + syst)

228

CHAPTER 4. NEUTRON MULTIPLICITY MEASUREMENT
Additionally, we performed the fit on single-ring e-like and µ-like subsamples,with the reconstructed lepton momentum as the horizontal axis, as displayed in
Figure 4.41. Comparison between single-ring e-like and µ-like subsamples reveals
that µ-like events exhibit a larger intercept. This observation is attributed to lowenergy muons captured by 16 O before decaying into electrons, especially within theproximity of the neutrino interaction point.

Figure 4.41: Average neutron capture multiplicity observed in single-ring e-like (top)and µ-like (bottom) events, as a function of reconstructed lepton momentum.

229

4.4. Average neutron capture multiplicity

4.4.3

Various hadronic interaction models

In neutrino event simulation, discussed in Sections 2.3.2 and 2.3.3, and illustrated in
Figure 2.46, the process is divided into two main components. The neutrino event
generator models outgoing particles from the nucleus, while the detector simulator
handles the propagation of these particles and the subsequent detector response.
Observed neutrons can originate from direct recoil in neutrino CCQE scattering, hadronic re-interactions within the target nucleus (final-state interactions, FSI),or hadronic showers of energetic outgoing nucleons within water (secondary interactions, SI). For neutrino reconstruction, primary interaction (PI) neutrons are ofprimary interest, while, for background rejection using neutrons, all three sources
are significant. Neutrino event generators handle PI and FSI, while the detector
simulator manages SI.
The contribution of these components varies with interaction kinematics. The
number of neutrons from PI and FSI is relatively independent of transferred energy,
especially dominant in low-energy events where outgoing hadrons lack the energy toknock out multiple neutrons. Therefore, the intercept measured in Section 4.4.2 isparticularly relevant to PI and FSI.
On the other hand, the number of neutrons from SI is highly dependent ontransferred energy, as energetic hadrons can produce more hadrons through showers
until they fully deposit their energy in the large detector. This quantity of neutrons
can serve as a calorimetric measure of the hadronic invariant mass, as demonstrated
in Section 4.4.2. The expected proportion of PI/FSI and SI in total observed signals
is illustrated in Figure 4.42.
There exist several models in the market that address each part of the simulation,
and a significant prediction variability was observed in the choice of the neutrino
event generator in an earlier study [34]. In this study, we particularly focus on the
FSI and SI parts, where uncertainties are not quantified, and differences in neutron
production are not well understood.

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CHAPTER 4. NEUTRON MULTIPLICITY MEASUREMENT

Figure 4.42: Mean neutron capture multiplicity plotted against visible energy, extracted from default simulation models (NEUT 5.4.0 for PI/FSI and SK detector
simulator skdetsim in Geant3.21 with SK-IV default model configuration for SI).
The blue histogram displays the number of neutrons outgoing as a result of PI and
FSI simulation, while the red bars represent the average of total neutron captures
per bin, including all three contributions: PI, FSI, and SI.
In this analysis, we examined six different neutrino event generator options,each with varying neutrino interaction and FSI models, along with five sets of SI
models implemented in our detector simulator. In total, we compared predictions forthe average neutron multiplicity across 30 different combinations of neutrino event
generator and detector simulator options.
The atmospheric neutrino flux model remained fixed; specifically, the HKKM
2011 atmospheric neutrino flux model [55] was used for all model combinations.

231

4.4. Average neutron capture multiplicity
Neutrino event generator options• NEUT 5.4.0
This neutrino event generator was utilized in our default atmospheric neutrino
event simulation. Details on the modeling of neutrino interaction and final
state interactions within NEUT 5.4.0 are provided in Section 2.3.2.
• NEUT 5.6.3
This version of NEUT includes a bug fix in nucleon binding energy, resulting ina deeper nuclear potential that reduces low-energy cross sections.
• GENIE 3.4.0 (specific to G 18a 10x 02 11b tune, x∈{a,b,c,d})
GENIE [157] stands out as a widely used neutrino event generator beyond SK
and T2K. This version of GENIE has incorporated various FSI model options,enabling a comparison of FSI models. The neutrino interaction modeling islargely similar to the aforementioned NEUT versions, including the modeling of
the nucleus ground state using the Local Fermi Gas (LFG) model and almost
identical treatment of QE and 2p2h interactions. Differences lie in resonant
pion production as well as inelastic scattering involving hadronization.
GENIE also provides a wealth of FSI models so that comparing various FSI
options within GENIE allows us to disentangle FSI from the PI/FSI convolution
in the outcome.

The differences among the primary neutrino event models are summarized in Table 4.9. Figure 4.43 provides a comparison of the fraction of simulated atmospheric
neutrino interactions in water, for each neutrino event generator. NEUT 5.4.0 predicts
a slightly higher fraction of CCQE events, attributed to a bug in nuclear binding
energy calculation. This is also evident in Figures 4.44 and 4.45, showing a more
pronounced low-energy peak below 200 MeV with NEUT 5.4.0. On the other hand,
NEUT 5.6.3 and GENIE 3.4.0 exhibit a similar CCQE fraction. NEUT 5.6.3 predicts
a slightly larger fraction of CC inelastic interactions, while GENIE predicts a larger
fraction of NC interactions. Generally, there is reasonable agreement among thethree primary neutrino event models.

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CHAPTER 4. NEUTRON MULTIPLICITY MEASUREMENT

Nucleusground state
QE
2p2h
RES
Cross sections

COH
SIS/DISNEUT 5.4.0

NEUT 5.6.3

GENIE 3.4.0

LFG

LFG

LFG

+ Binding E
correction
Nieves (2013)
Nieves (2013)
Berger-Sehgal(+Graczyk-Sobczyk)
Berger-Sehgal
Custom/Pythia/
GRV98LO(+Bodek-Yang)

Nieves (2004)
Nieves (2012)
Berger-Sehgal
Berger-Sehgal
Pythia/
GRV98LO(+Bodek-Yang)

Table 4.9: Summary of neutrino interaction model differences.

Figure 4.43: Comparison of the predicted fractions of atmospheric neutrino interactions in water among various neutrino event generators.

233

4.4. Average neutron capture multiplicity

Figure 4.44: True neutrino energy distributions in fully contained atmospheric neutrino events in water, simulated using different neutrino event generators.

Figure 4.45: Comparison of the distributions of reconstructed neutrino visible energy(left) and lepton zenith angle (right) among different neutrino event generators.

234

CHAPTER 4. NEUTRON MULTIPLICITY MEASUREMENT
The following are descriptions of the FSI models available with GENIE 3.4.0:• INTRANUKE/hA 2018 [158]This model, while not a cascade model, effectively calculates outgoing secondaries using pion/nucleon-to-nucleus cross-section data up to 1.2 GeV. Above
1.2 GeV, cross section data points are extrapolated.
• INTRANUKE/hN 2018 [158]This is a space-like intranuclear cascade (INC) model (see Section 2.3.3 fordescription) capable of working on most hadrons. This model was validated
against external data.
• Geant4 Bertini Cascade model [159]
Based on the Geant4 adaptation of Bertini [95], this is also an INC modelthat relies on CERN-HERA compilations of hadron–nucleon interaction data[160]. Associated cross section errors range from 10-30%. The model includes
a native evaporation and break-up model.
• INCL [161]This cascade model stands out from those mentioned earlier in a few ways.
Firstly, it’s a time-like INC model. All participants (incident projectile andtarget nucleons) and spectators (non-target nucleons) are treated as point
particles and tracked in both space and time. Additionally, each nucleon issituated in a potential well whose depth depends on its momentum and position. Collisions occur when two particles approach within a certain distance,
determined by parametrized cross sections specific to the type of hadron andcollision. Pauli blocking of a collision with a nucleon having less than Fermi
energy is determined based on the probabilistic occupancy of energy states in
the Fermi sphere. Tracking stops after a predetermined time, and the residual
nucleus at that point is passed to the de-excitation routine. In GENIE, it is coupled with ABLA07 [162], known to provide a good explanation for spallation
data when combined with INCL [163]. The model has undergone validation
across various observables in a wide energy range [164].

235

4.4. Average neutron capture multiplicity
The differences among the aforementioned FSI models, including the one native
to NEUT, are summarized in Table 4.10. Figure 4.46 shows an example of FSI model
predictions (G4Bertini not shown) in the estimated p-12 C reaction cross section as
a function of proton kinetic energy. While the agreement above 150 MeV is good,at lower energies the model predictions diverge.

Item / Model
Nucleus model
Collision
Propagation
Cross section
Stopping criterion
Pauli blocking
Medium effectFormation zone
Nuclear de-excitation

NEUT

hA hN
Continuum
Mean free path
Space
Tabulated
Nucleus radius
StrictNone
Yes (π only [165])
Yes
Yes (π only [166])
Yes (16 O only)
None

GENIE
G4Bertini

INCL
Particles
Distance
Time
Parametrized
Time
StrictProbabilistic
None
Yes
None
Yes [167]
Yes [162]

Table 4.10: Summary of FSI model differences in neutrino event generators.

Figure 4.46: Comparison of p-12 C reaction cross section predictions as a function of
proton kinetic energy. (Reprinted from [168])236

CHAPTER 4. NEUTRON MULTIPLICITY MEASUREMENT
Figure 4.47 illustrates the predicted outgoing neutron momentum distributions
for atmospheric neutrino interactions in water, using the previously mentioned neutrino event generator options. Notably, the proton and neutron multiplicities, aswell as the kinematics at low energies, showed significant variations among the different model options. Another factor contributing to these differences, in addition
to the scattering models themselves, was the distinct nuclear de-excitation (preequilibrium/compound) routines that come into play after all secondary particles
were tracked in the FSI models. These de-excitation models are responsible for theemission of very low-energy nucleons.

Figure 4.47: Comparison of FSI model prediction of outgoing neutron (left) and proton (right) momentum distributions for atmospheric neutrino interactions in water.

237

4.4. Average neutron capture multiplicity
Our detector simulators, skdetsim in Geant3.21 and SKG4 in Geant4.10.5.p01,
offer a range of hadronic secondary interaction models:

Detector simulator options• GCALOR
This model is available in skdetsim. For propagation of hadron with kinetic
energy below 10 GeV, GCALOR uses HETC [108] implemented based on Bertini[95]. For thermalization and captures of neutrons with kinetic energy lower
than 20 MeV, GCALOR uses MICAP which estimates outgoing particle multiplicity and kinematics based on the evaluated nuclear data ENDF/B-V.
• SK-IV/V default
This model is set as the default in skdetsim for simulating the pure waterphase. In the default settings of SK-IV/V, pions with momentum lower than
500 MeV/c (which is relevant to the ∆ resonance) are processed using the NEUT
pion FSI model. This model incorporates medium corrections, as outlined in[165], with parameters fine-tuned to pion-nucleus scattering data [100].
• SK-VI default
This model is set as the default in skdetsim for the Gd-loaded phase simulation. In addition to the SK-IV/V default, NeutronHP package based on
ENDF/B-VII.1 [31] from Geant4.10.5.p01 is used to propagate neutrons below
20 MeV kinetic energy.
• Geant4 Bertini cascade modelThis model is available in SKG4. This is basically the same as Geant4 Bertini
cascade model implemented in GENIE.
• Geant4 INCL
This model is available in SKG4. In Geant4.10.5.p01, it is coupled to the default
Geant4 precompound model [169] rather than ABLA [162] as in GENIE.

238

CHAPTER 4. NEUTRON MULTIPLICITY MEASUREMENTTo determine the average neutron capture multiplicity per outgoing hadron produced via neutrino-nucleus interaction within the ID water volume, we conductedsimulations using our detector simulators, skdetsim and SKG4. The simulations employed a particle-gun setup, wherein 10,000 single nucleons (hadrons) were randomly
fired at the tank center, covering a kinetic energy range of 0-10 GeV. Subsequently,
for each interaction model and momentum bin, we calculated the average numberof neutron captures.
Figure 4.48 shows a comparison of the predictions made by different secondaryinteraction models for the average neutron capture multiplicity in water, plottedagainst the incident nucleon momentum. Model variations were observed in therange of approximately 10% to 50%, with the INCL model notably predicting 2030% fewer neutron captures than the other models based on the Bertini cascade
model.

Figure 4.48: Comparison of predictions from different secondary interaction models
for the average neutron capture multiplicity as a function of incoming neutron (left)
and proton momentum (right) within a water volume.

239

4.4. Average neutron capture multiplicity

4.4.4

Generation of model predictions

Conducting a full detector simulation for all 30 model combinations is computationally expensive, and it is unnecessary for obtaining neutron capture counts. Instead,
we utilize tabulated average neutron capture multiplicity per incident hadron as
visually represented in Figure 4.48. We assign a weight to each outgoing particle∈n, p, π ± , µ− from the neutrino event generator, based on the tabulated average
neutron capture multiplicity to calculate the total expected signals for each event.

This approach is justified for two main reasons: firstly, the ”average” neutron
capture multiplicity from secondary interactions should primarily depend on theoutgoing particle multiplicity and kinematics, independent of the neutrino interaction itself. Secondly, most secondary neutrons are produced by inelastic interactions
or nuclear captures of one of {n, p, π ± , µ− }, as illustrated in Figure 4.49. However, it

to note that for higher energy events, kaon inelastic scattering becomes 5# isofimportantsecondaryneutrons
non-negligible, reaching the order of O(1)%.

• Checked physics processes that generated secondary neutrons→ # of secondary neutrons generated by neutron inelastic scattering is largely different among physics lists

neutron inelastic
proton inelastic!#/!$ inelastic"$ capture!$ capture
others# of secondary neutrons / event(The result of GCALOR is not included because the definition of physics process is different in GEANT3)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

GC

ALO

R

0.7366
SK Collaboration Meeting

FTF

P_B

QG

ERT

0.7806

_HP

SP_

BIC

0.6926

_HP

INC

INC

0.60790.6081

L++

L++

_HP

Figure 4.49: Breakdown of neutron production sources in atmospheric neutrino
interactions (below 2 GeV) in water, categorized by interactions and analyzed using
five distinct hadronic inelastic scattering models. The stacked bars are color-coded
to represent the specific neutron production source interaction. (Note: GCALOR does
not offer a breakdown by interactions.) The category labeled as “Others” includes
interactions such as kaon inelastic interactions, with a larger fraction in higher energy
events. (Reprinted from [170])240

CHAPTER 4. NEUTRON MULTIPLICITY MEASUREMENT
Furthermore, we conducted full detector simulations exclusively using the default
secondary interaction model for each neutrino event generator option. The purpose
was to obtain the reconstructed visible energy, which serves as the horizontal axis
in our measurements. The reconstructed visible energy for every event, spanning all
secondary interaction model choices, was derived from this comprehensive detector
simulation using the default secondary interaction model.
In this context, we made the assumption that reconstructed visible energy is
independent of the secondary interaction model. This assumption holds true tosome extent since the reconstructed visible energy is primarily influenced by charged
leptons resulting from neutrino interactions, and the variability in pion production
among different SI models is rather small (see Appendix E). Impact of large variability in nucleon production of SI models on visible energy is expected to be negligible
as nucleons are mostly invisible in the detector.
Figure 4.50 presents a comparison of predictions for the average neutron capture
multiplicity across all 30 model combinations as a function of event visible energy.
The variability in model predictions hovers around 30-40%, particularly noticeable
for higher event visible energy. Notably, a clear distinction exists between model
combinations featuring INCL as the secondary interaction model and those withalternative secondary interaction models. As observed in Figure 4.48, INCL resulted
in fewer neutron captures, translating to a 30-40% reduction in signal multiplicity
compared to other secondary interaction models such as the Bertini cascade model.

2414.4. Average neutron capture multiplicity

Figure 4.50: Comparison of the average neutron capture multiplicity predictions foreach model combination. Contributions from neutron, proton, and pion interactions
in water, along with negative muon captures on 16 O, were taken into account.

242

CHAPTER 4. NEUTRON MULTIPLICITY MEASUREMENTIn Figure 4.51, we break down the overall model prediction variability into twosources: neutrino event generation (PI/FSI) and detector simulation (SI). When
comparing model choices within each origin while keeping the choice fixed for theother origin, we observe that PI/FSI model prediction variability is pronounced atlower energy events, while SI model prediction variability becomes prominent at
higher energy events. This aligns with expectations, given that FSI model prediction variability is associated with low-energy nucleon kinematics and multiplicity,whereas higher energy nucleons play a more substantial role in SI model predictions. Additionally, SI occurs more frequently at higher energies, obscuring theoriginal hadron content more strongly in higher energy events. The default neutrino
event generator option (NEUT 5.4.0) showed the smallest neutron production, while
the default detector simulator option (SK-VI default) showed relatively larger neutron production.

Figure 4.51: Comparison of the impact of different neutrino event generator options(left) and detector simulator options (right) on the predicted average neutron capture multiplicity. In the left figure, each neutrino event generator option is pairedwith the nominal secondary interaction model (SK-VI default). In the right figure,each detector simulator option is paired with the nominal neutrino event generator
option (NEUT 5.4.0).

243

4.4. Average neutron capture multiplicity
It is important to note that the derived model predictions take into account tabulated contributions from a specific set of particle interactions only, omitting a small
contribution from kaon interactions, which contribute around O(1)% for higher energy events as mentioned earlier. Instead of directly comparing the derived model
predictions presented in Figures 4.50 and 4.51 with the data outlined in Section
4.4.2, we calculated the ratio between each model combination prediction and theprediction from the default model (NEUT 5.4.0 and SK-VI default options). We
then multiplied this ratio by the prediction based on full detector MC simulation
with the default model combination, to compensate for the absence of kaon contributions in the model predictions at higher energies.

244

CHAPTER 4. NEUTRON MULTIPLICITY MEASUREMENT

4.4.5

Comparison of model predictions with data

Figure 4.52 shows the comparison of the observed average neutron capture multiplicity with the prediction range, spanning the minimal to maximal predictions for each
visible energy bin. We observe agreement between the data and the predictions of
the nominal simulation in the low-energy range below 0.1 GeV and the higher energy
range above 1 GeV. However, in the intermediate energy range of 0.1 < E < 1 GeV,
the observed average neutron capture multiplicity is slightly smaller than our mostconservative prediction (NEUT 5.4.0 and INCL). This particular combination yielded
the best agreement in the fitted slope and intercept (Figure 4.53, Table 4.11). Figure
4.54 provides a visual comparison of the data with the linear model, assuming fitted
slope and intercept from the best model combination.

20
Mean (n, ) / event

15

Data (syst+stat)
Nominal MC
Varying FSI models only
Varying FSI & SI models

Data/MC

10

5
4
3
2
1
1.40
1.2
1.0
0.8
0.6

102

103
Visible energy [MeV]

104

102
103
104
Neutrino event visible energy [MeV]

Figure 4.52: Comparison of predicted average neutron capture multiplicities for different model combinations. The nominal MC results shown in red dashed line isfrom the full detector MC simulation of SK-VI atmospheric neutrino events (corresponding to NEUT 5.4.0 with SK-VI default).

245

4.4. Average neutron capture multiplicity

Figure 4.53: Comparison of fitted linear slope and intercepts between combined data
and model combinations.

246

CHAPTER 4. NEUTRON MULTIPLICITY MEASUREMENT

Rank
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30

PI/FSI
NEUT 5.4.0
GENIE INCL
NEUT 5.6.3
GENIE Bertini
GENIE hA
GENIE hN
GENIE INCL
NEUT 5.4.0
NEUT 5.4.0
GENIE INCL
NEUT 5.6.3
NEUT 5.6.3
NEUT 5.4.0
NEUT 5.4.0
GENIE INCLGENIE hA
GENIE Bertini
NEUT 5.6.3
GENIE INCL
NEUT 5.6.3
GENIE hN
GENIE hA
GENIE Bertini
GENIE hA
GENIE Bertini
GENIE hN
GENIE hN
GENIE hA
GENIE Bertini
GENIE hN

SI
INCL
INCL
INCL
INCL
INCL
INCL
SK-IV/V default
SK-IV/V default
SK-VI default
SK-VI default
SK-IV/V default
SK-VI default
GCALOR
G4Bertini
GCALOR
SK-IV/V default
SK-IV/V default
GCALOR
G4Bertini
G4Bertini
SK-IV/V default
SK-VI default
SK-VI default
GCALORGCALOR
SK-VI default
GCALOR
G4Bertini
G4Bertini
G4Bertini

Reduced χ20.394461
0.490170
0.614911
1.403655
1.527829
1.570021
2.526251
2.599332
2.778316
3.282918
3.423120
3.722445
3.853385
3.923076
3.981559
4.463219
4.816538
4.942095
4.949126
5.069997
5.522949
5.909420
6.151070
6.371587
6.883173
6.931615
7.712574
7.755188
8.219881
9.031177

p-value
1.00
1.00
0.99
0.05
0.02
0.01
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0

Table 4.11: Model combinations ranked by the reduced-χ2 between data (all events)
and predictions assuming linear model based on the fitted slope and intercepts shownin Figure 4.53. The χ2 and p-values are roughly estimated based on the assigned
total errors in our data.

247

4.4. Average neutron capture multiplicity

20.0
17.5

GENIE hN + Bertini
NEUT 5.4.0 + INCL
Data

Mean (n, ) / event

15.0
12.5
10.0
7.5
5.0

Data / Pred.

2.5
0.0
1.5

102

103

104

1.0
0.5
102

103
Visible energy [MeV]104

Figure 4.54: Data comparison with the linear model predictions assuming fitted
slope and intercept from (NEUT 5.4.0 + INCL) and (GENIE hN + Bertini) model
combinations.

248

CHAPTER 4. NEUTRON MULTIPLICITY MEASUREMENT
Let us summarize the implications of our findings:• SlopeAs shown in Figures 4.51 and 4.52, the slope of INCL was notably lower thanthat of other SI models, aligning well with the observed slope in all combinations. In contrast, other SI models showed less compatibility with our data,with Geant4 Bertini, widely used in detector simulation, showing the poorest
performance.
• Intercept
While the distinction is not as pronounced as with SI models, different PI/FSI
model choices also exhibit differences in the intercept. For instance, GENIE hN
and hA models generally show larger intercepts compared to other models,
making them incompatible with the data. The use of INCL as an FSI model
reduces the intercept by 20-30%.
NEUT 5.4.0 generally shows the smallest neutron production among all PI/FSI
options. This is attributed to a larger fraction of CCQE events compared to
non-QE events in the lower energy end, as demonstrated in Figure 4.43 and
4.45. The almost identical nature of NEUT 5.6.3 and NEUT 5.4.0, except for the
nuclear binding energy correction, allows us to attribute the 10% difference
between the two NEUT versions to the difference in the CCQE fraction.

Finally, we compared the average neutron “track length” or “travel distance” –the reconstructed distance between the neutrino interaction vertex and the subsequent neutron capture vertex – observed in the SK-VI (R) data with model predictions. The model predictions for different neutrino event generator options wereobtained from full detector simulation. However, due to the absence of full detector simulation for all model combinations involving different secondary interaction
model options, we conducted full detector simulations for the NEUT 5.4.0 option andfour distinct secondary interaction models – GCALOR, Bertini, INCL, and Geant4
Binary Cascade model (BIC) [171].

249

4.4. Average neutron capture multiplicity
Figure 4.55 illustrates the average neutron track length as a function of event
visible energy for the four secondary interaction models. Generally, for an energy
range over 100 MeV, the average neutron track length increases with the event visibleenergy, aligning with expectations. In the lower energy range, the length is larger due
to the poorer resolution of the neutrino interaction vertex, which competes with theactual distance traveled by the neutrons. Bertini-like models (GCALOR, Bertini)
predict approximately 5% larger average distances compared to more sophisticatedmodels (INCL, BIC) across all energy ranges. Based on this observation, we assumed
a uniform 5% uncertainty in the average distance, accounting for the uncertainty inthe secondary interaction model, for all energy ranges.

Figure 4.55: Comparison of average reconstructed neutron track length as a functionof event visible energy among different sets of full atmospheric neutrino detector
simulations with four different SI models. Only neutron captures with a distance ofless than 10 m are considered.

250

CHAPTER 4. NEUTRON MULTIPLICITY MEASUREMENT
Figure 4.56 presents a comparison of the observed average reconstructed distance
between the neutrino interaction vertex and the subsequent neutron capture vertex
with the corresponding model predictions using different neutrino event generator
options. The uncertainty in these predictions was added by a 5% uniform uncertainty
accounting for the variability in the secondary interaction model.
The observed average neutron track length was slightly smaller than the full MC
prediction of the default model combination, slightly favoring more sophisticatedmodels such as INCL and BIC over Bertini-variants. Overall, the observation was inagreement with the model predictions within the assumed 5% secondary interaction
model uncertainty and 5-10% statistical uncertainties. The variability in predictions
within different neutrino event generator options (FSI models) had a smaller impact(∼ 3%) compared to the secondary interaction model.

2514.4. Average neutron capture multiplicity

Figure 4.56: Comparison between SK-VI (R) observations and the predicted range
for the average neutron “travel” distance, which is the reconstructed distance between the neutrino interaction vertex and the subsequent neutron capture vertex.
Only neutron captures with a distance of less than 10 m are considered.
The red dashed line represents the full MC prediction with the default model combination (NEUT 5.4.0 and SK-VI default detector simulation), and the thicker red
shaded area represents the range of varying predictions due to changes in the neutrino event generator option, reflecting the variability in predictions due to PI andFSI model uncertainty. The lighter red shaded area represents the uniform 5% uncertainty due to SI model uncertainty, as observed in Figure 4.55.

252

Chapter 5
Discussion and conclusions
5.1

Discussion

What are the key factors that cause variations in model predictions?
The key distinctions among the models lie in their use of different INC models forthe simulation of hadron inelastic interactions. The inelastic interaction of nucleons,especially those of neutrons, is expected to be the predominant source of neutron
signal production (see Figure 4.49).
There is a notable discrepancy in the model predictions for outgoing nucleon
multiplicity and kinematics, particularly in the prediction of low-energy nucleons(specifically, below 300 MeV in kinetic energy or 800 MeV/c in momentum, as shownin Figure 4.47). In this energy range, incident nucleons near the 16 O Fermi momentum are likely to knock out another nucleon from the nucleus, as demonstrated by
the 16 O(n, 2n)15 O cross-section in Figure 5.1. Therefore, the INC model’s predictionof low-energy nucleon multiplicity as well as the target nucleon’s momentum distribution should be the primary determinant of the final neutron signal multiplicity.

253

5.1. Discussion
The semi-classical approach of INC models is generally not expected to perform
well for incident hadrons with kinetic energy less than 100 MeV (or momentum lessthan 450 MeV/c), where the incident nucleon wavelength becomes comparable tothe size of a nucleon. However, certain INC models, such as INCL, incorporate morequantum mechanical considerations compared to others, and it is worth exploring
how these considerations impact predictions. Here, we highlight unique features of
INCL that differentiate it from other models:• Tracking motion of all target nucleons
Besides the projectile, all nucleons in the target nucleus are tracked in both
space and time within a specific interaction time. The cascade stops evenwhen some particles remain within the nucleus.
• Position and momentum-dependent target nucleon potential
Nucleons are placed in a potential well whose size and depth depend on nucleon
position and energy, resulting in variable escape probability.
• Probabilistic Pauli blocking
Not every outgoing nucleon below the Fermi momentum pF is blocked; instead,
it is determined by taking into account the probabilistic occupancy of nucleon
energy states. This plays a crucial role in shaping INCL’s prediction of lowenergy nucleon production1 , placing it in a middle ground between models
that completely disregard Pauli blocking and those that strictly apply it for
all nucleons below pF .
• Coulomb repulsion
Coulomb repulsion of the nucleus affects the trajectories of incident protons(reducing effective cross section), but does not affect neutrons.

1

Recent studies indicate that INC models can effectively extend their validity to levels as low asa few MeV. For instance, incorporating the “discrete-level-constraint (DLC)” instead of the continuous distribution of nucleon energy states over the Fermi sea has been shown to be particularly
promising in this context [172, 173].

254

CHAPTER 5. DISCUSSION AND CONCLUSIONS• Nuclear de-excitation after the end of cascade
Nucleons or gamma-rays are emitted as a result of the residual nucleus deexcitation.
These characteristics of INCL can be succinctly described as capturing a more
realistic nuclear model that takes into account nucleon correlations, albeit in a semiclassical manner. The improved nuclear model is likely a contributing factor to theenhanced reproducibility of data, particularly in the low-energy region where simpler
versions of INC model without nucleon correlations are expected to fail.
16 O(n, 2n)15 O

50

EAF-2010
TENDL-2015
Brill (1961)
Yashima (2004)

Cross section [mb]

40
30
20
10
0

175

200 225 250 275 300 325
Incident neutron momentum [MeV/c]

Figure 5.1: 16 O(n, 2n)15 O cross-section as a function of incident neutron momentum.

What can be said about PI or FSI models?
The impact of PI/FSI models on the observed neutron multiplicity in atmospheric
neutrino events at SK is not as pronounced as the impact of SI models. Therefore,
it is hard to make definitive statements about the PI/FSI models. However, we caninfer that the native neutron production in GENIE is not favored by our measurement
results unless it is supported by INCL. Achieving a reliable estimation of neutron
detection efficiency in the low-energy range with less than a 10% uncertainty could
provide a basis for more definitive conclusions in the future.

255

5.1. Discussion
Implications on the neutron deficit observed in earlier studies
Previous studies conducted by SNO [33], T2K [34], and MINERvA [35, 36] all
employed the Geant4 Bertini cascade model (SNO, MINERvA) or Geant3 GCALOR(T2K), both of which were shown to overpredict neutron production in this study.
Utilizing INCL as the secondary hadron inelastic scattering model could potentiallyresolve this issue.
The overprediction issue observed in earlier studies was particularly pronounced
at lower energy transfer or visible energy, aligning with the observations in thisstudy. MINERvA also noted a deficit in low-energy recoil proton signals comparedto Geant4 predictions, which might be linked to the Geant4 Bertini model overpredicting low-energy neutrons compared to other models, as illustrated in Figure 4.47.

Unaccounted sources of model prediction variability
In addition to the discussed factors, there might be additional variability in hadronization models that determine outgoing hadron kinematics and multiplicity between
neutrino interaction and FSI. Moreover, potential differences between INC and nonINC models could contribute to the overall variability in model predictions.

256

CHAPTER 5. DISCUSSION AND CONCLUSIONS

5.2

Summary

Precisely predicting the number of neutrons produced in neutrino interactions is
crucial for advancing future projects involving large water Cherenkov detectors,
such as Super-Kamiokande (and potentially Hyper-Kamiokande). These projects
include the analysis of neutrino oscillation and the search for rare events like proton
decay and the diffuse supernova neutrino background (DSNB).
This thesis presents the measurement of the average neutron multiplicity observed in atmospheric neutrino interactions within the water volume of the SuperKamiokande detector. The measurement is provided as a function of the event’s
electron-equivalent “visible” energy, a semi-calorimetric measure of the neutrino
momentum transfer. The primary motivation behind this study was to validate theobserved neutron deficit from previous studies and evaluate the uncertainty in total
neutron production relative to kinematic observables.
The key advancements from previous studies include:• Extensive dataset spanning 12 years of atmospheric neutrino events in water• Broad energy range spanning 30 MeV to 10 GeV,
made possible by accurate modeling of neutron detection efficiency.
• Measurement of neutron track length as a proxy for neutron kinematics,
made possible through recent Gd-loading.
• Inclusion of INCL, an intranuclear cascade model with a realistic nuclear model,
which has not been previously compared in neutron production measurements.

257

5.2. Summary
For this purpose, a neutron detection algorithm was established based on a simple
low-energy trigger and a neural network binary classification, calibrated with an
Am/Be neutron source. The thesis presents an extensive study on the uncertaintyof neutron detection performance, revealing a noteworthy bias stemming from thecalibration setup and thermal neutron capture simulation errors in Geant4. Thesefindings may prove beneficial for future water Cherenkov detectors.
Neutron multiplicity was estimated through an event-by-event prediction of neutron detection efficiency using a multivariate non-linear regression technique applied
to simulation data. The observed neutron multiplicity showed a linear correlation
with the visible energy of the neutrino event. Multi-ring events, characterized bya higher fraction of inelastic interactions, exhibited larger multiplicities comparedto single-ring events. The results were compared with predictions from various
hadronic interaction models. While a deficit of neutrons was observed compared
to traditional, default semi-classical hadronic interaction models, better agreement
was achieved when using the INCL — a similar semi-classical model but with moresophisticated description of nucleus. This highlights the significance of nuclear modeling in predicting low-energy nucleon production and kinematics.

258

CHAPTER 5. DISCUSSION AND CONCLUSIONS

5.3

OutlookIn the coming years, the ANNIE experiment [174] is set to measure neutron production and relevant cross-sections using a Gd-loaded water Cherenkov detector situatednear low-energy neutrino beamline at Fermilab. T2K has recently integrated a newfine-grained scintillator tracker into their near detector complex, allowing for theobservation of neutron signals [175], similar to MINERvA. The WCTE experiment[176], a fine-grained compact water Cherenkov detector near the secondary hadron
test beamline at CERN, aims to measure neutron production from hadron-waterinteraction. This measurement will directly constrain hadronic interaction models
and disentangle FSI and SI effects.
This study, combined with these forthcoming efforts, has the potential to advance future investigations. A notable impact is expected in improving the analysis
of atmospheric neutrino oscillations targeting neutrino mass ordering and CP violation. The key lies in the differences in oscillation probabilities between νe andν̄e as illustrated in Figure 1.8. The classification of νe and ν̄e is imperative, andfor atmospheric neutrino analysis, this can be accomplished by tagging recoil neutrons from ν̄e . Thus, precise estimation of neutron production in neutrino eventsand associated uncertainties is crucial. This study is expected to significantly reduce these uncertainties, thereby enhancing the overall analysis. To illustrate, we
provide a simplified demonstration of how a more precise measurement of neutron
production, as achieved in this study, can enhance the sensitivity — for δCP as an
example — in the analysis of atmospheric neutrino oscillations at SK.
We focus on the impact of δCP sensitivity primarily in low-energy (<500 MeV)
νe /ν̄e events. For simplicity, we only examine the sub-GeV single-ring e-like sample,
and compare predictions of the following three neutron production models: a) a
“no-neutron” model (i.e., there is no use of neutron information in the analysis), b)
the nominal model based on our default simulation setup, and c) an updated modelwith a 20% reduction in neutron production.

259

5.3. Outlook
We assigned 1σ uncertainties in the predicted neutron multiplicity as 0%, 50%,
and 10% for each respective model. Figure 5.2 illustrates the predicted mean neutron
multiplicity as a function of reconstructed electron momentum.

Mean neutron multiplicity

Sub-GeV single-ring e-like
2.0
1.5

Nominal model (50% uncertainty)
Updated model (10% uncertainty)
No neutron model

1.0
0.5
0.0

200 400 600 800 1000 1200
Reconstructed ring momentum [MeV/c]

Figure 5.2: Predicted mean neutron multiplicity as a function of reconstructed electron ring momentum in the sub-GeV single-ring e-like sample, for three different
neutron production models described in the text.
To enhance the purity of νe CC interactions and eliminate ν̄e events, we may
reject events with one or more identified neutron signals. We assume a consistent
80% neutron detection efficiency across all models. Figure 5.3 compares the distributions of detected neutrons and the fractions of true νe events within 0-neutronsubsamples for three different models. The figure illustrates that the fraction of trueνe events improves with an increased number of detected neutrons.

Figure 5.3: Expected Sub-GeV single-ring detected neutron distribution (left) andthe fraction of true νe events in the 0-neutron subsample (right), comparing thethree models described in the text.

260

CHAPTER 5. DISCUSSION AND CONCLUSIONS
Then, we can compare the expected number of events in each momentum andzenith angle bin by varying oscillation parameters, as is typically done in the SK
atmospheric neutrino oscillation analysis. In this example, we compare the expected
event counts with varying δCP (other parameters taken from Table 4.1). Figure 5.4
compares the event counts in momentum and zenith angle bins expected by thenominal model between CP conserving and violating scenarios. We see that the
number of expected low-energy νe events is enhanced with assuming CP violation.2

Figure 5.4: Sub-GeV single-ring e-like 0-neutron events per year in each reconstructed electron momentum (prec ) bin (left) and zenith angle bin (right) expected by
the nominal model, for CP conserving (δCP = 0, brown) and violating (δCP = 3π/2,cyan) neutrino oscillation.
To quantify the sensitivity to δCP , we may calculate thePχ2 (δCP ) ≡ 2

Pi (ei (δCP )−oi + oi ln(oi /ei (δCP ))) based on the likelihood of the observed (in this example, expected by ground truth) bin count oi in the ith bin, based on the statistical uncertainties of the expected bin count ei (δCP ) that varies with assumed δCP .

2

CP violation has opposite-signed effects on P (νµ → νe ) and P (ν̄µ → ν̄e ), and the low-energy ν̄eevents are expected to decrease with CP violation in the normal mass ordering scenario. We mayleverage this effect further by purifying ν̄e events, however, purifying ν̄e is more challenging thanpurifying νe . This is because νe CC events also produce neutrons through hadronic re-scatterings,
requiring additional considerations beyond a simple cut on neutron counts for purifying ν̄e .

261

5.3. Outlook
Comparing with the ground truth histograms, assuming CP violation (δCP =P
3π/2), Figure 5.5 shows the distribution of χ2 as a function of the assumed δCP for
the three models with distinct uncertainties. We observe that purifying νe events byrequiring no tagged neutrons has an impact on the CP-symmetry (δCP = 0) rejection
power. Additionally, the size of the uncertainty in neutron production assumed by
the models is directly translated to the variation in the expected sensitivity. Whilethe updated model assumes reduced neutron production compared to the nominalmodel, the uncertainty in the nominal model is large, so the negative-1σ-varied
sensitivity of the updated model actually becomes larger than that of the nominalmodel, as shown in Figure 5.6, albeit by a small amount (∼5%).

Ground truth CP = 3 /2

12.5

2

10.0
7.5
5.0

Nominal model
Updated modelNo neutron model

2.5
0.0

0

1

2

3
Fit CP

4

5

6

P 2
Figure 5.5:
χ for the three neutron production models assuming distinct uncertainties, compared with the ground truth histograms assuming CP violation, asa function of assumed
P δ2CP values. The red dashed line indicates the ground truth
δCP = 3π/2. The
χ is calculated for the momentum and zenith angle bins of
sub-GeV single-ring e-like 0-neutron events as shown in Figure 5.4. The error bandfor each model was calculated based on model predictions of neutron production
varied by 1σ uncertainty.
This simplified demonstration provides one example of how a reduced neutron
production uncertainty can enhance the sensitivity of neutrino oscillation analysis
using neutron counts. In the actual analysis, we should utilize the other subsamples
and employ more sophisticated techniques to enhance neutrino reconstruction (e.g.,ν/ν̄ purification and missing neutrino momentum carried away by hadrons) basedon observed neutron counts and kinematics. Additionally, we need to consider othersystematic uncertainties that may wash out some of the sensitivity.

262

CHAPTER 5. DISCUSSION AND CONCLUSIONS

Ground truth CP = 3 /2, Fit CP = 0

Minimum

2

8
6

8.3

8.6

Nominalmodel

Updated
model

7.1

4
2
0

No neutron
model

P 2
Figure 5.6: Minimum
χ between the CP violating and conserving scenarios foreach model, considering neutron production uncertainties.
Apart from the analysis on atmospheric neutrino oscillation, we anticipate advancements in the following research areas due to reduced uncertainty in neutron
production and improved selection of the hadron re-scattering model:• Accelerator neutrino oscillation analysis
Optimal choices of hadronic interaction models are expected to improve thecapability of neutrino event generators in estimating outgoing particle kinematics and addressing contamination from hadronic, non-CCQE backgroundinteractions that is crucial for the test of neutrino CP symmetry.
• Rare event searches in water Cherenkov detectors
This study may expedite rare event searches in water Cherenkov detectorsthat utilize neutron-tagging to reject atmospheric neutrino backgrounds. This
study establishes a robust neutron detection method, reducing uncertainty
in neutron multiplicity, potentially improving planned rare event searches,
especially the search for proton decay.
Looking ahead, the future Hyper-Kamiokande, featuring an eight times larger
fiducial volume, aims to provide a definitive test of the neutrino mass ordering andCP symmetry, as well as the evidence for the aforementioned rare events. This study
provides a robust foundation for such future studies that rely on our understandingof secondary hadronic neutron production following neutrino interactions.

263

Appendix A
Reconstruction algorithm ofneutrino interaction
A.1

Vertex Reconstruction

We first find the vertex ⃗x that maximizes the following estimator:
X
1(ti − ∆Ti (⃗x) − T0 )2g(⃗x, T0 ) = P
exp −2
2(1.5σi )2i (1/σi ) i(A.1)

where σi is the timing resolution, ti is the hit time of the ith PMT, and ∆Ti (⃗x)
is the estimated photon time-of-flight from the assumed vertex ⃗x to the ith PMT.
T0 is the assumed signal generation time.
The initial Cherenkov ring direction is determined by the normalized sum of vectors connecting the initial guess vertex to each hit PMT, weighted by the corrected
hit charge qicorr (⃗x), given by:

qicorr (⃗x) = qiobs exp

264

|⃗xPMT − ⃗x| cos θ(⃗x)− iLf (θ(⃗x))



APPENDIX A. RECONSTRUCTION ALGORITHM OF NEUTRINOINTERACTION
is the position of the ith PMT, L is theHere, qiobs is the observed charge, ⃗xPMT
ilight attenuation length in water, θ(⃗x) is the incident angle assuming photon origin
at ⃗x, and f (θ) is the PMT angular acceptance.
The ring direction undergoes further refinement by testing various Cherenkov
angles. The Cherenkov angle, typically 42◦ for particles moving close to the speedof light in water, may vary for particles with lower momenta. A new goodness-of-fit⃗ θc ), is introduced to achieve three goals: (i) penalize small, nonparameter, G(d,Cherenkov-like angles, (ii) match the opening angle to the expected value, and (iii)maximize the charge contained in a ring of the assumed angle.



1
(θ
−
θ
)
c
exp
⃗ θc ) =G(d,exp −sin θc2σθ2⃗ θ)∂Qcorr (d,
∂θ
θ=θc

!2 Z

θc

⃗ θ)dθ
Qcorr (d,0Here, d⃗ is the assumed ring direction, θc is the tested Cherenkov angle, andθexp is the expected Cherenkov angle, chosen from 42◦ assuming near-speed-of-lightparticles, the angle assuming a muon, or the angle assuming an electron with energies
to produce the observed charge. σθ is the expected resolution of the Cherenkov angle.
The first term addresses goal (i), the second term addresses goal (ii), and the third
term addresses goal (iii).
⃗ θc ) refines the ring direction and Cherenkov angle. The initial
Minimizing G(d,guess vertex is then further refined by minimizing the estimator in Equation A.1
with a test vertex constrained to be on the line defined by the fitted ring direction
and the Cherenkov angle.

265A.2. Ring counting

A.2

Ring counting

Hough transform
For ring counting, we employ the Hough transformation technique [177]. This
technique aims to identify the optimal set of parameters of a function, typicallyforming a geometrical shape in the coordinate space, based on a limited set of available coordinates modeled by the function with a given parameter set. For instance,
when dealing with a few points on a line in a 2D Cartesian coordinate system, theHough transform helps identify the most suitable set of slope and intercept to best
characterize the given set of points. Similar to conventional regression tasks, thekey elements for estimating parameters using the Hough transform include the formof the function and the available data points.
In this technique, each data point is treated equivalently, representing all possible
value sets of parameters that can model the given data point. Essentially, each data
point in the coordinate space is transformed into all possible sets of parameters in

CS 376: space.
Computer
Vision - thelecturethe parameter
Subsequently,
set of7parameters that receives the most“votes” from the data points is likely the optimal parameter set for describing thedata points. Figure A.1 provides an illustrative example of circle center estimation
given a set of points on a circle.

Hough transform for circles• Circle: center (a,b) and radius r

Hough transf• Circle: center (a,b) and ra( xi  a ) 2  ( yi  b ) 2  r 2( xi  a ) 2  ( y• For a fixed radius r, unknown gradient direction• For an unknown radius r,

Intersection:
most votes forcenter occur
here.

?

Image space
Hough space

Image space

Figure A.1: Locating the center of a circle. (Reprinted from [178])

266

Hough transform for circles• Circle: center (a,b) and ra• Circle: center (a,b) and radius r( xi  a )  ( yi  b )  r
2

Hough transf

22• For an unknown radius r, unknown gradient direction( xi  a ) 2  ( y• For an unknown radius r,

APPENDIX A. RECONSTRUCTION ALGORITHM OF NEUTRINOINTERACTION
This method for estimating circle centers can be readily extended to the estimation of multiple circle centers. The distribution of weighted parameters in Hough
space will exhibit multiple modes if there are several underlying circles. In the context of Cherenkov ring counting at SK, we can initially project all PMT hits onto
a 2D plane represented by polar-azimuthal coordinates (Θ, Φ) calculated from thereconstructed vertex, given the cylindrical shape of our detector. Subsequently, theCherenkov rings in this plane will appear as circles with a known radius (determined by the Cherenkov angle and the reconstructed vertex). The PMT hits thenbecome our data points, each of which can be transformed into circles in the Hough
space coordinates (Θ′ , Φ′ ). The circles in Hough space are further weighted by theobserved charge of the corresponding PMT hit. An illustration of charge-weighted
PMT hits in Hough space is presented in Figure A.2.

hit PMT

600
400
200
0
150

Cherenkov ringθ(de 100
gr
ee 50
s)

300

rees)
g
e
d
(
φ

200
100
00

Figure A.2: Charge-weighted PMT hits in Hough space (Reprinted from [118])

267

A.3. Particle identification
The highest peak in the Hough space, as illustrated in Figure A.2, is identified
as a Cherenkov ring. Consequently, every event is guaranteed to have at least one
recognized ring, regardless of its faintness or potential absence. Then, rings arecounted iteratively. We assess whether considering the next ring candidate (thenext highest peak in the Hough space) enhances the likelihood of PMT hits, defined
by:
ring
NPMT

L(Nring ) =Xlog P (qi ,i,θi,j <1.2θc,j

Nring

X
j



exp
qi,j
)

(A.2)

ring
where Nring is the number of rings, NPMT
is the number of PMTs within (ornear the edge of) the assumed Cherenkov rings. The index i iterates over such
PMTs, and the index j iterates over the assumed Cherenkov rings. qi is the obexp
served charge of the ith PMT, and qi,jis the expected charge of the ith PMT due

to the j th Cherenkov ring. P (q1 , q2 ) is the probability of observing q1 when q2 is
expected. For low expected charge (q2 < 20 photoelectrons), P is tabulated basedon the observed single photoelectron charge peak of individual PMTs and a Poisson
likelihood of observing multiple photons. For higher expected charge, P is sampledfrom a Gaussian distribution. Although the actual implementation of Equation A.2
is intricate, the fundamental concept is to iteratively include Cherenkov ring candidates from the Hough space if L(Nring + 1) > L(Nring ).

A.3

Particle identification

Cherenkov rings are categorized as either e-like or µ-like. Electrons, with a lower
critical energy of 70 MeV in water, exhibit a distinct behavior. Beyond this energy
threshold, electrons tend to lose a significant amount of energy through bremsstrahlung,
resulting in the production of gamma-rays and subsequent electrons. This processleads to an electromagnetic shower that makes the ring appear fuzzier. In contrast,
muons, being much heavier, primarily lose a small amount of energy through ionization, preserving the sharpness of the ring edges. Figure 2.6 provides an illustrationof e-like and µ-like rings.

268

APPENDIX A. RECONSTRUCTION ALGORITHM OF NEUTRINOINTERACTION
For each identified Cherenkov ring, a likelihood of the PMT hit pattern is established based on both electron and muon scenarios. The lepton l ∈ e, µ scenario

pattern likelihood for the j th ring, denoted as Lpattern, is given by:j

Lpattern j(l) =

ring
N PMT
Y

i, θi < 1.5θc P

X exp
qi , qi, j exp (l) +
qi,k
k̸=j

!

(A.3)

In this equation, the product is taken over all PMTs contained within the j th ringwith an opening angle given by 1.5 times the Cherenkov angle θc . The probability
P (q1 , q2 ) retains the same definition as in Equation A.2. The last summation term
adds the contribution to the total charge from other rings in the event without
exp
assuming a particle type. The expected charge qi,j(l), assuming lepton type l forthe j th ring, is calculated based on tabulated values of MC-simulated charge for each
PMT. This calculation considers the lepton momentum and the distance between
the lepton vertex and the PMT. In the actual implementation, Equation A.3 andexp(l) take into account the contribution from scattered andthe expected charge qi,jreflected light, with the detailed process outlined in [118].
From Equation A.3, we can construct a probability assuming a χ2 distribution,
given by:
χ2j (l) = −2 log Lpattern(l)j"Pjpattern (l) = exp −(A.4)

χ2j (l) − min(χ2j (e), χ2j (µ))
ring
4NPMT

#

(A.5)

We can also consider the potential differences in Cherenkov angles (for singlering events only) between electron and muon scenarios by introducing P angle j(l):
P

angle



(θc, j − θjexp (l))2j(l) = exp −
2σθ



(A.6)

where θc,j is the reconstructed Cherenkov angle of the j th Cherenkov ring, andθjexp (l) is the expected Cherenkov angle under the lepton type l scenario. σθ is the269

A.4. Momentum reconstruction
uncertainty in the Cherenkov angle reconstruction. For multi-ring events, we simply
set Pjangle (l) = 1. Then, the total probability for each lepton type l scenario for thej th Cherenkov ring is given by Pj (l) = Pjpattern (l)Pjangle (l). The preference for the
scenarios can be quantified by the following single parameter, which we call “PID
likelihood” or “PID parameter”:
LPID
=
j

q
q
− log Pj (µ) − − log Pj (e)(A.7)

If this parameter is positive, the ring is classified as e-like, and if negative, µ-like.
Scaling the parameter by a constant does not affect the classification performance.
The distribution of this parameter is shown in Figure 4.9 in Section 4.2.1 (For T2K
data and simulation, it is shown in Figure 2.6, although using a different algorithm[43]). The overall mis-identification probability is at O(1)% level.

A.4

Momentum reconstruction

Each Cherenkov ring corresponds to a particle with a particular momentum, computed from the sum of the observed charge within the Cherenkov ring. In the case ofmulti-ring events, rings may overlap, so the observed charge must be divided among
each ring. The observed charge on the ith PMT from the j th ring is defined using
exp
the fraction of the expected charge from the j th ring (qi,j), such that:
exp
qi,j
P exp
j qi,jqi,j = qi

!

(A.8)

Then, the total charge contribution Qtotal(also known as RTOT in SK) from thejj th Cherenkov ring is quantified as:
Qtotal
=
j

270

GMC
GData

X
i

corr
qi,j
−

X
j

Qscat
j

!

(A.9)

APPENDIX A. RECONSTRUCTION ALGORITHM OF NEUTRINOINTERACTION
Here, GMC and GData are the overall gain scaling factors assumed in MC andobserved in data, respectively. The first term in the parentheses is the sum of thecharge contribution of the individual PMT hits of the j th Cherenkov ring, correctedby the PMT angular acceptance and the light attenuation in water. The second
term is the contribution by the expected scattered light due to all Cherenkov rings.
Specific conditions imposed on PMTs when calculating Equation A.9 can be foundin [118]. Qtotal
is converted into lepton momentum using a lookup table based onjMC simulation.

271

Appendix B
Data selection stages
B.1

FC1: Initial cuts to reject evident backgrounds

Low-energy radioactivity rejection• PE300 > 200 p.e.
PE300 , the maximum total ID PMT charge in a sliding 300-ns window, must
be greater than 200 photoelectrons (equivalent to 22 MeV electron).

Cosmic ray rejection• The OD trigger must be off, OR• NHITA800 ≤ 50
NHITA800 , the number of OD PMT hits within [-500, +300] ns window fromthe event trigger, must be equal to or smaller than 50 hits.

The first cut removes O(1) MeV radioactivity from our data. These cuts reduce
the event rate from O(106 ) to O(103 ) per day.

272

APPENDIX B. DATA SELECTION STAGES

B.2

FC2: Subsequent cuts to further reject backgrounds

Flasher and radioactivity rejection• PEmax /PE300 < 0.5
The maximum charge in any single PMT (PEmax ) must be smaller than thanhalf the maximum total ID PMT charge in a sliding 300-ns window (PE300 ).

Further cosmic ray rejection• The OD trigger must be off, OR• NHITA800 ≤ 30, OR
The number of OD PMT hits within [-500, +300] ns window from the event
trigger must be equal to or smaller than 30 hits.
• PEtotal > 100,000 p.e.
The total charge deposited in the ID must be greater than 100,000 photoelectrons. For such events, OD PMT can register fake hits due to crosstalk withnearby ID PMTs, so they should be exempt from the above OD selections.

These cuts reduce the event rate down to O(102 ) events per day.

273

B.3. FC3: Cosmic ray rejection based on fit

B.3

FC3: Cosmic ray rejection based on fit

Even at this point, a significant portion of the remaining data events consists ofcosmic ray muons and low-energy noise events that resemble signals. These events
typically manifest as having relatively small OD activity and non-negligible ID activity.
To further reject cosmic ray muons, we use a specialized muon fitter for eventswith over 1,000 ID PMT hits, where more than 230 photoelectrons are deposited.
This fitter determines the entry time (T0 ) and the entry and exit (or stop) vertices(⃗xentry and ⃗xexit/stop) that maximize the “goodness” parameter g according to
the following expression:X(ti − ∆Ti (⃗xentry , ⃗xexit/stop ) − T0 )21g(⃗xentry , ⃗xexit/stop , T0 ) = P
exp −2
2(1.5σi )2i (1/σi ) i(B.1)
Here, σi represents the timing resolution, ti is the hit time of the ith PMT, and∆Ti (⃗xentry, ⃗xexit/stop) is the estimated photon arrival time on the ith PMT since
T0 . The estimation of ∆Ti involves determining the photon production vertex on themuon track based on the initial and final muon vertices, as well as the Cherenkov
angle, and then summing the muon and photon time-of-flight.

Through-going muon rejection• g < 0.75, OR
The maximal through-going muon goodness g must be less than 0.75.
• NHITAentry < 10 and NHITAexit < 10
The number of hit OD PMTs located within 8 m from the entry (exit) point
in a [-500, +300] window NHITAentry (NHITAexit ) must be smaller than 10
PMTs.

274

APPENDIX B. DATA SELECTION STAGES
Stopping muon rejection• The goodness fitter fails to find the appropriate set of muon entry and stopping
vertices, OR• NHITAentry < 10
The number of hit OD PMTs located within 8 m from the entry point in a[-500, +300] window NHITAentry must be smaller than 10 PMTs.

Cable-hole muon rejection
On top of the SK detector tank, there are twelve “cable-holes” designed for thepassage of signal and high voltage supply cables. As illustrated in Figure B.1, no
OD PMT is positioned at these hole locations. When cosmic ray muons traverse
these openings, they enter the ID region without hitting OD PMTs. Consequently,
these events might be mis-identified as contained neutrino events due to the absenceof an OD signal. To prevent such mis-identifications, veto counters (consisting of a2 × 2.5 m plastic scintillation counter and a PMT) were strategically installed atfour out of the twelve cable-holes. Figure B.1 provides a schematic representationof the cable hole and the associated veto counter. Events with no veto counter hits
are selected for further analysis.

Low-energy event rejection• NHITtotal ≥ 500 hits, OR
The total number of ID PMT hits must be greater than or equal to 500 hits.
• NHIT50 ≥ 50 hits
The maximum number of ID PMT hits in a sliding 50-ns window (NHIT50 )must be greater than or equal to 50 hits. Before counting the number of hits
in the sliding window, the photon time-of-flight (TOF) is corrected for basedon the vertex that minimizes the variance of the TOF-corrected hit times.
NHIT50 of 50 hits corresponds to visible energy of 9 MeV.

275

B.3. FC3: Cosmic ray rejection based on fit

Figure B.1: A bird’s-eye view of the SK tank indicating the locations of cable holes(top) and a sectional view showing a cable hole equipped with the veto counter(bottom).

276

APPENDIX B. DATA SELECTION STAGES
Flasher rejection
Flasher events often have a broad timing distribution compared to events producedby charged particles. The hit time distributions of typical flasher events and neutrino
events are shown in Figure B.2.
• NHIT100 < 20
The minimum number of ID PMT hits in a sliding 100-ns window within[+300, +800] µs range from the event trigger must be less than 20 hits.

Figure B.2: PMT hit timing of flasher (top) and neutrinos (bottom)

Accidental coincidence rejection
If a low-energy radioactivity triggers an event without OD signature and a cosmic
ray accidentally comes in and fires both the OD and ID PMTs within O(100) ns
from the event trigger, such events can elude the previous OD cuts designed to rejectOD PMT hits near the event trigger.
• PElate < 5,000 p.e., OR

277

B.3. FC3: Cosmic ray rejection based on fit
The OD PMT charge within [+300, +800] ns from the event trigger (NHITAlate )must be smaller than 5,000 photoelectrons.
• NHITAlate < 20
The number of OD PMT hits within [+300, +800] ns from the event trigger(NHITAlate ) must be smaller than 20 hits.

The event rate at this point is around 45 events per day.

278

APPENDIX B. DATA SELECTION STAGES

B.4

FC4: Flasher rejection based on likelihood

Events triggered by PMT flashers, i.e., the internal discharge of PMTs, tend to
exhibit a single PMT hit with a very large charge deposit and small, isotropically
distributed nearby PMT hits due to the light emitted by the discharge. One of
the major characteristics of these events is that a PMT that “flashes” tends toflash repeatedly for a certain period of time. As a result, multiple events with very
similarhit patterns—SETboth in charge and positional distributions — may pass the
CHAPTER
5. DATA
three aforementioned data selection stages. Typical hit patterns of flasher events
are shown in Figure B.3.

1990/00/00:NoYet:NoYet
1990/00/00:;R= 0:NoYet
R :Z : PHI : GOOD
0.00: 0.00: 0.00:0.000
CANG : RTOT : AMOM : MS

NUM
6
RUN
65939
SUBRUN
5
EVENT
809997
DATE 2009-Nov-17
TIME
5:43:16
TOT PE:
1364.2
MAX PE: 1039.2
NMHIT :
270
ANT-PE:
6.3
ANT-MX:
1.6
NMHITA:
6

NUM
22
RUN
65939
SUBRUN
16
EVENT
2988426
DATE 2009-Nov-17
TIME
5:56: 3
TOT PE:
1994.0MAX PE: 1039.1
NMHIT :
587
ANT-PE:
16.0
ANT-MX:
3.1
NMHITA:15

1990/00/00:NoYet:NoYet
RunMODE:NORMAL
1990/00/00:;R= 0:NoYet
TRG ID :00000111
R :Z : PHI : GOOD
T diff.: 0.00
us
0.00: 0.00: 0.00:0.000
CANG : RTOT : AMOM : MSFEVSK :88010803
nOD YK/LW: 2/ 1
SUB EV : 0/ 0
Dcye: 0( 0/ 0/ 0/ 0)

RunMODE:NORMALTRG ID :00000111
T diff.: 0.00
us
FEVSK :88010803
nOD YK/LW: 4/ 2
SUB EV : 0/ 0
Dcye: 0( 0/ 0/ 0/ 0)

Comnt;

Comnt;

Figure B.3: Typical recurring flasher events.
We take advantage of this characteristic to reject flasher events. In simple terms,
Figure 5.14: Typical flasher event displays. They are diﬀerent events but caused by the samedetectsa largeamountof photons(the large circle).
we PMTrejectwhich
eventsthathavemultipleoccurrenceswith recurring hit patterns. To

achieve this, we calculate two metrics that quantify the “similarity” of a tested
event displays are shown in Fig. 5.14. The algorithm tries to find two diﬀerent events in which

event
with
the othercompared
events.
PMT
hit (charge)
patterns
are similar.
The ID is divided into 2 m × 2 m patches when comparing
the PMT charge profiles in the two diﬀerent events, event A and event B. Each patch has about
8 PMTs, and their charge sum is calculated. The following correlation coeﬃcient, r, is defined:

One metric is the “patch-by-patch charge correlation” r given by:r=B
1 ! (QiA − ⟨QA ⟩) × (QBi − ⟨Q ⟩)
r=X
,B
Aσ σ
B
1 N (QA
A ×
B (Qi − ⟨Q ⟩)i − ⟨Q ⟩)i

NσA × σB

(5.4)

(B.2)

B
i QA
where N is the number of patches,i (Qi ) is the ith patch’s total observed charge in event
AB
A (B), ⟨Q ⟩ (⟨Q ⟩) is the average charge over all patches in event A (B), and σA (σB ) is thestandard
deviation
the observed
chargein event
A (B).
SK-I-II,squares,
this singleandparameter
where
we dividetheof ID
wall into
1,450
patches
ofFor
2m
×and
2m
N is ther is used at this reduction step. A(B)thnumberSinceof patches(1,450). Qi
represents
the totalchargein thecharge
i patchof theSK-III, a Kolmogorov-Smirnov
test is performed
for theaccumulated
distribution normalized by the total observed charge in each event. The maximum distance between
the two accumulated charge distributions in event A and B, dKS , is introduced. Furthermore,probability density functions are made using atmospheric neutrino MC events and these twoparameters, r and dKS , are combined into a single likelihood parameter. The combined likelihood value, prob, is used to select events at this reduction step. A higher prob value indicates 279“well-matched” event with another event, and events with more than 3 of prob are rejected. The
prob distributions are shown in Fig. 5.15.

5.6

Fifth Reduction

B.4. FC4: Flasher rejection based on likelihood
two compared events A and B, ⟨QA(B) ⟩ is the average, and σ A(B) are the standardA(B)deviation of Qi.

The other metric is the “overall PMT-by-PMT charge discrepancy” dKS , whichis the Kolmogorov-Smirnov test statistic for the accumulated PMT-by-PMT charge
distributions of the two compared events.

Figure B.4: r and dKS distributions (left) and flasher likelihood distributions (Equation B.3, right) of flasher and neutrino events.

Figure B.4 illustrates the distinguishability of flasher events from neutrino events
through two metrics capturing the similarity in hit patterns between two events.
When flasher events are compared with a recurring event, they typically show ahigh value of r and a low value of dKS — a scenario rarely observed when comparing
two neutrino events.
To exploit this distinguishability, the probability density of obtaining r and dKS
for atmospheric neutrino events was established based on values calculated with
MC-simulated atmospheric neutrino events. Utilizing this probability density, we
can quantify how “unlikely” it is to obtain a specific set of r and dKS if an event
under test were a neutrino event. This is accomplished by defining the “unlikeliness”
metric L(r, dKS ):280

APPENDIX B. DATA SELECTION STAGES"
1 1 X
L(r, dKS ) =2 10 x∈r,dKS

Top 10−2Xiln p(x, qi ) − 2

10
X

ln

i



i
N αx

!

+C

#

(B.3)

Here, p(r, Q) and p(dKS , Q) denote the probability density of two atmospheric
neutrino events having r and dKS , respectively, when the average total charge of thetwo events is Q. Only the top 10 events with the largest r values and the smallest
dKS values are considered for a given event. The coefficients αr and αdKS and C are
tuned heuristically to align the peak of the L values for MC-simulated atmospheric
neutrino events to zero. Each event in the database is assigned a single value of L.
Figure B.4 compares the distribution of L between atmospheric neutrinos (peaking
close to zero) and PMT flashers (with larger values of L).
Events with L < 3 are selected, while events with L > 3 are considered flashers
and subsequently rejected.
The event rate at this stage is approximately 18 events per day.

281

B.5. FC5: Additional cuts to reject remaining backgrounds

B.5

FC5: Additional cuts to reject remaining backgrounds

At this stage, additional intricate criteria are applied to further eliminate cosmic
rays, flashers, and low-energy-cosmic-ray coincidences. Refer to [118] for details.
Another category of background events, “invisible muons,” involves cosmic-ray
muons losing energy below the Cherenkov threshold, becoming visible only when
stopping in the ID to produce Michel electrons. To distinguish these events fromlow-energy electron neutrino events, we rely on OD hits preceding the ID event
trigger, represented by the variable NHITAearly . Specifically, invisible muon events
are rejected using following complex data selection cuts:• NHITAearly < 50 hits, AND• PEtotal > 1,000 p.e., OR
Michel electrons have maximum energy of around 50 MeV, so that events withthe total ID PMT charge (PEtotal ) larger than 1,000 photoelectrons are very
unlikely to be originated from Michel electrons.
• NHITAearly < 5 hits, OR• dOD > 5 m AND NHITAearly < 10 hits, OR
Here, dOD is the distance between the position of the OD PMT hit clusteraround the event trigger and the earlier time range of [-9, -0.2] µs from thetrigger.
• dOD < 5 m AND NHITAearly + NHITA500 < 10 hits
Here, NHITA500 is the number of OD PMT hits within [-0.2, +0.3] µs range
from the event trigger.

The event rate at this stage is around 16 events per day.

282

APPENDIX B. DATA SELECTION STAGES

B.6

FCFV: Events fully-contained in fiducial volume

Following are the standard cuts to select fully-contained events in fiducial volume(FCFV) and further reduce the following sources of background:• No particle exit from the ID
The number of OD PMT hits in the largest cluster (NHITAC, explained in
Section 3.3.4), must not exceed 15.
• Low-energy backgrounds
The visible energy must be greater than 30 MeV.
• Background radioactivity near the tank wall
The reconstructed interaction vertex must be more than 1 m away from the
ID tank wall.
The event rate at this stage is around 9 events per day. Over 99% of theselected events are expected to be atmospheric neutrino interactions.

283

SK-VI / SK-IV

284

1
2
3
4
5
6

Sub-GeV MR

Multi-GeV 1R e-like

Multi-GeV 1R -like

Multi-GeV MR

Total

1.0
0

Sub-GeV 1R -like

01.1

Sub-GeV 1R e-like

Events / dayAppendix C

Comparison of atmospheric

neutrino data: SK-IV with SK-VI

8
SK-IV
SK-VI

6

4

2

Figure C.1: Event rates.

APPENDIX C. COMPARISON OF ATMOSPHERIC NEUTRINO DATA: SK-IV
WITH SK-VI

All

2.1

All
SK-VI data
SK-IV data

1.00

1.9

Events / day

Events / day

2.0

1.8
1.7

0.95
0.90
0.85

1.6
1.1 0.0

0.5

1.0

pvr2

1.0

1.5

2.0

2.5
1e6

0.9
0.0

0.5

1.0

1.5
R 2 [cm2 ]

2.0

0.80
1.1

Data / MC

SK-VI / SK-IV

SK-VI data
SK-IV data

1.05

1500

1000

500

1500

1000

500

1.0

0
pvz

500

1000

1500

0

500

1000

1500

0.9

2.5
1e6

z [cm]

Figure C.2: Reconstructed vertex distributions R2 (left), z (right)

All

All
SK-VI data
SK-IV data

4

SK-VI data
SK-IV data

6

3

Events / day

Events / day

5

2

0
1.5

3
2
1

0

2000

1.0

4000

EVis

6000

8000

10000

0.5
0

2000

4000
6000
Visible energy [MeV]

8000

10000

SK-VI / SK-IV

SK-VI / SK-IV

1

4

0
1.2

0

1

0

1

1.0
0.8

2

3

NRing

4

5

2
3
4
5
Reconstructed Cherenkov rings

6

7

6

7

Figure C.3: Event visible energy (left) and number of reconstructed Cherenkov rings(right).

285

Single-ring
SK-VI data
SK-IV data

Events / day

3.8
3.6
3.4

SK-VI / SK-IV

3.2
3.0
1.05 10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5 14.0
FirstRingType
1.00
0.95
e-likeRing typeµ-likeFigure C.4: Single-ring event visible energy (left) and ring PID (right).

286

APPENDIX C. COMPARISON OF ATMOSPHERIC NEUTRINO DATA: SK-IV
WITH SK-VI

Single-ring (Sub-GeV e-like)Single-ring (Sub-GeV -like)
SK-VI data
SK-IV data

0.625

0.45
Events / day

0.575
0.550
0.525

0.40
0.35
0.30

0.475

0.25

1.1 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00
cosz
1.0
0.9

SK-VI / SK-IV

0.500

SK-VI / SK-IV

Events / day

0.600

1.1 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00
cosz
1.0
0.9

1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00

1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00

cos zenith

cos zenith

Single-ring (Multi-GeV e-like)Single-ring (Multi-GeV -like)
0.18

SK-VI data
SK-IV data

0.18

Events / day

0.12

0.14
0.12
0.10

0.10

0.08

0.08

0.06

1.25 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00
cosz
1.00

1.25 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00
cosz
1.00

0.75
1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00
cos zenith

SK-VI / SK-IV

Events / day

0.14

SK-VI data
SK-IV data

0.16

0.16

SK-VI / SK-IV

SK-VI data
SK-IV data

0.50

0.75
1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00
cos zenith

Figure C.5: Cosine zenith angle distributions

287

Appendix D
Statistical uncertainties
D.1

Statistical inference

Statistical uncertainty stems from Nidetected — the number of detected signals in theith event — in Equation 4.1. We assume that Nidetected follows a binomial distribution
with Nisignal trials and the binomial probability ϵi , where Nisignal is the total numberof produced signals, and ϵi is the signal efficiency of the ith event.
If we assume that, for sufficiently small visible energy bins, both the number of
produced signals (N signal ) and the signal efficiency (ϵ) remain nearly constant acrossvarious events within the bin, then the standard deviation of N detected is:∆N detected ∼pp
N signal ϵ(1 − ϵ) ∼ N detected (1 − ϵ)(D.1)

Thus, the uncertainty in the average of N detected /ϵ across all events in the bin isgiven by:

N detected
δ
ϵ



1
∼√
Neventeventp
N detected (1 − ϵ)
ϵ

(D.2)

The reliability of the expression ⟨N detected /ϵ⟩ itself, as defined in Equation 4.1,

is rooted in the Central Limit Theorem. This is further supported by Figure 4.31,

288

APPENDIX D. STATISTICAL UNCERTAINTIES
where the signal multiplicity estimated using Equation 4.1 aligns closely with theactual signal multiplicity in the simulation.
The statistical uncertainty, as given by Equation D.2, holds true even in the more
realistic scenario where Nisignal follows a Poisson distribution, and ϵi is sampled froma normal distribution for each ith event.
Our event statistics are sufficiently large, so that we can easily adjust each visibleenergy bin to contain at least 20 neutrino events. Therefore, it is reasonable to use
Equation D.2 as an estimate for the statistical uncertainty across all bins.

D.2

Signal detection performance spread

The spread or fluctuation of the signal detection performance around the mean is
influenced by several factors. As previously discussed, the effect of these factors on√
the “average” signal multiplicity per event is mitigated by the factor of 1/ Nevent .

289

D.2. Signal detection performance spread
Detector modeling parameters
The temporal variations in the estimated signal efficiencies derived from Am/Be
data were at a maximum of 9.5% for SK-IV, 9.4% for SK-V, and 4.6% for SK-VI.
MC regression
The systematic uncertainty in the efficiency spread is derived from the 1σ prediction intervals of the signal efficiency obtained through GAM regression. On average,
the fraction of this interval relative to the GAM predictions was 6.2% for SK-IV,
7.5% for SK-V, and 2.3% for SK-VI. The 1σ prediction interval of the false positive
rate was 40.7%, 50.4%, and 37.6% for SK-IV, SK-V, and SK-VI, respectively.
Signal vertex
The proximity of the signal vertex to the vertex used for photon TOF correction plays a crucial role in determining the efficiency of signal detection. There
should be a certain level of uncertainty in our simulation regarding this distance.
To assess the impact, we compared the reconstructed distance from the neutrino
interaction vertex to the neutron capture vertex in both SK-VI (R) data and simulation, as illustrated in Figure 4.23. Understanding how the detection efficiency in
the INIT-NEUTRON vertex mode is influenced by distance, as shown in Figure 3.15,
allows us to extrapolate the differences in the distance distributions between data
and MC simulation to the signal efficiency. The expected fractional changes were3.5% for SK-IV, 4.0% for SK-V, and 2.3% for SK-VI.
Neural network credibility
The credibility of a neural network is inherently influenced by the diffuse boundary between class labels used during training, leading to inherent uncertainties inboth the weights and outputs of the trained network. Consider a scenario wheretraining data is randomly sampled from a larger dataset with a vague boundary between two binary classes, and a network of the same architecture is trained multiple
times. In such cases, one can anticipate a distribution of weights for the network.

290

APPENDIX D. STATISTICAL UNCERTAINTIES

Figure D.1: Variance in dropout-sampled neural network signal classification efficiency (left) and false positive rate per 500 µs (right).
For each unseen candidate, sampling from this ensemble of trained networks produces a distribution of network outputs. Gal and Ghahramani [179] demonstrated
that for networks trained with dropout, as used in this analysis, network outputs
can be sampled by activating dropout during inference without the need to train
network ensembles. Figure D.1 illustrates the distributions of signal efficiency andfalse positive rates sampled by activating dropout during inference.
The evaluated uncertainty in signal efficiency was 0.2%, 0.3%, and 0.2% for SKIV, SK-V, and SK-VI, respectively, while the uncertainty in false positive rates was
4.5%, 4.8%, and 0.9% for the same SK phases.
Event reconstruction
The uncertainties associated with the reconstructed event variables used as input
for the fitted GAM can influence the expected output performances. In this analysis,the 2% uncertainty in the visible energy scale was the only non-negligible factor
impacting the final estimates.
For signal efficiency, the assigned uncertainties were 0.3%, 0.2%, and 0.3% for
SK-IV, SK-V, and SK-VI, respectively. Also, the uncertainties for false positive
rates were 1.2%, 1.1%, and 1.3% for the corresponding SK phases.

291

D.2. Signal detection performance spread
Muon-induced backgrounds
The quantity of Michel electrons and nuclear de-excitation gamma-rays resulting from negative muon capture on nuclei is dependent on the number of muons
generated in an event. This quantity introduces a degree of uncertainty due to thevariability in pion production, a precursor to muon generation. By analyzing thetime distribution of both Michel electron and neutron capture candidates, we determined the maximum and minimum potential levels of muon-induced backgrounds.
The uncertainty in false positives due to the uncertainty in muon-induced backgrounds was estimated to be 2.5% for SK-VI.
The total systematic uncertainty in the signal detection performance spread
is summarized in Table D.1. The total uncertainty was scaled by the factor of√
1/ Nevent where Nevent represents the number of neutrino events in each visible energy bin.
SourceDetector response
MC regression
Signal vertex
Neural network
Event reconstructionµ-induced BG
Total

Signal efficiency
SK-IV SK-V SK-VI
9.5%
9.4%
4.6%
6.2%
7.5%
2.3%
3.5%
4.0%
2.0%
0.7%
0.8%
0.3%
0.2%
0.3%
0.2%

False positive rate
SK-IV SK-V SK-VI

11.9%

12.7%

5.5%

40.7%

50.4%

37.6%

4.5%
1.2%

4.8%
1.1%

41.0%

50.6%

0.9%
1.3%
2.5%
37.7%

Table D.1: Systematic uncertainties relevant to signal detection performance spread.

292

Appendix E
Comparison of neutrino event
generators

293

0.08
0.06

Model
NEUT 5.4.0.1
NEUT 5.6.3
GENIE hA
GENIE hN
GENIE INCL++GENIE G4Bertini

0.08

0.06
n / event

Model
NEUT 5.4.0.1
NEUT 5.6.3
GENIE hA
GENIE hN
GENIE INCL++GENIE G4Bertini

0.10

0.04

0.04
0.02

0.02
0

1.5
1.0
0.5

0

500

1000

1500

2000

500
1000
1500
2000
Outgoing proton momentum [MeV/c]

0.00
1.5
1.0
0.5
0.0

Ratio to
NEUT 5.4.0.1

Ratio to
NEUT 5.4.0.1

Outgoing protons / event

Atmospheric MC (FCFV): Evis [30, 10000] MeV

Flux: Atmospheric / Target: Water

0.12

0

250

500

0

250

500

Atmospheric MC (FCFV): Evis [30, 10000] MeV
0.012

0.008
0.006

0.001

Ratio to
NEUT 5.4.0.1

250

500

750

1000 1250 1500 1750 2000

1.0
0.5

0

250

500

750

1000 1250 1500 1750 2000

+ momentum [MeV/c]

2000

750

1000

1250

1500

1750

2000

n momentum [MeV/c]

Model
NEUT 5.4.0.1
NEUT 5.6.3
GENIE hA
GENIE hN
GENIE INCL++GENIE G4Bertini

0.003

0.002
0

1750

0.004

0.002

1.5

1500

0.005

0.004

0.000

1250

0.006

0.000
Ratio to
NEUT 5.4.0.1+ / event

0.010

1000

Atmospheric MC (FCFV): Evis [30, 10000] MeV/ event

Model
NEUT 5.4.0.1
NEUT 5.6.3
GENIE hA
GENIE hN
GENIE INCL++GENIE G4Bertini

0.014

750

0

250

500

750

1000 1250 1500 1750 2000

0

250

500

750 1000 1250 1500 1750 2000
momentum [MeV/c]

1.5
1.0
0.5

Figure E.1: Outgoing hadron multiplicity and kinematics comparison of different
neutrino event generator options.

294

APPENDIX E. COMPARISON OF NEUTRINO EVENT GENERATORS

Figure E.2: Outgoing hadron multiplicity and kinematics comparison of different
neutrino event generator options.

295

Figure E.3: Outgoing hadron multiplicity and kinematics comparison of different
neutrino event generator options.

296

APPENDIX E. COMPARISON OF NEUTRINO EVENT GENERATORS

Figure E.4: Muon multiplicity

297

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